Philosophies of Mathematics

[Horatio Parker]

The alternative to Platonism and the existence of absolutes is the reality that things only exist in relation to other things. That would include mathematics and geometry. A point is only meaningful if it symbolizes something. Anything more than that and it becomes a crutch, imho.

Except you need Plato, or something similar, to have a thing. Without things there are no relationships. Without one there is no two, no integers to relate.

 

[Phil Scott]

In any case, I will address the rest of your points, though I have to admit I consider the matters less interesting.

Perhaps we can leave it there, then. On the topic that apparently interests you, I only care to discuss the contrast between Hilbert and Euclid, as I thought was clear from the beginning. I feel your line of inquiry here is moving very much in the opposite direction. Besides which (and apologies for saying this), I find this style of philosophy incredibly tedious.

 

[Angra Mainyu]

Perhaps we can leave it there, then. On the topic that apparently interests you, I only care to discuss the contrast between Hilbert and Euclid, as I thought was clear from the beginning. I feel your line of inquiry here is moving very much in the opposite direction. Besides which (and apologies for saying this), I find this style of philosophy incredibly tedious.

While you only care to discuss the contrast between Hilbert and Euclid, you did claim that (for example)

That said, I think I can argue quite well that there's something blatantly platonistic in the language used by modern day classical mathematicians, while there is something blatantly intuitionistic about the language used by the ancient Greeks.

I'm pretty sure that I can counter argue quite well against any argument that there is something blatantly Platonistic in the language used by modern day classical mathematicians, and in fact, I'm pretty sure I've been doing just that. Now, you might find the replies tedious, that's alright, but they were properly addressing the above claim of yours, and other claims you make, such as the claim that "on the face of it" the language used by modern classical mathematicians is Platonistic. In short, my replies are on point.

Aside from that, I'd like to say a little bit more about a point you made earlier:

As for the intuitiveness of excluded-middle, it's absolutely absent if you phrase things like Euclid does.

It is absent but not because it's counterintuitive, but rather, because in such contexts, excluded middle does not come up (usually; if it did come up, it would be intuitive, as it is in all contexts in which it comes up, at least by the intuitions of nearly all of us). But for that matter, in that context P->¬¬P (purely for example, and like a zillion other things) does not come up, either. It's not a good reason to reject P->¬¬P, and similarly, it's not a good reason to reject (P v ¬P).

All that said, I'm okay with leaving it at that if you like - or to go on if you prefer.

 

[ryan]

Let me take a stab at this.

So for some number or mathematical expression, I think most of us would agree that there is some brain or conscious state correlated to each. Like the number 3+4 is brain/conscious state A, or the number 11 is also some other brain/conscious state. Then, doesn't this mean that there is a "scientific Platonism" for all numbers or expressions? Of course this would leave in question all expressions or numbers that haven't been thought of by brains or whatever it is that can think of such things.

 

[fromderinside]

Those marks Syrians put on clay indicating commerce were objective illustrations of one's math. The knots tied in Egyption surveyor ropes are expressions of one's math. Point here is math may be mental but it's application is material. Math predates AND serves as the basis for written language, science, probably philosophy and religion as well. Math is a bit like music as we are finding out over the past couple huyndred years of objective psychology and biology. The brain has special capacities when comes to these that don't seem well attached to sense, precept, or reason and especially not emotion.

If I might be so bold I think the relation of math with rationality is similar to the relation between core memory and graphics memory. Both get the job done but one is an overall tool whilst the other is specialized to visualization and sequence.

That doesn't explain math, describe it, nor make it all fluffy in one's mind; but it is a bit like how things are about math.

 

[fast]

Let me take a stab at this.

So for some number or mathematical expression, I think most of us would agree that there is some brain or conscious state correlated to each. Like the number 3+4 is brain/conscious state A, or the number 11 is also some other brain/conscious state. Then, doesn't this mean that there is a "scientific Platonism" for all numbers or expressions? Of course this would leave in question all expressions or numbers that haven't been thought of by brains or whatever it is that can think of such things.

