Is Mathematics Really Platonic or a Human Invention?

[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.

 

[Functor97]

This topic really boils down to the old parable "If a tree in the forest falls, and no one is around to hear, does it make a sound?"
We use empirical physics to derive the basis for our mathematics, the conections we draw and rules that the patterns adhere to we refer to as mathematics.
If the question is, "is pure mathematics the purest form of knowledge" as some mathematical platonists would have you believe, i would say it probably depends on how you define pure. All our knowledge must adhere to our physical reality, or else we must accept unreasonable hypothesis such as deities, or some metaphysical nonsense. I believe, and this is just that, that reality is dependent upon our consciousness, we in effect create reality by observing it, Wheeler wrote much on this idea. Mathematics accesses the deepest connections we humans can draw from our creation and thus it is quite pure, and the heart of all other connections we see. So i do believe that mathematics is the most basic of knowledge. However i do not believe the abstract structures would exist outside our observation. I do not however believe this consciousness is dependent upon humanity, rather on the nature of sentient beings, for example marsians and E.T both see the same connections and thus mathematics. The interesting part is the nature of consciousness, what the hell is it anyway? I have no idea, and i doubt any human really does. If all sentient beings recognise the same reality, that is case in point that they have evolved and continued to, because our vision of reality let's us. If this means that we are sensing a deeper reality, or just rigging the game, and many such constructions are possible, i am not sure. I do not believe this is a mathematical question, but as our knowledge is based upon mathematical reasoning, can we answer it, should we try? I am not sure.

Maybe we should move this to the philosophy forums?

 

[Functor97]

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

I beg to differ

 

[ThomasT]

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.

Everything depends on the world in some sense. Doesn't it?

I have to join the chorus in saying that I also don't fully understand exactly what mathematical Platonism refers to. Here's a breakdown of it, Platonism in the Philosophy of Mathematics , from the Stanford Encyclopedia of Philosophy.

There's other good sources on the web, and probably some good stuff at the Philosophy Forums in the subforum, Logic and Philosophy of Math .

Also, there's the locked thread in this forum (just a bit down the list), "Is math physically real?", which might provide some insights.

 

[Willowz]

Everything depends on the world in some sense. Doesn't it?

What do you mean?

 

[Evo]

This was moved to philosophy in error.

Ok, someone asked me nicely to re-open, so I'll let you guys play in here a bit longer.

 

[ThomasT]

What do you mean?

What things don't, in any sense, depend on the world?

 

[SixNein]

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.

I believe mathematics is platonic, but I have a difficult time explaining why. Mathematics describes the relationships of physical phenomenon, but those relationships are not physical in and of themselves. So do those relationships actually exist? I think they do in a platonic world.

 

[SixNein]

Post this in philosophy This has nothing to do with mathematics

I dunno. Mathematics and philosophy are very closely linked. I would call them sisters.

 

[SixNein]

Everything depends on the world in some sense. Doesn't it?

I have to join the chorus in saying that I also don't fully understand exactly what mathematical Platonism refers to. Here's a breakdown of it, Platonism in the Philosophy of Mathematics , from the Stanford Encyclopedia of Philosophy.

There's other good sources on the web, and probably some good stuff at the Philosophy Forums in the subforum, Logic and Philosophy of Math .

Also, there's the locked thread in this forum (just a bit down the list), "Is math physically real?", which might provide some insights.

The question is probably mathematics greatest philosophical question. I'm kind of surprised to see such a lack of interest in it. The answer to this question implys some important things. For example, if mathematics is just an invention not tied to the universe, some of the theorems like say Godel's theorem of incompleteness will be meaningless to scientific study. If mathematics is tied to the universe, Godel's theorem could effect scientific study.

 

[Willowz]

What things don't, in any sense, depend on the world?

Those things of which cannot be spoken of. But, we are talking about mathematics here.

 

[Willowz]

BTW, thanks to the person who asked to reopen the thread. Yay!

ON second thought I think it's better if this thread is locked.

