Does Math Exist Independent of Our Minds?

[Polymath257]

Nothing I said--e.g., "the relation a2 + b2 = c2 is not true of a plane triangle depending on whether or not some human is thinking it at any given point. That relation was true for plane triangles before humans came along to think it, and it will remain true after humans blow themselves up. Therefore, the thesis of mathematical realism is true"-- suggests that there are no non-Euclidean geometries. Nor did I say or suggest anything about "selecting Euclidean geometry over the non-Euclidean geometries." There definitely are non-Euclidean geometries, and unless one is specifically referring to Euclidean geometry, there is no reason to "select" it over non-Euclidean geometries. These facts definitely do not suggest that the thesis of mathematical realism is false.

Except that the statement 'a^2 +b^2 =c^2 ' is NOT true independent of the assumptions of Euclidean geometry. So it is NOT the case that it is true independent of our thoughts.

When you say 'plane triangle', you are making the assumptions of Euclidean geometry or else your statement is false.

So, does the game of chess exist outside of our minds?

 

[Jumi]

Mathematics exists outside our minds, just like other ideas do. They also exist outside reality. Good thing we can still use some of it in our modeling of the world around us.

 

[Polymath257]

I can understand where Pi comes from though. Whereas complex numbers are just weird.

What's interesting with this is that, for E&M, the complex numbers are not really essential. Everything can be described using just real numbers, although maybe not as compactly.

But, starting with quantum mechanics, the use of complex numbers becomes essential: quantum wave functions have complex amplitudes by necessity.

 

[Nous]

Exacept that the statement 'a^2 +b^2 =c^2 ' is NOT true independent of the assumptions of Euclidean geometry. So it is NOT the case that it is true independent of our thoughts.

So apparently you don't disagree with anything I said that you quoted: "the relation a2 + b2 = c2 is not true of a plane triangle depending on whether or not some human is thinking it at any given point. That relation was true for plane triangles before humans came along to think it, and it will remain true after humans blow themselves up. Therefore, the thesis of mathematical realism is true."

 

[Polymath257]

So apparently you don't disagree with anything I said that you quoted: "the relation a2 + b2 = c2 is not true of a plane triangle depending on whether or not some human is thinking it at any given point. That relation was true for plane triangles before humans came along to think it, and it will remain true after humans blow themselves up. Therefore, the thesis of mathematical realism is true."

No, I completely disagree with it. The question itself only makes sense after *we* decide what the rules are for plane triangles. if we choose the rules of Euclidean geometry, the statement becomes true. If we choose the rules of non-Euclidean geometries, the statement becomes false.

Our choice.

Once again, does chess exist independent of our minds?

 

[Willamena]

Does math exist independent of our minds? Why or why not?


In the philosophy of mathematics, there are basically two positions on the subject of whether or not mathematics exists independent of our minds.

Naturally, these positions date back over 2,000 years to the ancient Greeks. The first position is called Platonism, after the famous Greek philosopher, Plato. Essentially, Platonism holds that mathematics exists apart from out minds.

On the other hand, the second position -- called Formalism -- holds the opposite. Mathematics is a construct of our minds.

What think you?

This begs the question: does a mind exist for things to be independent of it?

 

[Altfish]

Not that weird if you use them with things like ac electricity.

So I believe

 

[Nous]

No, I completely disagree with it. The question itself only makes sense after *we* decide what the rules are for plane triangles.

What question are you referring to? I didn't state a question.

Show that a2 + b2 = c2 is not true for plane triangles, or that there are "rules for plane triangles" where a2 + b2 = c2 is not true.

 

[Polymath257]

What question are you referring to? I didn't state a question.

Show that a2 + b2 = c2 is not true for plane triangles, or that there are "rules for plane triangles" where a2 + b2 = c2 is not true.

Easy. The plane triangles of non-Euclidean geometry do not obey that equation.

 

[exchemist]

As a mathematician, I have thought about this issue quite a bit.

The first issue is defining mathematics. To include modern math, I define it as the study of abstract formal systems.

So, the second issue: Equations like the Pythagorus identity (that a^2 +b^2 =c^2 for a right triangle) are NOT true without some assumptions. In particular, this equation requires the assumptions of Euclidean geometry. But we *know* that there are other geometries that are equally consistent to Euclid's. In such geometries, this relation is false. So, there is no 'a priori' reason to think the Pythagorus identity is 'true': it is false in other systems that are just as 'valid' a priori as Euclidean geometry.