I don't think there is some brainstate correlation to mathematical expressions. But then again, what do I know...

 

[steve_bank]

Ryan has as his mission in life to conflate mysticism with math and science.

I never had a philosophy or way of looking at it.

I leaned it and through some trial and error experience learned to apply it. For me anything else was excess baggie. Learning and applying always felt good. Same went for people I knew. There was nothing beyond that.

 

[ryan]

Ryan has as his mission in life to conflate mysticism with math and science.

I never had a philosophy or way of looking at it.

I leaned it and through some trial and error experience learned to apply it. For me anything else was excess baggie. Learning and applying always felt good. Same went for people I knew. There was nothing beyond that.

This seems more about whether or not perfect forms or numbers exist in the mind assuming there is a mind.

But we don't have to allow minds. We can just find the physical correlation to something like, for example, 7. Unromantically, this amazingly concept of 7 will probably just correlate to some brain goo, which is probably not going to seem a lot like 7.

Either way, 7 as we know it exists! But upon inspection, the goo itself just looks different from the outside.

 

[Kharakov]

So the mathematical statement "The full decimal expansion of the number π must contain at least one of the digits 0,1,2,...,9 infinitely many times" would have an easy proof in classical mathematics -- if each digit only shows up finitely many times then π would be rational, but π is irrational so they can't. A constructive proof would need some way of explicitly showing which digit appears infinitely often to be acceptable under the intuitionist system, and I don't think that argument exists (yet?).

Most mathematicians accept that "they can't all appear finitely many times" is enough to conclude that "at least one appears infinitely many times", but intuitionists hold that the former statement is actually weaker than the latter, and that we wouldn't be able to claim the latter without a better argument.

Ok, if only one digit appears infinitely many times, the only irrational you can get would be an infinitesimal or some digit/9 *10^-whatever, and I'm not sure if this is counted as a number? Don't you need 2 digits to appear infinitely many times to have an irrational?

I wish Phil, and you, hadn't been driven away. I certainly hope I had little part in it.

 

[lpetrich]

Some philosophies of mathematics affect which kinds of proof are valid, and thus affect which mathematical results are valid.

Constructivism states that the only valid mathematical results are constructed ones.

Finitism is a subset of constructivism that states that the only valid mathematical results are those constructed with a finite number of steps.


Here is a simple constructive proof: that the sum of every pair of even numbers is also an even number. An even number n has form 2*k, where k is an integer. Here is the proof. Start with even numbers n1 = 2*k1 and n2 = 2*k2. Add them:

n1 + n2 = 2*k1 + 2*k2 = 2*(k1+k2)

Since the sum of two integers is also an integer, and since 2*(integer) is even, that proves the result.


A non-constructive proof works by contradiction, by showing that if some mathematical result is false, then it results in a contradiction.

Every proof that there is no largest prime number is non-constructive, by showing that the existence of such a prime leads to a contradiction. Here is a rather simple such proof.

Construct the function P = (product of (first n primes)) + 1

P is clearly not divisible by any of the first n primes, so the primes that divide it must be greater than p. Thus, for every purported largest prime, there is always a larger one.

Note that this proof does not require P itself to always be a prime. It also works for all its prime factors if it is composite. In fact, P is a prime for n = 1, 2, 3, 4, 5, but P(6) = 30,031 = 59*509, a composite number. Yet its prime factors are nevertheless greater than the first six primes: 2, 3, 5, 7, 11, and 13.


Turning to finitism, it makes anything involving infinite sets to be invalid. For instance, here is a proof that there is the same number of nonnegative integers as positive ones. The proof consists of finding a bijection between the two sets. Here it is:

For every positive integer n, there is a nonnegative integer n-1.

For every nonnegative integer n, there is a positive integer n+1.


When Georg Cantor published his theory of transfinite numbers, one of his colleagues, Leopold Kronecker, rather strongly opposed it on finitist grounds. That theory remained controversial until the early 20th cy.

