Is Mathematics Truly Platonic? Exploring the Debate and Evidence

[Willowz]

I don't understand why many/some mathematicians believe that mathematics is platonic. I mean, how would they know if mathematics is platonic? Surely, mathematics does depend in some way on the world.

How do you convince a mathematician that mathematics is not platonic.

 

[micromass]

Post this in philosophy This has nothing to do with mathematics

 

[SteveL27]

How do you convince a mathematician that mathematics is not platonic.

Poke 'im in the eyes. No wait, that's the answer to "How to you make a Venetian blind?"

Define your terms! Wiki gives seven different meanings to the word "Platonic," ranging from Platonic love to Platonic solids.

http://en.wikipedia.org/wiki/Platonic

If you mean, does math necessarily have to correspond to the real world, then how can non-Euclidean and Euclidean geometry both be logically consistent? They can't both be true about the world.

 

[Willowz]

Post this in philosophy This has nothing to do with mathematics

Well, I'm directing this question at mathematicians who happen to believe in platonism.

 

[SteveL27]

Well, I'm directing this question at mathematicians who happen to believe in platonism.

Can't you at least tell us what you mean by that? Here is the Wiki definition of Platonism:

Platonism is the philosophy of Plato or the name of other philosophical systems considered closely derived from it. In a narrower sense the term might indicate the doctrine of Platonic realism. The central concept of Platonism is the distinction between that reality which is perceptible, but not intelligible, and that which is intelligible, but imperceptible; to this distinction the Theory of Forms is essential. The forms are typically described in dialogues such as the Phaedo, Symposium and Republic as transcendent, perfect archetypes, of which objects in the everyday world are imperfect copies.

It keeps going on like that in a much longer and totally incomprehensible run-on paragraph. I for one can't make heads or tails out of it. Are you saying that math is perceptible but not intelligible? And what on Earth does that mean?

My freshman calc professor was perceptible, in the sense that he was physically present; but he was hardly intelligible!

http://en.wikipedia.org/wiki/Platonism

 

[pmsrw3]

Mathematical Platonism is the idea that mathematical things have an existence that is independent of the physical universe. (At least, that's what I think Willowz is referring to.) The main proponent of this idea whom I've read is Roger Penrose, but there are others. I don't personally think there's any way to KNOW whether or not this is true, but I find it a congenial viewpoint.

Steven Weinberg http://www.physics.nyu.edu/faculty/sokal/weinberg.html"

When I was an undergraduate at Cornell I heard a lecture by a professor of philosophy (probably Max Black) who explained that whenever anyone asked him whether something was real, he always gave the same answer. The answer was "Yes." The tooth fairy is real, the laws of physics are real, the rules of baseball are real, and the rocks in the fields are real. But they are real in different ways.

It seems clear to me that http://en.wikipedia.org/wiki/Monster_group" is real in a different way than Mars is real. That belief may not be exactly Mathematical Platonism, but it's close enough.

 

[disregardthat]

Mathematical platonism is not something that is wrong per se, it's in my opinion quite meaningless. Stating that mathematical truths are true independent of the physical universe can be interpreted in two ways. The harmless way; mathematical statements are not statements of the physical world, hence independent of the physical world. It's basically admitting that mathematics is something different than physics.

The more harmful way, is believing that mathematical truths are truths about something. One posits the actual existence of objects for which mathematical statements are about, which is not only completely meaningless (what are these things?), but it does redirect the focus of mathematics as an intuitive but rule-based activity to the illusion of mathematics as a sort of descriptive language of something mysterious or in some cases divine. It creates the mental picture of a mathematical universe which we connect to through mathematics, and makes us believe we are describing it. This point of view is in my opinion complete nonsense. I don't mean that it's implausible or unlikely, but that it is meaningless in a very fundamental manner.

 

[pmsrw3]

The more harmful way, is believing that mathematical truths are truths about something...

How is this "harmful"? What, precisely, is the damage done by this belief?

 

[Willowz]

Mathematical platonism is not something that is wrong per se, it's in my opinion quite meaningless. Stating that mathematical truths are true independent of the physical universe can be interpreted in two ways. The harmless way; mathematical statements are not statements of the physical world, hence independent of the physical world.

