Applicability of Maths to the Universe

[The Grand Nudger]

Something to ponder over -

The computers that we're using to have this discussion all have some limited ability to describe the objects in my metaphor. The task manager window can tell you, for example, what percentage of it's processing capacity is focused on a given task. We don't have anything like that.

There are external diagnostic tools and equipment that can provide even more information than whatever internal monitoring might be in place. This much is, thankfully, also true of us. We're not very good at it yet, but when we're loading up a brain with reactive dyes and watching it go off like fireworks, we are very much watching thoughts, which do exist in the way that any other thing exists as we use those terms, as objects.....and even further, as material objects.

The reason that dualists were so enamored of mind in the past boils down to our brains inability to communicate this information to us, and having absofuckinglutely no clue how brains worked, they decided that minds actually were just the incomplete description that is apprehend by our focused attention.

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Building from the above, we can tackle Sherlock Holmes. Sherlock Holmes is an object, that does exist, either on page or in memory. We may not actively apprehend the contents of that memory at all times, but it's still there, just as the video game I'm not playing right now is still on my pc. Sherlock isn't created when I access the file, and the file doesn't go away when I close the window. The active focus of attention has shifted to some other object, some other location.

If we burned those books and actively wiped those memories, sherlock would, indeed, disappear - in the reverse of the process by which he was created and maintained, because it does not..in reality, have any independent existence whatsoever. If you're starting to think to yourself, at this point, "do I exist in any way different from sherlock"...well, bingo. We're also stories that we tell ourselves

Aaaaaaall of that tedium finished, we can wonder whether or not this description of mind is true, and it may not be - but even if it were not true, it demonstrates that the semantics required to insist that thoughts are material objects do exist, and because of this, that the existence of thoughts doesn't actually pose any specific difficulty for materialism. If there is Other Stuff™, then we're going to need a better candidate for it than thoughts, or numbers, or characters from stories that we tell ourselves. We'll need a better candidate for it than anything in human experience.

[Jehanne]

Of course, the cardinality of the set of prime numbers is a countably infinite set.  Thousands of mathematical proofs exist that prove such; of course, an acceptance of ZFC and the Axiom of Infinity is necessary to get the balling rolling, so to speak.  Professor Wes Morriston has written extensively in response to WLC, and even quotes Craig who once stated that his ideas will make sense to an individual at least until that person has taken a course in elementary number theory, complex analysis, etc.

I keep hearing about this ZFC stuff but never getting around to reading up on it. ELI5 (or perhaps ELI15): What's it about?

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Read Dr. James A. Lindsay's book, Dot, Dot, Dot...Infinity Plus God Equals Folly, which is available on Audible, also (read by Dr. Lindsay himself).  He addresses Craig's argument.  That the number "2" (or, its square root) exists in some Platonic realm is silly; does the transcendental number "pi" exist there, also?  But, read (or listen) to Lindsay's book for more details.

[polymath257]

Of course, the cardinality of the set of prime numbers is a countably infinite set.  Thousands of mathematical proofs exist that prove such; of course, an acceptance of ZFC and the Axiom of Infinity is necessary to get the balling rolling, so to speak.  Professor Wes Morriston has written extensively in response to WLC, and even quotes Craig who once stated that his ideas will make sense to an individual at least until that person has taken a course in elementary number theory, complex analysis, etc.

I keep hearing about this ZFC stuff but never getting around to reading up on it. ELI5 (or perhaps ELI15): What's it about?

It's Zormelo-Frankl set theory with the axiom of Choice. This is the current standard set of axioms for set theory and is the basis of almost all modern mathematics. it came about because of inconsistencies in 'naive set theory' that were discovered at the end of the 1800's.

The basic ZF axioms say that there is an empty set, that you make a set that is the set containing any two objects, that the set of subsets of any given set exists, that you can take the union of any collection of sets and get a set, etc. The only undefined concept is that of set membership.

