Is Mathematics Truly Platonic? Exploring the Debate and Evidence

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In summary, the conversation discusses the belief of mathematical platonism, which is the idea that mathematical objects have an existence independent of the physical universe. This belief can be interpreted in two ways: the harmless way, which states that mathematical statements are not statements about the physical world, and the more harmful way, which posits the actual existence of mathematical objects. The harmful interpretation is seen as meaningless and carrying metaphysical baggage. The conversation also mentions the connection between mathematics and the physical world, and how some believe that mathematics is different from physics. Overall, the conversation raises questions about the nature of mathematics and its relationship with reality.
  • #36
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?
 
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  • #37
DaleSpam said:
Well, mathematics is neither erotic nor romantic, so it must be platonic.

I beg to differ :tongue:
 
  • #38
Willowz said:
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.
 
  • #39
ThomasT said:
Everything depends on the world in some sense. Doesn't it?
What do you mean?
 
  • #40
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. :wink:
 
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  • #41
Willowz said:
What do you mean?
What things don't, in any sense, depend on the world?
 
  • #42
Willowz said:
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.
 
  • #43
micromass said:
Post this in philosophy :smile: This has nothing to do with mathematics

I dunno. Mathematics and philosophy are very closely linked. I would call them sisters.
 
  • #44
ThomasT said:
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.
 
  • #45
ThomasT said:
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.
 
  • #46
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.
 
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  • #47
By coincidence, the Aug 2011 SciAm has an interesting artical about this -- Page 80
 
  • #48
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.
 
  • #49
Oldfart said:
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 ...
 
  • #50
I wish this stayed in general math. It was much more delightful with a conversation that made sense.
 
  • #51
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?
 
  • #52
Willowz said:
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.
 
  • #53
disregardthat said:
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.
 
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  • #54
Willowz said:
Actually, I think the his proof is a rational basis for believing in platonic forms.

What are you talking about? His proof of what?
 
  • #55
Erm, I mean his incompleteness theorem.
 
  • #56
Willowz said:
Erm, I mean his incompleteness theorem.

And how does that support platonic forms?
 
  • #57
By the fact that the mind perceives truth beyond formal systems.
 
  • #58
Willowz said:
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.
 
  • #59
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.
 
  • #60
Willowz said:
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.
 
  • #61
Willowz said:
Whereof one cannot speak thereof, one must be silent.

Wittgenstein never saw the Internet.
 
  • #62
Willowz said:
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.

If you have read Wittgenstein on mathematics at all, you will see he is a strong opponent of platonism. As far as my view is concerned and as I have expressed several times, I don't consider it a dichotomy at all. Platonism isn't false, it's meaningless in a very fundamental manner.
 
  • #63
Willowz said:
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.

Have you ever seen a perfect right-angled triangle?

A perfect sphere? Have you ever had an accurate measure of anything?

Mathematics is simply a tool we create to model reality. The concept of quantities and shapes are ingrained in us, but they are merely evolutionary products.

My 4 cents. (not directed to you btw, sorry)
 
  • #64
disregardthat said:
If you have read Wittgenstein on mathematics at all, you will see he is a strong opponent of platonism.
Why so?

As far as my view is concerned and as I have expressed several times, I don't consider it a dichotomy at all. Platonism isn't false, it's meaningless in a very fundamental manner.
I don't know how you can prove that. Maybe it's just another conventionalist interpretation that has falsely become an explanation that nobody really bothers with any more.
 
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  • #65
Willowz said:
Why so?

I don't know how you can prove that. Maybe it's just another conventionalist interpretation that has falsely become an explanation that nobody really bothers with any more.

In the book dictated from his lectures, "Lectures on the foundations of mathematics" he will often discuss (and dismiss) ideas related to platonist ideas much better than I can explain his views.
 
  • #66
I wonder if his views are shared today. Maybe it'd be better if I just look at Quine's, Putnam's views on platonism/ect.
 
  • #67
Willowz said:
I wonder if his views are shared today. Maybe it'd be better if I just look at Quine's, Putnam's views on platonism/ect.

I for one read Wittgenstein's views with enthusiasm. I think he has invaluable insight to the nature of mathematics, and I'd recommend the book I mentioned for anyone who are interested in these kinds of discussions. I don't see how you'd be better off ignoring certain kinds of perspectives.

I don't think his views are widely shared, but that is no argument against his positions.
 
  • #68
Thank your for your time in this thread, disregardthat. It's been a pleasure.

EDIT: I'm sorry if I could not answer any questions directed at me. This may have happened for two reasons, 1) I did not have an answer, 2) I did not want to unknowingly spread falsehoods. (Be it, knowingly or unknowingly I wouldn't want to spread them).
 
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  • #69
The problem is, how do we separate our minds from reality?
The world of abstract mathematics is one we access through our minds, so we can simplify this question by asking how did our minds develop, what exactly is thought?

The case for platonism seems to be weaker, compared to what it was a couple of centuries ago. Our minds are physical objects, our thoughts are biological processess which must be built from the laws of physics, or they would be distinct from the physical world. So if our thoughts are formed from within the laws of physics, how is it that our mathematics allows so much more? it can provide the laws of physics for an infinite number of possible universes. It is clearly an emergent phenomenon, the question is how does it emerge?
 
  • #70
Hm. I've been enjoying reading the book "Is God a Mathematician?" which essentially deals with this subject matter.
 

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