Yet, this doesn't explain why mathematics describes the physical world so well. It actually denies that claim. I assume that you don't take this position nor platonic realism for the matter.

It's basically admitting that mathematics is something different than physics.

I don't know what this means or could mean.

 

[Willowz]

How is this "harmful"? What, precisely, is the damage done by this belief?

It's harmful in that it carries a lot of metaphysical baggage. Meaningless statements perpetuating meaningless statements.

 

[pmsrw3]

It's harmful in that it carries a lot of metaphysical baggage. Meaningless statements perpetuating meaningless statements.

I don't see it. To be clear, I'm not talking here about whether Platonism is harmful. I'm trying to understand disregardthat's claim that it is harmful for people to believe that mathematical objects exist.

Let's take an example. I believe that there exists a simple group of order 60. My group theory textbook says so, and produces arguments that I find utterly persuasive. What harm, exactly, am I doing by believing this? What is the "lot of metaphysical baggage" caused by this belief, which you find so harmful? And do you really believe that "There exists a simple group of order 60" is a "meaningless statement perpetuating meaningless statements"?

 

[pmsrw3]

I don't know what this means or could mean.

Interesting. Do you understand the statement, "Biology is different from physics"? How about "General Electric is different from Pixar"? Do you deny the meaningfullness of differences between all conceptual entities, or is there something special about the relationship between the disciplines of mathematics and physics that makes you unable to comprehend how there could possibly be even the slightest distinction between them?

 

[disregardthat]

Don't take harmful literally, of course it won't physically harm anyone. But as Willowz says it is a lot of metaphysical baggage, for which one has no reason to believe in-while it distorts what mathematics really is about. Yet some people feel inclined to believe in such things as platonic entities.

 

[SteveL27]

Mathematical Platonism is the idea that mathematical things have an existence that is independent of the physical universe. (At least, that's what I think Willowz is referring to.)

That's my guess too, but what about Plato's cave? We see shadows of things that are real. So even though we can only see shadows, the real things are still out there.

So does that mean that mathematical and other abstract things exist but we only see their shadow? Or what does it mean, exactly?

I think the OP is not clear on the issue. The Wiki links were so complicated that if the issue were really that simple, Wiki would say so. If all Platonism is saying that a novel exists but its plot doesn't -- since the novel is a physical thing, but the plot is only a human mental abstraction -- then that seems like a rather trivial point.

So I honestly wonder if we are working from a clear definition of Platonic. I was hoping to hear from the OP with a few examples. Is the number 5 Platonic? Is the "completed infinitey" of the natural numbers? The uncountably infinite of the real numbers that are not definable, and cannot ever be named?

We're not all talking about the same thing until someone actually tells us what Platonism is. I tried to look it up but I could not understand what Wiki was saying; it seemed to presuppose formal knowledge of technical terms in philosophy. And the story of the cave seems to have an ambiguous meaning when applied to abstract things.

 

[pmsrw3]

Don't take harmful literally, of course it won't physically harm anyone. But as Willowz says it is a lot of metaphysical baggage, for which one has no reason to believe in-while it distorts what mathematics really is about. Yet some people feel inclined to believe in such things as platonic entities.

So, you agree with Willowz that my belief that there exists a simple group of order 60 causes me to carry around "a lot of metaphysical baggage"? You believe that I have "no reason to believe" this? You believe that this belief of mine "distorts what mathematics really is about"?

I have to disagree with you on all three points.

 

[pmsrw3]

That's my guess too, but what about Plato's cave? We see shadows of things that are real. So even though we can only see shadows, the real things are still out there.

So does that mean that mathematical and other abstract things exist but we only see their shadow? Or what does it mean, exactly?

I think the OP is not clear on the issue. The Wiki links were so complicated that if the issue were really that simple, Wiki would say so. If all Platonism is saying that a novel exists but its plot doesn't -- since the novel is a physical thing, but the plot is only a human mental abstraction -- then that seems like a rather trivial point.

