Is Mathematics Really Platonic or a Human Invention?

[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.

 

[SteveL27]

Whereof one cannot speak thereof, one must be silent.

Wittgenstein never saw the Internet.

 

[disregardthat]

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.

 

[Hells]

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)

 

[Willowz]

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.

 

[disregardthat]

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.

 

[Willowz]

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.

 

[disregardthat]

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.

 

[Willowz]

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).

 

[Functor97]

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?

 

[shelovesmath]

Hm. I've been enjoying reading the book "Is God a Mathematician?" which essentially deals with this subject matter.

 

[qsa]

nature is made of a very simple statistical math, no PDE or non-commutitive geometry. It may be very hard to believe, but only one possible design of a dynamic universe is possible with such simple math, and it is our reality. No wonder why we are so astonished.

see my profile for a glimse of such a fact.

 

[PlatosHeaven]

It seems like a lot of people in this thread are conflating the metaphysical theory of mathematical platonism and its epistemic implications.

The metaphysical version is simple to define. It says that mathematical objects exist, they are abstract (i.e. causally inefficacious), and they exist independently of any intelligent being's belief or action.

This is quite independent of whether we can have knowledge of the objects that are implied by mathematical platonism.

Most people accept the claim that at least some mathematical theorems are true. If you believe in a correspondence theory of truth, or at least that the truth of sentences in the language of mathematics depends on the success or failure of reference, then it's hard not to accept that these theorems are ontologically committed to the existence of the abstract objects they refer to.

But it's also widely held that these objects exist, but the independence clause doesn't hold. The sentences of mathematical language refer to something, but that something depends on the existence of minds. This doesn't sit well with me. It means that we "construct" the truth of mathematical sentences. This goes against our common sense notion of truth, where the truth about objects out there in the world depends on those objects, not us. If the truth of mathematical statements isn't independent, then it is ultimately arbitrary, since it depends on the axioms we choose (and since axioms are true by definition it's an empty construction). Furthermore, either it's an entirely different kind of truth than the truth about physical objects, or the truth about physical objects also depends on the existence of minds.

If you're willing to accept this relativist position with respect to truth, then I suppose it's a consistent position. However, I believe that the enormous empirical success of math (and thus physics) would be a miracle if they were simply constructions without any external notion of truth.

 

[apeiron]

If you're willing to accept this relativist position with respect to truth, then I suppose it's a consistent position. However, I believe that the enormous empirical success of math (and thus physics) would be a miracle if they were simply constructions without any external notion of truth.

Why is it a miracle that if we are free to model reality, that our models might not approach some consistent state? It is what we should expect of modelling.

Equally, why would it be a miracle that a reality also approaches some self-consistent state? To persist long enough to have observers, a reality would have to be well-behaved. It would have to fall into the patterns we call lawful.

So we have two processes going on - the epistemic (our invention/discovery of mathematical truths), and the ontic (reality's development/discovery of its own persisting equilibrium balance).

Conflation here is to conflate the two - epistemic discovery and ontological self-invention. Although they are certainly parallel stories. There is a modelling relation that connects them.

 

[Willowz]

So, what you are saying is that whatever the mathematical statement, we can make it refer to something? Isn't this a quasi-distinction between applied math and pure maths.

 

[apeiron]

So, what you are saying is that whatever the mathematical statement, we can make it refer to something? Isn't this a quasi-distinction between applied math and pure maths.

I agree that I'm am stressing useful maths that actually talks about the world and with syntax it is always possible to generate pure nonsense.

This is a standard point in linguistics - "Colorless green ideas sleep furiously."

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

 

[PlatosHeaven]

Why is it a miracle that if we are free to model reality, that our models might not approach some consistent state? It is what we should expect of modelling.

Equally, why would it be a miracle that a reality also approaches some self-consistent state? To persist long enough to have observers, a reality would have to be well-behaved. It would have to fall into the patterns we call lawful.

So we have two processes going on - the epistemic (our invention/discovery of mathematical truths), and the ontic (reality's development/discovery of its own persisting equilibrium balance).

Conflation here is to conflate the two - epistemic discovery and ontological self-invention. Although they are certainly parallel stories. There is a modelling relation that connects them.

Modelling is not the same thing as reference. Arguably, a mathematical model refers to mathematical objects--which are non-physical and causally inefficacious--and draws a comparison between them and physical objects. Of course, this is a simplification. The success of a model doesn't necessarily imply truth, but I think it does imply some sort of reference.