The the axiom in dispute here is the axiom of infinity: essentially that there is an infinite set. This is used in an essential way to define the set of natural numbers and, later on, the set of real numbers.

The axiom of choice is one many people have heard of. It essentially says that if you have a collection of non-empty sets, then there is another set that has one element in common with each of the original sets.

The main difference between ZFC set theory and Cantor's original is that you cannot define arbitrary collections as sets. So, for example, the collection of all sets is not a set. Properly speaking, it simply does not exist at all in ZFC. In  sense, it is 'too big' to be a set. This avoids the paradoxes of Cantor's original system while keeping the material on one-to-one correspondences and cardinality.

Pretty much all of modern mathematics can be constructed inside of the ZFC axiom system. It is a rather lengthy process to get to the set of real numbers simply starting with the notion of set membership, but it is possible. And, from there, it is possible to construct all of calculus, differential equations, etc as well as group theory, graph theory, etc.

The one place where ZFC fails to be helpful is when discussing certain aspects of category theory and hence, of algebraic topology. The standard way of doing this is to allow for 'classes' which are 'too big to be sets'. But it is then impossible to talk about the collection of 'all classes'.

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I keep hearing about this ZFC stuff but never getting around to reading up on it. ELI5 (or perhaps ELI15): What's it about?

---

Read Dr. James A. Lindsay's book, Dot, Dot, Dot...Infinity Plus God Equals Folly, which is available on Audible, also (read by Dr. Lindsay himself).  He addresses Craig's argument.  That the number "2" (or, its square root) exists in some Platonic realm is silly; does the transcendental number "pi" exist there, also?  But, read (or listen) to Lindsay's book for more details.

There is an even deeper level. Alan Turing defined the concept of a 'computable real number'. In essence, it is one for which some computer program can compute the digits in the decimal expansion of that number.

So, for example, pi *is* a computable number: we have programs that will give as many digits of pi as you want. SImilarly for e and sqrt(2), and pretty much any number you have ever heard of.

But what Turing discovered (and it is not difficult to prove) is that there are real numbers (in the usual axiom system) that are not computable: no computer program can compute their decimal expansions. And, in fact, 'most' real numbers are not computable.

In what sense do these uncomputable real numbers exist in some Platonic realm?

[GrandizerII]

There is an even deeper level. Alan Turing defined the concept of a 'computable real number'. In essence, it is one for which some computer program can compute the digits in the decimal expansion of that number.

So, for example, pi *is* a computable number: we have programs that will give as many digits of pi as you want. SImilarly for e and sqrt(2), and pretty much any number you have ever heard of.

But what Turing discovered (and it is not difficult to prove) is that there are real numbers (in the usual axiom system) that are not computable: no computer program can compute their decimal expansions. And, in fact, 'most' real numbers are not computable.

In what sense do these uncomputable real numbers exist in some Platonic realm?

This brings me to a somewhat related question I've had on my mind for a while. Why is it that such special numbers "found in nature" such as pi and e are irrational (and seemingly "arbitrary") as opposed to "clean" rational numbers? Is it to do with the particular number system being used (the decimal)?

[polymath257]

There is an even deeper level. Alan Turing defined the concept of a 'computable real number'. In essence, it is one for which some computer program can compute the digits in the decimal expansion of that number.

So, for example, pi *is* a computable number: we have programs that will give as many digits of pi as you want. SImilarly for e and sqrt(2), and pretty much any number you have ever heard of.

But what Turing discovered (and it is not difficult to prove) is that there are real numbers (in the usual axiom system) that are not computable: no computer program can compute their decimal expansions. And, in fact, 'most' real numbers are not computable.

In what sense do these uncomputable real numbers exist in some Platonic realm?

This brings me to a somewhat related question I've had on my mind for a while. Why is it that such special numbers "found in nature" such as pi and e are irrational (and seemingly "arbitrary") as opposed to "clean" rational numbers? Is it to do with the particular number system being used (the decimal)?