Next, we can say similar things for many other 'clearly true' statements. For example, the statement that 5 is prime depends on the number system used. If we use the Gaussian integers, it is no longer the case that 5 is prime (it can be factored as (2+i)(2-i) ).

Another piece of the puzzle: Godel showed that any system of axioms that is strong enough to talk about the positive integers *cannot* be proven internally to be consistent. In addition, such systems will *always* have statements that can neither be proved nor disproved in that system (such statements are said to be independent). For an independent statement, we can literally choose whether it is true or false and neither way will produce new contradictions.

So, as a thought experiment: is the game of chess independent of our minds? We certainly invented it. But, once we have chosen the rules, are the winning strategies, the solutions of various problems, etc determined? Or are they just part of our mental world with no 'outside' component?

Now, in the way I think about it, we invent chess. But once we have decided on the rules, we discover the consequences of those rules. The game of chess still only exists in our minds, but there is a component that is now to be discovered.

As I see it, the exact same thing is true of mathematics: we invent the rules and discover the consequences of those rules. Furthermore, unlike with chess, we choose the rules of mathematics to be maximally expressive of abstract relations.

What that means is that once we have chosen the rules we want (those of set theory, for example), we discover the consequences of those rules. Math still only exists in our mental worlds, but because we chose the rules to align with our intuitions *and* to be as expressive as possible, we can use math as a language to describe the world around us.

As to why math is as good as it is in describing the world, I see this as partly by our design of the math. We *chose* the basic rules to allow us to model the world. Much of math was specifically invented to help us describe aspects of the world. So is it really so surprising that a language invented to describe certain aspects of the world actually manages to do so?

And this ignores the wide areas of math that have *nothing* to do with describing anything about the 'real world'. While the funding tends to go for those topics relevant to the outside world, there are very large areas of math where no expectation of applicability is held and where the internal beauty of the ideas is paramount. Such areas are NOT descriptive of the 'real world' and are not meant to be. They are more like 'science fiction' if you want.

As you can see, I am not a Platonist. I am very much a formalist when it comes to math. BTW, there are other views on how math works: look up intuitionism and intuitiionist logic at some point.

What I particularly like about your description is the synthesis of human invention (of the rules) with discovery (of the resulting consequences). It seems to me it is the discovery element that can mislead people into thinking that maths was always "out there" in the universe, in some way, before mankind came along.

 

[exchemist]

I can understand where Pi comes from though. Whereas complex numbers are just weird.

Yes. But so powerful, for modelling any kind of periodic phenomenon....not to mention connecting pi to e......

 

[Vouthon]

@Polymath257 next week, when my hectic work schedule permits me some free time (I'm a commercial lawyer and Brexit is possessing me at the minute, sapping all my intellectual energy), I would very much like to weigh into this debate with you in a more intricate manner. Your posts are always so expertly and beautifully argued

For now though (because I lack the time to discuss this topic in any degree of depth), I'm curious what you think about your colleagues - many of them, as I shave shown in this thread, ranking amongst the most eminent luminaries of the contemporary fields of mathematics and theoretical physics - who have become convinced "mathematical realists"?

According to these theorists, mathematical platonism is the most widely held interpretation - even though this often surprises non-mathematicians - indeed, the encyclopedia of philosophy even states that it is (as opposed to formalism or intuitionism). Why would this be so?

Why are the likes of Andrei Linde, Laura Mersini-Houghton, Roger Penrose, George Ellis, Edward Frenkel, Alexander Vilenkin, Alain Connes and numerous other prominent mathematicians/cosmologists/theoretical physicists persuaded by platonism? Even the mathematician Professor Peter Woit, who is famously critical of such speculative theories as the multiverse and the 'pure mathematics' of String Theory (which, as yet, shows no connection with conventional physics i.e. no super-symmetry at the LHC) has been open about the fact that he is himself a type of mathematical platonist (he said in 2014, "I myself favor some form of “Platonism” or realism about mathematics", while being critical of Tegmark).

Connes goes so far as to claim, as do Penrose and Frenkel, that "the great discoveries of the 20th century, in particular Godel’s work, have shown that the formalist viewpoint is unsustainable". These are bold statements for such high-profile figures to make, but they nevertheless do.

I would be most intrigued to watch a discussion between yourself and one of these scientists/mathematicians.

They're are evidently convincing arguments on both sides.

