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Offline WillLem

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WillLem's Blog: Permission or Forgiveness?
« on: April 14, 2020, 09:17:16 AM »
Topics:

Infinity (from post 1 onwards)
Expanding Earth Theory, Truth, Simulation Theory, Ice Cream and Mathematics...
Tetris
The Perfect Game Controller!
Retrobriteing
Paradoxes
Permission or Forgiveness?



On Infinity.

EDIT: Given the misunderstandings that have arisen from the initial version of this post, I just want to start this off by clarifying a couple of things:

1. I have nothing against mathematics, mathematicians, or indeed any school of abstract thought (actually, I happen to be subscribed to Numberphile on YouTube and I regularly watch and enjoy their videos, and find them very fun and educational). I am simply interested to understand it better, and have come up against questions to which the answers have caused further confusion, in some areas. In other areas, I now feel more enlightened thanks to people's explanations and the progress of this discussion.

2. I know that infinity is not a number. I am asking about the relation of the concept of infinity to the mathematical system of numbers.

---

In trying to understand infinity, we often use numbers to demonstrate its size and shape. For example, there are an infinite number of natural numbers given that it's always possible to add 1 to any natural number. However, there are also an infinite number of real numbers between 0 and 1.

The problem is, that in attempting to count from 0 to anything, you must first define an arbitrary "counting distance."

In counting from 0-10, for example, we might impose a "counting distance" of (1). So:

1 (1) 2 (1) 3 (1) 4 (1) 5 (1) 6 (1) 7 (1) 8 (1) 9 (1) 10

You could just as easily make the counting distance (2) or (5), though.

So, then, we might want to know what the distance between 0 and 1 is. How do we count that? Well, we could use a counting distance of (0.1). So:

0 (0.1) 0.1 (0.1) 0.2 (0.1) 0.3 (0.1) 0.4 (0.1) 0.5...

But, then what about the distance between 0 and 0.1. Now we need a yet smaller counting distance. Let's try (0.01):

0 (0.01) 0.01 (0.01) 0.02 (0.01) 0.03 (0.01) 0.04 (0.01) 0.05...

Because of this tendency towards smaller and smaller decimal numbers, mathematicians have concluded that there is a larger infinity between 0 and 1 than the size of an "infinite list" of natural numbers. In fact, the understanding seems to be that that the numbers between 0 and 1 are "unlistable", in that you can always generate a real number that isn't on a given list by simply changing one digit within any real number relative to its position on the list.

This may be true, but it's a truth that seems somewhat arbitrary, imposed as it is by the limitations of a digital counting system.

In order to define anything using numbers, it's necessary to choose a digital "counting distance," which is always arbitrary and finite when observed from analog reality.

Infinity is an analog concept, and is therefore incompatible with any digital counting system.

Hence, the mathematical notion of "different sized infinities" (such as Aleph 0, Aleph 1, etc) seems necessarily and conceptually flawed; defining their "size" in such a way succeeds only in placing yet another arbitrary finite value on the idea of infinity.

Infinity is not finite. It is in-finite.

Therefore, I would ask the question: is Mathematics the best tool we have for understanding and interacting with the infinite? Perhaps Music and Art are better, but could there even be something that we as humans can't even conceive that would truly allow us to comprehend the incomprehensible?
« Last Edit: June 16, 2021, 01:02:30 AM by WillLem »

Offline Proxima

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Re: WillLem's Blog
« Reply #1 on: April 14, 2020, 02:59:45 PM »
Because of this tendency towards smaller and smaller decimal numbers, mathematicians claim that there is a larger infinity between 0 and 1 than the size of an "infinite list" of natural numbers. In fact, they claim that the numbers between 0 and 1 are "unlistable", in that you can always generate a real number that isn't on a given list by simply changing one digit within any real number relative to its position on the list. This may be true, but it's a truth that's arbitrary, imposed as it is by the limitations of a digital counting system.

No, that's not how it works.

Mathematicians have a precise definition that they use to say that two infinite sets are of the same size, or are of different sizes. Two sets are of the same size if they can be put into one-to-one correspondence: pairing every element of one set with a unique element from the other.

Code: [Select]
{England, France, Germany}
    |        |        |
    v        v        v
{magenta, orange, lavender}

This works for infinite sets as well, provided you can pair the elements in a patterened way so that it's clear the pattern continues for ever:

Code: [Select]
{1, 2, 3, 4,  5,  6...}
 |  |  |  |   |   |
 v  v  v  v   v   v
{1, 4, 9, 16, 25, 36...}

In particular, the process of pairing elements with the natural numbers {1, 2, 3...} is called counting, and any set for which this can be done is countable.

Now, despite the "tendency towards smaller and smaller decimal numbers", the terminating decimals between 0 and 1 are countable. By "terminating decimal" I mean a number like 7/25 whose decimal expansion (in this case 0.28) comes to an end.

Let's try to prove this. You can't list them in ascending numerical order, because before 0.1 would come 0.01, and before that would come 0.001, and before that... you can't even get started.

But you can list all the decimals of length 1, then all those of length 2, then all those of length 3, and so on. Because each list is finite, you can complete it and go on to the next.

However, some numbers are non-terminating, for instance 1/3 = 0.3333333.... Cantor's claim is that the set of all numbers between 0 and 1 is non-countable (too large to be counted). You cannot put them into a list, in any order, than includes every number.

This is not because our decimal system of notation is limited. It's obvious that I can't physically write an infinite list, but Cantor's claim is much stronger: even granting me the mathematically idealised power to create any list of numbers I like, in any notation, it is still impossible to list all numbers between 0 and 1.

This is proved by contradiction. First, suppose we have such a list. Every number has a decimal representation (maybe terminating, maybe not), so we imagine that the list is in decimal form. Take the first digit of the first number, the second digit of the second number, the third digit of the third number and so on, and change each digit to something else. This creates a new number that isn't anywhere on the list (it can't be, because it differs in at least one digit from every number on the list). Suppose I try to get round this by creating this "diagonal" number and adding it to the list? Well, I can't add it "at the end", because the list is infinite. If I add it at, for example, the 1000th place, that changes what number is in each place from that point onward, so we can create a new diagonal number that also isn't on the list.

Let me know if you have any more questions or need me to explain some of this better :P

Offline WillLem

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WillLem's Blog: On Infinity
« Reply #2 on: April 14, 2020, 03:24:49 PM »
This is proved by contradiction. First, suppose we have such a list. Every number has a decimal representation (maybe terminating, maybe not), so we imagine that the list is in decimal form. Take the first digit of the first number, the second digit of the second number, the third digit of the third number and so on, and change each digit to something else. This creates a new number that isn't anywhere on the list (it can't be, because it differs in at least one digit from every number on the list). Suppose I try to get round this by creating this "diagonal" number and adding it to the list? Well, I can't add it "at the end", because the list is infinite. If I add it at, for example, the 1000th place, that changes what number is in each place from that point onward, so we can create a new diagonal number that also isn't on the list.

I understand this perfectly, I just don't think it's an effective way of interacting with infinity. The very idea of creating an infinite list of anything is absurd: if it's a truly infinite list, you will always be able to add something to that list that isn't already on it.

I understand the difference between listing integers and real numbers, as demonstrated by Hilbert's Infinite Hotel paradox and indeed Cantor's list of non-terminating decimals. This isn't what I'm disputing at all: these guys are far more mathematically sound than I am and have probably spent far more time thinking about this sort of thing, so all respect to their theories.

What I'm saying is that any attempt to demonstrate, interact with, or otherwise understand infinity which involves anything that is finite in its nature (such as digital numbers) is flawed.

The fact is, you can't make an infinite list: you can only infer an infinite list using ellipses, e.g.:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10...

Is not an infinite list. If implies that it could be infinite if it were to continue in the same manner, but who is to say that it does?

What if a revelation of the rest of this list manifested the following result:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 56.

Again, just so there's absolutely no misunderstanding here:

I understand that Cantor's discovery was that the "infinity" as represented by a list of non-terminating decimals seems to be larger than the "infinity" of, say, a list of natural integers with a "counting distance" of (1).

N.B. (I'm not sure what the actual mathematical term for a "counting distance" is...).

However... and this is my point: both lists would only go on forever if they actually existed, but they are imaginary, and can only ever be so. Neither adequately represents infinity as a concept. Let's suppose that it were possible to make such a list: which would be the longer list? If one of the lists is longer even by 1 digit, then the other is not an infinite list.

Here's another list:

1, 2, 3, 4, 5, 6, 7, 8, 9, orange, 10, 11, 12, 13, 14, 15, The Simpsons, 16, 17, January, 18...

What do the ellipses even mean?
« Last Edit: June 24, 2020, 02:31:06 AM by WillLem »

Offline ∫tan x dx

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Re: WillLem's Blog
« Reply #3 on: April 14, 2020, 03:28:43 PM »
I'm sorry bud, but what you've written here is complete nonsense. I'm going to give you a breakdown of what's wrong bit by bit.

Quote
In trying to understand infinity, we often use numbers to demonstrate its size and shape.
Nope. Mathematicians generally use set theory to discuss the kind of infinities you are talking about. We use sets of numbers to give examples of the results, so that it is somewhat easier to process.

Quote
The problem is, that in attempting to count from 0 to anything, you must first define an arbitrary "counting distance."
First of all, the notion of "counting distance" has no real meaning in terms of actual mathematics. If you want to get technical, when defining what "counting" is, then the Peano axioms (https://en.wikipedia.org/wiki/Peano_axioms) are a nice way to define the Natural numbers (denoted N = {0,1,2,...}). There are the natural numbers and there is addition, neither are used or defined in the way you are claiming. The Natural number known as "1", "one", "unity" (or whatever other notion you wish to describe it) is most certainly not arbitrary. Its existence is in fact a logical consequence of the axioms by which the Natural numbers are defined. The name we give to it, as well as the symbols we use to denote it are arbitrary, but what these symbols represent in a formal context is not arbitrary.

Quote
Because of this tendency towards smaller and smaller decimal numbers, mathematicians claim that there is a larger infinity between 0 and 1 than the size of an "infinite list" of natural numbers.
This is utterly wrong. Mathematicians know that the set of real numbers between 0 and 1 has a greater cardinality than the Natural numbers, but their arguments have nothing to do with what you've written here. In fact, this statement says very little other than saying that there is no smallest real number strictly greater than zero. In fact the sequences that you have given all feature Rational numbers. It is an established fact that the set of Rational numbers (denoted Q) is of the exact same cardinality as N, and both have smaller cardinalities than R.

Quote
In fact, they claim that the numbers between 0 and 1 are "unlistable"...
This is true, and I have to say that the term traditionally used to describe this - "Uncountable" - is not exactly the most illuminating. Unlistable is a better word for it in my opinion, since the elements of an unlistable set cannot be put into one-to-one correspondence with the Natural numbers. That is, they cannot be listed in their entirety.

Quote
... in that you can always generate a real number that isn't on a given list by simply changing one digit within any real number relative to its position on the list.
What you are referring to here is Cantor's Diagonal Argument (https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument). This is a famous mathematical proof which shows that there can be no bijection (one-to-one correspondence) between the Natural numbers and the Real numbers; the Real numbers have a larger cardinality.

Quote
This may be true, but it's a truth that's arbitrary, imposed as it is by the limitations of a digital counting system.
It is true. We've proved it to be true. It's not an arbitrary truth, any more than it is true that 1+1=2. The fact that we use the decimal counting system has absolutely nothing to do with whether or not it is true. I agree that the exact machinations of the proof in its common presentation does rely on base-10 notation, but there are several ways to restate and prove the result without referring to decimal notation. An analogous statement works in binary, ternary, base-67325469253, and indeed any base you care to think of. And if that's still not enough, then I can give a proof that does not rely on any system of "decimally" bits. You can use the continued fraction expansion (https://en.wikipedia.org/wiki/Continued_fraction) of a real number instead of its decimal expansion, and continued fractions do not require decimal points.

All of this is of course largely meaningless, since the proof still stands regardless of how you wish to represent numbers on paper. The proof simply says that no matter how you try to pair up the Natural numbers with the set (0,1), there will always be an infinite amount of real numbers left out.

In order to define anything using numbers, it's necessary to choose a digital "counting distance," which is always arbitrary and finite when observed from analogous reality.
False. Not that this statement actually has any rigorous meaning anyway. As stated before, mathematicians do not define numbers in this manner, and the term "counting distance" has no meaning in this context whatsoever. It is a term you have invented which bears no resemblance to any terminology used in a formal context. And mathematics is all about formal contexts.

Quote
Infinity is an analogous concept, and is therefore incompatible with any digital counting system.
This statement has no real meaning either. Are you trying to say that it is impossible to write down infinity as a number? Of course it is impossible; infinity is not a number. It is not an element of the Naturals, Rationals or Reals. The term "Infinity" is actually much more nuanced than you give it credit, and has various differing meanings based on the context in which it is being used.

For example, in analysis the term "as x tends to infinity" refers to the behaviour when the variable x grows without limit. In systems such as the surreal numbers or hyperreal numbers, infinity is actually used as a number, and you can perform arithmetical operations with it. The context that you are referring to in your post is set theory, however. Specifically the properties of cardinalities of sets.

Either way, your statement is once again meaningless.

Quote
Hence, the mathematical notion of "different sized infinities" (such as Aleph 0, Aleph 1, etc) is necessarily and conceptually flawed; defining their "size" in such a way succeeds only in placing yet another arbitrary finite value on the idea of infinity.
Nope. First off, the mathematical notation (denoting cardinalities as Aleph 0, Aleph 1, etc) has nothing to do with whether or not the theory is valid. We could have called them "Zoobie", "Clomb", "Spoozle" for all the difference that it makes. Which is none; the notation that humans assign to them is only a human-readable representation to the underlying mathematical objects.

You also seem to think that such objects are not definable, even though they most certainly are. Aleph 0 is defined as the cardinality of the Natural numbers. Aleph 1is the cardinality of the set of all countable ordinal numbers. These concepts are indeed well defined; mathematicians wouldn't use them otherwise. Furthermore, we have not "placed yet another arbitrary finite value on the idea of infinity". We have simply given names to the different cardinalities. There is a hierarchy amongst these objects, with aleph 0 being the smallest.

Quote
Infinity is not finite. It is in-finite.
This is completely redundant... Nobody was even saying this anyway???

Quote
Therefore, Mathematics is not the best tool we have for understanding and interacting with the infinite.
Wrong. Completely and utterly wrong. Mathematics is the BEST tool for understanding infinity.

Quote
Perhaps Music and Art are better...
Nope. Music and art lack the rigor and preciseness to even come close.

Quote
... there could even be something that we as humans can't even conceive that would truly allow us to comprehend the incomprehensible.
And this is where I think the core problem lies. You assume that infinity is this magical, mystical, inconceivable force of reality.

It really isn't.

Infinity is actually completely understandable from a mathematical context, as long as you understand the rules and know how to use them appropriately. It seems that you do not, however. Either way, what you have written here shows a complete lack of understanding of infinity from any formal point of view, as well as a poor understanding of mathematics in general.

Mathematicians have been working on this stuff for centuries. We know what we are doing. You do not.


I apologise if my rant comes off as being confrontational or overly hostile. But I cannot abide when people who clearly have no idea what they are talking about decry well established results in any given area. It's this kind of self assured superiority that causes flat earthers, and other nonsensical beliefs.

Offline Proxima

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Re: WillLem's Blog
« Reply #4 on: April 14, 2020, 03:38:36 PM »
What do the ellipses even mean?

That is a gold-standard excellent question, and deserves a much longer answer that I don't have time to attempt right now :P

However, I think it's not the main issue here.

It is of course true that you can't have an infinite list of numbers in the real world. It's equally true that you can't have a perfectly cubical object, or a perfectly straight line. When mathematicians talk about the properties of a cube, they mean an imaginary, idealised cube. You can't touch one or interact with one, but you can prove that the length of the diagonal is sqrt(3) times the length of the side.

Infinite lists fall into the same category. If you don't like my use of ellipses, suppose I say "the set of all natural numbers". That is a well-defined set: for any mathematical object, we know how to test whether it is a natural number or isn't. Then, by convention, I want to use "1, 2, 3, 4..." not to mean "some list of numbers beginning 1, 2, 3, 4" but specifically "the set of all natural numbers". Since the set is well-defined, we may want to talk about it; therefore it helps to have a notation allowing us to do so. It feels like maybe your problem is with the notation rather than the concept?

Offline WillLem

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Re: WillLem's Blog
« Reply #5 on: April 14, 2020, 04:21:40 PM »
First of all, the notion of "counting distance" has no real meaning in terms of actual mathematics.

You're correct here, this is a term I used for want of a more established technical term that I couldn't find whilst writing this post. To explain it, what I mean is the distance between any given number and the next number in a pattern or series of numbers.

For instance:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 has a "counting distance" of (1).

0, 2, 4, 6, 8, 10, 12, 14, 16, 18 has a "counting distance" of (2).

0, 0, 1, 2, 3, 5, 8, 13, 21 has a "counting distance" of (the previous two digits added to each other).


Mathematicians know that the set of real numbers between 0 and 1 has a greater cardinality than the Natural numbers, but their arguments have nothing to do with what you've written here.

Maybe I've misunderstood then: does "cardinality" refer to a group of numbers' tendency towards an infinite value? This is a term I'm unfamiliar with. It's always been my understanding that the Aleph groups are a way to try to define different infinite series?

An analogous statement works in binary, ternary, base-67325469253, and indeed any base you care to think of.

I know what you're saying, and counting systems can be extremely accurate. However, by its definition, an analogous system is continuous. The simplest example I can think of is a circle. Mathematics uses 3.14159 to approximate circular measurements, but pi is yet another example of an ultimately digital, finite number that's used to interact with an analogous shape that's infinite in its nature. Granted, it's accurate enough for everyday purposes, but it's still a digital representation of an analogous truth.

All of this is of course largely meaningless, since the proof still stands regardless of how you wish to represent numbers on paper. The proof simply says that no matter how you try to pair up the Natural numbers with the set (0,1), there will always be an infinite amount of real numbers left out.

But what exactly does this prove, other than: infinite lists are infinite?

It is a term you have invented which bears no resemblance to any terminology used in a formal context. And mathematics is all about formal contexts.

Hmm, this is probably the reason I was never any good at it! :crylaugh:

Quote
Infinity is an analogous concept, and is therefore incompatible with any digital counting system.
This statement has no real meaning either. Are you trying to say that it is impossible to write down infinity as a number?

No, I'm not saying that, but I can see where the misunderstanding may have occurred. I suppose a better way to put it might be: infinity is a concept greater than that which can be accurately represented and/or understood using a digital counting system. But even that doesn't really do justice to what it is I'm thinking. It's difficult to put these things into words sometimes.


In systems such as the surreal numbers or hyperreal numbers, infinity is actually used as a number, and you can perform arithmetical operations with it. The context that you are referring to in your post is set theory, however. Specifically the properties of cardinalities of sets.

I've never heard of surreal numbers (or indeed hyperreal ones), that sounds interesting!

I do appreciate you clarifying the context of my post, though. I was only vaguely aware of set theory and number cardinality, and didn't really know what they meant. I have a better understanding of it now, and realise that this, indeed, is likely what I'm referring to as far as the maths goes. Regarding infinity though, I still believe that this is a concept that's ultimately beyond mathematics.


Either way, your statement is once again meaningless.

You've said this quite a lot in your post. I realise it was something of a rant, and clearly something you're passionate about, and ultimately it's good to have your engagement. However, since you feel quite happy to reiterate this statement again and again, I'll take issue with it: if you have failed to recognise the meaning in something that someone has taken the time to formulate and share, that's on you. Discussion often helps to reveal meaning, especially concerning topics that rely very heavily on terminology, level of education, abstract perspective, and/or ongoing investigation.


Quote
Hence, the mathematical notion of "different sized infinities" (such as Aleph 0, Aleph 1, etc) is necessarily and conceptually flawed; defining their "size" in such a way succeeds only in placing yet another arbitrary finite value on the idea of infinity.
Nope. First off, the mathematical notation (denoting cardinalities as Aleph 0, Aleph 1, etc) has nothing to do with whether or not the theory is valid. We could have called them "Zoobie", "Clomb", "Spoozle" for all the difference that it makes. Which is none; the notation that humans assign to them is only a human-readable representation to the underlying mathematical objects.

My issue isn't necessarily with the notation here, it's with the concept of "different sized infinities" or "different sized infinite groups". If something is infinite, it has no finite limit. Therefore how can it have a "size" or be put into a "group", as we understand it? If it can be grouped/sized, then it isn't infinite. I could be wrong about this, but it doesn't seem like I am. It's this aspect of it that I'm looking to discuss and understand better, really.


You also seem to think that such objects are not definable, even though they most certainly are. Aleph 0 is defined as the cardinality of the Natural numbers. Aleph 1is the cardinality of the set of all countable ordinal numbers. These concepts are indeed well defined; mathematicians wouldn't use them otherwise. Furthermore, we have not "placed yet another arbitrary finite value on the idea of infinity". We have simply given names to the different cardinalities.

OK, fair enough: cardinality is a term that's new to me. My understanding was that the Aleph groups referred to the infinite nature of what they are used to define. Clearly, I have misunderstood this.


Mathematics is the BEST tool for understanding infinity.

Why?


Music and art lack the rigor and preciseness to even come close.

Show me a hypothetically infinite list of seemingly meaningless numbers, and I'll initially feel confused until either someone explains them, or I apply pre-existing knowledge to make sense of them. Then, I'll probably "understand" the concept at least in a basic, cognitive sense. Show me a picture of a corridor that seems to go on forever, or a video of an endlessly spinning globe, or even just a picture of a circle, and I'll have a far better, deeper comprehension of what "infinity" is.


And this is where I think the core problem lies. You assume that infinity is this magical, mystical, inconceivable force of reality.

Perhaps this is true. But my point is more that I do not believe that mathematics is the best way to interact with it, whatever it is, whether it's a magical, mystical force of reality or a boring, easy-to-understand force of reality.


Either way, what you have written here shows a complete lack of understanding of infinity from any formal point of view, as well as a poor understanding of mathematics in general.

I'll grant you that, my understanding of mathematics is likely to be far inferior to yours. I am interested though, and I do try.


But I cannot abide when people who clearly have no idea what they are talking about decry well established results in any given area. It's this kind of self assured superiority that causes flat earthers, and other nonsensical beliefs.

By the same sword, I cannot abide when people hold onto "established results" so rigidly and stubbornly that they refuse to open their mind up to any possible alternative perspective, however reasonable/absurd it may be. So, at least we feel the same way about that, if only from different perspectives! ;P

For the record, though: I am not a flat Earther! There is far too much evidence to prove that Earth is a sphere, if not a perfect one. However, I do like the idea of Antarctica being a huge ice wall surrounding the Earth, and explorers seeking to find what's on the other side of it. That kind of idea yields great stories!

I can see the opportunity to learn a lot here. Maths was never my strong point, and every time I venture towards it I find something I either don't agree with, don't understand, or that just seems like it's trying to use tangible, rigid concepts and theories to express the intangible. I don't dislike maths though, there's massive beauty and order in a lot of it, and I continue to find that attractive. I think my brain is just wired to question established formalities, and this can sometimes prevail in any internal dialog or learning process whether I like it or not! :lemcat:
« Last Edit: April 14, 2020, 05:11:32 PM by WillLem »

Offline Proxima

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Re: WillLem's Blog
« Reply #6 on: April 14, 2020, 05:42:19 PM »
You're correct here, this is a term I used for want of a more established technical term that I couldn't find whilst writing this post. To explain it, what I mean is the distance between any given number and the next number in a pattern or series of numbers.