So I honestly wonder if we are working from a clear definition of Platonic. I was hoping to hear from the OP with a few examples. Is the number 5 Platonic? Is the "completed infinitey" of the natural numbers? The uncountably infinite of the real numbers that are not definable, and cannot ever be named?

We're not all talking about the same thing until someone actually tells us what Platonism is. I tried to look it up but I could not understand what Wiki was saying; it seemed to presuppose formal knowledge of technical terms in philosophy. And the story of the cave seems to have an ambiguous meaning when applied to abstract things.

The OP was definitely not clear. But "Mathematical Platonism", if that's what he's talking about, is a much narrower position than "Platonism". It's still not completely defined -- there are several somewhat different versions -- but it isn't all as incomprehensible as that Wiki quote on Platonism you posted (which I agree is impenetrable). Take a look at http://www.iep.utm.edu/mathplat/" . It's fairly comprehensible, at least in the beginning.

 

[SteveL27]

So, you agree with Willowz that my belief that there exists a simple group of order 60 causes me to carry around "a lot of metaphysical baggage"? You believe that I have "no reason to believe" this? You believe that this belief of mine "distorts what mathematics really is about"?

I have to disagree with you on all three points.

No, I didn't say that at all. I said that I personally have no idea what Platonism is. And that nothing I read in this thread has convinced me that anyone else does either.

Personally I believe in the mathematical existence of uncountable sets. However I have no idea if that makes me a Platonist or an anti-Platonist. I gather that makes me an anti-Constructivist, or a Cantorian, or something.

I already gave the example of the logical consistency of both Euclidean and non-Euclidean geometry. To me that shows that mathematical objects need not have physical existence. I thought that single example would either refute or confirm Platonism ... if only someone actually knew for certain what Platonism means with respect to mathematical existence.

I didn't make the baggage remark. Did I say something that looked like I was agreeing with it? I'm perfectly fine with mathematical existence of nonmeasurable sets and all the other stuff constructivists dislike.

I just don't know what a Platonist is -- even after trying to figure it out.

 

[spamiam]

I think this is a pretty interesting question, but I don't think it has a simple yes or no answer. Certainly mathematics was created to solve real world problems. All the typical field operations of $+, -, \times,$ have very clear real world applications.

But on the other hand, if we look at things like the Banach-Tarski paradox, certainly their sphere doubling method exists as a mathematical object, but it almost certainly does not exist as a real world method.

However, if we consider probability, the real world applications are undeniable. As Sheldon Ross points out in his First Course in Probability, "It should be noted that it is an empirical fact that events having mathematical probability 1, do, in fact, occur in practice with certainty." Furthermore, he points out that, if it is unfeasible to calculate a probability mathematically, we can simply use a computer simulation to give us a very good approximation for the actual value.

If I had to choose a side, I would probably say that mathematics exists independently of the real world. Will I ever see a large cardinal walking down the street? Doubtful. Do objects of infinitesimal size exist in the real world? Who knows. But mathematically, both of these objects exist without any doubt.

 

[pmsrw3]

No, I didn't say that at all.

I was replying to disregardthat.

 

[pmsrw3]

Will I ever see a large cardinal walking down the street? Doubtful. Do objects of infinitesimal size exist in the real world? Who knows. But mathematically, both of these objects exist without any doubt.

Hey, buster, you'd better watch out. By this statement you're in direct danger of a lot of metaphysical baggage and distorting what mathematics really is about.

 

[SteveL27]

I was replying to disregardthat.

Oops. Oh well I read the IEP link and I'm more confused than ever. The second sentence in that article is:

For example, a platonist might assert that the number pi exists outside of space and time and has the characteristics it does regardless of any mental or physical activities of human beings.

How could anyone ever answer that definitively one way or another? It's clear to me that pi is a human abstraction; that pi does not exist in the physical universe, but does have a definite mathematical existence that we can use to our advantage in the physical world.

But if you ask, does pi exist if no human being ever lived to think about it? Well, who can ever answer a question like that?