 

[Kangaroo Feathers]

Does math exist independent of our minds? Why or why not?


In the philosophy of mathematics, there are basically two positions on the subject of whether or not mathematics exists independent of our minds.

Naturally, these positions date back over 2,000 years to the ancient Greeks. The first position is called Platonism, after the famous Greek philosopher, Plato. Essentially, Platonism holds that mathematics exists apart from out minds.

On the other hand, the second position -- called Formalism -- holds the opposite. Mathematics is a construct of our minds.

What think you?

All rather depends if you believe the universe would exist more or less the same if there was no one here to see it, doesn't it?

 

[Altfish]

....not to mention connecting pi to e......

My all time favourite equation...

It doesn't get much better than that.

 

[Nous]

Mathematics is the basic nature of Bramhan'
The thing is, it only shows its colors in Bramhan's dream-world which is this universe

I take it that you are referring to Brahman.

I must say that's the first time time I've ever come across the statement about mathematics being "the basic nature of Brahman". I'm glad to hear it.

 

[Polymath257]

@Polymath257 next week, when my hectic work schedule permits me some free time, I would very much like to weigh into this debate with you in a more intricate manner. Your posts are always so expertly and beautifully argued

For now though, I'm curious what you think about your colleagues - many of them, as I shave shown in this thread, ranking amongst the most eminent luminaries of the contemporary fields of mathematics and theoretical physics - who have become convinced "mathematical realists"?

According to these theorists, mathematical platonism is the most widely held interpretation - indeed the encyclopedia of philosophy even states that it is. Why would this be so?

Why are the likes of Laura Mersini-Houghton, Roger Penrose, George Ellis, Edward Frenkel, Alexander Vilenkin, Alain Connes and numerous other prominent mathematicians/cosmologists/theoretical physicists persuaded by platonism?

Connes goes so far as to claim, as do Penrose and Frenkel, that "the great discoveries of the 20th century, in particular Godel’s work, have shown that the formalist viewpoint is unsustainable".

It's interesting that the reason I go for formalism is precisely because of Godel's work.

When doing math, there is certainly a *feeling* of discovery. It feels like the conclusions arrived at are, in some sense 'out there'.

I think that is largely an illusion because we tend to all agree on the basic axioms. Most of the people you mention don't work with how we choose the basic axioms for math. Instead, they are using the already agreed upon axioms to describe higher level phenomena.

And, truthfully, there is a belief among many mathematicians that the basic axioms are irrelevant. My advisor certainly thought that way. But, after looking at the results that have come out over the last few decades, I just don't hold that to be justifiable.

According to Godel, it is *possible* that our current axioms are inconsistent. if they are shown to be, how will we choose the *new* rules for mathematics? Well, there are certain results that *every* axiom system we choose will allow: for example, some version of the fundamental theorem of calculus will remain. That theorem and its results have proven too useful for the modeling of the real world to be discarded in toto. But are there aspects of modern math that may well go away? Yes, indeed.

Ultimately, we choose the axioms of math based on two criteria: the expressiveness (which allows for modeling) and aesthetics (how much they agree with our intuitions and our sense of beauty). this both explains why such axioms as the Axiom of Choice have been accepted and why others, like the Continuum Hypothesis remain in limbo. The negation of the Axiom of Choice leads to ugly math: the arguments are forced and rather stilted. The assumption of AC, on the other hand, leads to very beautiful math. In contrast, neither the assumption of the Continuum Hypothesis nor its negation leads to obviously superior mathematical results.

 

[Polymath257]

My all time favourite equation...

View attachment 25338
It doesn't get much better than that.

Often considered one of the most beautiful equations in all of mathematics.

 

[ameyAtmA]

I take it that you are referring to Brahman.

I must say that's the first time time I've ever come across the statement about mathematics being "the basic nature of Brahman". I'm glad to hear it.

However Brahman' is Consciousness first. There is Brahman' and then Nature of Brahman' (svabhAv)

This post (#10) is what I mean by that

Who Created the Universe? God or Mathematics?

 

[metis]

To the question of the OP, yes and no. Math is merely numbers that stand for various forms of sequencing and we know that such changes do indeed exist.

 

[Nous]

However Brahman' is Consciousness first. There is Brahman' and then Nature of Brahman' (svabhAv)

This post (#10) is what I mean by that

Who Created the Universe? God or Mathematics?

I like everything you say here about Brahman and mathematics.