For instance:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 has a "counting distance" of (1).

The term you want is first differences. ("First" because we can then go on to talk about second differences, i.e. the distance between each first difference and the next, and so on. Thus, the sequence 1, 4, 9, 16, 25... has first differences of 3, 5, 7, 9... and second differences of 2, 2, 2, 2....)

Quote
Maybe I've misunderstood then: does "cardinality" refer to a group of numbers' tendency towards an infinite value? This is a term I'm unfamiliar with. It's always been my understanding that the Aleph groups are a way to try to define different infinite series?

"Cardinality" just means "size", as in the size of a set, and it's used because it's more specific, the word "size" having many possible shades of meaning.

The cardinality of a set is the number of elements it contains. For example, the set of all even primes has cardinality 1 (there is only one even prime, namely 2).

Aleph-0 is defined as the cardinality of the set of natural numbers. Note that I say "set" here and not "series". These are two different concepts. A set is a collection of objects, independent of ordering.

Quote
I know what you're saying, and counting systems can be extremely accurate. However, by its definition, an analogous system is continuous. The simplest example I can think of is a circle. Mathematics uses 3.14159 to approximate circular measurements, but pi is yet another example of an ultimately digital, finite number that's used to interact with an analogous shape that's infinite in its nature. Granted, it's accurate enough for everyday purposes, but it's still a digital representation of an analogous truth.

Again I think you are confusing notations with the objects being referred to. Let's take 1/3 first as a simpler example. The number 1/3 is well-defined: it's the result of dividing 1 by 3, or the number that represents how much of a cake you have if you divide it into three equal slices. I could also say "the number that is the solution to 3x - 1 = 0", and mathematicians would happy with that, but one should be wary of that sort of definition unless it's really clear that there is exactly one such number (which, in this case, there is).

To write 1/3 as a decimal, we first ask: How many times does 1/10 go into it? Three. The amount left over, (1/3 - 3/10), can be calculated precisely by fraction arithmetic, and is exactly 1/30. How many times does 1/100 go into this remainder? Three. And so on. This is just the standard way of converting any number to a decimal. The end result, 0.333333..., means that 1/3 is made up of three tenths, three hundredths, three thousandths, and so on.

I say "and so on" because there is a clear pattern here, but it's important to note that if you carry out this process literally, it would never end. You can never express 1/3 exactly as a sum of tenths, hundredths, thousandths and so on; there will always be a little bit left over. Still, as soon as we see that the pattern is "every digit is 3", we are happy to put "..." at the end. In this case, it can be proved that the process will in fact continue in the same way for ever; every single digit will always be 3; there are no hidden surprises a few million digits down the line.

That doesn't mean the number 1/3 is ill-defined, though! It just means it doesn't fit neatly into our decimal system.

Pi is a similar case, but a bit more complicated. There is a precise number, which we call pi, that is the ratio of the circumference to the diameter of any circle (or any number of other equivalent definitions). If you try to work out its decimal representation, you get "3.141592652589793238462643383279502884197169399375105820974944592307816406...." -- a higgledy piggledy jumble of digits with no pattern. That doesn't mean the number pi itself is ill-defined. It just means that when we say "pi" or use it in a formula, we mean the exact number, not any of the decimal, fractional or other approximations.

(On a side note: you mean "analogue". "Analogous" is a separate word that means "behaving in a similar way to something, so that you could draw an analogy between them".)

Quote
But what exactly does this [the diagonal argument] prove, other than: infinite lists are infinite?

It proves that some infinite sets are larger (have greater cardinality) than others. Again, think of the set itself -- the mathematical collection of all numbers between 0 and 1 -- not any physical list.

Quote
My issue isn't necessarily with the notation here, it's with the concept of "different sized infinities" or "different sized infinite groups". If something is infinite, it has no finite limit. Therefore how can it have a "size" or be put into a "group", as we understand it? If it can be grouped/sized, then it isn't infinite. I could be wrong about this, but it doesn't seem like I am. It's this aspect of it that I'm looking to discuss and understand better, really.

Please go back to my first post. I explained that size (or cardinality) is defined in terms of being able to put a set in one-to-one correspondence with another. Any two sets with cardinality 3, for example, can be put in one-to-one correspondence with each other. No set with cardinality 3 can be put in one-to-one correspondence with cardinality 4; if you try to pair the elements, you will always have one left out.

Quote
By the same sword, I cannot abide when people hold onto "established results" so rigidly and stubbornly that they refuse to open their mind up to any possible alternative perspective, however reasonable/absurd it may be. So, at least we feel the same way about that, if only from different perspectives! ;P

Well, this is one of the advantages of mathematics. Any result we can rigorously prove to be true is established for all time.

Offline ∫tan x dx

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Re: WillLem's Blog
« Reply #7 on: April 14, 2020, 06:42:28 PM »
First off, I want to apologise again for any insulting statements I made in my previous post. My intention was not to belittle you for not having the same kind of formal training that I or others my have.

It is certainly true that mathematics can be an especially opaque subject, especially regarding topics that have little or no basis in the everyday world around us - such as infinity. I would therefore like to take this opportunity to teach you about some of the concepts involved.

I think what is best here is to start with some explanations of some of the terminologies mentioned in this thread. That way we are all at least talking about the same things.

Quote
... what I mean is the distance between any given number and the next number in a pattern or series of numbers
A sequence of numbers is generally given by what is known as a formula, or perhaps a function.

A quick aside, functions are themselves mathematical objects. They are sometimes referred to as mappings, since in a technical sense they map between sets. That is, a function is fed elements from one set, and spits out an element from another set. (Some functions can actually have multiple inputs, but for the sake of argument, we'll restrict ourselves to single input -> single output.)

Now, consider a function called "icecream". This function takes a single input of a person, and outputs a flavour of ice cream which that person thinks is best. So the input set for this function (called the Domain in mathematical terms) is the set of all People (Denoted here as P). Likewise, the output set (the Range in mathematical terms) is the set of all ice cream flavours (Denoted here as F).

Thus we may define:
Code: [Select]
icecream : P -> FSo icecream maps People to Flavours.

Let's plug in a few values:
icecream(tan x dx) = Chocolate.
icecream(Proxima) = Vanilla.
icecream(WillLem) = Mint.
(I apologise in advance for any inaccuracies in the output of this function).

Now this is all very well and good, but it is by no means exhaustive. In order to fully define this icecream function, we would have to poll every single person on earth - or to put it another way, each element of the input set has to be special cased. Which is why we generally prefer true mathematical functions that are given by a formula - we don't have to do a lot of silly work for every possible number, simply plug into the formula and there is the answer.

Let's get back to maths.

For example, the sequence
0, 1, 2, 3, 4, 5, ...
may be defined as
Code: [Select]
f : N -> N(where in this case, N stands for the Natural numbers: N = {0, 1, 2, 3, ...})
So f is a function (or mapping) from the Naturals to the Naturals. The function can be evaluated using the formula
Code: [Select]
f(n) := n
Let's pick apart this notation. The n in the parenthesis is the parameter - the input to the function. The ':=' bit says 'this is how to calculate the result', and the n at the end is what the output is. In this case, we simply return the input. Thus the zeroth term is 0, the first is 1, and so on.

We may do this since the set of Natural numbers N is in fact a well defined mathematical object. It is an infinite set which contains every single non-negative integer by definition. There are no qualms about compiling an infinite list in a finite universe, because we are working with abstract mathematical objects and not any kind of actual physical object. That's one of the things you can do in mathematics; define something rigorously and you can use it as you please, even if it is an infinite set.

As for your other examples:

0, 2, 4, 6, 8, ...
Define
Code: [Select]
g : N -> Nwhere g(n) := 2n
So this function g doubles the input.

0, 1, 1, 2, 3, 5, 8, 13, 21, ...
Define
Code: [Select]
fib : N -> N           fib(0) := 1
where{ fib(1) := 1
           fib(n) := fib(n-1) + fib(n-2), for n >=2

Now this one is obviously trickier to define, largely because of its recursive nature. But it is a valid definition; it results in a single unambiguous answer for each of the inputs. Like with the icecream function, the inputs for 0 and 1 are special cased - both are defined to specifically yield a value of 1. But for all other input values, we recursively call the fib function over and over again until it eventually terminates with a fixed answer.

There is obviously more nuance to functions than just this, but as an introduction I think this will do.

Now, let's move onto cardinality.
https://en.wikipedia.org/wiki/Cardinality

The cardinality (sometimes called "Size") of a set S is defined as the "number of elements" of S.

Wait a minute, we can make a function out of this!
Code: [Select]
card : {Sets} -> {Cardinalities}Huh. This is kind of weird, isn't it?
So, the first part of the mapping (the Domain) is
Code: [Select]
{Sets}That is the Set of all sets. Now this is indeed a nebulous concept; sets containing sets??? Whatever next?!1
And as for the output we have
Code: [Select]
{Cardinalities}Which seems even more esoteric.

So... What is a cardinality then?

It's like a natural number. But different. And there are infinite cardinalities.

Well that's not really helpful, is it?

We'll start somewhere simpler. Consider the following set:
Code: [Select]
S1 = {13, 652, 7632411}What is the cardinality of this set? Three, right? Simple.
What about this one:
Code: [Select]
S2 = {Chocolate, Vanilla, Mint}Also three. Easy enough.
This one?
Code: [Select]
S3 = {Hurgle, Blorpo, Snrouse}I have no idea what those things are, but I know there are three of them.

So... What's the point?
Well the point is that even though these sets are all utterly different, they still have something in common with each other - their cardinality. They are the same size.

This gives us a way to compare sets. If we have a set with 100 elements, then that set is larger than a set with 20 elements. We don't necessarily need to compare the sets directly, we can instead compare their cardinalities.

But how can we do this for arbitrary sets? As Proxima suggested, we look for bijections.

Think of an ordinary movie theatre, with some unknown number of seats. Now imagine that a crowd of cinema patrons enters the theatre. (I realise how unrealistic such a thing would be in the modern day COVID-19 world, but bear with me here). Even if we do not know the exact numbers involved, either seats or patrons, we can still make some true statements about whether or not the cinema is full.

Suppose an usher looks into the theatre and makes some observations about the relationships between seats and patrons:
  • If every patron has a seat, but there are seats without patrons, then we know that the theatre is not at max capacity - there are more seats than patrons. That is, the cardinality of seats is greater than the cardinality of patrons.
  • If every seat has a patron, but there are still patrons standing, then we know that the theatre is overloaded. The cardinality of patrons is greater than the cardinality of seats.
  • If every seat has a patron and every patron has a seat, then we are at the exact capacity. The cardinality of patrons is equal to the cardinality of seats.

This is the power of bijections! A bijection between two sets is a pair of mappings (oh hey, these are functions!) which have certain properties.
  • Both mappings must be injective.
  • Both mappings must be surjective.
Uh... more mathematical terminology.

Injective:
Suppose we have a one-way mapping between two sets f : A -> B.
f is injective if no two elements of A are mapped to the same element in B.
For example, consider the following two functions:
Code: [Select]
g : N -> N, g(n) := 2nand
Code: [Select]
h : Z -> N, h(n) := n*nNote, Z here means the set of all integers, positive and negative. Z = {... -3, -2, -1, 0, 1, 2, 3, ...}

Now, can you see why g is injective but h is not?
g is a "doubling" function. It simply doubles the input. It is easy to see that no two distinct inputs are mapped to the same output - if you take any two different natural numbers (like 5 and 6) and double them, you ain't going to get the same answer (10 and 12).

Whereas h is a "squaring" function. What's the problem here? Look at the elements +2 and -2:
Code: [Select]
h(-2) = (-2)*(-2) = 4
h(2) = 2*2 = 4
Ah. Two distinct elements are both mapped to the same result. The function h is not injective.

Surjective:
This one is more subtle. Our mapping f : A -> B is surjective if for each element b in B, there is at least one element a in A that is mapped onto it.

For example:
Code: [Select]
k : N -> N, k(n) := n + 50The function k simply adds 50 to each input. Now, both the Domain (the set of valid inputs) and the Range (the set of possible outputs) of k are defined as N. So let's ask a question:
Is there any input which is mapped onto the value 42?

The answer is no, since the input set only contains non-negative integers, and adding 50 to any of those cannot result in a value of 42. Thus k is not surjective.

If you look back to the movie theatre example, we tried to construct simple mappings between patrons and seats.
Remember that there are two mappings, one going each way
Patrons -> Seats (does each patron have a seat?)
Seats -> Patrons (is each seat occupied by a patron?)

It was only when the number of seats and patrons matched that we had a bijection. You can check this against the definitions of injective and surjective as above.


Okay. So... What are we doing again?


Quick recap:
  • We want to rigorously talk about sizes of sets, including infinite sets.
  • Cardinalities are ways of comparing sizes of sets without directly caring about the specific contents of each sets.
  • If we can construct a pair of mappings (a bijection) between two sets, we know those sets have the same cardinality (same size).

As Proxima mentioned in their post, this is how one-to-one correspondences work. A bijection between two sets is a one-to-one correspondence!

Proxima also gave an excellent example of a bijection between two distinct infinite sets,
Code: [Select]
{0, 1, 2, 3, 4,  5,  6...}
 |  |  |  |  |   |
 v  v  v  v  v   v
{0, 1, 4, 9, 16, 25, 36...}

Here the input set is N, the natural numbers, and the output is the set of square numbers (denoted here as Sq). Both sets are infinite, but they are of the same cardinality.
The bijections involved are to do with squaring/taking square roots. You can check that the functions
Code: [Select]
f1 : N -> Sq, f1(n) = n*n
f2 : Sq -> N, f2(n) = sqrt(n)
Are indeed both valid functions in this context.


So onto the main result - Cantor's diagonal argument.
https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

Cantor's diagonal argument that there is no bijection between the Natural numbers N, and the Real numbers between 0 and 1 - denoted (0,1).
It is certainly possible to create arbitrary mappings between these sets, but the point is that no matter how you do it, the functions involved will fail in at least one of injectivity or surjectivity.

And this is the beauty of the argument: by attempting to list every real number we are implicitly trying to create a mapping between the two sets! We say that the first element is such-and-such, and the second is blah-blah, and so on! But no matter how clever we try to be when listing the reals, we can never create a bijection, there are always an infinite number of Reals still not on the list! No function f : N -> (0,1) can be both injective and surjective! Thus we are forced to conclude that the set (0,1) is in fact, larger than N!


I hope you find this illuminating. Yes, the concepts involved can be tricky to wrap your brain around, but there really is no mystery about infinity or infinite sets. And of course, reality is under no obligation to coincide with your personal interpretations of it.
The fact that infinite sets can be larger than one another is a well established consequence of mathematical logic. Just because you do not fully understand the concepts involved, does not mean that the mathematicians are wrong. In fact, the reverse is almost certainly true. The fact that you made your original post with such blatant misunderstandings suggests to me a great deal of arrogance. This is what I take issue with; this is what irritates me.

Quote
Show me a picture of a corridor that seems to go on forever, or a video of an endlessly spinning globe, or even just a picture of a circle, and I'll have a far better, deeper comprehension of what "infinity" is.
I find it ironic that you admit your picture only "seems to go on forever". Even your video of the spinning globe is necessarily finite in length. Even then, with the endlessly spinning globe, what is infinite about this? The number of revolutions? Number. Hmm. That's mathematics, then. Same for the circle. Infinity is an inherently mathematical concept, and your suggestion that mathematics is not the best tool to understand it smacks of willful ignorance. What would be the best tool then?


*** Note 1: Actually it gets worse. The "set of all sets" isn't technically a set; such a collection is too big to be a set! It is instead an example of what is known as a Proper Class, which I'm not going to go into here because that would be opening up a whole other can of worms. That's the thing about mathematics: the more you dig, the worse (or better) it gets.

Offline kaywhyn

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Re: WillLem's Blog
« Reply #8 on: April 14, 2020, 09:40:39 PM »
How awesome there are other math enthusiasts like me here in Proxima and integral of tan x dx (love your user name, btw)! I have two math degrees,  so you can imagine the numerous amount of math courses I had to take as a math major in the last 11-12 years. In particular, my math degrees are specifically for the teaching field (i.e, not applied or pure math). However, set theory was something that I didn't really study in college/university, so I'm not as well-versed in it as tan x dx is in it, though I'm familiar with it enough to be able to follow his entire explanation completely. Also, even with two math degrees, I'm definitely not an extreme expert on the matter. I cannot do the Putnam problems, for example.

I think I can see where you're confused, WillLem. I, too, initially had trouble with the concept of how some infinite sets can have the same size when I first read about this on my own years ago, which I believe is the source of your confusion? However, with tan's explanation, I hope it's clear to you that the set of natural numbers and the set of even positive integers have the same cardinality (size), even though you would think the set of even numbers is smaller due to how it misses some numbers, eg, {1, 3, 5, ...} while the set of natural numbers includes the missing odds and the evens together. As he has said, the correct way is to look at it as a bijection, as we say infinite sets have the same cardinality if there exists a bijection between them by definition. Yes, both sets are infinite, and so it's impossible to say what the exact cardinality of each set is with a specific number, but the fact that a bijection exists means both sets are still countable, countable because a bijection exists (definition of countable set). Even more, they're countably infinite.

In contrast, it has been proven that the set of real numbers is uncountable because no matter how you do it, you cannot make a bijection with the set N = {0, 1} and R. In other words, it does not satisfy the definition of what it means to be countable, and so the reals is uncountable, uncountably infinite to be exact.

This is what I love about mathematics: the results are intriguing and have beauty to them. In contrast to other subjects, it's easy to check people's work, as most problems you either have the correct answer or you don't. The results aren't going to change on a daily basis thanks to mathematical properties.
https://www.youtube.com/channel/UCPMqwuqZ206rBWJrUC6wkrA - My YouTube channel and you can also find my playlists of Lemmings level packs that I have LPed
kaywhyn's blog: https://www.lemmingsforums.net/index.php?topic=5363.0

Offline grams88

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Re: WillLem's Blog
« Reply #9 on: April 14, 2020, 11:04:57 PM »
Yes we are getting more people doing their own blogs, I like that. L) We get to see the person behind the lemmings player. I'm not a flat earth guy myself and have to say the discussion here seems quite in-depth.

Offline WillLem

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Re: WillLem's Blog
« Reply #10 on: April 15, 2020, 08:33:14 PM »
The term you want is first differences

...

"Cardinality" just means "size", as in the size of a set, and it's used because it's more specific, the word "size" having many possible shades of meaning.

Thanks! This makes sense.

So then, the question I still have is... how can one infinite set be bigger than another infinite set if both are infinite? Would this be one way of understanding it:

You could, in theory, have a Lemmings level that is infinitely long but still 160px high, whilst at the same time having a Lemmings level that is infinitely long and infinitely high. Whilst both have infinite properties, one is bigger than the other.

I say "and so on" because there is a clear pattern here, but it's important to note that if you carry out this process literally, it would never end. You can never express 1/3 exactly as a sum of tenths, hundredths, thousandths and so on; there will always be a little bit left over... That doesn't mean the number 1/3 is ill-defined, though! It just means it doesn't fit neatly into our decimal system.

I have a question about this that I can't quite put into words, but... basically, doesn't this prove that number systems are flawed?

(On a side note: you mean "analogue". "Analogous" is a separate word that means "behaving in a similar way to something, so that you could draw an analogy between them".)

Right you are, I've corrected the OP.

It proves that some infinite sets are larger (have greater cardinality) than others.

How can a set have a definable size if it is infinitely large?

First off, I want to apologise again for any insulting statements I made in my previous post. My intention was not to belittle you for not having the same kind of formal training that I or others my have.

That's OK, I understand it can be frustrating when people seem to be ignorant and you certainly seem passionate about this subject, which is great! I appreciate you taking the time to write such a substantial reply, as well.

Thus we may define:
Code: [Select]
icecream : P -> FSo icecream maps People to Flavours.

Let's plug in a few values:
icecream(tan x dx) = Chocolate.
icecream(Proxima) = Vanilla.
icecream(WillLem) = Mint.

I do love a bit of Mint ice cream. ;P

I understand this very well, so thank for the explanation. One question that arises might be: how could this function be expanded to account for variations within the same ice cream flavour, or the fact that any given person might choose a different flavour as their favourite on a given day?


There are no qualms about compiling an infinite list in a finite universe, because we are working with abstract mathematical objects and not any kind of actual physical object. That's one of the things you can do in mathematics; define something rigorously and you can use it as you please, even if it is an infinite set.

I can certainly see why that's incredibly useful. The question here would be: if mathematics is a system of understanding and interacting with reality in the abstract, as useful and accurate as it tends to be, is it not still dependent on its creators (i.e. the human race, specifically mathematicians) for its development, definition and maintenance?; and, that being the case, is it not therefore limited by current existing limits of our understanding?

If so: then, surely, further investigation and development is needed - and this may involve consideration of radical or seemingly nonsensical ideas (although not necessarily) in order to expand.

If not: then how can we prove it?

There is obviously more nuance to functions than just this, but as an introduction I think this will do.

I understood your explanations perfectly, thanks for that! :thumbsup:

This gives us a way to compare sets. If we have a set with 100 elements, then that set is larger than a set with 20 elements. We don't necessarily need to compare the sets directly, we can instead compare their cardinalities.

How can there be a cardinality of infinite size if infinity itself is not a number. I think this is where my main question is coming from: I understand that a set of 3 things, whether it's fruit, numbers, snorples or anything else, has a cardinality of 3. Easy enough.

But a set of infinite things suddenly doesn't match with the use of numbers to define its size, and is therefore incompatible. I may be wrong about this, of course, but this is just how it appears to me given the information I have.

A bijection between two sets is a one-to-one correspondence!

This is another thing I'm struggling to get my head around: my above question regarding Lemmings level sizes states it well enough, but to refer to bijections specifically:

Take an infinite set, A. It has infinity things in it (which is already a slightly flawed statement since infinity is not a number, but bear with me...)

We then get another infinite set, B. It also has infinity things in it.

The bijection rule would have me understand that both sets must be the same size, with each value in set A corresponding to a value in set B. Even in the case of the two Lemmings levels (one Infinity x 160 pixels, the other Infinity x Infinity pixels) has some bijection: each 160-pixel-tall vertical column of pixels in the first level corresponds to an infinite column of pixels in the second level, so in this sense, they have bijection, since we aren't concerned with what the value is, just whether it has correspondence.

Therefore, it seems that both levels have the same cardinality. If it works like this for an example of a rectangle 160 x infinity and a square infinity x infinity, then why doesn't it work for natural numbers and real numbers?

And this is the beauty of the argument: by attempting to list every real number we are implicitly trying to create a mapping between the two sets! We say that the first element is such-and-such, and the second is blah-blah, and so on! But no matter how clever we try to be when listing the reals, we can never create a bijection, there are always an infinite number of Reals still not on the list!

Again, totally understand what's happening here. However: if the list can always have an extra value added to it, then it is indeed infinite. But the same is true of the infinite list of natural numbers. So:

For example, let's just say that the 337th number on Cantor's list is:

0.947374092730918749703986518724977...