Rationality is not affected by the number system used. A number is either rational or irrational. It is rational if it is a 'ratio' of two integers and irrational if not.

So, sqrt(2) is irrational. You cannot write it as ratio of two integers. The ancient Greeks knew this well before the decimal system for *describing* numbers.

Now, pi and e are more than just irrational; they are transcendental. In other words, they are not solutions of any polynomial with integer coefficients.

Well, it turns out that there are only countably many numbers that are *not* transcendental and uncountably many that are. So, in a sense, the vast majority of real numbers are transcendental. An 'arbitrary' real number is likely to be transcendental (and hence, irrational).

[Fireball]

This brings me to a somewhat related question I've had on my mind for a while. Why is it that such special numbers "found in nature" such as pi and e are irrational (and seemingly "arbitrary") as opposed to "clean" rational numbers? Is it to do with the particular number system being used (the decimal)?

Rationality is not affected by the number system used. A number is either rational or irrational. It is rational if it is a 'ratio' of two integers and irrational if not.

So, sqrt(2) is irrational. You cannot write it as ratio of two integers. The ancient Greeks knew this well before the decimal system for *describing* numbers.

Now, pi and e are more than just irrational; they are transcendental. In other words, they are not solutions of any polynomial with integer coefficients.

Well, it turns out that there are only countably many numbers that are *not* transcendental and uncountably many that are. So, in a sense, the vast majority of real numbers are transcendental. An 'arbitrary' real number is likely to be transcendental (and hence, irrational).

Bolding mine. Can you run that past me, but at a slow walk? I never would have thought that this would be the case. I suspect that if I go look it up in a text the discussion will be way over my head.

[GrandizerII]

This brings me to a somewhat related question I've had on my mind for a while. Why is it that such special numbers "found in nature" such as pi and e are irrational (and seemingly "arbitrary") as opposed to "clean" rational numbers? Is it to do with the particular number system being used (the decimal)?

Rationality is not affected by the number system used. A number is either rational or irrational. It is rational if it is a 'ratio' of two integers and irrational if not.

So, sqrt(2) is irrational. You cannot write it as ratio of two integers. The ancient Greeks knew this well before the decimal system for *describing* numbers.

Now, pi and e are more than just irrational; they are transcendental. In other words, they are not solutions of any polynomial with integer coefficients.

Well, it turns out that there are only countably many numbers that are *not* transcendental and uncountably many that are. So, in a sense, the vast majority of real numbers are transcendental. An 'arbitrary' real number is likely to be transcendental (and hence, irrational).

Right, a number is more likely to be transcendental and therefore irrational. Fair enough. But when we speak of numbers like pi and e, we speak of numbers that are very special numbers that can be "captured in nature". If you divide the circumference of a circle by its diameter, you get this special number called pi. Why is such a special number not a "neat" rational number? More specifically, why is pi the exact value as it is?

[polymath257]

Rationality is not affected by the number system used. A number is either rational or irrational. It is rational if it is a 'ratio' of two integers and irrational if not.

So, sqrt(2) is irrational. You cannot write it as ratio of two integers. The ancient Greeks knew this well before the decimal system for *describing* numbers.

Now, pi and e are more than just irrational; they are transcendental. In other words, they are not solutions of any polynomial with integer coefficients.

Well, it turns out that there are only countably many numbers that are *not* transcendental and uncountably many that are. So, in a sense, the vast majority of real numbers are transcendental. An 'arbitrary' real number is likely to be transcendental (and hence, irrational).

Right, a number is more likely to be transcendental and therefore irrational. Fair enough. But when we speak of numbers like pi and e, we speak of numbers that are very special numbers that can be "captured in nature". If you divide the circumference of a circle by its diameter, you get this special number called pi. Why is such a special number not a "neat" rational number? More specifically, why is pi the exact value as it is?