 

[disregardthat]

So, you agree with Willowz that my belief that there exists a simple group of order 60 causes me to carry around "a lot of metaphysical baggage"? You believe that I have "no reason to believe" this? You believe that this belief of mine "distorts what mathematics really is about"?

Believing that there exists a group of order 60 doesn't say much, that's the way we talk in mathematics whether one is a platonist or not, so I don't see your point at all. If you believe there is some platonic entity corresponding to the simple group of order 60, and that it exists in some sense independently from the physical world, then that's an entirely different thing, and closer to what mathematical platonism is all about.

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

My take on the reason why platonism might seem appealing is how we use our language when talking about mathematics. It resembles the way we speak of physical objects in that we say "there exists" for mathematical objects as well as for physical objects. But the relation is non-existent and one is tricked into believing that since the way of talking are so similar, they must in some sense have similar meanings, and thus imagining a mathematical universe consisting of mathematical objects analogous to how the physical world consist of physical objects.

 

[Willowz]

So I honestly wonder if we are working from a clear definition of Platonic. I was hoping to hear from the OP with a few examples.

There is no clear definition of Platonic Realism apart from itself (just like Mathematics). He made a (false?) equivocation between mathematics and his Platonic Realism. Most of Plato's philosophy was taught by amazing allegories/parables/stories. I recommend reading the https://secure.wikimedia.org/wikipedia/en/wiki/Allegory_of_the_Cave" .

 

[pmsrw3]

But if you ask, does pi exist if no human being ever lived to think about it? Well, who can ever answer a question like that?

Yes, I agree, sort of. Although I think we can decide how to answer that question, simply by deciding what we mean by the word "exist".

A (slightly) more concrete question, which I think gets at this, if not right at the heart of it, is this: Suppose one day we establish contact with an extraterrestrial intelligence, which has developed entirely independently from us. If we beam them pi and e (let's do it in binary -- don't want to assume they have 10 fingers), will they recognize it? I have not the slightest doubt that they will. I think if we beam them the primes, they'll know exactly what we're doing. I think that if we beam them the orders of simple groups, they'll recognize that, too. I think that gets at the existence, in some non-trivial sense of a pi and an e independent of human rationality. Of course, it doesn't settle the question of whether they exist independent of rationality (full stop), or the physical universe.

 

[SteveL27]

If I had to choose a side, I would probably say that mathematics exists independently of the real world. Will I ever see a large cardinal walking down the street? Doubtful. Do objects of infinitesimal size exist in the real world? Who knows. But mathematically, both of these objects exist without any doubt.

Unfortunately, Platonism seems to demand much more than that. According to the IEP link supplied by pmsrw3 at
http://www.iep.utm.edu/mathplat/

it says that Platonism requires

Independence: Mathematical ontology is independent of all rational activities, that is, the activities of all rational beings.

In other words to be a Platonist, you have to believe that not only does a large cardinal have mathematical existence; but also, that a large cardinal would exist even if no humans had ever lived to think about the subject.

That, I find rather difficult to believe.

In other words Platonism doesn't say that mathematical objects have abstract existence. It says they exist even in the absence of mathematicians.

Perhaps the natural numbers have such existence. At least the finite ones. But large cardinals? Those require mathematicians.

Looks like I'm not a Platonist!

Ummm ... ps ... infinitesimals? Only in nonstandard analysis. That's yet another interesting case. Infinitesimals don't even have mathematical existence in standard analysis as practiced by the overwhelming majority of working mathematicians. But they are logically consistent. If no mathematicians had ever lived, would infinitesimals both exist and not exist?

I don't think Platonism -- if it means existence of ideas in the absence of human beings to have the ideas -- is tenable.

 

[disregardthat]

Analogies can be persuasive, but unfortunately metaphysics and analogies do not go hand in hand. Metaphysical truths is not something that can be established on the basis of appealing to intuition by the usage of analogies.

 

[pmsrw3]

Believing that there exists a group of order 60 doesn't say much, that's the way we talk in mathematics whether one is a platonist or not, so I don't see your point at all. If you believe there is some platonic entity corresponding to the simple group of order 60, and that it exists in some sense independently from the physical world, then that's an entirely different thing, and closer to what mathematical platonism is all about.