The 1,689,236,899,172th number is:

0.74928370942803642739287097209847098...

The 1,456,423,978,354,535,764,234,980th number is:

0.370927309487203947209387409286906902...

I could go on... both lists can always be added to, so how can they not have the same cardinality, as we understand it? And... if they can be different sized sets, then this is what leads me to question the integrity of the idea of "infinite sets", particularly with reference to their size/cardinality.

I hope you find this illuminating. Yes, the concepts involved can be tricky to wrap your brain around, but there really is no mystery about infinity or infinite sets. And of course, reality is under no obligation to coincide with your personal interpretations of it.

Haha - believe me, I know it doesn't! ;P

However, I would disagree that there is no mystery about infinite sets: it's a mystery we're currently engaging in discussion about. If it wasn't a mystery, there would be nothing to explain from either side.

The fact that you made your original post with such blatant misunderstandings suggests to me a great deal of arrogance. This is what I take issue with; this is what irritates me.

I apologise for this, but there may have been a misunderstanding: I'm not saying that mathematics or mathematicians are wrong, I'm saying that I am investigating the concept of infinity from a mathematical point of view and finding only questions, rather than answers. I've updated the OP so it's worded more carefully and in a more open, question-asking tone so as not to cause further irritation or misunderstanding.

Even your video of the spinning globe is necessarily finite in length. Even then, with the endlessly spinning globe, what is infinite about this? The number of revolutions? Number. Hmm. That's mathematics, then.

If the globe were a uniform colour and moving at an inconstant speed, it would be impossible to visually measure the number of revolutions. The fact that it is spinning at all is what suggests the concept of infinity, not that the spinning can be measured.

Infinity is an inherently mathematical concept, and your suggestion that mathematics is not the best tool to understand it smacks of willful ignorance. What would be the best tool then?

Again, I'm not suggesting that maths isn't the best tool, I'm simply asking the question as to whether or not it is. Perhaps I worded the question as a somewhat ignorant-sounding statement. My bad, I apologise if that's how it came across.

Since you ask, I would say: I don't know what the best tool is, that's why I'm investigating the subject.
« Last Edit: April 15, 2020, 08:41:58 PM by WillLem »

Offline Proxima

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Re: WillLem's Blog
« Reply #11 on: April 15, 2020, 09:22:15 PM »
I'll quote things a bit out of order, since it might help to clarify the basic concepts before using them to build more complex ones.

Quote
However: if the list can always have an extra value added to it, then it is indeed infinite. But the same is true of the infinite list of natural numbers. [...]

I could go on... both lists can always be added to, so how can they not have the same cardinality, as we understand it?

Not entirely sure what you mean by "can be added to". A set of any size can be added to, in the sense that you can add another element to it. But of course, for a set of size 3, adding an element makes it size 4. Indeed, for any finite size, adding an element increases the cardinality by 1. But for an infinite size (cardinality), adding an element does not change the cardinality, and in fact this is precisely the mathematical definition of "infinite".

In case this isn't clear, here's an example. Take the set of positive integers {1, 2, 3, 4...} and add the element 0. The two sets can be put into one-to-one correspondence:

Code: [Select]
{1, 2, 3, 4, 5...}
 |  |  |  |  |
 v  v  v  v  v
{0, 1, 2, 3, 4...}

So, both sets have the same cardinality, and this cardinality is infinite. Remember that "infinity" is not the name of a number; it's a description of a cardinality. When we say a cardinality is infinite, we mean that adding a single element does not change the cardinality.

Quote
How can a set have a definable size if it is infinitely large?

The size (cardinality) of the set of positive integers {1, 2, 3, 4...}" is clearly larger than 1, 100, 1000000, or any other finite number. So we need a new name for it: aleph-0. We define aleph-0 to mean "the cardinality of the set of positive integers". Any set that can be put into one-to-one correspondence with the set of positive integers has this cardinality. There you go, I defined it 8-)

Quote
So then, the question I still have is... how can one infinite set be bigger than another infinite set if both are infinite? Would this be one way of understanding it:

You could, in theory, have a Lemmings level that is infinitely long but still 160px high, whilst at the same time having a Lemmings level that is infinitely long and infinitely high. Whilst both have infinite properties, one is bigger than the other.

That's a resonable guess, but no. Firstly, note that the even positive integers can be put into one-to-one correspondence with all positive integers:

Code: [Select]
{1, 2, 3, 4, 5...}
 |  |  |  |  |
 v  v  v  v  v
{2, 4, 6, 8, 10...}

So, (2 x aleph-0) = aleph-0.

Similarly, 160 x aleph-0 = aleph-0, so your first level (160 by aleph-0) has aleph-0 pixels. To envisage this, suppose you try to count the pixels by starting from one corner, going down the first column, then moving to the next column and going up, and so on... this counting process would include every pixel on the level, so the number of pixels is countable (and remember that "countable" is the same as "has cardinality aleph-0").

Now, you might think the second level (aleph-0 by aleph-0) has greater cardinality, but in fact it doesn't. Suppose it has a top-left corner and continues right and down to infinity. We can count the pixels in a pattern like this:

Code: [Select]
1--2  9--10
   |  |  |
4--3  8  11
|     |  |
5--6--7  12
         |
16-15-14-13

Since the pixels can be counted, i.e. can be put into one-to-one correspondence with the positive integers, the set of pixels on this level is countable (aleph-0). In other words: aleph-0 x aleph-0 = aleph-0.

Quote
I have a question about this that I can't quite put into words, but... basically, doesn't this prove that number systems are flawed?

It proves that one system can't do everything. Our decimal system is perfectly adequate for its intended purpose: expressing real numbers to any desired degree of accuracy and making it easy to compare the sizes of numbers and perform arithmetic with them. There are simple operations we sometimes want to do that can't be done within the decimal system, such as adding 1/3 and 1/6. To get around this, we can either add to the decimal system (by adding the recurring notation, together with rules for performing arithmetic on recurring decimals), or use a different system.

Similarly, our decimal system can't express all possible cardinalities of sets, which is why we add the notation "aleph-0". That doesn't mean aleph-0 is ill-defined, because I have given a precise definition: it's the cardinality of the set of positive integers. It also doesn't mean the decimal system is "flawed", because expressing the cardinalities of infinite sets is outside its intended purpose in the first place.

Offline Proxima

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Re: WillLem's Blog
« Reply #12 on: April 15, 2020, 09:40:10 PM »
This is another thing I'm struggling to get my head around: my above question regarding Lemmings level sizes states it well enough, but to refer to bijections specifically:

Take an infinite set, A. It has infinity things in it (which is already a slightly flawed statement since infinity is not a number, but bear with me...)

We then get another infinite set, B. It also has infinity things in it.

The bijection rule would have me understand that both sets must be the same size, with each value in set A corresponding to a value in set B. Even in the case of the two Lemmings levels (one Infinity x 160 pixels, the other Infinity x Infinity pixels) has some bijection: each 160-pixel-tall vertical column of pixels in the first level corresponds to an infinite column of pixels in the second level, so in this sense, they have bijection, since we aren't concerned with what the value is, just whether it has correspondence.

Therefore, it seems that both levels have the same cardinality. If it works like this for an example of a rectangle 160 x infinity and a square infinity x infinity, then why doesn't it work for natural numbers and real numbers?

You start by talking about two sets of pixels, but then you say that each column in one set corresponds to a column in the other. That's not a bijection. A bijection is specifically a correspondence whereby each element (here, each pixel) in one set is paired with a unique element of the other.

Now, it is possible to set up a bijection between a 160 x infinity level and an infinity x infinity level. I showed how in my post above, but to reiterate:

Code: [Select]
1--2--3--4...159--160
                   |
320...--163--162--161
 |
321--322--323...

Code: [Select]
1--2  9--10  25--26
   |  |  |   |   |
4--3  8  11  24  27
|     |  |   |   |
5--6--7  12  23  28
         |   |   |
16-15-14-13  22  29
|            |   |
17-18-19-20--21  30...

It doesn't work for natural and real numbers because there is no way to set up a bijection between the natural numbers and all real numbers. This is what the diagonal argument proves. If there were such a bijection, we could turn it into a list of all real numbers (by listing whichever real number is paired with 1, then the real number paired with 2, and so on), and then use the diagonal argument to construct a real number that is not on the list. That's a contradiction; therefore there is no such bijection.

Offline ccexplore

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Re: WillLem's Blog
« Reply #13 on: April 15, 2020, 10:44:43 PM »
Finally took a dip here.  So I guess this is the "WillLem is insulting all the mathematicians out there" topic. ;P

So basically, just because you seem to personally have trouble grasping the mathematical treatment of infinity, therefore they must be "necessarily and conceptually flawed".  I can't even find a smiley for this. :XD:

Quote
Show me a picture of a corridor that seems to go on forever, or a video of an endlessly spinning globe, or even just a picture of a circle, and I'll have a far better, deeper comprehension of what "infinity" is.

That's completely equivalent to mathematicians giving the set of all natural numbers ({1, 2, 3, …}) as an example of infinite set.  You seem to argue the set of all natural numbers is abstract and does not "exist in the real world", but does either of your examples actually exist in the real world either?  The picture of the corridor is almost certainly of a real corridor that is in fact not infinite, you are only imagining in your mind that it goes on forever.  The endlessly spinning globe in the real world will likely be rudely terminated at some point in time far in the future, be it due to our Sun going nova, the heat-death of the universe, etc.  So they are actually just as abstract as talking about a set of natural numbers created simply by endlessly adding 1 to a previous number to get more and more numbers into the set.  It is quite fair to argue that infinity is intrinsically an abstract concept.  You can try to ground it against "real-world examples" that are approximations, but they are still approximations that require some mental idealizations to be absolutely true to the properties of infinity.

And don't be so sure about the world being analog either.  Some physicists postulates that space and time themselves might be subjected to the laws of quantum mechanics and therefore may actually be quantized, such that there may actually be a minimum length/duration enforced by the laws of physics.  In effect, our physics may well turn out to be digital and not truly analog.  There's just so much we don't yet fully understand about the physics of our world.

You mentioned music and art, but I'm having trouble following what examples you have mentioned that are actually coming from those disciplines and relating to infinity.  I'm rather surprised you didn't mention philosophy.  The nature of infinity had actually been historically discussed as philosophy far, far longer than the modern mathematical set-theoretic treatment had existed.

If the globe were a uniform colour and moving at an inconstant speed, it would be impossible to visually measure the number of revolutions. The fact that it is spinning at all is what suggests the concept of infinity, not that the spinning can be measured.

It may not be visually measurable, but if there is truly no way at all (not even non-visually) to measure the spinning, then how do you prove that it is even spinning at all?

Anyway, the spinning globe is no better than, say, a light beam moving forward forever unimpeded.  The only difference is that the former only assumes time is infinite, while the latter would require both space and time to be both infinite (although it could also be that maybe the universe actually has a looping topology, and the light beam could potentially loop back to a formerly visited position in space after traveling forward a sufficient distance, in which case we don't actually need space to be infinite.  Again, so much we don't know about the universe.).

Offline Dullstar

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Re: WillLem's Blog
« Reply #14 on: April 16, 2020, 03:45:28 AM »
This topic is certainly producing some interesting discussion.



Now, it is possible to set up a bijection between a 160 x infinity level and an infinity x infinity level. I showed how in my post above, but to reiterate:

Code: [Select]
1--2--3--4...159--160
                   |
320...--163--162--161
 |
321--322--323...

Code: [Select]
1--2  9--10  25--26
   |  |  |   |   |
4--3  8  11  24  27
|     |  |   |   |
5--6--7  12  23  28
         |   |   |
16-15-14-13  22  29
|            |   |
17-18-19-20--21  30...

I'm not really following this argument (that, of course, doesn't necessarily mean it's wrong). To give an idea of what my current understanding is: If I had to compare the area of the levels, my instinct would be to say that the infinity*infinity level is larger than the infinity*160 level (I'm intentionally avoiding terms like "cardinality" here, since I'm not quite sure I understand how it works with infinite sets). My reasoning would be that, suppose we compare an infinity*160 level with an infinity*320 level. In the x-direction, each level would be infinite. Pixels are countable; if we are at pixel n, there exists a pixel n+1 and there aren't any pixels in between (real numbers, by comparison, are uncountable; while you can compare which of two real numbers is larger, there is no "next" real number, since if you tried to define one, you'd always be able to insert another one in between). So there should be a one-to-one correspondence between pixels in each level in the x-direction. But the second level has twice as many pixels in the y direction for each pixel in the x-direction, thus the second has a greater area. An infinity*infinity level would have an infinitely greater area than either.

Other possibly wrong thoughts about infinity:
 - Suppose we have an infinite set that represents real numbers between 0 and 1. We have a second infinite set representing the real numbers between 0 and 2. Both sets are, of course, infinite. But the first set is a subset of the second set - any number that belongs in the first set will also belong in the second set, but a number that is in the second set might not belong in the first set. Thus, the second set is larger than the first set.
 - But maybe this argument breaks down if we look at natural numbers and then use the doubling function. If we have the set of all natural numbers, we could define a second set that is the numbers from the first set doubled. Each number in the first set has exactly one number in the second set that it corresponds to. But every number in the second set would also exist in the second set, so, arguably, the second set is therefore a subset of the first set and thus it is smaller...
 - I'm not really sure what's a correct statement here.

Offline Proxima

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Re: WillLem's Blog
« Reply #15 on: April 16, 2020, 04:24:50 AM »
I'm not really following this argument (that, of course, doesn't necessarily mean it's wrong). To give an idea of what my current understanding is: If I had to compare the area of the levels, my instinct would be to say that the infinity*infinity level is larger than the infinity*160 level (I'm intentionally avoiding terms like "cardinality" here, since I'm not quite sure I understand how it works with infinite sets). My reasoning would be that, suppose we compare an infinity*160 level with an infinity*320 level. In the x-direction, each level would be infinite. Pixels are countable; if we are at pixel n, there exists a pixel n+1 and there aren't any pixels in between (real numbers, by comparison, are uncountable; while you can compare which of two real numbers is larger, there is no "next" real number, since if you tried to define one, you'd always be able to insert another one in between). So there should be a one-to-one correspondence between pixels in each level in the x-direction. But the second level has twice as many pixels in the y direction for each pixel in the x-direction, thus the second has a greater area. An infinity*infinity level would have an infinitely greater area than either.

Yes, this is very tempting but wrong. :P

Forget about area for now and concentrate on numbers, because that's simpler.

Suppose you have a list of items and you want to know how many items are on the list. You can write "1" against the first item, "2" against the second, and so forth. If you eventually stop -- for example at "8" -- and that is the last item, then you know there were eight items on the list.

A variant of the same process works for infinite sets. With infinite sets, of course, there is no "last item" and the process will never stop. But if I can list the items of some set in a patterned way, so that I can see that I can label the first item 1, the second item 2, and so on, and match every single item with a unique positive integer, then the set must be the same size as the set of positive integers.

Because we can pair the positive integers with the even positive integers (1 -> 2, 2 -> 4, 3 -> 6 and so on), the set of positive integers and the set of even positive integers are the same size. You might think the set of positive integers is larger, since it includes all the even positive integers and also the odd positive integers, but they are the same size.

Another way to put this is that an infinite set can be the same size as a subset of itself. In fact, that is how mathematicians prefer to define "infinite".

Now, how about the set of all integers (positive and negative)? Is this larger than the set of positive integers, or the same size?

We can try to answer this by listing the integers: 0, 1, 2, 3, 4, 5... But we'll never get to the end of the positive integers and get started on the negative ones, so this list doesn't include all integers. Instead, we should list them in an order like 0, 1, -1, 2, -2, 3, -3... Now all integers are on the list, so we can see the set of all integers is indeed the same size as the set of positive integers. In other words, when enumerating the items in an infinite set, the order can matter.

Similarly, your 160 x infinity and 320 x infinity levels are the same size. You can't prove this by looking along the x-direction first, because you'll never get to the end of the first row and get started on the second. Instead, go down the first column, then down the second column, then the third, and so on. Each column is finite, so in both cases, this order of counting the pixels will take in all of them, proving that in both cases, the set of pixels in the level is the same size as the set of positive integers. And two sets that are both the same size as N must be the same size as each other.

What about the infinity x infinity level? In this case, we can't go down the columns either, but we can start from a corner and enumerate the pixels in a zigzag pattern (see my post on the previous page). Therefore this level, too, is the same size. (This supposes that the level has a corner and continues to infinity only rightward and downward. What if it's an infinite plane with no edges at all? Well then, we can start from an arbitrary pixel and enumerate the pixels in an outward spiral.)

* * *

You mention the fact that the real numbers are dense (between any two distinct reals, there is another real). This is true, but oddly enough it's a red herring. The rational numbers (numbers that can be expressed as fractions) are also dense: between any two distinct rationals, a and b, there is the number (a + b)/2, which is also rational. You would think that there must be more rationals than integers...

Amazingly enough, this is wrong, the rationals are countable. This is quite easy to understand if you think again about the infinity x infinity level. Suppose we label one axis "numerator" and the other "denominator"; then any rational can be paired with a unique pixel on the level, and we already showed how to count those.

Or we could just enumerate the rationals directly: 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1.... In other words, I start by listing the only rational with numerator + denominator equal to 2, then those with numerator + denominator equal to 3, and so on. This list will hit every positive rational sooner or later, and it's easy to tweak this method to cover the negative rationals as well if you like.

* * *

As for real numbers between 0 and 1, compared with real numbers between 0 and 2, both sets are uncountable, so we can't compare their sizes by enumerating them. However, we can still put them in correspondence with each other, showing that both sets have the same size as each other (and both are larger than the set of positive integers). To do this, pair every real number x between 0 and 1 with the number 2x between 0 and 2. Again, this illustrates that an infinite set can be put into correspondence with a subset of itself.

Offline ccexplore

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Re: WillLem's Blog
« Reply #16 on: April 16, 2020, 05:11:56 AM »
Cantor's cardinality measure is a specific way to extend the intuitive concept of "size of a set" from finite sets to infinite sets, where human intuition is far less reliable and certainly lacks rigor.  It specifically avoids the concept of "counting" (which is obviously problematic for infinite sets) in favor of a simpler notion of bijection, aka 1-to-1 mapping.  Two sets, be it finite or infinite, have the same cardinality if there exists a 1-to-1 mapping between elements of one set to those of the other.  For finite sets, Cantor's cardinality is equivalent to the intuitive notion of the size of the set as measured by number of elements.

Cantor's cardinality measure does have the property that for infinite sets, it is indeed possible for a proper subset of the set to have same cardinality as the entire set itself, something that's not possible with finite sets.  At the same time, Cantor also proved that not all infinite sets will have the same cardinality--the set of real numbers is for example provably not possible to create a 1-to-1 mapping with the set of natural numbers; any attempts to do so will always result in a real number that is not included in the mapping (eg. via the diagonal argument).

Cantor's cardinality measure works well because it assumes very little about the set.  Your example with level areas for example is in effect imposing a two-dimensional arrangement of the elements of your set of pixels, and similarly puts restrictions on how you perform the mapping of pixels between the two arrangements.  I'm not saying you can't necessarily come up with a mathematically consistent measure in your case, but even if you can, it will likely be more restricted in applicability compared to Cantor's cardinality.

Offline ccexplore

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Re: WillLem's Blog
« Reply #17 on: April 16, 2020, 06:43:19 AM »
You've said this quite a lot in your post. I realise it was something of a rant, and clearly something you're passionate about, and ultimately it's good to have your engagement. However, since you feel quite happy to reiterate this statement again and again, I'll take issue with it: if you have failed to recognise the meaning in something that someone has taken the time to formulate and share, that's on you.

Maybe "meaningless" is too strong a word, but I think the main point is that you are trying to refer to various concepts and terminologies that you have little understanding of.  It is a little hard to take your initial post seriously when it demonstrates a great deal of confusion and misunderstanding about the mathematical things you try to refer to, and at the same time phrased in such ways as to seem to assert decades of work by world-class mathematicians are complete garbage.

And I'm sorry to say, just because someone happened to spend time to formulate or share something doesn't mean the thing that was formulated or shared necessarily had to make much or indeed any sense.  Example.  Of course I'm not saying what you're doing here is anything like that, but it's just as much on you as it is on the readers to help come to a mutual understanding.  And poor uses of mathematical concepts and terminologies is unsurprisingly likely to become a roadblock to such mutual understanding.

The question here would be: if mathematics is a system of understanding and interacting with reality in the abstract, as useful and accurate as it tends to be, is it not still dependent on its creators (i.e. the human race, specifically mathematicians) for its development, definition and maintenance?; and, that being the case, is it not therefore limited by current existing limits of our understanding?

Doesn't that go for any human discipline?  "Understanding" for humans is fundamentally a mental process.  It will always be limited by our brains.

If so: then, surely, further investigation and development is needed - and this may involve consideration of radical or seemingly nonsensical ideas (although not necessarily) in order to expand.

If not: then how can we prove it?

As I recall, the historical development of set theory and its treatment of concepts like infinity was specifically in service of mathematics.  At the time, mathematics like calculus already had concepts that touched on infinity, but the treatment of those things tended to be very hand-wavy.  The development of set theory came about partly to help addressed those parts of mathematics that lack rigor.

Mathematics is based on proving other statements to be true starting from a small set of statements accepted as "ground truth", aka axioms.  One unfortunate property of logic is that if you ever end up with a statement that can be demonstrated as both true and false, then the entire system falls apart because you can take that true-and-false statement, and applying the standard operations of logic (deduction etc.) will actually allow you to then prove any other statement to be both true and false.  And thus the system becomes useless.  So mathematicians need to be much more conservative--the number of axioms should be kept low to help avoid tripping yourself over a self-contradiction, and then try to prove everything else as either true or false as followed from the small set of axioms plus the application of standard logic like deduction etc.

That's not to say you can't end up with results that seem "radical" or "nonsensical".  Mathematics explore a lot of things that are so far beyond our everyday experiences that it is expected to be counterintuitive from time to time as a result.  Here's an example.

It's good to recognize that because understanding is limited by our brains, it is quite easy to arrive at counterintuitive results especially when exploring areas of study that are so far removed from everyday experiences, but just because our intuition may be ill-adapted at processing the results doesn't mean it's false or incomplete.  After all, a dog tends to chase its own tail because perhaps it doesn't understand that tail is his, but that obviously don't mean the tail is in fact not the dog's; it just means his brain is not equipped to properly process the concept of self.

I'm saying that I am investigating the concept of infinity from a mathematical point of view and finding only questions, rather than answers. I've updated the OP so it's worded more carefully and in a more open, question-asking tone so as not to cause further irritation or misunderstanding.

Again, I'm not suggesting that maths isn't the best tool, I'm simply asking the question as to whether or not it is. Perhaps I worded the question as a somewhat ignorant-sounding statement. My bad, I apologise if that's how it came across.

Since you ask, I would say: I don't know what the best tool is, that's why I'm investigating the subject.

<snip>

Perhaps this is true. But my point is more that I do not believe that mathematics is the best way to interact with it, whatever it is, whether it's a magical, mystical force of reality or a boring, easy-to-understand force of reality.

This feels like a very personal question.  Understanding by whom and for what purposes?  Everyone's brains work differently and I don't really know what works for you and to what degree.  At the same time, set theory and its treatment of certain concepts like infinity is simply what mathematicians have found useful over the years as applied to their every day work on mathematics.