 

[Oldfart]

By coincidence, the Aug 2011 SciAm has an interesting artical about this -- Page 80

 

[kings7]

Not everything depends on the physical world; however, this depends on a whole brain/mind debate. Depending on what side you fall on, "non-physical" things can exist or not exist.

This conversation has so many definitions that people aren't agreeing on that it makes it hard to follow :/ But it definitely is interesting.

 

[SteveL27]

By coincidence, the Aug 2011 SciAm has an interesting artical about this -- Page 80

But it was moved to an obscure journal of philosophy and then locked ...

 

[disregardthat]

I wish this stayed in general math. It was much more delightful with a conversation that made sense.

 

[Willowz]

An interesting fact. Godel was a platonist.

So, if mathematical existence is not in it's axioms or postulates, then where is it?

I still don't think it is platonic. But, who am I to say?

Can anyone explain the present day outlook in the field of mathematics?

 

[disregardthat]

An interesting fact. Godel was a platonist.

Gödel was also highly religious and believed in a close connection between mathematics and the divine. It isn't really relevant what other mathematicians themselves believed (not that you implied that), since for these kinds of beliefs it does not take much to convince anyone of anything (because it's simply an appeal to imagination). It's not something you can rationally convince yourself of, it's just neat. And that's the problem.

Asking where is mathematical existence is exactly like asking "where is "1"?". Of course we can't give a sensible answer to this, only make cop-outs like "in a platonic world of mathematical objects", or "in your mind" (or worse: "in your brain"). Obviously, the problem here is not the location of 1, but the fact that we have tricked ourself into believing mathematical existence has anything to do with physical existence, or that they will have similar properties since they are both called "existence". This is not so, and mathematical existence, or, the usage of the word "exists" in mathematics, is much more like any other mathematical rule of engagement, like the word "implies", "equals", "contradicts", etc... It's as any of these words used in a certain way, but does not imply the outer existence of anything.

 

[Willowz]

It's not something you can rationally convince yourself of, it's just neat. And that's the problem.

Actually, I think the his proof is a rational basis for believing in platonic forms.

Anyways, my point is that this is the strongest case for platonism that there is. But, again math does or must in some way or another depend on the world.

 

[disregardthat]

Actually, I think the his proof is a rational basis for believing in platonic forms.

What are you talking about? His proof of what?

 

[Willowz]

Erm, I mean his incompleteness theorem.

 

[disregardthat]

Erm, I mean his incompleteness theorem.

And how does that support platonic forms?

 

[Willowz]

By the fact that the mind perceives truth beyond formal systems.

 

[disregardthat]

By the fact that the mind perceives truth beyond formal systems.

"Truth" as a mathematical concept has a very specific definition (which is calculated from the construction of a string sentence), and applied to mathematical statements will give certain results, but the proof of that some mathematical statements can be formed that cannot be proven nor disproven isn't exactly perceiving truth beyond formal systems, but rather showing a limitation to proofs with respect to their relation with the mathematical definition of the truth of a mathematical statement.

Truth in this sense is a formal mathematical notion as any piece of mathematics, and should not be compared to e.g. truth of physics (which has a categorically different aspect to it, whether you take the realist stance or not). It doesn't give any support for platonic forms, for you will have to assume that true statements in mathematics are true about something in order to see how Gödel's theorem can relate to it in the first place. Thus one is basically assuming platonic forms (or something that which mathematical statements refer to), and then interpreting Gödel's theorem in relation to this view. The formalities has nothing to do with platonic forms whatsoever, and much less give support to it.

 

[Willowz]

Anyway, all that I have done in this thread is draw a false dichotomy between mathematics being platonic or non platonic. It is what it is. Whereof one cannot speak thereof, one must be silent.

 

[Char. Limit]

Anyway, all that I have done in this thread is draw a false dichotomy between mathematics being platonic or non platonic. It is what it is. Whereof one cannot speak thereof, one must be silent.

Eloquent.