 

[lpetrich]

Ultrafinitism goes every further than finitism. It states that every positive integer greater than some large but finite one is meaningless.

I think that we have a sort of intuitive ultrafinitism. Our short-term memory can hold around 4 items, for instance.

There is also practical ultrafinitism, how great a range of positive integers can be that we can memorize or record. That varies with recording medium, for obvious reasons. Here are computer-hardware limits on unsigned-integer size:

• 8-bit: 255
• 16-bit: 65,535
• 32-bit: 4,294,967,295
• 64-bit: 18,446,744,073,709,551,615

One can get around that limit by doing manipulations in software, in effect, doing hardware manipulations only on subsets of the numbers' digits. I find these powers of 10:

• 1 kilobyte: 924
• 1 megabyte: 946,958
• 1 gigabyte: 9.70*10^8
• 1 terabyte: 9.93*10^11

I'm using binary instead of decimal ones, ones that are sometimes called kibibyte, mebibyte, gibibyte, etc.

Since the observable Universe has around 10^86 elementary particles, one bit per particle yields a maximum positive integer of 10^(10^85).

 

[lpetrich]

Turning to the issue of how many decimal digits, this question can be answered more generally for place representations. For base B, the digits are 0, 1, 2, ..., up to B-1. Not surprisingly, B >= 2. For a place-system version of a decimal representation, there will always be an infinite number of digits after the point, though if all after some digit are all 0, they are almost always omitted.

Here is a proof that if there is an infinite number of only one digit, then the number is rational. Since there is only a finite number of all the other digits, there will be a maximum value of the location of each of these digits. Call it m. Split the number into two parts, one before and including digit m, and the other after digit m.

The before part can be multiplied by B^m to give an integer, therefore, that part is rational. The after part can be expressed as
\( \sum_{k=m+1}^\infty D B^{-k} = \frac{1}{B - 1} \frac{D}{B^m} \)
This is very obviously a rational numer, and since the sum of two rational numbers is rational, then this number must be a rational number.

 

[Speakpigeon]

Here's a thread for discussing the philosophies of mathematics, their merits and faults. Constructivism, formalism, logicism, platonism, etc. What do you think?

IMO, at the surface level, the majority of working mathematicians think of their day-to-day mathematics platonistically, with the understanding that their real foundation lies elsewhere. I would say that most would probably end up backing some flavor of logicism or formalism, but there are a variety of other philosophies that I've seen from the seemingly reasonable constructivism to the seemingly unreasonable ultrafinitism.

I think Phil Scott mentioned that he would be interested in giving his opinions on constructivism here. I'd be interested in seeing that, or any other well thought out opinions.

As I understand it, what mathematicians actually do is an empirical science, irrespective of how they themselves construe their work. The main reason to accept that it is, is that most mathematicians throughout history have been able to agree among themselves as to what demonstrations were acceptable as proofs.

It's also interesting to note that mathematics has grown ever more abstract, starting essentially with two theories, arithmetic and geometry, which were very close to the empirical understanding of the natural world at the time. However, even with Cantor's work on infinite sets, mathematics remains essentially an empirical science. So, I would say that the "real foundation" of mathematics is in the data collected necessarily empirically and that mathematics is growing more mature by going more abstract.

What you seem to be talking about seems nothing like proper philosophies to me. Rather, constructivism, logicism, formalism, seem to be codes of best practice, or more simply, methods of work, conceived as such in the absence apparently of any real understanding of the "real foundation" of mathematics, only possibly inspired by various informal philosophical intuitions.
EB

 

[Speakpigeon]

The assertion that there is in fact a line at all can be spooky to the trusting person hearing the assertion who can yet use no instruments of science to positively identify the line's presence.

In character, a student at the front row may chuckle and say: oh teacher(!), why in the world would you even tell us such a thing (that there is a line) when we can all plainly see what is so blindingly obvious for ourselves. But then as he turns to point to this line that exists, his face turns pale upon recognizing that the other students can't see it either. He so very much trusts the teacher. Where? Where or where can it be?