I'm not even sure I understand what it means to say to "interact" with "infinity".  Isn't it just an abstract concept?  What does it mean for you to "interact" with "fairness" for example?  In our everyday physical world we do not actually directly experience anything infinite--the examples you come up with invariably relies on infinite time or possibly infinite space, neither of which any human can directly experience.  Though perhaps they are easier or less confusing to mentally process, but probably only if you avoid thinking too deeply about it.

I'm also not sure why you want to be picking on mathematics on this.  There is a much richer, longer history and literature from philosophy on the exploration and discussion of infinity, that you seem completely oblivious to.  Maybe explore that a little bit too and see if it is more useful for your personal understanding?

By the same sword, I cannot abide when people hold onto "established results" so rigidly and stubbornly that they refuse to open their mind up to any possible alternative perspective, however reasonable/absurd it may be. So, at least we feel the same way about that, if only from different perspectives! ;P

For the record, though: I am not a flat Earther! There is far too much evidence to prove that Earth is a sphere, if not a perfect one. However, I do like the idea of Antarctica being a huge ice wall surrounding the Earth, and explorers seeking to find what's on the other side of it. That kind of idea yields great stories!

Kinda funny you mentioned "flat earther", because in some ways you kind of sound like one, and I think you recognized it or you wouldn't have brought up the flat earther thing even though that had never come up in the discussion.

The problem is this is only a matter of perspective in the sense that your perspective of the mathematical treatment seems so incomplete and confused.  In some cases I think you just read way more into something than what it actually means.  Cantor's cardinality measure is just some measure you can apply to both finite and infinite sets.  It is defined based on the notion of 1:1 mapping (aka bijection) as opposed to any intuitive notion of "counting".  For finite sets it happen to be equivalent (ie. produces the same results so to speak) to counting number of elements.  For infinite sets, it is not surprising that you will end up with results unique to infinite sets, it's nothing more than infinite sets being fundamentally different from finite sets.  Maybe it's less confusing for you if you just forget Cantor's cardinality measure as being somehow equivalent to "size", and just note that it happens to behave like "size" when it comes to your everyday finite sets, but otherwise just think of it as a specifically defined measure in mathematics and don't insist on trying to mesh it against your everyday intuition of size when it comes to infinite sets.  After all, seems rather a little rigid maybe to insist on a perfect mesh with intuition for a concept like infinity that is already so far removed from everyday experience?

it's trying to use tangible, rigid concepts and theories to express the intangible.

I don't understand why you seem so rigidly insist on infinity necessarily being "intangible", or why the mathematical treatment is "rigid" (or even if it is based on whatever definition you are applying, why is that a problem).  There seems to be a mental bias in your current thinking that I'm not completely grasping. ???

Offline namida

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Re: WillLem's Blog
« Reply #18 on: April 16, 2020, 08:05:27 AM »
Quote
Kinda funny you mentioned "flat earther", because in some ways you kind of sound like one, and I think you recognized it or you wouldn't have brought up the flat earther thing even though that had never come up in the discussion.

WillLem wasn't the first to bring this up; ∫tan x dx was:

I apologise if my rant comes off as being confrontational or overly hostile. But I cannot abide when people who clearly have no idea what they are talking about decry well established results in any given area. It's this kind of self assured superiority that causes flat earthers, and other nonsensical beliefs.
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Offline kaywhyn

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Re: WillLem's Blog
« Reply #19 on: April 16, 2020, 08:31:52 AM »
Wow, all the fellas above can definitely explain all this way better than I can. Did you gentlemen study math in college/university? Or is math just something that you always had a passion for and love studying in your free time? I have always loved math as a child, and I was a math major in both my undergrad and in grad school. I was trying to respond earlier with an example for WillLem on how infinite sets can have different cardinalities. The one I used was the set of natural numbers and integers, only to find out in my research, to my surprise, that they have the same cardinality. So, it looks like I don't know as much about set theory as I thought I do. Indeed, Proxima just explained why they do by means of a bijection between the two existing, so by definition they have the same cardinality. The example I used in my only post to this topic so far was the set of naturals and the even positive integers having the same cardinality, as you can simply take an element from N and pair it with the even integer from Z that's twice it. So, looks like I explained that correctly as confirmed by Proxima. To be fair, I did mention that set theory was something I didn't take or really study as a math major. When it comes to set theory, I only know the basics, i.e, definition of a set, element of a set, what it means for two sets to be equivalent, and cardinality, among other things. I'm only vaguely familiar with infinite sets and proving their cardinalities are the same or not depending on if a bijection exists through reading about it in my free time and not through an instructor in a math course. Proxima, tan, and cc did a great job with their explanations, so I'm still able to follow pretty much all of them.
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Offline Flopsy

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Re: WillLem's Blog
« Reply #20 on: April 16, 2020, 10:42:27 AM »
I also did Mathematics at University, I graduated back in 2008 kaywhyn.

It's nice to see who knows about the subject.

I don't really want to get involved in this discussion however because it's not really something I understand myself to be honest.

Offline Proxima

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Re: WillLem's Blog
« Reply #21 on: April 16, 2020, 04:15:13 PM »
Just a hobby for me. I also loved mathematics when I was a child, and grew up reading popularisations such as the books of Martin Gardner and Raymond Smullyan, and later Hofstadter's Gödel, Escher, Bach. In particular, since we've been talking about infinity, Smullyan's book Satan, Cantor and Infinity contains a lot of these arguments, presented through conversations between a Sorcerer and his pupils, who start off not knowing much about the subject and go through the common misunderstandings.

When I reached 16, everyone around me expected me to go on to study maths at university, but I was clear that it wasn't for me. I didn't want to become too specialised; I wanted to learn more about other aspects of the world. I chose to study Oxford's famous PPE degree, and went on to an MA in Philosophy.

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Re: WillLem's Blog
« Reply #22 on: April 16, 2020, 06:54:32 PM »
I also did Mathematics at University, I graduated back in 2008 kaywhyn.

It's nice to see who knows about the subject.

I don't really want to get involved in this discussion however because it's not really something I understand myself to be honest.

That's awesome, Flopsy. I was in my 2nd year of college when you graduated from university. That was also the year that I switched my major from biochemistry to math. I thought I wanted to do pharmacy, but I gave up on that once I took my second chemistry college course. I enjoyed chem a lot in high school, but it was just too much for me in college. Math was probably more difficult, but at least I was studying something I've been enjoying since childhood.

I know what you mean, though, as I'm not that well-versed in set theory, in particular proving things about infinite sets. I'm just familiar with them enough to be able to follow the explanations given.

Just a hobby for me. I also loved mathematics when I was a child, and grew up reading popularisations such as the books of Martin Gardner and Raymond Smullyan, and later Hofstadter's Gödel, Escher, Bach. In particular, since we've been talking about infinity, Smullyan's book Satan, Cantor and Infinity contains a lot of these arguments

When I reached 16, everyone around me expected me to go on to study maths at university, but I was clear that it wasn't for me. I wanted to learn more about other aspects of the world. I chose to study Oxford's famous PPE degree, and went on to an MA in Philosophy.

How awesome, Proxima! Thanks for clarifying. I would had guessed you studied math in college/university due to your depth of knowledge in the subject when in fact that you didn't. It's great that you were able to take something you read as a kid and apply it here. I definitely commend you for being able to take up reading. I never was, and still aren't, much of an avid reader myself. I just can't bring myself to read books.

I would like to point out that besides infinity, there are plenty of other nuances in math, but we all probably already know this. I have taken so many math courses and seen numerous examples in various mathematical topics in the last 12-13 years (I studied math for both my undergrad and grad school) to know that mathematicians definitely mean business when it comes to being precise in mathematical definitions, especially relating to mathematical pedagogy, the art of teaching math. In my experiences, students will take what the teacher says very carefully, and this can lead to confusion. For example, think back to how you were taught multiplication in elementary. It's likely that you were taught multiplication to mean repeated addition. 3 x 5, for instance, simply means 3 + 3 + 3 + 3 + 3. So, multiplication is simply a faster way of doing repeated addition. Similarly, -3 x 5 means to add -3 five times, i.e, -3 + (-3) + (-3) + (-3) + (-3). Ok, all well and good, but can you apply this same definition to, say, -3 x -5? Try as you might, you won't succeed, because what exactly does it mean to add -3 negative 5 times? So, we can't use the elementary definition of repeated addition that we learned for multiplication for this problem. This simply means that while the common definition of multiplication that we learned in elementary applies to many problems, it doesn't work for all, especially when negatives are involved. Since we only learn about positive integers at first, this isn't a problem, but once you start learning about negatives, this is where previously learned mathematics possibly start breaking down, like it does here for -3 x -5.

One way is to use a pattern. What's 3 x -5? That's -15. How about 2 x -5? That's -10. Now 1 x -5. That's -5. If you look at the answers, every time we decrease the first number by 1, the answer increases by 5. Continuing this pattern, you can conclude that -3 x -5 = 15. However, it is important to note that this is not a mathematical proof that two negatives multiplied together yield a positive. Rather, it is a way to convince oneself that - x - = a positive. If I remember correctly, the proof relies on the fact that 0 times any number equals 0 and the distributive property. Similarly, there is a proof that 0 times any number equals 0 (I saw this proof in an upper division math course in college), so this isn't something special that mathematicians agreed on just for the sake of it and that we should just take for granted. There's an actual mathematical proof of this fact!

Another good example is when we were taught how to order numbers. We all agree that 2 < 5, since when we learned to count, we know that 5 comes after 2 on the number line. However, when it comes to fractions, 1/2 > 1/5 even though we know that 2 < 5. This right here seemingly violates what we were taught previously on counting and ordering and will definitely confuse a child (I know I was when I was first taught fractions). Fractions is where previously taught mathematics breaks down. Once you consider fractions are different, eg, a special kind of number with two parts (right now I'm blanking out on how my college instructors taught us how fractions are defined, I think they might had used the word "number" in there), a numerator and denominator, then you'll see that once again that you can't apply similar concepts that you learned about ordering the counting numbers (the natural numbers) to them. What exactly does it mean to order things with two parts in them that mean different things (numerator is only a part, while denominator is the whole)? So, yet another instance where previously taught math concepts break down!

Seeing how this topic started with infinity, it can be seen the same way. However, as some have correctly pointed out, infinity is not any particular number, but rather a concept of things going on and on without end. You can treat infinity as though it is a number so that you'll be able to apply the same mathematical properties as you do with actual numbers, but infinity is a special case where known mathematical properties need to be slightly tweaked so that we can more easily work with infinity but at the same time they still hold true for numbers.
« Last Edit: April 16, 2020, 09:59:00 PM by kaywhyn »
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Offline WillLem

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Re: WillLem's Blog
« Reply #23 on: April 17, 2020, 12:42:55 AM »
We define aleph-0 to mean "the cardinality of the set of positive integers". Any set that can be put into one-to-one correspondence with the set of positive integers has this cardinality. There you go, I defined it 8-)

A basket can be made that is large enough for 3 oranges, so I can understand 3 oranges belonging to a set. However, there is no basket that can be made that is big enough for (every-positive-integer) oranges, so how can every positive integer belong to a "set", as we understand it?

Suppose it has a top-left corner and continues right and down to infinity.

I do understand your explanation, and can reasonably visualise how each pixel in one example would have a corresponding pixel in the other, the same as how positive integers can be mapped to positive even numbers 1-1. I totally understand that part of it.

What I don't understand is... how can something that is infinite have "a top-left corner"?

So I guess this is the "WillLem is insulting all the mathematicians out there" topic. ;P

It's kind of annoying that it's been taken that way, I didn't set out to insult anyone; such is the wonder of internet communication. For what it's worth, I apologise for any offense I may have unintentionally caused.

You seem to argue the set of all natural numbers is abstract and does not "exist in the real world", but does either of your examples actually exist in the real world either?  The picture of the corridor is almost certainly of a real corridor that is in fact not infinite, you are only imagining in your mind that it goes on forever.

Mathematics only implies that the list of positive integers goes on forever, usually using an ellipsis(...) It, in fact, doesn't, as such a list only exists in the imagination, which makes it exactly as valid as an imaginary infinite corridor.

And don't be so sure about the world being analog either.  Some physicists postulates that space and time themselves might be subjected to the laws of quantum mechanics and therefore may actually be quantized, such that there may actually be a minimum length/duration enforced by the laws of physics.

I'm guessing you're referring to Planck length here. This is something I don't even pretend to fully understand, and I'm sure that greater minds than mine have come up with plenty of reasons why this should be accepted as the smallest possible length in the observable physical world. No dispute there at all; I can accept that there is a possibility that the world is indeed digital and not analog, and that this would render certain aspects of my initial premise somewhat redundant.

You mentioned music and art, but I'm having trouble following what examples you have mentioned that are actually coming from those disciplines and relating to infinity.  I'm rather surprised you didn't mention philosophy.  The nature of infinity had actually been historically discussed as philosophy far, far longer than the modern mathematical set-theoretic treatment had existed.

Absolutely - reference to philosophical concepts as well as mathematical ones is equally valid and interesting for this discussion. As for artistic examples, I'd say Escher has produced some stunning works that are both mathematically and artistically concerned with the concept of infinity. Specifically, Relativity and Waterfall would be great examples of this, but there are others.

It may not be visually measurable, but if there is truly no way at all (not even non-visually) to measure the spinning, then how do you prove that it is even spinning at all?

This is a brilliant question, and one that I can't answer. It's fascinating though, and I love to think about questions like this from time to time.

Suppose we have an infinite set that represents real numbers between 0 and 1

What causes my confusion about real numbers belonging to a set at all is this:

Start at 0. Now count to 1, but don't miss out any numbers in between.

Where do you start?

This is what I meant by "arbitrary counting distance" in my OP: we can manually create a starting point for such a process, because we can define numbers such as 0.5, 0.237987, 0.09719682649862389749283496 and so on.

But... where exactly is the first of these numbers?

We also use 0 as a convenient starting point for a lot of things, for example:

Instead, we should list them in an order like 0, 1, -1, 2, -2, 3, -3... Now all integers are on the list

But this brings me back to my question as to how something that's infinite can have a "top left-hand corner"...

Cantor's cardinality measure is a specific way to extend the intuitive concept of "size of a set" from finite sets to infinite sets, where human intuition is far less reliable and certainly lacks rigor.

I can't find a smiley for this. Cantor was a human, so any ideas he may have established are, by definition, based on human intuition!

It is a little hard to take your initial post seriously when it demonstrates a great deal of confusion and misunderstanding about the mathematical things you try to refer to, and at the same time phrased in such ways as to seem to assert decades of work by world-class mathematicians are complete garbage.

I have taken care to retract any such perceived assertions, as they were not intended to have been taken this way at all. I am not so arrogant as to suggest that mathematics is "complete garbage" - your words, not mine - in fact, I am certain that far greater minds than mine have spent far more time than me thinking about these things.

Is it wrong for me to question their findings, though, limited though my own understanding may be?

set theory and its treatment of certain concepts like infinity is simply what mathematicians have found useful over the years as applied to their every day work on mathematics.

The longer something has been established as "truth", the more necessary it is to question and re-evaluate it.

I'm not even sure I understand what it means to say to "interact" with "infinity".  Isn't it just an abstract concept?

I guess I mean how can we make sense of it, explore it, when it's seemingly unexplorable. As you've touched upon: nothing in the physical world, particularly humans, can experience the infinite except in our imaginations. We seem to agree about this.

I'm also not sure why you want to be picking on mathematics on this.  There is a much richer, longer history and literature from philosophy on the exploration and discussion of infinity, that you seem completely oblivious to.  Maybe explore that a little bit too and see if it is more useful for your personal understanding?

I'm not "picking on mathematics", and have already apologised for and edited the tone of my original post. Please can we move on from that aspect of this discussion now.

Philosophy is certainly an area I'm also interested in. I began this discussion from a mathematical point of view because that's the side of it I understand the least, but I'm more than happy for this discussion to develop into something more philosophically-orientated.

Kinda funny you mentioned "flat earther", because in some ways you kind of sound like one, and I think you recognized it or you wouldn't have brought up the flat earther thing even though that had never come up in the discussion.

Actually (as Namida also pointed out - thanks!), ∫tan x dx was the one who mentioned flat earthers, in relation to the regrettable way I initially worded my questions about mathematical understanding of infinity. To re-interate: I am not a flat-earther, nor do I subscribe to any other such belief system.

I have to say, I had no idea that my post would come across this way or cause people to take offence. I am simply asking questions, that's all. I do not claim to have greater knowledge, or assuredness of a particular belief system - far from it: I'm essentially open-minded in my thinking on most subjects, and merely wish to seek further understanding.

The problem is this is only a matter of perspective in the sense that your perspective of the mathematical treatment seems so incomplete and confused.

Correct: that's why I'm discussing it - to further complete and clarify my perspective. So far, the discussion is having that effect. I now understand far more about concepts such as bijection, cardinality and set theory than I did before.

I'd just like to close this post by clarifying a few things for the purposes of the discussion moving forward:


1. I have nothing against mathematics, mathematicians, or indeed any school of abstract thought (actually, I happen to be subscribed to Numberphile on YouTube and I regularly watch and enjoy their videos, and find them very fun and educational). I am simply interested to understand it better, and have come up against questions to which the answers have caused further confusion, in some areas. In other areas, I now feel more enlightened thanks to people's explanations and the progress of this discussion.

2. I know that infinity is not a number. I am asking about the relation of the concept of infinity to the mathematical system of numbers.

3. I think you're all awesome. :lemcat:
« Last Edit: April 28, 2021, 02:50:40 AM by WillLem »

Offline Proxima

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Re: WillLem's Blog
« Reply #24 on: April 17, 2020, 02:51:18 AM »
A basket can be made that is large enough for 3 oranges, so I can understand 3 oranges belonging to a set. However, there is no basket that can be made that is big enough for (every-positive-integer) oranges, so how can every positive integer belong to a "set", as we understand it?

A set, being a mathematical object, doesn't have to exist in the real world. As I said earlier, when mathematicians talk about a cube, for example, they mean an idealised cube with perfectly straight and exactly equal edges, even though such an object can't exist in reality.

Quote
What I don't understand is... how can something that is infinite have "a top-left corner"?

Imagine an infinite grid of squares. Imagine that one particular square is black. Then imagine that every square, continuing rightwards and downwards from that square, is black. Now the black area continues to infinity in two directions (right and down) but has a definite top-left corner; an ant walking over the black area leftward or upward would eventually reach the border with the white area.

This is the 2-dimensional equivalent of the distinction between the positive integers (starting at 1 and continuing to infinity in one direction) and the set of all integers (continuing to infinity in both directions).

Quote
Mathematics only implies that the list of positive integers goes on forever, usually using an ellipsis(...) It, in fact, doesn't, as such a list only exists in the imagination, which makes it exactly as valid as an imaginary infinite corridor.

Everything mathematicians talk about is imaginary. If you don't accept imaginary things as valid, you can't begin to do mathematics.

You also can't begin to make bank transfers, because that is asking someone to manipulate imaginary concepts on your behalf. Your bank account isn't a Gringotts vault containing actual physical money....

Quote
What causes my confusion about real numbers belonging to a set at all is this:

Start at 0. Now count to 1, but don't miss out any numbers in between.

Where do you start?

You are confusing the concepts of set and series or sequence.

A set is a collection of objects, without regard for order. The set {5, 6} containing only the numbers 5 and 6 is the same as the set {6, 5}.

A sequence is a progression that starts with a first term, then a second term and so on. For example, the sequence 1, 4, 9, 16... of square numbers, in which the nth term is n^2.

In many cases, these concepts overlap. We can talk about the sequence of square numbers when we think of them (or perform mathematical operations on them) as a sequence, such as finding a formula for the nth term. We can also talk about the set of square numbers (the collection of all square numbers, without regard for order) when we want to think about them as a set, for instance to ask "what is the cardinality of this set?" (how many square numbers are there?)

In other cases, it's important to draw the distinction. The real numbers are a set, but not a sequence. There is no "first term" or "second term". You cannot count them.

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The longer something has been established as "truth", the more necessary it is to question and re-evaluate it.

Mathematicians are not unfamiliar with this principle.

Offline Simon

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Re: WillLem's Blog
« Reply #25 on: April 17, 2020, 10:19:27 AM »
Happy to see so many here with a grounding in set theory.

I've enjoyed formal mathematical education. Still, our uni never offered even a basic course on axiomatic set theory. This baffled me to no end, for several reasons:

1. Set theory (really the study of infinite sets) is enormously interesting by itself. Fascinating questions arise immediately and naturally, already during the first few weeks of first-semester mathematical training.

2. The tooling is surprisingly useful in related research. Topology, group theory, abstract algebra, ... everywhere, there will be questions of cardinality. Occasionally, transfinite induction or Zorn's lemma appear in proofs, and it becomes much more natural to use them yourself.

In universities, it's common to spoon-feed every first-semester mathematics student Zorn's lemma without proof. But it's not satisfying.

Much more rarely but not impossibly, independence results appear in the related fields during research, but attacking those would be beyond a basic course in set theory anyway, so their proofs are perhaps rightfully skipped.

3. Such internet discussion 20 years ago was one of the many reasons I studied mathematics and not physics. The problems in mathematics sounded much more interesting.

Given the lack of courses at university, I've read set theory books in the evenings during my undergrad studies. In 2008, Deiser's Einführung in die Mengenlehre was an ideal entry-level text, it read like a criminal novel despite staying 100 % formal in its proofs. Around 2012, this dovetailed into Kunen's Set Theory with my goal of at least a rough understanding of Gödel's and Cohan's CH results.

In 2014, I sacrificed time off my PhD to hold my own one-semester course in axiomatic set theory. It was a seminar with the students giving 90-minute talks that I would design and select literature for. The seminar became a smash hit, I had to assign two students to every talk to accomodate everyone. By the middle of the seminar, we had introduced von-Neumann ordinals and cardinals, and my wish is that at least these ideas have stuck with all participants.

To this day, I get questions for when that seminar repeats, and it's always sad to tell them it was a one-shot. >_>



The answers in this topic give solid explanations. I have practically nothing to add yet that wouldn't go over the top or be confusing. Still, I'll happily follow this.

I had been reluctant to reply because the unedited OP reeked like a troll. It wished to discuss the real numbers, noble goal. Then the obvious attack fails, normal and interesting. Then it suggests that mathematics is not the best tool for a problem that can't even be stated outside mathematics.

There is no slur at all in pursuing art. That's merely unlikely to yield understanding of the reals, which seemed like the point. :P

It's possible to reject infinite sets and still do mathematics, but that didn't become popular in 20th-century-mathematics. The theorems of ZFC would then still be true, but considered uninteresting.

-- Simon
« Last Edit: April 17, 2020, 12:57:04 PM by Simon »

Offline WillLem

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Re: WillLem's Blog
« Reply #26 on: April 17, 2020, 01:04:45 PM »
Thanks for the explanation Proxima, that helps to clear a few things up. It's obvious to me now that I was perhaps visualising it differently, but I understand the crucial differences between a "set" and a "series" better now. :thumbsup:

Then it suggests that mathematics is not the best tool for a problem that can't even be stated outside mathematics.

Philosophy?

The concept of infinity was philosophical before it was mathematical. Granted, mathematics has so far been the tool of choice for discussing, comprehending and otherwise dealing with the concept, but I am asking whether or not it's the best choice.