Well, pi can also be defined as twice the value of the first zero of the cosine function or as the value of certain integrals. In more formal mathematics, that is more often the definition.

The number e comes up most naturally in the solution of differential equations (y'=y) and is thereby related to 'constant growth or decay rates'. That is why it 'comes up naturally'. That particular differential equation is picked out not only for simplicity, but because it is translation invariant.

But, for example, we could study Bessel functions and ask for the value of the first zero of J_0(x). That will also be a transcendental value. Such functions come up naturally when looking at certain spherically symmetric situations (as the radial function). and the values of the roots of the Bessel functions show up 'naturally' in nature as well for such cases.

And, of course, all of these are related to each other. The cosine is the real part of the complex exponential, so pi and e are related via Euler's formula. The Bessel values are related to Fourier integrals, which involve either complex exponentials or trig functions.

And, Fourier integrals and series very naturally have both pi and e arise as scaling factors.

As for your last question, I'm not sure how to answer a 'why' question when it comes to math. The value of pi is what it is because we define trig functions the way we do or we need a circumference or surface for a symmetric figure (which naturally involves trig functions and hence complex exponentials). Finally, exponentials are homomorphisms from the additive reals to the multiplicative positive reals, so we selected algebraically.

[polymath257]

Well, it turns out that there are only countably many numbers that are *not* transcendental and uncountably many that are. So, in a sense, the vast majority of real numbers are transcendental. An 'arbitrary' real number is likely to be transcendental (and hence, irrational).

Bolding mine. Can you run that past me, but at a slow walk? I never would have thought that this would be the case. I suspect that if I go look it up in a text the discussion will be way over my head.

Ok, let's go through a few of the basics first. We identify cardinality by looking at one-to-one correspondences between sets. We say that a set is countably infinite if it can be put into one-to-one correspondence with the set of natural numbers, N={0,1,2,..}. We say a set is countable if it is either finite or countably infinite.

There are three basic facts about countable sets that are important:

1. Any subset of a countable set is countable.
2. If you have a countable collection of sets, and each set is itself countable, then the union is countable.
3. Any set that is one-to-one correspondence with a countable set is itself countable.

Almost all countability arguments are based on using these three facts repeatedly. If you need me to prove them, I can, but for this, I'll take them as known.


We proceed by several steps:

Step I: the set of integers (positive, negative, and zero) is countable.In fact, the positive integers with zero is the same as the natural numbers and the negative integers can be put into a one-to-one correspondence with the naturals. So the collection of integers is the union of two countable sets, so by 2  above, it is countable.

Step II: The collection of linear functions with integer coefficients is countable. Such a function is of the form mx+n where m and n are integers. For each value of m, there are countably many values of n (corresponding with the integers). And the collection of all linear functions is the union as m varies, so is the union of countably many countable sets. Again, by 2, this is countable.

Step III: The collection of quadratic functions with integer coefficients is countable. Such a function is of the form kx^2 +mx+n where k, m, and n are integers. Now, for each value of k, there are only countably many values of m and n (linear functions!) and the collection of all quadratic functions is then the countable union (one  set for each value of k) of countable sets (the linear piece). So, again by 2, the whole is a countable set.

Step IV: For each n, the collection of polynomials with integer coefficients of degree n is countable. This is done by induction on n. We have n=1 and n=2 by steps II and III. But the same argument as in step II shows that if the collection of polynomials of degree n is countable, so is the collection of degree n+1. Just take the union over the first coefficient of the possible following terms. So the induction works.

Step V: The collection of ALL polynomials with integer coefficients is countable. This is just the union over the the sets in step IV, one for each degree n. This is a countable union, and once again the result is countable.

Step VI: For each polynomial with integer coefficients, there are only finitely many roots. If the polynomial has degree n, then there are at most n roots. This is a fact from algebra. if you need me to prove it, let me know.