As I said earlier, I was not talking about mathematical platonism. I'm disputing your assertion of a raft of "harmful" consequences that arise from my believing that there exists a simple group of order 60.

My take on the reason why platonism might seem appealing is how we use our language when talking about mathematics. It resembles the way we speak of physical objects in that we say "there exists" for mathematical objects as well as for physical objects. But the relation is non-existent and one is tricked into believing that since the way of talking are so similar, they must in some sense have similar meanings, and thus imagining a mathematical universe consisting of mathematical objects analogous to how the physical world consist of physical objects.

I think you've got this exactly backward. A mathematical platonist is not one who believes that mathematical things exist in the same way physical things do. The platonist, in fact, believes exactly the opposite of this: he believes that mathematical things exist in a quite distinct way from physical things.

A platonist accepts that there are different kinds of existence, as in the Weinberg quote I posted above. You may consider this to be "a lot of metaphysical baggage", but it appears to me unavoidable in any case. I cannot see how anyone can believe, for instance, that the presidency of the USA exists in the same way as the chair you're sitting in. And yet most people would agree, if only as a matter of linguistic convenience, that both exist, and that when we say they exist, there is enough similarity in what we mean to justify using the same word.

 

[disregardthat]

As I said earlier, I was not talking about mathematical platonism. I'm disputing your assertion of a raft of "harmful" consequences that arise from my believing that there exists a simple group of order 60.

You should read my comment where I point out that you shouldn't take the harmful bit literally. I could more appropriately call it misleading, as one would call any belief in illusions. You should also know that platonic existence and existence is not interchangable terms. Saying a simple group of order 60 exists is not equivalent to saying that it exists platonically.

I think you've got this exactly backward. A mathematical platonist is not one who believes that mathematical things exist in the same way physical things do. The platonist, in fact, believes exactly the opposite of this: he believes that mathematical things exist in a quite distinct way from physical things.

I did not say that platonists believes that mathematical things exists in the same way as physical things do, I said that the platonists have a picture of mathematical existence analogous to physical existence, in so far physical existence is independent of human beings. In this sense there is no denying of the similarities, and calling it an "opposite" type of existence is digging oneself deeper into the metaphysical pile of garbage.

A platonist accepts that there are different kinds of existence, as in the Weinberg quote I posted above. You may consider this to be "a lot of metaphysical baggage", but it appears to me unavoidable in any case. I cannot see how anyone can believe, for instance, that the presidency of the USA exists in the same way as the chair you're sitting in. And yet most people would agree, if only as a matter of linguistic convenience, that both exist, and that when we say they exist, there is enough similarity in what we mean to justify using the same word.

I don't see how you can justify the existence of mathematical objects as platonic entities by some linguistic similarity. This is exactly my point however, that people do, but it is a very obvious logical error. No one believes that the presidency of the USA exists platonically just because we say that "it exists" in the same way as I say my chair exists (who in their right mind would claim that a precidency exists independently of human beings?). Justifying platonic existence by linguistic similarities is, as I said, a logical error.

 

[pmsrw3]

You should read my comment where I point out that you shouldn't take the harmful bit literally.

Yes, I read it -- you clarified that you didn't mean it was physically harmful. You then proceeded to explain the ways in which you thought it was harmful. You said the belief in the existence of mathematical things results in "a lot of metaphysical baggage". You also said that this belief "distorts what mathematics really is about".

I believe that there exists a simple group of order 60. I do not believe that this belief of mine harms me, or misleads me, or causes me to carry a lot of metaphysical baggage, or distorts my understanding of what mathematics is really about.

I did not say that platonists believes that mathematical things exists in the same way as physical things do, I said that the platonists have a picture of mathematical existence analogous to physical existence, in so far physical existence is independent of human beings. In this sense there is no denying of the similarities, and calling it an "opposite" type of existence is digging oneself deeper into the metaphysical pile of garbage.

But I didn't, did I? Read what I wrote again. You will see that I neither said nor implied that mathematical existence is an opposite type of existence to physical existence.