Note: asking, not suggesting either way.

Anyway, if people are just going to continue to take what I've said as an attack, belittle me for my lack of formal mathematics education, and ignore the fact that I have edited my OP having learned from my mistake, then I no longer have any interest in discussing it here.
« Last Edit: April 17, 2020, 01:28:46 PM by WillLem »

Offline Simon

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Re: WillLem's Blog
« Reply #27 on: April 17, 2020, 01:58:27 PM »
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Philosophy? The concept of infinity was philosophical

I'm happy to learn more here. I merely didn't see any examples and don't have much experience here myself. I hope that explains my math-slanted reply.

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take what I've said as an attack

With "attack", I meant: method to solve the given problem (which was to show the uncountability of the reals).

Reworded: The first attempt to solve the question about the real numbers fails, which I call "normal" because I expect this at that time halfway reading through OP given the methods of the attempt, and which I call "interesting" because I expect good discussion following. I did not take edited OP as a potshot against established mathematics, you had made that clear already in the days since.

80 % of the edited-as-is OP is still math. Thus, please make it very clear where that mathematical context is over.

The problem is that otherwise, reading OP allows one to assume you wish to solve the uncountability of the reals with music/art, when instead you wish to learn how music/art/philosophy have dealt with not-necessarily mathematical inifinity, which indeed happens to fascinate also non-mathematicians.

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and ignore the fact that I have edited my OP

I have seen the edit. I wrote that the unedited OP discouraged me from replying on day 0.

It still sounds to me like you want to understand the uncountability of the reals. Your replies to other people also suggest interest in distinguishing countable and uncountable sets.

-- Simon

Offline ccexplore

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Re: WillLem's Blog
« Reply #28 on: April 17, 2020, 04:13:58 PM »
Mathematics only implies that the list of positive integers goes on forever, usually using an ellipsis(...) It, in fact, doesn't, as such a list only exists in the imagination, which makes it exactly as valid as an imaginary infinite corridor.

It goes on forever in that for any positive integer you can come up with, I can add 1 to it, to get a different positive integer that is bigger than the one you give me.  This is what mathematics really mean by the list "go on forever".

The ellipsis is just a shorthand notation.  It is actually not how one would rigorously formulate all the positive integers; after all, the next number after "1, 2, 3, ..." might not be 4, it might be 5 for example if you are actually talking about the Fibonacci numbers.  The notation using ellipsis may be convenient, but clearly imprecise.  Instead of example members plus ellipses, to be rigorous you would just provide a formula that says you start with 1, and then you can keep creating the next integer by adding 1 to the previous.  You don't need any ellipses and there is no ambiguity.

Yes, you can call it imaginary in the same way as the imaginary infinite corridor.  My question is, it seems like you object to the imaginary status of the list of all positive integers more than you object to the imaginary infinite corridor, is that so and if yes, why is that? ???

Suppose we have an infinite set that represents real numbers between 0 and 1

What causes my confusion about real numbers belonging to a set at all is this:

Start at 0. Now count to 1, but don't miss out any numbers in between.

Where do you start?

Ok, I think you have a confusion between the concept of a sequence vs a set.  A sequence is ordered and countable.  A set is not ordered and doesn't have to be countable.  You can define a set purely based on a formula that can describe, given something, whether that something belongs to the set or not.

As already explained by others, it is actually impossible to list out all the real numbers, or even just the ones between 0 and 1, in any kind of lists.  Any attempt to do so will provably result in some real number that is not part of the list you try to make.  The set of real numbers can be defined simply by the number being expressible as, for example, a regular integer before the decimal point followed by an infinite sequence of digits 0-9 after the decimal point.  Any number expressible in that form is a member of the set.

This is a completely understandable confusion given that the informal ellipsis notation for numeric sequences, is often used in place of a formal formulation, as a way to describe the membership criteria for the set.  That is, the set of positive integers is often written down as {1, 2, 3, ...}, but as explained above, it is actually rather informal and imprecise to use this ellipsis notation.  You don't have to use ellipses to define an infinite set, and for something uncountable like the real numbers, such a notation is completely useless anyway.

What I don't understand is... how can something that is infinite have "a top-left corner"?

The sequence of natural numbers (1, 2, 3, ...) has a start but no end.  It has a minimum member but no maximum; any number you can think of, I can add 1 to it to make a bigger number.  So it is unbounded (ie. infinite) but in one direction only.

Other examples abound, such as Dullstar talking about levels being infinite in one direction but finite in another.

Cantor's cardinality measure is a specific way to extend the intuitive concept of "size of a set" from finite sets to infinite sets, where human intuition is far less reliable and certainly lacks rigor.

I can't find a smiley for this. Cantor was a human, so any ideas he may have established are, by definition, based on human intuition!

Right.  At the same time, he also took care to ensure his formulations are precise and logically consistent, so that it is usable for further mathematical inquiry.  As I mentioned before, a logical inconsistency/paradox (ie. a statement both true and false) is bane to mathematical proofs, because the operations of logic can take even just one inconsistency and allow it to "prove" any other statement in the system to be both true and false as well.  A system with unresolved paradoxes cannot be used as the basis for further mathematical inquiry, especially for something like set theory that was considered for possible use as foundational basis of all of mathematics.

It dismays me that somehow the only thing you seem to read into the sentence is that I'm somehow denying the role of human intuition in any discipline.  Of course human intuition plays a part in the exploration and understanding of any topic, no one is trying to deny that.  But at the same time, surely you can agree that human intuition isn't infallible either?  That's really all I'm saying, and somehow you seem to be reading a completely different meaning into it apparently. ???

Is it wrong for me to question their findings, though, limited though my own understanding may be?

No, but your initial attempts, at least in their wordings, seem to show you haven't even bothered questioning your own understanding of those findings.  Shouldn't one first "question and evaluate" one's own understanding of the things one is trying to challenge, lest the attempt only wound up challenging a completely distorted version not congruent to what the findings were actually saying?

It also doesn't help much that between your questions, you talked about things like this

Quote
By the same sword, I cannot abide when people hold onto "established results" so rigidly and stubbornly that they refuse to open their mind up to any possible alternative perspective

Somehow seeming to insinuate that anyone who don't share the same confusion as you had on the details of set theory's treatment of infinity, must only be because they are "rigid" and "stubborn", as opposed to maybe other possibilities like, I don't know, maybe they understood it a little better?  Interesting that Dullstar also asked some questions here that basically seek for the same clarifications on some of the same things you were a little confused about, and yet somehow he managed to not have to bring up anything about anyone being rigid and stubborn, which is neither here nor there and certainly not helping to actually addressed the questions being raised.

The longer something has been established as "truth", the more necessary it is to question and re-evaluate it.

Sure, but at the same time, it's not like this topic is something that's decreed as unquestioned truth one day by some king or emperor, and then blindly accepted for here on out.  That's never how things work in any academic disciplines.  Right from the beginning as the initial version of the theory or framework is still being formulated and explored, things are already questioned, discussed and critiqued, and things get revised multiple times in the journey towards the formulation you see today.  People often lose sight of this, because from a textbook, of course they skip over right to what ends up being the current "final" formulation of the topic; they are not going to waste ink elaborating on all the various initial earlier formulations, the things that were questioned or critiqued and thus revised, possibly even some things that proved to be false starts and thus abandoned.  Even if you read up on additional reference materials that go more into the history and evolution of the topic, they too are liable to summarizing and not necessarily provide the full scope of everything that had happened throughout the evolution of the topic or area of study.  So far from it being unquestioned, what you see today generally had already endured years of questions and evaluations; you wound up with what had stood the tests of time.

For a topic that had been around for so long and exposed to so many people, it would be patently silly to assume that the questions one can raise haven't likely already been asked in a similar way before by someone else and then addressed, especially questions from someone who is still learning about the topic.  At the same time, don't assume that there aren't questions raised today on the topic either.  It's just that given the things that had already stood the test of time, one would likely need to question at a deeper level, at a much more particular part of the theory, in order to truly be asking something that hadn't already been asked and addressed before.  It might be hard for the average person to find examples of such questions as they likely resides in relatively obscure academic journals devoted to rather deep specializations into the topic or area of interest, but that doesn't mean no such questions are raised.

I'm not even sure I understand what it means to say to "interact" with "infinity".  Isn't it just an abstract concept?

I guess I mean how can we make sense of it, explore it, when it's seemingly unexplorable. As you've touched upon: nothing in the physical world, particularly humans, can experience the infinite except in our imaginations. We seem to agree about this.

I guess I don't really see how it is "unexplorable".  Really, if we can talk about it, isn't that enough to make it explorable?  Academic studies are full of things that don't have direct physical manifestations in the real world anyway.  Geometry may be based on our experiences in the real world, but there is no actual physical manifestation of an ideal geometric line of 0 thickness and infinite length.

At the same time, just because something is grounded in the physical world, don't assume that our intuition and initial understanding of it will necessarily be more reliable and accurate.  After all, Aristotle the ancient Greek philosopher believed that it is the natural order for heavier things to fall faster than lighter things.  And honestly, that matches our everyday intuition pretty well, so pretty much everyone quickly believed it to be true.  It took a surprisingly long time before people finally discovered that, no, in fact that's not true.  In the absence of air, all objects would actually fall at the same speed regardless of their weight.  It is actually the air providing air resistance that results in the differences one observes in the speed of falling, and in turn the air resistance depends on other properties of the object besides its weight/mass.

Offline ccexplore

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Re: WillLem's Blog
« Reply #29 on: April 17, 2020, 04:41:20 PM »
It's possible to reject infinite sets and still do mathematics

Thanks Simon for bringing this up, I guess there's something for me too to read up more on.

So as one can see, academics are far from being rigid unquestioning sycophants, when it comes to complex topics like infinite sets.  The above topic actually also touches nicely on the fuller history of the development of mathematics in the 19th century.  Read the "history" section of the article, and you see that the development was far from a few person coming up with some ideas, and then everyone else just blindly accepts from here on out.  (Then again, no one can possibly seriously believe that's how academic pursuits ever work in the real world, right?)

The topic of philosophy of mathematics might also be of interest, given some of the things that have been brought up.  I'm certainly no expert in this area though, I can only offer the Wikipedia article as a starting survey for people to further explore as they see fit.

Offline Forestidia86

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Re: WillLem's Blog
« Reply #30 on: April 17, 2020, 05:08:30 PM »
Something like a philosophical wiki is the Stanford Encyclopedia of Philosophy: Entry for Philosophy of Maths
There are more entries for more specific subjects like Intuitionism in the Philosophy of Mathematics etc.

This thread reminded me somehow of the Zeno's Paradoxes.

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Re: WillLem's Blog
« Reply #31 on: April 17, 2020, 11:44:59 PM »
I'm happy to learn more here. I merely didn't see any examples and don't have much experience here myself. I hope that explains my math-slanted reply.

I'm equally interested in both sides: mathematical and philosophical. I regret the way this conversation started really.

It still sounds to me like you want to understand the uncountability of the reals. Your replies to other people also suggest interest in distinguishing countable and uncountable sets.

Yes, that's definitely a big part of it. I want to know how to get from 0 to 1 without missing a decimal, which is of course impossible. This idea both fascinates and confuses me.

Ok, I think you have a confusion between the concept of a sequence vs a set.  A sequence is ordered and countable.  A set is not ordered and doesn't have to be countable.

I think I understand this now; Proxima has also explained it the same way.

As already explained by others, it is actually impossible to list out all the real numbers, or even just the ones between 0 and 1, in any kind of lists.

In that sense, it seems like every number is a potential infinity in itself, given that decimal places can theoretically go on forever.

surely you can agree that human intuition isn't infallible either?

Absolutely: this thread is a fine example of my own human tendency to not be infallible! :crylaugh:

No, but your initial attempts, at least in their wordings, seem to show you haven't even bothered questioning your own understanding of those findings.  Shouldn't one first "question and evaluate" one's own understanding of the things one is trying to challenge, lest the attempt only wound up challenging a completely distorted version not congruent to what the findings were actually saying?

---

That's never how things work in any academic disciplines.  Right from the beginning as the initial version of the theory or framework is still being formulated and explored, things are already questioned, discussed and critiqued, and things get revised multiple times in the journey towards the formulation you see today.

I think there's been a very unfortunate misunderstanding in all of this, and that's the notion that I somehow reject academic rigour.

I absolutely do not: I have a Master's degree in Music, and wouldn't have got it without embracing the academic side of that particular discipline.

Similarly, I have nothing but the utmost respect for mathematicians, and Mathematics as an academic discipline. I regret the way I have approached this topic because mathematical logic, infinity and other such concepts are things I've always been fascinated by; I really don't know what made me think that coming at it with a seemingly anti-maths/anti-discipline angle would be a good idea.

I guess that it's a side of the argument that I felt needed some exposure for whatever reason, and it was somewhat spontaneous on my part. As ccexplore has said: I should have examined my own understanding of that side of the argument before posting about it.

Anyway, what's done is done. I appreciate the time that people are taking to reply to my posts and explain the various points and help to fill the gaps in my knowledge and understanding. Having settled from the misguided thoughtgasm that was my original post, I can now see the subject with a lot more informed clarity.

I'm lucky, really, that this forum attracts a lot of clearly very educated people with whom this discussion is possible at such a depth.

This thread reminded me somehow of the Zeno's Paradoxes.

Zeno's Paradoxes are great fun to mentally wrestle with every now and again. In fact, it was these that first introduced me to the possible confusions surrounding the concept of infinity in the first place.
« Last Edit: April 19, 2020, 12:40:28 AM by WillLem »

Offline Simon

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Re: WillLem's Blog
« Reply #32 on: April 20, 2020, 02:38:14 PM »
Quote from: ccx
Instead of example members plus ellipses, to be rigorous you would just provide a formula that says you start with 1, and then you can keep creating the next integer by adding 1 to the previous.  You don't need any ellipses and there is no ambiguity.

Right, it's important that one may define ℕ without resorting to "...". Only once everybody agrees what this set should be and that it exists, we can write ℕ = {0, 1, 2, 3, ...} as shorthand or as a reminder, not as a definition.

In the beginning, it's acceptable to "just believe" that ℕ and ℝ exist, and later replace them with more rigorous definitions.

When one wants a completely rigorous definition of ℕ within ZFC, the most popular system of axioms for set theory, it gets elaborate: It's common to define ℕ as the smallest infinite von-Neumann ordinal. This happens to be the set of all finite von-Neumann ordinals, which we then interpret as natural numbers. But that requires an understanding of ordinals first, and why a definition as "the smallest ordinal that satisfies X" is sound.

Quote from: WillLem
I can understand 3 oranges belonging to a set. However, there is no basket that can be made that is big enough for (every-positive-integer) oranges, so how can every positive integer belong to a "set", as we understand it?

Does a train suffice instead of a basket? :lix-suspicious: But more seriously:

Naively, a set X is a mathematical object such that, given any mathematical object y, the statement yX "makes sense", i.e., it is either true or false. Also, sets may not be "too big" such that one runs into Russel's paradox or similar problems.

Thus, the set ℕ of all natural numbers exists; you can tell me with a straight face that 3, 5, and 329 belong to ℕ (because they are natural numbers), and you can tell me that ♥ and M aren't natural numbers, thus don't belong to ℕ.

As long as one accepts the existence of infinite mathematical objects, one can feel reasonably safe that the existence of ℕ doesn't yet trigger any paradoxes or contradictions: Decades of work haven't found any. We can't prove that it's really paradox-free, but the fundamental obstacle is not infinite sets, it is that for any sufficently rich system of axioms, one cannot prove from itself that it's free of contradictions. So this is really the best that we can get.



If one is not satisfied with naive definitions and wants something more rigorous and founded on classical logic, ZFC is the most popular formal system of set-theoretic axioms.

In ZFC, everything that exists is a set. The axioms force that some sets exist, such as the empty set ∅, an infinite set (doesn't matter which), the two-element set for any two given elements, and some more things. Note that the existence of infinite sets is explicitly forced in ZFC, it's part of the design of this system of axioms.

Because everything is a set, but we still want to do mathematics similarly to how we're used to, we'll model our desired mathematical things using sets. E.g., we define the natural number 0 to be the empty set ∅, the natural number 1 is the 1-element set {∅}, the number 2 is {∅, {∅}}, an ordered pair of two things x and y is the set {x, {x, y}}, and a function, a.k.a. a mapping, is a set of such ordered pairs.

It's similar to how everything in a computer (text, numbers, images, sound) is only a sequence of bytes under the hood. It's rarely necessary to consider the bytes: We don't talk about bytes much at all, we talk about numbers and texts and images, and it all makes sense to both of us. But if the need arises, we know how to look under the abstraction and examine the raw bytes.



Infinite sets are exactly those sets X that admit injections XX that miss elements. (An injection is a function that never takes the same value at two different inputs.) For example, ℕ is infinite because the function nn + 3 takes only different values for different inputs n and misses (never takes as value) the first three natural numbers 0, 1, 2.

Finite sets are exactly those such that their size is expressible using a natural number.

It's not obvious that every set satisfies exactly one of these two, but it's provable in ZFC.

The nature of infinity as a philosophical or artistic idea will likely be fundamentally different to such a property of a set to admit certain functions to itself.



If one rejects the readily-existing infinite sets, one can still consider ℕ as building instructions to produce ever more mathematical objects, each itself finite, beginning with 0, and call them numbers. This is the basic idea of Finitism.

When one dives into formal logic, one will eventually separate two languages:
  • In the finitistic metalanguage, we conduct proofs. Each proof can only have a finite number of steps and argue about a finite number of symbols. (It doesn't matter that, inside the theory, that symbol means something infinite. The proof treats it as one thing that has properties, e.g., being infinite, whatever that may mean in the theory.)
  • In the domain-specific language that only makes sense when talking about objects of the theory, such as ZFC, we can use the term "infinite" even though that makes no sense in the metalanguage.
E.g., we can argue about the magic in the Harry Potter books, and our argument will use magic-related words that have meaning within that theory, even though nothing magic-related makes sense in our outside world.

It's useful to keep the metalanguage as weak as possible, to avoid introduction of inconsistencies. Reason is again: We can't prove that the metalanguage is consistent merely by using the metalanguage.

It's similar to how, in software, we don't want features unless there are good reasons to have them, as every feature has the potential to introduce bugs. :lix-grin:

Quote from: WillLem
I have a Master's degree in Music

Hats off, then. It's one of the hardest subjects to even get admitted.

Regarding music, the "most infinite" thing that comes to my mind is the Shepard glissando. But maybe you have even better examples. :lix-grin:

-- Simon
« Last Edit: April 20, 2020, 06:30:18 PM by Simon »

Offline ccexplore

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Re: WillLem's Blog
« Reply #33 on: April 20, 2020, 09:36:51 PM »
Quote from: WillLem
I have a Master's degree in Music

Hats off, then. It's one of the hardest subjects to even get admitted.

Interesting.  I don't have any experience to say anything about this one way or another; any guesses as to why it seems to be so difficult to get admitted?  Are they just pickier?

I tend to think of law and medicine as common examples of degrees that are often considered difficult to get into and just as difficult to finish.

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Re: WillLem's Blog
« Reply #34 on: April 20, 2020, 11:13:40 PM »
Quote from: WillLem
I have a Master's degree in Music
Hats off, then. It's one of the hardest subjects to even get admitted.
why it seems to be so difficult to get admitted?  Are they just pickier?

Likely, it varies by university. Still, music and sports entrance exams are highly skill-based. I think it's common that the student needs to play two different musical instruments really well, and pass other skill tests about for musical intuition.

Law and medicine require excellent grades in high school, but I don't think they commonly have exams for entry otherwise. skill. At least from personal experience, good grades are easier to get than getting good at several musical instruments.

Medicine in Germany occasionally requires the best possible high school grade, 1.0 after ceil-rounding to tenths. If you don't make this, you can still get in, but you're placed in a queue first. Each year, a fixed ratio of slots are filled from the queue's front, and another fixed (small) ratio are filled from the queue at random. It may take 5 to 10 years to be admitted for medicine via queue. But at least you can get in even with decent grades if becoming physician is your major goal in life.

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Offline kaywhyn

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Re: WillLem's Blog
« Reply #35 on: April 21, 2020, 09:23:24 AM »
It can certainly depend on the school, as some schools have a more rigorous curriculum than others and therefore will be harder, but I think the difficulty of your major will be about the same regardless of which college/university you attend. My alma mater UCLA is highly prestigious and has in recent years finally beaten out the oldest UC school, UC Berkeley, out of the top spot on the list of top public schools in U.S News and Times Report, when we been 2nd place on the list behind Berkeley for many years. At UCLA, the graduation rate for math majors is only 2/10, i.e, 20%, and so I was one of the few 2/10 who graduated with a math degree. Math is definitely a hard major, moreso at UCLA, but I'm sure if I had studied math at another university in California it would still be up there high in difficulty, although maybe not as much. It depends on the school. Besides UC's, another major school system are the CSU's (California State University). CSU's aren't as rigorous as the UC schools, but they're still widely regarded as some of the best schools in California. In particular, CSUF and CSULB are one of the most prestigious in this system. The former was where I attended grad school and did my Master's in math after I graduated from my undergrad at UCLA, while the latter was where I did the teaching credential program to get my teaching credential in math after I graduated with my master's from CSUF. When I did my master's in math, it was nowhere near as difficult, but I still didn't consider it a complete pushover, as I still had to study in order to do well in my math classes there, although I didn't study anywhere near as much at CSUF as I did at UCLA. At the same time, I think master's degrees in the same major as the bachelor's are easier in general. In particular, all of my math classes at CSUF were pretty much all review of the math I studied at UCLA, hence played a big part of why my Master's was easier.

Regarding your Master's in music, WillLem, that's awesome and congrats. I've loved music a lot as a kid, and I still do. I consider the arts essential to have in school, so it would disappoint me greatly if music and other electives, like band, which I didn't take, were taken away from schools. I took choir every single year in middle and high school, so yes, ladies and gentlemen, I love to sing, and I know the basics of music theory (how many beats each note is, the letter names of notes on the staff, etc). I do not know how to play any instrument, although I did take a basic piano course during summer school one year back when I was still at UCLA for my undergrad. I've always wanted to learn how to play the piano, though. Nowadays, I listen to radio music a lot, so I always have music playing in my car and while I'm working at my desk (I have an iHome). Perhaps I should consider finding a Zoom course that people are doing during quarantine from all this COVID madness. Or perhaps I should get back into recording singing videos, something which I only did for a few songs and for only a few weeks in college before I stopped completely. 
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Offline ccexplore

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Re: WillLem's Blog
« Reply #36 on: April 21, 2020, 10:31:30 AM »
I guess things vary from country to country.  In the US, law and medicine are generally graduate/post-graduate programs to be entered after you got your bachelor's (which doesn't necessarily have to be from the same school), and do have entrance exams.  I didn't consider in other places they may start you off at your undergraduate studies and include those 4 or so years (plus the BA) as part of the program.

So I guess for music, what we're discussing would basically be about getting admitted to a conservatory, which I imagine is pretty competitive too.

To be clear, when considering difficulty of getting admitted, I'm mainly considering within the pool of applicants, as opposed to something like the average theoretical difficulty measured over any random person.  So yes, for the many people who have no interest or little aptitude in music to begin with, it would be hard if they were to apply, but the people who actually applied are more likely to already be both more interested and more talented than the average person.  Even so, within the applicants I can still see it being quite competitive to successfully get admitted.