It must be in Platonic land. Platonicville?

But wait, although the language we use is most helpful in explaining our points, the literalists become quite conflicted. To say that there is something when there is no detection of it is quite bothersome for some.

Well, your notion of a Platonic line seems logically incoherent on the face of it. Where would be the line, which would have to exist, between our world and this Platonic world?
EB

 

[Speakpigeon]

Let me take a stab at this.

So for some number or mathematical expression, I think most of us would agree that there is some brain or conscious state correlated to each. Like the number 3+4 is brain/conscious state A, or the number 11 is also some other brain/conscious state. Then, doesn't this mean that there is a "scientific Platonism" for all numbers or expressions? Of course this would leave in question all expressions or numbers that haven't been thought of by brains or whatever it is that can think of such things.

Very good point.

Also, I would guess that this is the intuitive basis for some mathematicians to reject the notions of infinite sets, assuming that no brain could possibly think up each element of any infinite set. We couldn't even think up all infinite sets to begin with...

So, this contradicts a Platonic view of mathematics, and yet, we need to find a way to explain how the human mind can come up with the notion of infinite set.
EB

 

[Speakpigeon]

Let me take a stab at this.

So for some number or mathematical expression, I think most of us would agree that there is some brain or conscious state correlated to each. Like the number 3+4 is brain/conscious state A, or the number 11 is also some other brain/conscious state. Then, doesn't this mean that there is a "scientific Platonism" for all numbers or expressions? Of course this would leave in question all expressions or numbers that haven't been thought of by brains or whatever it is that can think of such things.

I don't think there is some brainstate correlation to mathematical expressions. But then again, what do I know...

No correlation is needed. All we need is that different brains understand each other talking mathematics.

Compare this with a situation were you would have one mechanical device to count the number of litres in a tank, and one electronic device doing the same. I would expect no correlation between the states of the two devices but if they had to communicate, they could get to agree on how many litres there are in the tank.

That being said, it would still be very surprising if all human brains didn't have some physical organisation in common allowing all of them to conceive of numbers and other mathematical concepts.
EB

 

[Speakpigeon]

Don't you need 2 digits to appear infinitely many times to have an irrational?

Without going so far as to say that this is irrational, I would certainly be very surprised if that was true.

We do need at least two digits to represents numbers, for example in base 2. But even in base 2, I can't intuit why an irrational number could not have an infinite number of 1s in its "decimal" part. Oh wait, such a number would be 0.111... minus some finite decimal part number, and so would be a rational. Right, you may well be right...

I wish Phil, and you, hadn't been driven away. I certainly hope I had little part in it.

I certainly have no part in it!

I understand Phil Scott is a bit wary of much of philosophical talk of the kind Angra Mainyu favours.
EB

 

[steve_bank]

Some philosophies of mathematics affect which kinds of proof are valid, and thus affect which mathematical results are valid.

Constructivism states that the only valid mathematical results are constructed ones.

Finitism is a subset of constructivism that states that the only valid mathematical results are those constructed with a finite number of steps.


Here is a simple constructive proof: that the sum of every pair of even numbers is also an even number. An even number n has form 2*k, where k is an integer. Here is the proof. Start with even numbers n1 = 2*k1 and n2 = 2*k2. Add them:

n1 + n2 = 2*k1 + 2*k2 = 2*(k1+k2)

Since the sum of two integers is also an integer, and since 2*(integer) is even, that proves the result.


A non-constructive proof works by contradiction, by showing that if some mathematical result is false, then it results in a contradiction.

Every proof that there is no largest prime number is non-constructive, by showing that the existence of such a prime leads to a contradiction. Here is a rather simple such proof.

Construct the function P = (product of (first n primes)) + 1

P is clearly not divisible by any of the first n primes, so the primes that divide it must be greater than p. Thus, for every purported largest prime, there is always a larger one.