 

[apeiron]

Modelling is not the same thing as reference. Arguably, a mathematical model refers to mathematical objects--which are non-physical and causally inefficacious--and draws a comparison between them and physical objects. Of course, this is a simplification. The success of a model doesn't necessarily imply truth, but I think it does imply some sort of reference.

I'm not following you here.

A modelling relation would relate a model, a formal description of some system of causal entailments, to a world via a process of measurement, a feedback loop of predictions and tests. So the model would be referring to the world both in terms of its globally motivating concepts and in the localised measurements it suggests.

In this context, what is the difference between a mathematical object (ie: concept or quality) and a physical one?

A physical concept would be something like mass, energy, charge, spin, momentum. The general qualities that are variables in equations - the essential ideas in whose name crisp measurements can be made.

I suppose mathematical objects might be number, dimension, symmetry, limit? I'm not sure what you mean. But all these seem to be pretty physical notions too. Although less about substantial things and more about formal relationships. But still physically-inspired notions for all that.

 

[BWV]

Roger Penrose has offered the Mandelbrot set as proof of mathematical Platonism - the object, being infinite, must exist outside of ourselves and is no invention of the human mind - it has to be discovered, not merely thought up

 

[agentredlum]

The julia set was discovered long before computers.

http://library.thinkquest.org/26242/full/types/ch5.html

 

[disregardthat]

Roger Penrose has offered the Mandelbrot set as proof of mathematical Platonism - the object, being infinite, must exist outside of ourselves and is no invention of the human mind - it has to be discovered, not merely thought up

These kinds of arguments always make it hard to take a person serious. It seems as if, once convinced of a position, one will take any fact as evidence for it. Platonists insist on mathematics referring to something independent and ontological, but are never in the position to point at it, explain it (it just exists).

If one take a closer look at how the word "exists" in mathematics actually is used, one will notice that it is not very different from any other rule, like "is equal to", "implies" and so on.

 

[Willowz]

*In regards to the previous post*

It's true that the burden of proof is on the mathematicians side, as to why he or she may believe in Platonism. But, maybe it takes being a mathematician to believe in Platonism. Just to be fair.

 

[Kherubin]

I, as others before me have stated, have a paltry understanding of the philosophical underpinnings of this thread, however, if I could offer some simple definitions, which may or may not be useful by others' reckoning, but which I have seen stated numerous times in the subject literature:

Formalism:- School of thought suggesting mathematics is 'invented'
Platonism:- School of thought suggesting mathematics is 'discovered'

The most interesting views that I have come across regarding the subject are those of Stephen Wolfram, as expounded in a video I have posted previously:

http://www.closertotruth.com/video-profile/Is-Mathematics-Invented-or-Discovered-Stephen-Wolfram-/1384

His suggestions are all the more riveting considering the seemingly 'objective' viewpoint he has taken on the subject of mathematics, and the years he has spent studying this topic, if you will, from the 'outside'.

Now, regarding whether or not I understand his conclusions is another matter.

His initial statements, that our mathematics is an 'artifact', a product of human culture, and hence (as others have posited) would be, in some ways, markedly different from extraterrestrial 'mathematics' is, at first glance, Formalist.

However, all this argument apparently does is to push the debate back a 'step'. The 'Universe of Possible Mathematicses' which he introduces, could be thought of as Platonic in nature. Or, at this stage, with a sufficiently general definition of mathematics as a formal system composed of an arbitrary string of symbols, is the question of Formalism vs. Platonism defunct?

I'm interested to hear others' views on this,
Thank you for your time,
Kherubin

 

[Paulibus]

The trouble with many (I suppose not all) mathematicians and quite a slew of philosophy folk
seems to me to be that they're not quite up to speed about what we've learned about our origins over the last half-century or so. I recommend a dose of Bill Bryson's A Short History of Nearly Everything, Stephen Oppeheimer's Out of Africa's Eden and Robert Sapolsky's The Trouble with Testosterone to alleviate ignorance about our recently acquired understanding of what we are.

It seemsnow clear that we're the most articulate species among (some sadly extinct) African Apes. And chances are one of the most important skills evolution engendered in us is to communicate. We're the all-chattering,-talking, -writing, -calculating, -inventing, -arguing, -twittering kind of Simian (for those who don't appreciate being called monkeys). We've invented several thousand languages and dialects to help us describe the physical world, how it works and the sometimes dangerous contingent circumstances we find ourselves in. Communicating effectively has turned out to be a great way to keep our numbers growing relative to the competition. Good, because this is how Evolution seems to strive and thrive.