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Re: WillLem's Blog
« Reply #37 on: April 21, 2020, 07:25:12 PM »
In general, yes, many grad programs are ones that you start right after you finish your bachelor's, and it's not at all an unusual situation to do your bachelor's at one school while doing grad school at a different one. In my case, I didn't fare so well academically at the university I did my undergrad, and so I would not had been able to get into any master's program at my alma mater. Now that I've done both a master's and a credential program elsewhere at two different schools and did well with both, I should be able to come back and get accepted into some kind of graduate program at my original alma mater. That's what I'm planning to do sometime in the future, but I'm not exactly sure when.

There are some post-baccalaureate programs that you can do while you're still working on your undergrad, but there's not that many. A good example would be working on a teaching credential while doing your bachelor's. Depending on the school, it is also possible to work on a master's AND teaching credential at the same time (they call it a joint master's/credential program). Where I'm from, the CSU's don't offer such a program, but the UC's and some other non-CSU's do. In the CSU's, you can only work on your master's in education after you finish and obtain your teaching credential. So, if I wanted to do a joint master's/credential program, I would had needed to go to another school to do it. While killing two birds with one stone would had been nice with a joint program and get both a master's and a credential at the same time, it would had meant way more work than doing both programs separately. Personally, I would had gotten overwhelmed by the huge amount of schoolwork if I had done a joint program, so it's all good.

I think it shouldn't be difficult to be admitted even if you're not already into the subject matter, and if you do get accepted, it's generally difficult to do well in a program where one isn't already well-versed and not as enthusiastic and interested about the subject. Of course, requirements are different for every graduate program, but generally one has to have a solid undergrad GPA in order to be admitted into any master's program. It's still possible to get admitted even if you don't, but usually you'll have to appeal. It's generally not required to have a bachelor's related to the subject of the program you're doing for your master's in order to be admitted, at least here in the USA. It might be different abroad. Like you mentioned, it varies from place to place. Usually it's the job that you're applying for that requires that you did your degree related to the work required. This is especially true for teaching beyond the high school level, although I have several co-workers who did their bachelor's in a completely unrelated field to the subject they're teaching. Some of my math teacher friends were history majors or something non-math related, for example. At the same time, teachers come into the teaching field in different ways. Some have worked on their credential almost right away after graduating from their undergrad, while others have done work in other fields for years before making the decision to go into teaching. Like with doing your bachelor's and master's at different universities, it's also not uncommon to do a program in one field for your bachelor's but then do your master's in a completely different field than your bachelor's. I did both of mine in math, as I feel like I can't see myself doing anything else besides math, maybe other than engineering or possibly something like NASA, as I do love science as well, astronomy in particular. It hasn't been done yet, but I'm quite certain that I won't be doing math for my PhD like I did for my bachelor's, master's, and teaching credential. Instead, I plan to either do a 2nd master's in education, which was originally my plan after graduating from my undergrad, or a PhD in a TBD field.
« Last Edit: April 21, 2020, 07:52:53 PM by kaywhyn »
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Offline WillLem

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Re: WillLem's Blog
« Reply #38 on: April 24, 2020, 12:03:28 AM »
Regarding your Master's in music, WillLem, that's awesome and congrats. I've loved music a lot as a kid, and I still do. I consider the arts essential to have in school
---
perhaps I should get back into recording singing videos, something which I only did for a few songs and for only a few weeks in college before I stopped completely.

Thanks kaywhyn, congratulations to you also on your MA in Mathematics!

Music is a great thing to have in schools, even if it's only extracurriculur; quite a lot of British schools are dropping their music departments: it's usually the first thing to go when a school is struggling financially, and some schools are way more focused on the sciences so their Music departments are tiny and woefully underfunded.

Thankfully, private and independent music schools seem to be booming in the UK at the moment. I work for one, and it's a brilliant organisation. I hope to see even more of these as time goes on because they create great opportunities for young people who'd otherwise miss out on a formal musical education.

Coming back to the flat-Earthers thing for a moment: I have the ultimate proof that the Earth is not flat:

If it was, cats would have pushed everything off it by now! :lemcat:

Offline WillLem

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WillLem's Blog: On Expanding Earth Theory
« Reply #39 on: May 04, 2020, 07:46:38 PM »
I'm almost worried about doing this again, but... well, in the name of freedom of expression, here goes...

Planet Earth might be expanding.

Arguments for:

  • The continents fit together seamlessly on a smaller globe (this has been repeatedly proven and demonstrated by some of the world's finest scientists throughout history).
  • Planets are known to vary in size; the smallest being composed of rock, the largest being composed of rock surrounded by layer upon layer of gas (however, the exact reasons/causes for this are still under continuing investigation).
  • Planets have layers. Things that have layers tend to grow (onions, trees, symphonies, Lemmings levels ;P).
  • There is no satisfying explanation for why continents and oceans are massive. (This is my own thought added to this - I haven't yet read anything or seen anything that has really satisfied the question of why there are such equally enormous land masses and oceans, with such distinctive shapes. Emphasis on the word yet; I'm an open minded guy).

Arguments against:

  • There is nothing to adequately/conclusively explain the force behind, or cause of, this growth (but then - what about explaining the growth/variance in size of any planet?)
  • There is nothing to adequately/conclusively explain where all the water came from (but then - why is Earth the only observable planet with water? We know that gas comes from liquids, which comes from solids. This simple, provable fact could account for it in some small way - maybe Earth is entering the "liquid" stage of its development, and will next start to become a gas giant...).
  • Plate Tectonics Theory is grounded in years upon years of scientific discovery, thorough research from multiple branches of study, and physical exploration (OK, but humans have historically believed all manner of Theories with similar "certainty", and with similarly irrefutable evidence and conviction, only to later be corrected by another Theory).

You're all awesome. :lemcat:

Discuss! 8-)
« Last Edit: June 24, 2020, 02:28:22 AM by WillLem »

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Re: WillLem's Blog
« Reply #40 on: May 04, 2020, 11:50:41 PM »
Again, this is the kind of thing that scientists would likely know - or at least suspect - if there was much chance of it. It's not the sort of thing they'd overlook.

For an example of a very similar concept being considered - look into the scientific discussions on whether the universe may be expanding. ;)
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Offline ccexplore

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Re: WillLem's Blog
« Reply #41 on: May 05, 2020, 01:08:23 PM »
A quick perusal of Wikipedia yields this article: https://en.wikipedia.org/wiki/Expanding_Earth.  Basically the theory had been proposed back in the 19th century, but ultimately had not been found to line up with the current physical evidence, so it is currently rejected by the scientific community.  To quote the article:

Quote
Examinations of data from the Paleozoic and Earth's moment of inertia suggest that there has been no significant change of Earth's radius in the last 620 million years.

Technically speaking, the earth probably experiences microscopic amounts of expansions and contractions simply due to factors like being at slightly different distances from the sun as it orbits the sun.  The resulting small temperature and gravity changes will have some effect, though that's probably not what you are picturing in mind. ;P

The earth's internal temperature is expected to cool down over time, because heat radiates outwards and the internal sources of heat won't last forever.  The heat from the sun will not make up for this (I think the sun's heat mostly just drives the weather and the carbon cycle).  How the internal cooling affects the size of the earth I'm not sure, though usually solids contract as they cool.  Again, we're talking very tiny amounts here over very long stretches of time.

From a quick skim of the literature on Wikipedia, the prevailing plate tectonics theory explains that the continental plates movements over time can, from time to time, bring all the continents together into a supercontinent.  Such a movement would then force the colliding plate boundaries to fit against each other by force, and then later as the plate movements break them back apart again, you would see the separated continents with matching boundaries.  So instead of starting off unbroken, the matching boundaries actually occurred during the times when plates were pushed against each other.  The idea of an expanding earth breaking apart a mono-plate to form continents is a valid hypothesis, but not necessary to explain the matching continental boundaries, and apparently not supported by other current physical evidence.

=====================

There is nothing to adequately/conclusively explain where all the water came from (but then - why is Earth the only observable planet with water?

Not really sure what this have to do with the earth expanding or not expanding?  Anyway, one thing to keep in mind is the idea of a habitable zone.  Liquid water can only exists under fairly narrow ranges of temperature and atmospheric pressure.  Otherwise they are either locked away as ice, or has long since evaporated/sublimated as vapor.

You can browse Wikipedia for details, but I do believe most other locations in the solar system have some form of H20 (or at least are believed to have them--direct evidence would require sending out spacecrafts to explore, so it will take time to get good coverage of observations), just typically not in liquid form.   Comets are mostly ice and dirt if I recall correctly.  Some moons of the outer planets is theorized to have water deeper inside.  The cores of the gas giants may have a layer of ice, etc.  I don't even know for sure whether earth actually holds the most amounts of H20 (of any form) in our solar system, but even if it does, it could just reflect the planet being in the habitable zone where liquid water is stable.

maybe Earth is entering the "liquid" stage of its development, and will next start to become a gas giant...).

That's not expected to happen.  The earth start off hotter than today and will cool over time.  This is not conducive to turning things into gas.  Moreover, my understanding is that the general elemental makeups of the inner planets are too different compared to the outer gas giants to credibly be able to go from one to the other.

Offline WillLem

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Re: WillLem's Blog
« Reply #42 on: May 05, 2020, 05:01:19 PM »
The earth start off hotter than today and will cool over time.  This is not conducive to turning things into gas.

I read about this last night - apparently, there has been evidence to suggest that Earth started as a gas giant and that it all happened the other way around.

Coupling this with the hypothesis that the habitable zone may be something that could itself expand, this could mean that Jupiter and Saturn may one day be habitable!

Offline ccexplore

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Re: WillLem's Blog
« Reply #43 on: May 05, 2020, 06:02:14 PM »
Coupling this with the hypothesis that the habitable zone may be something that could itself expand, this could mean that Jupiter and Saturn may one day be habitable!

Yeah, I wouldn't necessarily celebrate that even if it were to happen.  Siberia and Death Valley are technically "habitable" but not really the kind of climate you as a human would want to live in year-round.  Significant changes to the current habitable zone will very likely push Earth towards either end, if not put it entirely out of the zone altogether.  Remember that there is vast distance between Earth and Jupiter--you are crossing both Mars and the asteroid belt.  A change large enough to put the outer gas giants into habitable zone will almost certainly be catastrophic to Earth.

From what I remember, our sun is not expecting to have significant changes for 4 or so billion years.  Eventually at some point, as it started running out of hydrogen on the surface, it will turn into a red giant and expand greatly in size.  That will change the habitable zone for sure--but we're also talking about an expansion of the sun so catastrophic that all the inner planets including Earth, are expected to be swallowed up by the expansion.  Before all that happens I don't believe the sun is expected to have a significant enough change in its output to change the habitable zone, not to the degree of the zone boundaries crossing one let alone multiple planetary orbits.

Offline mobius

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Re: WillLem's Blog
« Reply #44 on: May 20, 2020, 01:39:44 AM »
I don't have time to read through all this so I apologize if this was already covered but I don't think it was;

On Math:
Do you think numbers exist? Or to put it under another way which of the "three schools of thought" do you prefer? According to numberphile that is https://www.youtube.com/watch?v=1EGDCh75SpQ This video explains the three theories better than I can.

Platonism; Numbers are real; all mathematical claims are true; they exist; they're abstract objects.

Nominalism; Numbers are not "real" objects; they just describe things that exist. Math is only true claims about the world.

Factionalism; Numbers do not exist.

I'm torn between nominalism and factionalism. I'm not sure as it may depend on the definition which could get a bit squirrelly.

The first two fall short especially when pushed to limits. E.g. start talking about the more abstract and complex parts of math like infinity or imaginary numbers.
If numbers are real; that is some kind of objects that exist in the world; where are they, what are they?
If numbers are merely a language to describe reality what is the reality that is being described with a number like pi? It's just an approximation

Deep down I believe mathematical claims are 'true' merely because we say they are and we decide what truth is in the first place. We created this complex theory in our mind and it ends at our mind. I believe all concepts are 'fictitious' in that sense or illusory. And I don't mean we 'create' it in the same sense that you might create a song or work of art. It is created when you look at a mathematical problem. There is only one thing that can happen; one way it can work because you have the brain that you have that works in this way. Math is an aspect of our mind; a language, a code that describes the world or makes up the world. Perhaps it is universal. Or it might turn out that other intelligent life operates with a totally different set of mental tools instead....

--------------------

On Earth might be expanding;
Honestly I don't find these arguments very compelling sorry. I've never heard this but it doesn't make a lot of sense to me.
-I thought this was why the Pangaea theory was formed? And I thought Pangaea is/was proven by plant/animal fossils being similar in certain places like Africa and South America etc.
-I don't really see what's strange about continents and oceans being massive. Earth imo is basically a 'water world'. It's 78 (or around that?) percent ocean.

They basically know (imo) how planets form and how/why they grow. In our case there was the 'period of large bombardment' when the solar system was young and there was debris (asteroids large and small) everywhere flying around. Larger objects have more gravity so they attract the most stuff and grow over time.
Nowadays (the past million(s) of years) there's relatively a tiny amount of asteroids and debris flying around, not nearly enough (that I'm aware) to make planets grow.


Are you familiar with project Kepler space telescope? Over the past ~10 years Kepler has discovered (with the help of the so-called citizen scientist project) over 1,000 exo-planets. These are planets (mostly gas-giants but many are more like Earth's size) around other stars. Kepler has only looked at a tiny portion of sky so far. Scientists now estimate there may be more than twice as many planets as stars in the observable sky (which means a huge huge amount).

What's interesting to me is they've discovered many "super earths". These are earth like planets but larger and often around smaller cooler, older stars. And what they've found so far suggests these might be more abundant than planets more like earth AND they might have calmer more uniform temperature and weather and geothermal activity. And these stars live much longer than our sun will as well (as long as trillions of years...) Meaning life would have longer to get a start; and easier time surviving and longer to live and evolve.
In hubris we always say "Earth is perfect for life" But it may turn out we our unlucky and there are in fact BETTER places for life elsewhere.

Conversely with fewer upheavals and difficulties these planets may have a HARDER time producing life; as maybe evolution needs those challenges to adapt and produce more complex life. 

anyhoo It's a very exciting time for science!

What are you're thoughts on alien life?
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Offline ccexplore

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Re: WillLem's Blog
« Reply #45 on: May 20, 2020, 07:53:20 AM »
"Existence" is a pretty slippery concept anyway.  Simulation theory for example propose the possibility that everything we perceive may in fact just be part of a simulation.  If that is the case then you can probably argue that no object is any more "real" than numbers, that none of them are in fact "real".

Simulation theory actually can potentially help describe the difference between numbers and your everyday "real" objects.  Just like Lemmings, the simulation could consist of both code--programming that governs how the physics work in the simulated world, and data representing the actual objects or elements of the simulated world--just like the data loaded from a level file that describes the individual elements such as terrain, objects, etc., plus additional data in memory needed for things like lemmings that are introduced into the level while level is running.  Numbers seem more aligned to the programming of a simulation while "real" objects seem more aligned to the data.  Interestingly, outside of the simulation, they may not actually be all that different--in a computer for example, the code and the data actually are stored ultimately the exact same way (as bits); they are only different based solely on how the CPU is instructed to operate over them--either to read them as instructions to execute, or as data to be operated on by the executing instructions.

If numbers are merely a language to describe reality what is the reality that is being described with a number like pi? It's just an approximation

Pi is not an approximation.  "3.14" or "3.14159" are approximations of pi.  It so happens that Pi does not have a finite representation as a decimal number, but its definition is simply based on the ratio of circumference to diameter.  In a way, the circle is the embodiment of Pi.

Now, you can argue whether the perfect geometric circle is "real" or not in the same way you discuss whether numbers are platonic, nominal or factional.

It is true that it can be very difficult if not impossible, to ascertain how much of our perceived reality is just a product of our mind versus having an independent existence outside of the mind, though I feel like this goes for anything not just numbers though.

Offline Forestidia86

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Re: WillLem's Blog
« Reply #46 on: May 20, 2020, 10:01:01 PM »
I think the philosophical (ontological/metaphysical) question whether numbers or sets exist revolves around the question whether abstract objects exists. Abstract objects are such outside of space-time or outside the material world. At least sets as entities are not part of the material world in that sense.

Offline mobius

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Re: WillLem's Blog
« Reply #47 on: May 21, 2020, 01:00:35 AM »
Yeah what I'm actually trying to argue whether the perfect geometric circle is "real" or not. Just didn't think how to word it properly.
The 'actual' pi is as you describe of a "perfect" circle. IMHO perfection is another concept of the mind and as such isn't real in any ultimate sense.
Tell me if this makes any sense at all;
If you draw a circle on a piece of paper; even if you use tools and or calculations to get it as perfectly round as possible it's always going to be off; thousands if not millions of atoms in the lead or the paper etc can't be perfect (and then we could go down the the quantum level but that just opens up a whole other can of worms...). Therefore when calculating pi you get 3.1415... something but it's not real pi, right? The formal way pi is calcuatlated is really complicated; it uses math; no 'real' world objects (That is; not measuring something in the world). Doesn't this sort of dictate that pi itself is abstract in that sense? = not real.

Actually I do think that all objects (the "real" world) in some sense isn't real because reality itself is just another concept. What's reality? As opposed to what? What's in our mind? Is our mind not part of reality?
This may sound very strange but it's an idea I've had for quite a few years now but it was very much solidified/given more weight when I started meditating by actual experience. I can't explain it anymore than that however, without sounding crazy (as if I don't already) :P


I was never a huge fan of simulation theory. For one; it has the same problem the God problem has: you just ask who created/maintains the simulation of the simulator? Then you get a recursion. IMO there isn't as yet strict evidence the universe/life is super recursive like this. At least in size; life gets very very different the larger/smaller you get. In space life seems similar enough in distant galaxies but we're only scratching the surface of that exploration. And we're looking back in time (which is subject of another blog I want to write).

Secondly the theory assumes that life (in our sense of the word) arises when these simulations are made. That seems like a slippery slope.
As to the example of Lemmings;
Lemmings do exactly what the programmer tells them to do. When bugs occur it was because there was either a mistake in the code or the code combined in a way that was not foreseen producing an unexpected result.
but isn't it true that in our real world there are always exceptions. That is; our theories of the universes (including math) are only approximations; rules which make sense in the free and imaginative mind; have the ability to compute amazing results in reality; but cannot ultimately define it 100%.


I'm just arguing the same thing over again and it doesn't lead anywhere... I don't want to believe in Simulation theory very strongly and I'm not sure why. Maybe it's close relation to belief in God?
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Re: WillLem's Blog
« Reply #48 on: May 21, 2020, 04:43:12 AM »
Some excellent conversation going on here, I'm glad this topic is now thriving in the way I'd originally hoped. :lemcat:

Do you think numbers exist? Or to put it under another way which of the "three schools of thought" do you prefer?

I'm almost certainly a Mathematical Fictionalist, in that I do tend to question the existence of numbers, or at least what we understand about "numbers existing", and I do tend to regard them as being products of thought rather than original truths. However, they are incredibly useful, and number systems are responsible for many of world's great advancements.

What Numberphile says about the Fictionalist reconciling this dichotomy as "numbers are successful, but this doesn't make them true" resonates with me more than anything else that was mentioned in this video, so that would probably be where I stand. Of course, it's a very tentative standpoint based on relatively limited understanding, hence my fascination with the subject and desire to discuss and learn more.

I'm torn between nominalism and factionalism. I'm not sure as it may depend on the definition which could get a bit squirrelly.

Since squirrels are cute, fast, intelligent and resourceful, I wouldn't worry too much about things getting squirrelly.

The first two fall short especially when pushed to limits... If numbers are real; that is some kind of objects that exist in the world; where are they, what are they?... If numbers are merely a language to describe reality what is the reality that is being described with a number like pi? It's just an approximation

Agreed. This is the root of my fascination with the subject. In the video, the example was used of describing numbers to a child using objects: "here's a pencil, add another pencil and you have two pencils!"

I can't help but then think: yes, but a pencil is made up of 1 piece of wood and 1 piece of lead, so 1 pencil is equal to 2 parts. This would seem, on the surface of things, to prove that 1 = 2, but of course it doesn't prove that at all. The object of "1 pencil" is here being treated as a whole object, despite being made up of multiple elements.

However, it leaves the question behind: the fact that the pencil is being treated as 1 object has been decided - i.e. it is a product of human decision and system-making. It it not, therefore, a truth.

This thinking is not necessarily useful, of course, as it somewhat arbitrarily challenges a system that's proven to be successful, valid, and progressive. It merely asks the question: yes, but is it true? However, I keep coming back to that, and always have done. Maths is amazing, no doubt, but it has its limits.

Math is an aspect of our mind; a language, a code that describes the world or makes up the world. Perhaps it is universal. Or it might turn out that other intelligent life operates with a totally different set of mental tools instead....

That would be incredible: if we were to meet beings from other planets who also use Mathematics, or - better still - have some other way of understanding things that seems completely bizarre to us at first, but works for them!

I thought this was why the Pangaea theory was formed? And I thought Pangaea is/was proven by plant/animal fossils being similar in certain places like Africa and South America etc.

The evidence all points to the continents being joined at some point in history, and Pangea theory was developed in support of Plate Tectonics theory. However, ask yourself which is more plausible:

1. All of Earth's continents were once joined together one one side of the planet, whilst all of its water made up the other half. Because... reasons.

2. The Earth was once smaller, and as it grew the crust broke apart, new crust was formed, and the gaps filled with water (likely from comets).

OK, I've biased the second option slightly, but it's really hard not to when you describe each theory in these terms!

Are you familiar with project Kepler space telescope?... What's interesting to me is they've discovered many "super earths"... In hubris we always say "Earth is perfect for life" But it may turn out we our unlucky and there are in fact BETTER places for life elsewhere.

This is fascinating: I'll look into this, for sure! I have no doubt that there are other places in the Universe which humans may find more temperate and better suited to us than Earth. It's just an issue that we can't seem to stop squabbling with each other long enough to make significant progress in finding these places, but... it's only a matter of time.

What are you're thoughts on alien life?

Aliens exist. There simply must be other planets out there which support life. It simply wouldn't make sense that, in a seemingly infinite Universe, there is only 1 inhabitable planet with life on it. As far as I'm concerned, the very idea is proof enough of life elsewhere. It would be good to find some, though.

"Existence" is a pretty slippery concept anyway... Numbers seem more aligned to the programming of a simulation while "real" objects seem more aligned to the data.

So, God is a coder! Who'd have thought a programmer would take such a viewpoint? ;P

Seriously though, this is a very interesting idea and would indeed account for the difference between the existence of manufactured systems (languages, numbers, programming code, etc) and the existence of "real" objects (lemmings, humans, ice cream, etc).

My main objection to this theory would be that it's based on our recent understandings of science and technology, and isn't compatible with more historical thinking. The average human living a few centuries ago wouldn't have had any concept of what a "simulation" was. We only understand it now because we've brought about the possibility through our various technological and linguistic advancements. So, naturally, this leads us to question whether such advancements may explain our own reality. An interesting idea, but isn't independent enough to stand up against more constant, abstract ideas about truth and the nature of existence/reality.

I was never a huge fan of simulation theory. For one; it has the same problem the God problem has: you just ask who created/maintains the simulation of the simulator?

Yeah, exactly - that, too.