I don't see how you can justify the existence of mathematical objects as platonic entities by some linguistic similarity... Justifying platonic existence by linguistic similarities is, as I said, a logical error.

But I didn't do that, did I? I explained that the linguistic similarity arises from a similarity in the concepts, and that that similarity justifies calling them both existence.

No one believes that the presidency of the USA exists platonically just because we say that "it exists" in the same way as I say my chair exists (who in their right mind would claim that a precidency exists independently of human beings?).

No one that I know of. Did you see anyone claiming the platonic existence of the presidency? I didn't...

 

[disregardthat]

I believe that there exists a simple group of order 60. I do not believe that this belief of mine harms me, or misleads me, or causes me to carry a lot of metaphysical baggage, or distorts my understanding of what mathematics is really about.

We are discussing platonic existence, no one here is arguing against your claim that a simple group of order 60 exists. The point which I am getting tired of saying is that believing in platonic existence carries a lot more with it than just believing such-and-such exists.

But I didn't do that, did I? I explained that the linguistic similarity arises from a similarity in the concepts, and that that similarity justifies calling them both existence.

That has nothing to do with what we are talking about here. No one denies mathematical existence nor rejects the usage of the word. Platonic existence however is the subject here, but you seem to think that these are interchangable notions, which makes me wonder if you actually understand what the platonic view of mathematics asserts.

No one that I know of. Did you see anyone claiming the platonic existence of the presidency? I didn't...

Then what was your ramble about? That existence as a word is not used exclusively for physical objects-that there are different kinds? Please.

 

[Char. Limit]

I kinda skimmed this thread, so sorry if my comment here sounds stupid.

Find me a seven-dimensional exotic sphere in reality and I'll accept that mathematics is not platonic.

 

[Functor97]

Interesting topic.

In my opinion mathematics is deduction based upon empirical observation. If reality was not as it is, our mathematics would be quite different. If we came into contact with an alien intelligence, then of course they would be mathematical, that is how we define intelligence...but even yet they are from the same reality as us, and thus would have formulated a similar mathematics.

Are fairy tales real? Is there "some place" where the events of harry potter are currently taking place?

 

[Willowz]

Just a nice quote that I think may be relevant.

Using one sort of arithmetic of geometry, for example, we might find that our physics could be reduced to a logically neat and simple system, which is intellectually satisfying, whereas using different arithmetics and geometries, we should find our physics full of very complicated ad hoc hypotheses. But what we find neat, simple, easy, and intellectually satisfying surely depends rather on our psychological make-up, than on the behaviour of measuring rods, solids and fluids, electrical charges--the 'external world'.

 

[SteveL27]

A (slightly) more concrete question, which I think gets at this, if not right at the heart of it, is this: Suppose one day we establish contact with an extraterrestrial intelligence, which has developed entirely independently from us. If we beam them pi and e (let's do it in binary -- don't want to assume they have 10 fingers), will they recognize it? I have not the slightest doubt that they will. I think if we beam them the primes, they'll know exactly what we're doing. I think that if we beam them the orders of simple groups, they'll recognize that, too. I think that gets at the existence, in some non-trivial sense of a pi and an e independent of human rationality. Of course, it doesn't settle the question of whether they exist independent of rationality (full stop), or the physical universe.

What if we beam them

a) The definition of an inaccessible cardinal;

b) The plot of the novel Moby Dick.

Would we say that aliens must inevitably recognize one or the other?

Secondly, the definition of Platonism says that a thing exists independent of any rational process. So the fact that earthlings and aliens both recognize prime numbers, does not show that the primes have Platonic existence. It's this requirement that troubles me. Other than physical things such as the sun, moon, stars, rocks, trees, oceans, how can any product of the human mind be said to have existence independent of human or alien minds?

(ps) Let me put this another way. The primes are inevitable. But are the Peano axioms? Couldn't aliens recognize primes, but perhaps not have had the same historical development of foundations as earthlings? Our own foundations are barely 100 years old, after all. It's not even clear WE will have the same foundations in another hundred years.

 

[Dale]

Well, mathematics is neither erotic nor romantic, so it must be platonic.