Note that this proof does not require P itself to always be a prime. It also works for all its prime factors if it is composite. In fact, P is a prime for n = 1, 2, 3, 4, 5, but P(6) = 30,031 = 59*509, a composite number. Yet its prime factors are nevertheless greater than the first six primes: 2, 3, 5, 7, 11, and 13.


Turning to finitism, it makes anything involving infinite sets to be invalid. For instance, here is a proof that there is the same number of nonnegative integers as positive ones. The proof consists of finding a bijection between the two sets. Here it is:

For every positive integer n, there is a nonnegative integer n-1.

For every nonnegative integer n, there is a positive integer n+1.


When Georg Cantor published his theory of transfinite numbers, one of his colleagues, Leopold Kronecker, rather strongly opposed it on finitist grounds. That theory remained controversial until the early 20th cy.

Interesting and informative. I read a small book How To Read And Do Proofs years back to be able to follow mathematical proofs in engineering books.

From what you said it seems to follow as with a syllogism a mathematical proof can be logically valid, no ambiguities, but in the end useless.

That would make mathematics in the end proven empirically like hard science.

The proof that for every integer there is an integer plus 1 I is not empirically provable. And that leads to questioning the foundations of math.

Fourier Transforms and Laplace Transforms are every her in technology. There is a proof that shows there can only be one LaPlace Fourier Transform pair for any real signal. It is a foundation in all areas of technology.

The are proofs that reduce to the rules of algebra, but what about underlying proofs of algebra?

 

[Speakpigeon]

The proof that for every integer there is an integer plus 1 I is not empirically provable.

Could you explain clearly how any physical theory could possibly be empirically provable if "the proof that for every integer there is an integer plus 1 I is not empirically provable" as you claim here?

I'm sure we will all be very interested in you explanation.

Thanks.
EB

 

[Speakpigeon]

Some philosophies of mathematics affect which kinds of proof are valid, and thus affect which mathematical results are valid.

Constructivism states that the only valid mathematical results are constructed ones.

Finitism is a subset of constructivism that states that the only valid mathematical results are those constructed with a finite number of steps.


Here is a simple constructive proof: that the sum of every pair of even numbers is also an even number. An even number n has form 2*k, where k is an integer. Here is the proof. Start with even numbers n1 = 2*k1 and n2 = 2*k2. Add them:

n1 + n2 = 2*k1 + 2*k2 = 2*(k1+k2)

Since the sum of two integers is also an integer, and since 2*(integer) is even, that proves the result.


A non-constructive proof works by contradiction, by showing that if some mathematical result is false, then it results in a contradiction.

Every proof that there is no largest prime number is non-constructive, by showing that the existence of such a prime leads to a contradiction. Here is a rather simple such proof.

Construct the function P = (product of (first n primes)) + 1

P is clearly not divisible by any of the first n primes, so the primes that divide it must be greater than p. Thus, for every purported largest prime, there is always a larger one.

Note that this proof does not require P itself to always be a prime. It also works for all its prime factors if it is composite. In fact, P is a prime for n = 1, 2, 3, 4, 5, but P(6) = 30,031 = 59*509, a composite number. Yet its prime factors are nevertheless greater than the first six primes: 2, 3, 5, 7, 11, and 13.


Turning to finitism, it makes anything involving infinite sets to be invalid. For instance, here is a proof that there is the same number of nonnegative integers as positive ones. The proof consists of finding a bijection between the two sets. Here it is:

For every positive integer n, there is a nonnegative integer n-1.

For every nonnegative integer n, there is a positive integer n+1.


When Georg Cantor published his theory of transfinite numbers, one of his colleagues, Leopold Kronecker, rather strongly opposed it on finitist grounds. That theory remained controversial until the early 20th cy.

I don't think this is any problem at all. Different mathematicians start with different assumptions, so it's only proper that they should end up with different conclusions.

Again, this just shows that either mathematics is an empiric science or it's not a science at all, but if you want to say it's not a science then nothing is. Your choice.
EB