Pi is not an approximation.  "3.14" or "3.14159" are approximations of pi.  It so happens that Pi does not have a finite representation as a decimal number, but its definition is simply based on the ratio of circumference to diameter.  In a way, the circle is the embodiment of Pi.

But then - which came first, Pi or the Circle?

It is true that it can be very difficult if not impossible, to ascertain how much of our perceived reality is just a product of our mind versus having an independent existence outside of the mind, though I feel like this goes for anything not just numbers though.

Absolutely agreed. Numbers are a good place to start the discussion though, because they are a man-made system which holds up to extensive scrutiny, meaning that they could represent a doorway to truth.

To expand on this: if I say "flying unicorns exist", the response is either likely to be amusement, pity or ridicule. However, if I say "the number 7 exists", it stands up to further discussion. So, better to pursue this particular line of enquiry - even if we may not agree that numbers exist, we can agree that the concept of numbers may be a key to better understanding of reality. Rather than focus on the key though, I'm more interested in finding the door and seeing where it leads.
« Last Edit: May 21, 2020, 04:53:34 AM by WillLem »

Offline Forestidia86

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Re: WillLem's Blog
« Reply #49 on: May 21, 2020, 01:12:32 PM »
However, it leaves the question behind: the fact that the pencil is being treated as 1 object has been decided - i.e. it is a product of human decision and system-making. It it not, therefore, a truth.

Yeah it is an interesting question what we accept as distinct objects, we usually don't treat random parts of space-time as one object. But how we count is a matter of units I think, pencil is another unit than the lead core.

What is a truth for you, what entities are able of being true or false? One approach would be that (only (certain)) sentences can be true or false.
Imagine there is exactly one pencil lying on the table at a certain point. Would you see the sentence: "There is exactly one pencil on the table at that certain point." as true or as false or as neither? Is such a sentence for you able to be true or false?

Offline ccexplore

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Re: WillLem's Blog
« Reply #50 on: May 21, 2020, 07:08:45 PM »
I was never a huge fan of simulation theory. For one; it has the same problem the God problem has: you just ask who created/maintains the simulation of the simulator? Then you get a recursion.

Simulation theory doesn't claim it's an infinite recursion of simulations though.  It only suggests that we (and the specific universe we're in as we know it) might be in one, and some would argue that it might be more likely that we are in one than not (because there could be simulations within a simulation, so the argument goes that there are more worlds one could be in that are simulated versus not simulated).

Anyway, I don't have a strong preference for or against the theory, but I do feel it can be useful to at least consider some implications that could arise from the theory.  For example, the thing I pointed out about how the code and data of a running Lemmings game are in fact all just bits, even though within the simulated universe of the lemmings level they couldn't be more different in their natures.  Even if our world isn't actually simulated, it could still be the case that maybe both concrete and abstract objects can still in fact arise from the same common underlying elements, despite us experiencing them very differently within the universe.

Offline ccexplore

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Re: WillLem's Blog
« Reply #51 on: May 21, 2020, 09:55:54 PM »
I thought this was why the Pangaea theory was formed? And I thought Pangaea is/was proven by plant/animal fossils being similar in certain places like Africa and South America etc.

The evidence all points to the continents being joined at some point in history, and Pangea theory was developed in support of Plate Tectonics theory. However, ask yourself which is more plausible:

1. All of Earth's continents were once joined together on one side of the planet, whilst all of its water made up the other half. Because... reasons.

See, that's a common misconception that I thought I tried to dispel earlier in this thread but apparently not successfully.  To be fair, the way you stated it above actually is technically accurate, just I think there is a huge unspoken misconception that I need to emphasize again.

Specifically, plate tectonics do not actually posit that the Earth started off as Pangaea.  Instead, the continents of today were the result of the breaking apart of an earlier supercontinent that geologists named Pangaea, but other stuff happened well before you got to Pangaea.  Here's the introductory paragraph in Wikipedia's article for Pangaea (with emphasis added by me):

Quote from: Wikipedia
Pangaea or Pangea ( /pænˈdʒiːə/[1]) was a supercontinent that existed during the late Paleozoic and early Mesozoic eras.[2][3] It assembled from earlier continental units approximately 335 million years ago, and it began to break apart about 175 million years ago.[4] In contrast to the present Earth and its distribution of continental mass, Pangaea was centred on the Equator and surrounded by the superocean Panthalassa. Pangaea is the most recent supercontinent to have existed and the first to be reconstructed by geologists.

Plate tectonics merely posit that the continents will move over time, and periodically from time to time their trajectories will bring them all together, and then at those times you get a supercontinent.  Pangaea is the most recent incarnation of such a supercontinent, and as such, it is how the shapes of the continents of today fit together.  But it's not the first one, nor the only time Earth had a supercontinent, nor does the theory says the Earth must start off with one.

At the times in Earth's history when continents did come together, their old boundaries were effectively erased by great, violent force as the continents collide and smoosh together.  It's like if you take two balls of play-doh and smoosh them together into one, in the process the touching parts of the original balls' surfaces fuse together and effectively disappear as distinct surfaces.  So you lose much of the original shapes of whatever earlier continents there were before the formation of the supercontinent.  In contrast the process of the supercontinent breaking apart doesn't have anything causing the boundaries created from the break to diverge from one another, and so the resulting continental pieces will still retain a lot of the boundaries that fit against one another.

=====================

The average human living a few centuries ago wouldn't have had any concept of what a "simulation" was.

Yes and no.  Yes, modern technological advances had given specific ways to describe and talk about simulation theory, language that didn't exist for earlier philosophers.  But very similar ideas had existed long before computers were invented, just that they used different languages and concept instead to talk about it.  To quote the introductory paragraph on Wikipedia's article on simulation theory:

Quote from: Wikipedia
There is a long philosophical and scientific history to the underlying thesis that reality is an illusion. This skeptical hypothesis can be traced back to antiquity; for example, to the "Butterfly Dream" of Zhuangzi,[1] or the Indian philosophy of Maya. A version of the hypothesis was also theorised as a part of a philosophical argument by René Descartes.

The "Butterfly Dream" alluded to for example, basically boils down to the question:  "Is it the man dreaming he is a butterfly, or is it the butterfly dreaming he is a man?  Can you really tell which is which?"  If you consider that a dream is a mind's simulation of the "real" world, then effectively you have a form of simulation theory in another guise.

Offline mobius

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Re: WillLem's Blog
« Reply #52 on: May 22, 2020, 01:26:59 AM »
The way I see it math isn't exactly a language or description. It's part of the way our mind works. It's how we see the world. This is how we classify and define the world. Doing this leads to further ways of classifying and defining. Which leads to our imagining a different world (ideas) which leads to altering the outside world to change it (creating things). When creating a wooden table for example; you don't create the thing out of nothing. You alter existing materials to become the shape of the table. When calculating the trajectory of a spaceship; You didn't create the trajectory (that's an abstract thing anyway). You had an idea; then enacted it onto the world, altered the movement of the spaceship and results were had; which were most certainty not 100% like your idea; but similar (if successful). That's how success is generally defined (well one way).
----------
The only limits to math might be our own imagination ;)

---------
well this is certainty getting on quite a subjective soap box and I'm sure someone like Arty could speak better on this but I would think there's a big difference between the concepts of God and coder. Coders use a pre-existing or self make set of rules to create a system and are then bound by that system/must play within the rules of that system. "God"(s) are not bound by any system or anything at all. They can change the rules; Maybe (this is a recent thought I've had) they aren't even limited by an imagination (if that even makes any sense... not sure that it does).
That's what really limits us; if you didn't/can't think of it; it's not going to happen. Even if it's something really simple. If it doesn't occur to you; you can't do it, except by stumbling upon it by accident.

quote from WillLem;
[Seriously though, this is a very interesting idea and would indeed account for the difference between the existence of manufactured systems (languages, numbers, programming code, etc) and the existence of "real" objects (lemmings, humans, ice cream, etc).]

Still though I have trouble understanding the meaning of this when pushed to the limit: What truly is the difference between these things (real and inside our head) when our head is part of this reality in the first place? When you have a thought it is an image, voice or feeling that is a mixture or regurgitation of previously collected stimulus. The images in your head are "real" in that they are there, they exist.

This is part of my rejection of simulation theory. In order to support that you have to have a clear distinction between reality and simulation which imo; there isn't such a thing. That distinction itself is man-made and subjective. Maybe this is a circular argument I'm making idk; the more I think about it the less it makes sense.


[But then - which came first, Pi or the Circle?]

the answer is which ever first occurred to the first human (or perhaps non-human....) that thought of it. Which means essentially; circle. In the 'real world' there is no such thing as pi and there is no such thing as circles.

The concept of objects too again, like all other, is illusory. We can break up the universe into multiple 'things' and separate it and label it in any way that we want; but ultimately the universe is just one 'thing'; everything including you.

Joe Rogan said when congressman Anthony Wiener had his **** pic scandal that's when he started taking simulation theory seriously :laugh::laugh:

Oh btw; if you like this sort of thing try the book "Super Mind" by John Micheal Godier.
« Last Edit: May 22, 2020, 01:34:08 AM by mobius »
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Offline ccexplore

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Re: WillLem's Blog
« Reply #53 on: May 24, 2020, 10:42:12 AM »
If you draw a circle on a piece of paper; even if you use tools and or calculations to get it as perfectly round as possible it's always going to be off; thousands if not millions of atoms in the lead or the paper etc can't be perfect (and then we could go down the the quantum level but that just opens up a whole other can of worms...). Therefore when calculating pi you get 3.1415... something but it's not real pi, right? The formal way pi is calcuatlated is really complicated; it uses math; no 'real' world objects (That is; not measuring something in the world). Doesn't this sort of dictate that pi itself is abstract in that sense? = not real.

Yes, you may never find a perfect circle in the real world.  But you do find plenty of slightly imperfect circles that deviates slightly from pi, and the mathematics tell you how the amount of geometric imperfection translates to an amount of numeric deviation.  It also tells you how you could try to smooth out, reduce some of the geometric imperfections, and then you can get ever closer numerically to pi (even if never actually achieve it exactly).

So you're in the interesting position that for something that's abstract and thus one may argue as "not real", somehow you get plenty of "more real" things in the real world that still all behave rather closely to this possibly non-existent abstract thing, and can often be made to become ever closer to it even if never actually achieving the abstract perfect state.  It kind of makes the question of whether the abstract thing is "real" or not almost a little irrelevant?

I don't want to believe in Simulation theory very strongly and I'm not sure why. Maybe it's close relation to belief in God?

I'd rather leave religion out of this.  But I'm rather surprised actually, I'd kind of think that a belief in God could actually make simulation theory easier to accept, at least if you consider that God may be the maker of the simulation.  If you for example take Christianity's tenets about how man are created in God's image, how God is all-powerful and can make anything happen in our world, all that actually seem rather even more sensible if our world is a simulation and God has created it and is maintaining it from time to time.  Even things like death, heaven and hell can be part of the simulation and aren't invalidated in any way by the hypothesis.  It even boosts God's divinity status as something only He uniquely possesses, as everything else is in the simulation and only He exists outside of it.

In any case, I will point out that simulation theory has an obvious built-in weakness, in that the theory being true can also very well mean it can never be proven.  Just like there's no way for the lemmings to escape your computer into the world outside, most simulations would effectively trap their inhabitants within the simulation, possibly with zero observable interference from the "outside", making it impossible for the inhabitants to come to any definitive proof that they are in a simulation.

For me, simulation theory is more useful as a "what-if", a framework to talk about and further explore ideas and possibilities.  Whether we are actually in a simulation or not I almost couldn't care less.  For example, the analogy that data and code are both just bits, it may have started from the idea of how our computers operate, but can potentially translate even if our world isn't actually a simulation--the more general idea that even something as seemingly different as, say, laws of physics versus the atoms and particles they govern, may well in fact have some common underlying element/entity that embodies or explains both.

Math is an aspect of our mind; a language, a code that describes the world or makes up the world. Perhaps it is universal. Or it might turn out that other intelligent life operates with a totally different set of mental tools instead....

Here's the thing though.  Somehow math seems surprisingly good at describing the physics of our world.  It also seems to be the case that the laws of physics remain constant over the entire known universe.  Even if other intelligent life operates with some other mental language and process, if we expect that they would arrive at many of the same conclusions that we do with our math and physics, then it would seem to imply there'd actually be a translation (perhaps a very complex and elaborate translation, but one may exist nevertheless) between our math and physics and whatever their mental tools may be.  And so the "languages" are maybe not so different after all.  This actually makes the language analogy really apt here--it's just like how different human languages can be translated to one another, and while some things may be lost in translation, there's still enough common ground between diverse cultures and groups to make translations fairly effective.  The language analogy also shows that, even though we don't quite have a "universal language" today, it doesn't preclude common ground.  Even if math is not universal, at least some of it will likely still be translatable to whatever mental tools the aliens may be using.

Offline WillLem

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« Reply #54 on: May 24, 2020, 02:25:49 PM »
What is a truth for you, what entities are able of being true or false?

I suppose that one mark of truth is constancy - i.e. if something cannot be changed by time, perspective or physics, then it is "true". However, it is also possible for something to be true right now, but false tomorrow. However, such things are subject to the question of whether they were ever true, a question that can only be reconciled by evidence. And, such evidence needs also to be "true".

It's a great question though, and one I can't easily answer.

Imagine there is exactly one pencil lying on the table at a certain point. Would you see the sentence: "There is exactly one pencil on the table at that certain point." as true or as false or as neither? Is such a sentence for you able to be true or false?

Yes, but it would need to be verified. It's possible to see what appears to be a pencil on what appears to be a table seemingly in the present moment, but that requires three things to be confirmed: that what you're seeing is in fact a pencil, and that what it's on is in fact a table, and that you're seeing it now rather than remembering it later. For these, we largely rely on our senses, as well as prior knowledge of what the objects are, and awareness of the passage of time.

Pangea theory was developed in support of Plate Tectonics theory. However, ask yourself (if it's) plausible (that) all of Earth's continents were once joined together on one side of the planet, whilst all of its water made up the other half.

Specifically, plate tectonics do not actually posit that the Earth started off as Pangaea.  Instead, the continents of today were the result of the breaking apart of an earlier supercontinent that geologists named Pangaea, but other stuff happened well before you got to Pangaea.

Whether or not Earth's land masses started off as a super-continent is not what's in doubt here: Expanding Earth Theory questions whether they were ever in such a formation on one side on Earth at the size it is now whilst all of the water was on the other side. It seems to be more plausible that the continents were in fact joined on all sides, which of course would only be possible on a smaller planet, with or without water.

At the times in Earth's history when continents did come together, their old boundaries were effectively erased by great, violent force as the continents collide and smoosh together.  It's like if you take two balls of play-doh and smoosh them together into one, in the process the touching parts of the original balls' surfaces fuse together and effectively disappear as distinct surfaces.

I just want to take a moment to enjoy your use of the verb "smoosh" twice in as many consecutive sentences. :lemcat:

Smooshing, as I understand it, is what you do to dogs, cats or other small animals that are particularly cute, displaying extreme physical affection. i.e. "come here so I can give you a smoosh!" :crylaugh: :lemcat:

The "Butterfly Dream" alluded to for example, basically boils down to the question:  "Is it the man dreaming he is a butterfly, or is it the butterfly dreaming he is a man?  Can you really tell which is which?"  If you consider that a dream is a mind's simulation of the "real" world, then effectively you have a form of simulation theory in another guise.

That's true, I hadn't thought of it that way: dreams are, in a way, a simulation of reality. Descartes attempt to reconcile this with his oft-quoted "Cogito, ergo sum" - existence is certain as long as there is awareness. However, the nature of that existence is still, indeed, up for investigation.

Coders use a pre-existing or self make set of rules to create a system and are then bound by that system/must play within the rules of that system. "God"(s) are not bound by any system or anything at all.

Aren't they?

That's what really limits us; if you didn't/can't think of it; it's not going to happen. Even if it's something really simple. If it doesn't occur to you; you can't do it, except by stumbling upon it by accident.
...
When you have a thought it is an image, voice or feeling that is a mixture or regurgitation of previously collected stimulus. The images in your head are "real" in that they are there, they exist.

Is thought, therefore, God?

In the 'real world' there is no such thing as pi and there is no such thing as circles.

I disagree here: the observable world is full of naturally-occurring circular and spherical objects. I'd probably prefer to conclude that the shape came first, and pi is our way of understanding and measuring it. But then, it could also be reasoned that pi is the "natural code" that allows circular shapes to exist in the first place... hence the question.

Yes, you may never find a perfect circle in the real world. But you do find plenty of slightly imperfect circles that deviates slightly from pi, and the mathematics tell you how the amount of geometric imperfection translates to an amount of numeric deviation... It kind of makes the question of whether the abstract thing is "real" or not almost a little irrelevant?

How so? (i.e. how does it make it irrelevant?)

I will point out that simulation theory has an obvious built-in weakness, in that the theory being true can also very well mean it can never be proven. Just like there's no way for the lemmings to escape your computer into the world outside, most simulations would effectively trap their inhabitants within the simulation, possibly with zero observable interference from the "outside", making it impossible for the inhabitants to come to any definitive proof that they are in a simulation.

If you could enter a simulated world of your own making, design it to be exactly as you want it to be, but with the caveat that you would either:

a) Not be aware that it was a simulation after entering it, but if you ever realised it was then you would immediately leave and never be able to return to it.

or

b) Be aware that it was a simulation, but be unable to leave after you entered it.

Would you enter the simulation under either of these conditions? If so, which one, and why?
« Last Edit: April 28, 2021, 03:10:50 AM by WillLem »

Offline mobius

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Re: WillLem's Blog
« Reply #55 on: May 24, 2020, 02:28:57 PM »
I wrote this before Willem posted and I don't feel like reading his post and/or modifying mine right now :P

First of all just to be clear:
-I'm not trying to argue in any way that math isn't useful or doesn't describe the "world" well.
-I'm not exactly a religious person. (though sarcastic comments I've made during Lix and what-not may have confused people). While I do have interests in religion; religious people wouldn't call me religious. [this is a topic for a different conversation]

I don't believe in God; that's partly why I also don't like Simulation theory. Like I said earlier it has similar problems. The biggest one for me is ultimately it just raises more questions than answers:

-Who creates God/the simulation?
-when/where/why/how was God/this simulation made?
-ultimately what does this mean for my life here and now? If anything at all. And if nothing: then why even care?

-But more directly I think the main thing I dislike about the theory is it just seems like too big of a leap/assumption about too many unknowns about the universe. Like if we are ants inside a colony and we're trying to guess about how this glass container works and what exists outside of it; we're basing it all on the stuff we're doing inside our sand colony here and now. Clearly; that's not even remotely like what's going on outside... in more ways than one.

Don't get me wrong: it is fun to think about. That's why I recommended that book. And I agree with your (ccexplore's) points made.  But at the end of the day if I was asked if I believe that a monotheistic religion or simulation theory are likely good explanations of our universe: I would answer no, not at all.

quote from ccexplore:
[Here's the thing though.  Somehow math seems surprisingly good at describing the physics of our world....]

First of all I'm not arguing against this point. I agree to a point. However I feel like everyone who makes this argument misses a key point:
Math must describe the world; there's no other possibility. If it didn't explain the physics of our world; it wouldn't exist; and we wouldn't be having this conversation. If I propose something to you like 1+1=3; this doesn't work; doesn't lead to any new or interesting insights; doesn't lead to any new math or anything useful what-so-ever. Therefore you'd throw it out and say this is "wrong" or "pointless". So IMO; it's no shock or oddity that "math *somehow* explains our world to an amazing degree of "accuracy". We are deciding what *all* of these things mean; WE are saying this is how the world operates. So why are we so amazed that it works?

It's what I always think when people say things like "it's a miracle that we are here; that the universe exists and is so perfect for human life."

First of all; I wouldn't exactly call it perfect for human life when things like 'natural disasters' (which are just things about the way the earth operates normally and has to; or earth would be very different; in some cases in not so great ways) kill millions of people over the centuries.

But secondly the larger issue: why wouldn't it be a "miracle" we exist? Because if we didn't exist we wouldn't be able to say "it sucks we don't exist, what a shame we aren't alive to enjoy the fruits of a universe that isn't here for us... sad. ???

So my question to everyone raises this issue is what would life be like if math didn't work so well? Or if you think things came about "by chance" and they weren't orderly? Do you think life would be chaotic or something?
Personally I think there are an infinite number of ways life could be like for us; endless trillions upon trillions of possibilities of different, maths, languages, cultures etc. But if we had a similar thought process to what we're talking about here; then no matter what different possibility the other matters are; we'd be arguing the exact same thing. That math/science etc somehow "amazingly" describes our world.

It kind of makes the question of whether the abstract thing is "real" or not almost a little irrelevant?

Yes actually; that is kind of my point. "Real" is a concept that we humans create in our minds; it's part of the rule structure of our mind/society. It's required for our society to work the way it does today. But ultimately there's no reality and no illusion.
"The world is just the way that it is. It is neither good nor bad, right nor wrong." -Ajahn Sumedho
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Offline ccexplore

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Re: WillLem's Blog
« Reply #56 on: May 24, 2020, 11:04:17 PM »
Whether or not Earth's land masses started off as a super-continent is not what's in doubt here

But that's exactly the problem.  The statement "Earth's land masses started off as a super-continent." is false, the misunderstanding I kept trying to correct.  Plate tectonics actually posits that the Earth's land masses go through cycles of coming together and breaking apart.  It's not unlike how the hands of a (analog) clock are usually apart from one another, but every 24 or so times a day they would happen to be right on top of each other for an instant.  The hands don't have to start off together.  They can well be very far apart at the time you put the battery/power to turn on the clock, and over time the hands will still periodically meet one another.

According to Wikipedia, it appears the Earth has had 3 times so far where the land masses come together:

Quote from: Wikipedia
The movement of plates has caused the formation and break-up of continents over time, including occasional formation of a supercontinent that contains most or all of the continents. The supercontinent Columbia or Nuna formed during a period of 2,000 to 1,800 million years ago and broke up about 1,500 to 1,300 million years ago.[79] The supercontinent Rodinia is thought to have formed about 1 billion years ago and to have embodied most or all of Earth's continents, and broken up into eight continents around 600 million years ago. The eight continents later re-assembled into another supercontinent called Pangaea; Pangaea broke up into Laurasia (which became North America and Eurasia) and Gondwana (which became the remaining continents).

The age of the Earth is estimated to be around 4.5 billion years old.  So what may be the first time the Earth had a supercontinent was at least 2.5 billion years after the birth of the planet.  It is quite likely that the land masses were apart from one another during those long years before the first occurrence of a supercontinent.

Indeed, with a cycle of coming together and breaking apart, an expanding earth theory actually becomes less useful even without taking into consideration the contradicting of other physical evidence scientists have found.  For the land masses to come back together (as had occurred multiple times in the past and will do so in the future), it isn't quite enough to just say the earth shrinks back down--if you take the current arrangement of continents, merely shrinking the earth would still leave you with at least 2 distinct oceans; you need significant movement of the continents to end up with one pan-ocean.  So the earth changing size doesn't actually quite explain a repeating cycle of land masses coming together and breaking apart, while plate tectonics do, and can do so without having to posit the earth also changing size (though the theory doesn't preclude that either; it's only through physical evidence that scientists have determined there hadn't been any significant changes in size).

they were ever in such a formation on one side on Earth at the size it is now whilst all of the water was on the other side.