Step VII: the collection of all possible roots of all polynomials with integer coefficients (the collection of algebraic numbers) is countable: this is the union over the countably many polynomials of the finite (hence countable) set of roots for each.

The last step is *exactly* the set of numbers that are NOT transcendental.

Now, I assume you know that the set of all real numbers is NOT countable (once again, i can prove it if you want).

But this means that the collection of transcendental number is also NOT countable (if it was countable, its union with the algebraic numbers would be countable--but this is the whole set of real numbers).

If you have questions about any part of this, feel free to ask.

[Fireball]

Bolding mine. Can you run that past me, but at a slow walk? I never would have thought that this would be the case. I suspect that if I go look it up in a text the discussion will be way over my head.

Ok, let's go through a few of the basics first. We identify cardinality by looking at one-to-one correspondences between sets. We say that a set is countably infinite if it can be put into one-to-one correspondence with the set of natural numbers, N={0,1,2,..}. We say a set is countable if it is either finite or countably infinite.

There are three basic facts about countable sets that are important:

1. Any subset of a countable set is countable.
2. If you have a countable collection of sets, and each set is itself countable, then the union is countable.
3. Any set that is one-to-one correspondence with a countable set is itself countable.

Almost all countability arguments are based on using these three facts repeatedly. If you need me to prove them, I can, but for this, I'll take them as known.


We proceed by several steps:

Step I: the set of integers (positive, negative, and zero) is countable.In fact, the positive integers with zero is the same as the natural numbers and the negative integers can be put into a one-to-one correspondence with the naturals. So the collection of integers is the union of two countable sets, so by 2  above, it is countable.

Step II: The collection of linear functions with integer coefficients is countable. Such a function is of the form mx+n where m and n are integers. For each value of m, there are countably many values of n (corresponding with the integers). And the collection of all linear functions is the union as m varies, so is the union of countably many countable sets. Again, by 2, this is countable.

Step III: The collection of quadratic functions with integer coefficients is countable. Such a function is of the form kx^2 +mx+n where k, m, and n are integers. Now, for each value of k, there are only countably many values of m and n (linear functions!) and the collection of all quadratic functions is then the countable union (one  set for each value of k) of countable sets (the linear piece). So, again by 2, the whole is a countable set.

Step IV: For each n, the collection of polynomials with integer coefficients of degree n is countable. This is done by induction on n. We have n=1 and n=2 by steps II and III. But the same argument as in step II shows that if the collection of polynomials of degree n is countable, so is the collection of degree n+1. Just take the union over the first coefficient of the possible following terms. So the induction works.

Step V: The collection of ALL polynomials with integer coefficients is countable. This is just the union over the the sets in step IV, one for each degree n. This is a countable union, and once again the result is countable.

Step VI: For each polynomial with integer coefficients, there are only finitely many roots. If the polynomial has degree n, then there are at most n roots. This is a fact from algebra. if you need me to prove it, let me know.

Step VII: the collection of all possible roots of all polynomials with integer coefficients (the collection of algebraic numbers) is countable: this is the union over the countably many polynomials of the finite (hence countable) set of roots for each.

The last step is *exactly* the set of numbers that are NOT transcendental.

Now, I assume you know that the set of all real numbers is NOT countable (once again, i can prove it if you want).

But this means that the collection of transcendental number is also NOT countable (if it was countable, its union with the algebraic numbers would be countable--but this is the whole set of real numbers).

If you have questions about any part of this, feel free to ask.

OK, thanks, I appreciate that you did my homework for me.   I don't generally ask people to do that, but you've explained succinctly what I wanted to know. I'll trade you for an answer to a question you might have on auto repair, because I have a lot of experience with that. Not good with fuel injection, or many electronic ignition systems, because I graduated from uni and went to work as an engineer when those were becoming more commonplace.

I'll read this a couple more times so that the ideas are cemented into my head. It usually takes about three times before the bell goes, "ding!", but I get the gist of it. You've a clear expository style!