Mathematics with its many dialects is one such language invented by we Simians. Like French, Japanese, music and poetry, it didn't exist before we found out how effective Maths is in decribing and helping to manipulate our physical world and its inhabitants. Mathematics let's us describe stuff quantitatively, which is why numbers were invented,probably in the Middle East or Africa, not so long ago, to help keep account of resources. I don't believe Maths was discovered in academia, deserts, woods, or the heavens. I also think that while it's all very well to talk learnedly of the Mandelbrot set, groups and prime numbers as candidates for discovered mathematical objects, that this is learning to run before you can walk. I suggest it's better to first settle whether pedestrian objects, say like the number seven, was discovered or invented. In the meantime, until this thread has twittered to its conclusion, I'll go along with Hells who put it this way:

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.

 

[agentredlum]

Thank goodness for the black obelisk...otherwise we'ld still be chewing grass and slinging poop...

 

[Willowz]

Wittgenstein never saw the Internet.

lol, I just looked over the thread and got it. Good one!

 

[Willowz]

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

I just read it today. Nice read, but it was mostly history and some feeling about math in general from physicists. But, a nice read.

 

[MarcoD]

I, as others before me have stated, have a paltry understanding of the philosophical underpinnings of this thread, however, if I could offer some simple definitions, which may or may not be useful by others' reckoning, but which I have seen stated numerous times in the subject literature:

Formalism:- School of thought suggesting mathematics is 'invented'
Platonism:- School of thought suggesting mathematics is 'discovered'

The most interesting views that I have come across regarding the subject are those of Stephen Wolfram, as expounded in a video I have posted previously:

http://www.closertotruth.com/video-profile/Is-Mathematics-Invented-or-Discovered-Stephen-Wolfram-/1384

His suggestions are all the more riveting considering the seemingly 'objective' viewpoint he has taken on the subject of mathematics, and the years he has spent studying this topic, if you will, from the 'outside'.

Now, regarding whether or not I understand his conclusions is another matter.

His initial statements, that our mathematics is an 'artifact', a product of human culture, and hence (as others have posited) would be, in some ways, markedly different from extraterrestrial 'mathematics' is, at first glance, Formalist.

However, all this argument apparently does is to push the debate back a 'step'. The 'Universe of Possible Mathematicses' which he introduces, could be thought of as Platonic in nature. Or, at this stage, with a sufficiently general definition of mathematics as a formal system composed of an arbitrary string of symbols, is the question of Formalism vs. Platonism defunct?

As far as I remember, and I don't remember a lot about Platonism, the historical definition of Platonism is very different from what we currently understand it to mean. He lived more than two thousand years ago, and at that point in time, I think Platonism was a philosophy to 'unify' what is currently understood to be mathematics, logics, physics and even biology and sociology. For example, if I remember correctly, and I am sure I don't, he postulated a [real] mathematical world consisting of ideals where in the universe around us entities are imperfect approximations of that. I believe original Platonism is only interesting for philosophers interested in history.

I like your definitions of 'platonism' and 'formalism,' but you changed the subject somewhat.

Personally, I mostly unify both approaches: I think humans discover mathematical theorems within a formal system which is fixed.

What I find interesting is that if we recognize that math at the moment is a continuous search in a formal realm by imperfect humans, then the question becomes what we are missing, and how we can enhance humans to, for instance, discover new theorems through automation.

EDIT: There is something which makes an answer to whether mathematics is Platonic or formalist fundamentally applicable to law. If mathematics is discovered, then it is an intellectual achievement which should be patentable. Otherwise, mathematical theorems cannot be intellectual property of persons.

 

[BWV]

this is sort of an angels on the head of a pin argument that gets bogged down in semantics. At a very minimum, mathematics is discovered the same way that any number of things are discovered where a set of rules are created whose execution creates consequences that may not be readily apparent to human intuition. Think about taking bits of colored glass and making a kaleidoscope - you cannot really predict all the shapes that will ensue once you do this. Is this discovered or invented? Its a complicated question because a great deal of art & music gets created by similar procedures - how much did Pollock discover his painting style? how much of musical style was discovered by learning from experiment the sounds of particular combinations of pitches?