It seems like you're also picturing Pangaea somewhat wrong.  As you should know, the Earth is actually mostly ocean.  The supercontinent would not occupy anywhere remotely close to the full area of a hemisphere, it's more like around a quarter or so of the total surface area:
 


It seems to be more plausible that the continents were in fact joined on all sides

The Earth got all its water on its surface relatively early on.  That volume of water doesn't just disappear or reappear by magic even if the planet were to change size.  For the planet to have started off so much smaller that the current total area of land mass would actually occupy almost all of the planet's surface, the same volume of oceanic water we have today would pretty much completely cover up all land masses leaving the planet looking entirely ocean actually.

Which points to another thing actually.  The earth's crust still exists wherever there are ocean, it's only merely buried by a large quantity of water.  The earth's tectonic plates (not to be confused with the relatively small portions of visible land masses protruding above the oceans) are always "joined on all sides", just that a lot of their touching boundaries are actually at the deep, deep bottom of the ocean floor.

==============================

Yes, you may never find a perfect circle in the real world. But you do find plenty of slightly imperfect circles that deviates slightly from pi, and the mathematics tell you how the amount of geometric imperfection translates to an amount of numeric deviation... It kind of makes the question of whether the abstract thing is "real" or not almost a little irrelevant?

How so? (i.e. how does it make it irrelevant?)

I just mean that for a concept that might be "imaginary" and "non-existent", we still see and measure its effects on everyday things that are decidedly "real".

It's not unlike earlier when you brought up "flying unicorns exist" and contrasting with "the number 7 exists".  Unlike pi and circles, there isn't a large number of examples of "almost flying unicorns" in the world, nor can you modify or alter something to become ever more closer to a true flying unicorn.  (Well okay, I can see someone imagining some sadistic thing being done on some poor animals--um, please don't? :P).  Even if one thinks pi is as imaginary as flying unicorns, it doesn't invalidate that the former still applies a lot more to the physical world compared to the latter.

=========================
 
If you could enter a simulated world of your own making, design it to be exactly as you want it to be, but with the caveat that you would either:

a) Not be aware that it was a simulation after entering it, but if you ever realised it was then you would immediately leave and never be able to return to it.

or

b) Be aware that it was a simulation, but be unable to leave after you entered it.

Would you enter the simulation under either of these conditions? If so, which one, and why?

Well, first of all, most probably I would reject both given their restrictions on leaving.  It's on entirely practical grounds--the world outside of the simulation still exists and is very relevant to my safety and well-being amongst other things.  To put simply, I'm not willing to risk my house burning down with me in it while I'm in a simulation.

Now, to make it more interesting, perhaps we should posit that you can somehow arrange matters in the world outside of the simulation so that this primal safety concern is not applicable.  For example, maybe the simulation is capable of letting you experience millions of years of passage of time within the simulation while barely a nanosecond has passed in the world outside.  You're just not in danger of the house burning down within the duration of a nanosecond.

I have to say that A is still more preferable, given the possibility of escape compared to B.  After all, there are still people and things outside of the simulation that matter to me a great deal, and similarly in the world outside there are things and people for whom I matter.  B would seem to imply a clean, permanent break with all that of the outside world and I'm simply not ready and not motivated to experience such a clean, permanent break.  Even if we posit that for B you start off lose all memories of your life outside the simulation, I'm also not ready nor motivated for other people and things in the outside world to experience a clean, permanent break from me either.

So now, to make it more interesting again, let's say we have an arrangement where it's not just you, but also everyone and everything you hold dear, can also enter the same simulation with you, so that the concerns raised above no longer applies.  I still feel like I prefer A more.  The problem of B precluding any escape from the simulation still has the problem that, unless you're 100% confident the simulation is perfect, you may well be committing forever to a simulated world that fails to unfold the way you expected.  You may have thought you were entering heaven when you stepped into the simulation, but turns out it actually winds up being hell.  At least with A there's still hope of an eventual escape.

Anyway, while I don't really prefer either option, so far I like option A better than option B I guess?  Interesting to see what other people thinks.

Offline WillLem

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Re: WillLem's Blog
« Reply #57 on: May 25, 2020, 05:13:18 PM »
Plate tectonics actually posits that the Earth's land masses go through cycles of coming together and breaking apart.

Something about this seems contradictory... if the landmasses on the surface can continually join together and break apart in this way, how do we account for what's going on beneath the ocean's surface? As you've pointed out - the continents are much bigger than what we see above water. Why would oceanic crust continually change whilst continental crust stays mostly the same, just moving about on the surface?

unless you're 100% confident the simulation is perfect, you may well be committing forever to a simulated world that fails to unfold the way you expected.  You may have thought you were entering heaven when you stepped into the simulation, but turns out it actually winds up being hell. 

It's interesting that it can be interpreted this way, but let's suppose the simulation is 100% perfect, and completely of your own design. Ultimately, the danger with A is that you're living in a perfect world, blissfully unaware that it isn't real; so, returning to the real world upon realising it's a simulation and then having to face whatever reality is there has the potential to make normal reality seem unbearable.

In a way - we experience A already, every day. We do all kinds of things to enter different states of consciousness (get drunk, play games, watch movies, whatever) and then ultimately return to the reality of our lives. We are, hopefully, motivated to always make reality as good as it can be. But when things aren't going our way, that can be very difficult to come to terms with.

Situation B, on the other hand, is a state of full knowledge of the truth that you're in a simulation, but not being able to choose to return to reality. I think the real caveat with option B is not that there is no chance of escape (indeed, the motivation to escape might be fairly low if everything is perfect), but that there would always be the sense that you'd given up on reality and chosen to live in the dream. No matter how good it was, I don't think you could ever shake that regret.

So, logically - reality is always the best option.
« Last Edit: May 25, 2020, 05:21:08 PM by WillLem »

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WillLem's Blog: On Tetris
« Reply #58 on: June 15, 2020, 11:46:30 AM »
Tetris is a puzzle game!

Discuss. :lemcat:
« Last Edit: June 24, 2020, 02:29:46 AM by WillLem »

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Re: WillLem's Blog
« Reply #59 on: June 16, 2020, 11:22:16 PM »
Yes. It's not purely a puzzle game - it has significant elements of action and luck too - but it is still, and I would even say primarily, a puzzle game.
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Offline ccexplore

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Re: WillLem's Blog
« Reply #60 on: June 17, 2020, 07:07:41 AM »
It's a pretty broad category anyway, both video game puzzles as well as even traditional non-digital puzzles.  Tetris definitely leans quite heavily on luck and action though, in particular those are the sole elements of the game's difficulty curve.

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Re: WillLem's Blog
« Reply #61 on: June 19, 2020, 11:07:40 AM »
I feel like it's difficult to really categorize Tetris. It's kind of a puzzle game not entirely unlike how Portal is an FPS. But while categorizing Portal as a puzzle game is a pretty obvious choice even if it's a game where you walk around in first person and shoot things, I'm not sure the action label fits Tetris either. It's definitely more action than puzzle, though - fitting the pieces together is trivial - it's reacting to what's hapening that's the hard part. A Tetris game with NeoLemmix's control features wouldn't be a very compelling game.

Offline WillLem

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Re: WillLem's Blog
« Reply #62 on: June 20, 2020, 02:53:52 AM »
I've just recently bought Tetris Effect and started playing it (since my new laptop can keep up with its incredible graphical fireworks display) - it's definitely taking the game to a whole new level. It's still Tetris, but there's something more tangible, even emotional about it.

The music responds to your playing, as do the visuals, which makes it all the more immersive, and because each stage has a completely different setting (one minute you're playing alongside dolphins in a vast ocean, next minute you're playing in a rave club at 3AM, then you're playing in a tribal village sat around a fire, etc) - again, it's still Tetris, but the setting adds a whole new dimension which makes it feel almost like some sort of bizarre Quantum Leap-style RPG, in which you're jumping seamlessly from world to world, all the while keeping those tetrominos under control, and the character is you!

Its potential for being incredibly addictive and its innate connectiveness can actually be somewhat off-putting initially - I felt quite wary of it for the first half an hour or so, but once I realised just how rich and varied the game is in terms of its visuals (and, of course, music), I realised that the game is far from just another way to make Tetris even more addictive: it's an experience as therapeutic and self-exploring as walking through an art gallery or even travelling around the world. I'm trying to say these things without too much exaggeration, but I genuinely felt like I was on a journey whilst playing.

Thankfully, its familiar puzzle element and visual fireworks do keep it just superficial enough that it doesn't completely draw you in, and it's possible to stay fairly near the surface of it all for the most part; I was able to hold a conversation with someone whilst playing, which certainly can't be said of all video games.

I can imagine playing it in VR is a different story, though!

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WillLem's Blog: The Perfect Game Controller!
« Reply #63 on: June 24, 2020, 07:12:09 PM »
What is, in your opinion, the perfect game controller?

For me, the Top 3 would be:

1. Nintendo GameCube Controller


Far more than just a go-to Smash Bros. controller (I very very rarely play Smash and don't even own a copy of Melee yet!), the GameCube controller wins the top spot due to simply how it feels when it's in your hands. The designers knew what they were doing when they set out to make the perfect ergonomic controller, and the result is a chunk of loveliness that is now in its 4th generation of console compatibility, and for good reason. The perfect height of the joystick makes it a first choice for 3D platforming and RPGs, and the beautifully sprung trigger buttons make performing grabs in SSX Tricky an absolute joy. It is, of course, also fantastic for fighting and racing games.

It also looks the part, with its appealing colourful interface enduringly reminiscent of classic Nintendo gaming, and its unique, mesmerizing shape simply asking to be picked up and played with.

Downsides? OK, so it isn't wireless (unless you get the ugly and expensive Wavebird which isn't actually compatible with a lot of games and can't be used with GC-to-USB adapters, bafflingly), and the D-Pad leaves a lot to be desired. Also, it would have been nice to have twin shoulder buttons; I have no idea why Nintendo decided to only have one, it seems obvious that two would have been better, opening up more options and greater compatibility.

However, despite its flaws it still stands as my favourite controller: it looks great, feels even better and plays like a dream. Little wonder, then, that 4 generations on, Nintendo gamers are still reaching for this controller even in the presence of more technologically capable alternatives.

A deserved 8.5/10.


2. Nintendo Wii U Pro Controller


Whilst its surface may be somewhat more bland than the other entries in this list, this one is arguably the best at simply being a controller! It's wireless, superbly weighted, all the buttons and joysticks feel smooth and well-made and have perfect click travel, and its layout and decently-sized D-Pad make it ideal for many different types of game. What it perhaps lacks in aesthetics it more than makes up for in its premium build quality and usability. Nintendo were savvy enough to make it compatible with almost every Wii U game as well, making it a necessary and worthwhile accessory for the console.

Its only downsides are a lack of USB support (it would have been nice to be able to hook this up to a PC like you can with the X-Box and DualShock controllers), and some may not like the position of the analog sticks as compared to a DualShock. Personally, I like the layout and can switch between this and a DualShock pretty much seamlessly during a gaming session.

A solid 8/10.


3. Sony DualShock 3


Is the DualShock 4 better? Definitely. But, I don't have a PlayStation 4 and so would have felt a bit of a hypocrite placing that controller in my list! I have a PlayStation 2, and I use a DualShock 3 as a controller via an adapter.

Controller designers during the PS2/GameCube era knew what they were doing - the DualShock 2 (which is basically a wired 3) is a controller you can pick up and start using like you were never without it. Instantly familiar, instantly classic, it felt just how game controllers always should have felt, despite its initially intimidating design (intimidating, that is, to those of us used to the comparitively basic and cartoony SNES, Megadrive and N64 controllers).

It's a controller that does everything you need it to without showing off too much or demanding too much of you. It's quite neat and compact compared to other controllers, but it has a chunkiness and heft to it that lets you know it means business, and it'll handle pretty much anything you want to throw at it.

Available in a variety of colours to suit the player, usually with a tastefully premium metallic finish, it also looks the part as well.

The main downsides I find are the height of the joysticks, which stick out just a little bit more than I'd like, and the low button travel which can make them feel somewhat soft and mushy. Ultimately, a controller needs to have it down in these areas as much as possible, or more than make up for it in others.

A comfortable 7/10.
« Last Edit: June 26, 2020, 02:45:35 AM by WillLem »

Offline namida

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Re: WillLem's Blog: The Perfect Game Controller!
« Reply #64 on: June 24, 2020, 08:45:05 PM »
Quote
the DualShock 2 (which is basically a wired 3)

Not quite true. Although very few games use them, DS3 supports motion detection, whereas DS2 does not. The triggers are also redesigned on the DS3 compared to the DS2 - although they're actually capable of detecting a "partial press" on both, the DS3 has redesigned them so this is actually practical. (The L1, R1, and shape buttons are also pressure-sensitive on both controllers, but on both, they're not practical to really make use of this so basically nothing uses the feature. The DS4 realised this, and dropped the pressure-sensitivity on all buttons other than L2 and R2.)

DS4 is a great controller. It also works much more tidily with PC than the DS3 does, with some newer games even having native support (but if not - DS4Windows is much cleaner than the various solutions for DS3), right down to showing PS buttons instead of Xbox ones on-screen. My understanding is that DS4 is compatible with PS3 too, so even if you don't have a PS4, you still have two systems (three if your DS3 adapter for PS2 also supports DS4) you can use it on. ;)

I've never used either of those Nintendo controllers, but I can't say I like the look of either. Both have the Xbox style "swap the dpad and left stick" setup, which I'm very much NOT a fan of - to the extent that it's outright one of the reasons I prefer Playstation over Xbox. The Wii U one takes this even further and applies the same swap to the right half of the controller too (which IMO is even worse than doing it on the left); while the Gamecube one, what's wrong with the traditional diamond layout for four buttons (although this is a much lesser concern than the other points here; this one feels like the kind of thing I'd adjust to soon enough).
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Offline Dullstar

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Re: WillLem's Blog: The Perfect Game Controller!
« Reply #65 on: June 25, 2020, 05:22:46 AM »
The GC controller is great for 3D games, IMO. It feels very good to use.

The Dualshock has a really good D-pad in the most comfortable position on the controller, which makes it really nice for 2D games, which I play a lot of.

The Xbox controller makes me die inside because it uses the same button labels as Nintendo controllers, but they're arranged differently, which gave some bad first impressions to it. At least with learning the Playstation layout, it was more like, "Oh, I don't know where the Square button is, let me look at the controller" while Xbox was more "It says to press A and I'm pressing A, wtf? Oh wait, that's not actually A." Combine that with a few of those games that have stupid quicktime events where you have to remember which button is which, and that first impression stuck. I mean, I guess you could use the colors instead of the labels, but then it's still kind of like trying to do the thing where you have to say what color the word is and then get presented with a color name where the colors don't match the word, e.g. red blue, purple

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Re: WillLem's Blog: The Perfect Game Controller!
« Reply #66 on: June 27, 2020, 09:19:45 PM »
"Oh, I don't know where the Square button is, let me look at the controller" while Xbox was more "It says to press A and I'm pressing A, wtf? Oh wait, that's not actually A."

This is another reason I love the GC controller: basically no chance of hitting the wrong button!

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Re: WillLem's Blog: The Perfect Game Controller!
« Reply #67 on: June 27, 2020, 09:20:36 PM »
I've made a short but sweet video sharing my GC controller collection, if anyone's interested :lemcat:

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WillLem's Blog: Current Hobby - Retrobriteing old gaming gear!
« Reply #68 on: July 18, 2020, 01:40:08 AM »
Current Hobby: Retrobriteing!

I've recently got into Retrobriteing: this is the process of taking old grey/white plastics that have "yellowed" over time due to the ageing of the flame-retardent chemicals in the plastic, and restoring them to their former grey/white glory! It usually involves a general refurbishment of the items as well, which is something else I've always enjoyed.

So far I've done a grey GameCube memory card (which now looks brand new again!). Unfortunately I didn't think to take "before" shots, but here's a picture of the memory card as it is now after the RB process. I'd say the procedure was an emphatic success - the part of the cartridge that sticks out of the GC was noticeably more yellow and discoloured before, to the point that it looked like a yellowy-grey strip across the bottom. It's now virtually disappeared (this photo reveals a very faint line which is actually much more difficult to see irl), and the cartridge looks a much more uniform colour. It's also so much cleaner - it really pops, just like new. I'm super happy with the results:



And, here's my old DMG GameBoy shell - it had already been in the salon for a few hours by the time I realised I should probably take a picture so the process had already begun to take effect - but, you can see that the part of the shell that's protected by the screen cover is a much lighter grey than the part that's exposed:



And here it is in the salon itself - it's a small, shallow cardboard box with a closing lid:



As you can see, I've attached a UV strip light to the underside of the lid and coated the interior of the box with foil to reflect the light at all possible angles - standard Retrobrite procedure. The idea is then to coat the item to be RB'd with a 12% Hydrogen Peroxide solution and wrap it in cling film to ensure as even a coat as possible. The HP reacts with the UV light and creates some sort of magical chemistry combo which restores the original colour. It's also possible to achieve this by putting the item in direct sunlight, but concentrated UV light provides a much more efficient way of doing it, and - by most accounts - gets the most even results.

I'm going to leave it in overnight and see how it looks in the morning. If it's much better, I'll clean it up, put the system back together and share some photos of the finished project! It might need some time in the sun as well just to polish it up - this was very yellowed before I started.

The GameBoy is by far the most affected of the items I'm looking at doing. Next up is the grey controller-port-cover from my GameCube. This isn't too bad tbh, but there's a slight yellowing gradually spreading from one corner, and I'd like to give the fans a really good cleanout as well, so I'm looking forward to taking that on.

---

Footnote: My GameCube was given to me by a friend who didn't see the point of having it anymore after they bought a Wii, but I'd always been quite fond of the GC so they were kind enough to let me have it. It's a black one, which is super cool, but I've always really liked the classic purple one. If this whole RB thing turns out to be a success, I might get a cheap, dirty and discoloured purple GC from eBay and do a full clean-and-restore project. Discoloration of the unit very often lowers the price significantly, and I love the idea of giving old, unwanted and uncared-for items a new lease of glory!
« Last Edit: July 21, 2020, 10:36:34 AM by WillLem »

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Re: WillLem's Blog: Current Hobby - Retrobriteing old gaming gear!
« Reply #69 on: July 19, 2020, 12:08:30 AM »
Quick update:

The GameBoy shell is coming along very nicely, but the progress has now slowed down significantly. Yesterday, the first 6 hours or so showed the biggest amount of noticeable difference, but I'd say it's definitely going to need another 12 hours or so, maybe even longer, to finish the process:



You can see from the picture that the grey of the main body is now much closer to the grey of the screen area than yesterday's picture (see previous post) - but, it's now taking more and more hours to notice any difference. We're forecast some sunshine tomorrow, so I will give it a few hours on the parcel shelf in my car, see if a bit of natural sunlight helps to finish it up.

In the meantime, I've ordered a replacement glass screen protector for it and I've also picked up a purple GameCube for £25! - it's got quite a lot of discolouring to the controller socket cover and the lid, and it looks like it could do with a clean and polish. I'll make sure I take some before and after shots for this one; I plan to restore it as fully as possible.

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Re: WillLem's Blog: Current Hobby - Retrobriteing old gaming gear!
« Reply #70 on: July 21, 2020, 10:33:08 AM »
Got there at last!



This took 3 days and nights in the end. For anyone else interested in doing a bit of Retrobriteing themselves, UV lights is definitely the way to go rather than leaving it in the sun - I found that the latter method dried up the Peroxide solution and made it sticky. I managed to rescue it in time, but I think this would definitely have caused streaking if left for too long. Using UV lights is slower, but ultimately more controlled and you get nice even results.

Anyways, here is the freshly Retrobrited GameBoy in all its glory:



Again, I really wish I'd thought to take before pics of this so you could see how badly yellowed it was before I started. It looks better than ever now! Oh, and it works as well :lemcat:
« Last Edit: July 22, 2020, 12:34:52 AM by WillLem »

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Re: WillLem's Blog: Current Hobby - Retrobriteing old gaming gear!
« Reply #71 on: July 21, 2020, 09:28:42 PM »
I wasn't aware it was possible to get replacement screen protectors. I should probably look into that; I've got a Game Boy Color as well as an original Game Boy in my collection that have badly scratched up screens.

I had one of the later model GBA SPs that had a backlit (rather than a front-lit) screen, but it doesn't charge anymore. I also have a DS Lite with a similar-ish problem. I should probably look into that; getting them repaired would be quite nice.

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Re: WillLem's Blog: Current Hobby - Retrobriteing old gaming gear!
« Reply #72 on: July 22, 2020, 12:28:12 AM »
I wasn't aware it was possible to get replacement screen protectors. I should probably look into that; I've got a Game Boy Color as well as an original Game Boy in my collection that have badly scratched up screens.

I had one of the later model GBA SPs that had a backlit (rather than a front-lit) screen, but it doesn't charge anymore. I also have a DS Lite with a similar-ish problem. I should probably look into that; getting them repaired would be quite nice.

I picked mine up from SuperSmashMedia on Amazon, it was only £5. Most (if not all) of the available replacements are third party, but they look basically identical. The lettering on the one I got is slightly clearer and the colour of the stripes is a touch darker, but I actually prefer that. And - of course - it's brand new, so gives the GameBoy that pristine look. It's currently on display on my bookcase, I keep glancing over at it and feeling chuffed that I actually bothered to do it after years of thinking about it!

Have a look at Handheld Legend. These guys are USA-based so may charge shipping fees outside the US (and they're not the cheapest) but they sell pretty much everything for Nintendo hand-helds.

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Re: WillLem's Blog: Current Hobby - Retrobriteing old gaming gear!
« Reply #73 on: July 22, 2020, 12:46:52 AM »
Whilst I'm waiting on the purple GameCube arriving, I got to work on my existing black GC. I RB'd the controller port cover and memory card slots, plus took the whole thing apart, cleaned it, got rid of all the dust, and polished the jewel on the lid. Really happy with the results, and it's given me a lot of experience working specifically with the GC which I'll be able to apply to the purple one when it arrives.

Here's the black one, looking cooler than ever! 8-)



« Last Edit: July 22, 2020, 01:39:05 AM by WillLem »

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Re: WillLem's Blog: Current Hobby - Retrobriteing old gaming gear!
« Reply #74 on: August 03, 2020, 09:18:26 PM »
For anyone who's been reading the recent posts on retrobriteing: I've had the purple GC for about two weeks now and it's in progress: I'm waiting on another set of UV lights and still figuring out what to do about the base, which has a chipped bit of plastic.

I'm making a video documenting this one; it will be up when I've finished the project.

Thanks for reading! :)

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Re: WillLem's Blog: Current Hobby - Retrobriteing old gaming gear!
« Reply #75 on: April 28, 2021, 02:21:12 AM »
I completely forgot to upload the link to the GameCube refurbishing video!

Anyways, here it is! I hope you enjoy it :lemcat:

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Re: WillLem's Blog: Paradoxes
« Reply #76 on: April 28, 2021, 02:22:20 AM »
Paradoxes

Is it possible to read something that you haven't already read?

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WillLem's Blog: Permission or Forgiveness?
« Reply #77 on: June 13, 2021, 04:29:35 PM »
I've added a poll.

Which would you rather ask for: permission to do something, or forgiveness for having done it already?