[sciborg2]

Here's an intro into this idea from Sir Roger Penrose, who posits three worlds that I understand to originally belong to Popper.

While Plato's World o' Forms feels like it belongs more to D&D than conceptions of reality, the idea of Mathematical Platonism did find a sympathetic naturalist ear belonging to Massimo Pigliucci.

There is a difference between general Platonism and the mathematical flavor. For Plato, each apple, say, is but an imperfect example of the absolute (and perfect) Idea of an apple. But as Aristotle quickly realized, Plato has it exactly backwards: we arrive at the general idea of ‘apple’ by mentally abstracting a set of characteristics we think common to all actual apples. It is we who conjure the ‘perfect’ idea from the world, not the world copying the concept.

But now contrast the idea of an apple with the idea of a circle. Here Aristotle’s approach becomes more problematic, as we don’t find any true circles in nature. No natural object has the precise geometric characteristics of a circle, and in a very strong sense we can also say that the circles we draw are but imperfect representations of the perfect idea of a circle. Ah – but whence does such a perfect idea come from?

Consider another way to put the problem. One major difference between science and technology is that science discovers things, while technology is about human inventions . We discover the law of gravity; but we invent airplanes to allow heavier-than-air flight despite the law of gravity. But where do mathematical objects, like circles and numbers, or mathematical theorems like the Pythagorean one, or Fermat’s Last one, come from? Are they inventions of the human mind, or are they discoveries?

I hope you’re beginning to feel as queasy as I did when I started to take the matter seriously, because contrary to Aristotle’s approach to knowledge, my gut feeling was that mathematicians discover things, not invent them. This was a huge paradigm shift from my days as a scientist.

[Kellais]

This is much more my speed 

Problem with the argument is, that the basics of Math as we do it atm are Axioms...and Axioms are arbitrarily chosen (well ok that very much oversimplyfies it, but i needed a punch-line ) ...and on those you build up a coherent mathematics. As you could have chosen those starting Axioms another way, i'd say it is more of an invention than a discovery ... on the otherhand, if you postulate that every possible Axiom-set is kind of "out there" and we just uncovered one of them...well i guess we are discovering it.

All in all i guess i am more of an Aristotle-ian Mathematician ... because even with the example of the circle i think his observation is correct. Why? Because the circle is a very special polygon...namely the infinity-gon (a shape with infinity edges...a limes of an n-edge where n goes to infinity)...which is only an abstraction in our mathematical minds.
In any case, why should the circle be a perfect polygon, if you go with Plato. I mean...just imagine the pain of having to draw one

[sciborg2]

But did the concept of a perfect circle* come about due to this attempt at taking a polygon's number of sides to infinity? It seems to me the concepts of circle and polygon were thought up separately and then connected?

I'm not sure I understand why the choice of Axioms affects the consideration of math being invented or discovered?

In any case, Massimo lays out a more formal argument for the existence of Platonic Math here.

It's worth a read for curiosity's sake.

*Love the band by the way

[Kellais]

To be honest, i do not know. I guess the polygons came before the circle in everyday life but i don't know if that is true. I am sure that people were able to draw circles before they thought about it being a polygon with infinit edges, if that is what you are hinting at.

Well if you postulate that we discover math, shouldn't we be able to come to all conclusions without having to have a set set of "rules" before starting the process? I mean it is all already there, isn't it?!
If you do start by "inventing" a set first and then go from there...i think it is much more looking like you constructed it all then uncovered it.
On the other hand...this is a lot like the chicken egg thing...so what do i know. I just try to engage you in this thread I'm not saying i'm right.

So, sci, what's your opinion then? Care to elaborate?

[sciborg2]

I think both sides have interesting arguments. Maybe on some level Tegmark's Ultimate Ensemble comes down to the "Universe as Quantum Computer" jazz.

I haven't made up my mind yet, as the more serious philosophy dealing with this question is out of my intellectual grasp for now.

[sciborg2]

More on Mathematical Platonism from the Internet Encyclopedia of Philosophy

For the interested reader. Have not gone through it myself. May post some tidbits that interested me when I do.

[Wic]

What may be relevant to the discovery/invention question is what Wigner called the Unreasonable Effectiveness of Mathematics in the Natural Sciences.  There's something absolutely frightening about the fact that we have occasionally developed an entire mathematical model only to later find that it applies to some model of reality (quantum mechanics is what's in my head at the moment).

There are always limitations in the real world that add constraints to how much it acts like these models, but still, we can discover things within math that lead us to discover things in the universe.  Einstein thought the existence of black holes in his models was a flaw that would later be corrected - and then we found them.  It's fairly preposterous that these axiomatic abstractions should reveal anything about the world.

Think about an electron.  They are point particles - literally (we say with decent confidence) zero volume.  And yet in that zero volume, like the intersection of lines, exists things we call 'charge', 'mass', 'spin', 'quantum number', and so on.  It's something that is beyond all intuition, purely mathematical from our perspective, and yet it exists, unquestionably.

[sciborg2]

Einstein thought the existence of black holes in his models was a flaw that would later be corrected - and then we found them.  It's fairly preposterous that these axiomatic abstractions should reveal anything about the world.

Yeah, I asked my old discrete math professor if he thought math was created or discovered. He mentioned the paper you linked - thanks for that as I'd never actually read it. 

The effectiveness of digressions in the field becoming applicable is uncanny IMO. Beyond that I don't know what to make of it.

[sciborg2]

Closer To Truth asks Stephen Wolfram: Is Mathematics Invented or Discovered?

Wolfram concludes the mathematics we have today is a historical artifact, and so it seems to me that this would be anti-Platonic.

It's interesting to think of logic as a possible formal system. It's hard to understand exactly how this works.

[sciborg2]

The mathematical world -> Some philosophers think maths exists in a mysterious other realm. They’re wrong. Look around: you can see it.

Still, despite its clean lines and long history, Platonism cannot be right either. Since the time of Plato himself, nominalists have been urging very convincing objections. Here’s one: if abstracta float somewhere outside our own universe of space and time, it’s hard to imagine how can we see them or have any other perceptual contact with them. So how do we know they’re there? Some contemporary Platonists claim that we infer them, much as we infer the existence of atoms to explain the results of chemistry experiments. But that seems not to be how we know about numbers. Five-year-olds learning to count don’t perform sophisticated inferences about abstractions; their contact with the numerical aspect of reality is somehow more perceptual and direct. Even animals can count, up to a point.

Aristotelian realism stands in a difficult relationship with naturalism, the project of showing that all of the world and human knowledge can be explained in terms of physics, biology and neuroscience. If mathematical properties are realised in the physical world and capable of being perceived, then mathematics can seem no more inexplicable than colour perception, which surely can be explained in naturalist terms. On the other hand, Aristotelians agree with Platonists that the mathematical grasp of necessities is mysterious. What is necessary is true in all possible worlds, but how can perception see into other possible worlds? The scholastics, the Aristotelian Catholic philosophers of the Middle Ages, were so impressed with the mind’s grasp of necessary truths as to conclude that the intellect was immaterial and immortal. If today’s naturalists do not wish to agree with that, there is a challenge for them. ‘Don’t tell me, show me’: build an artificial intelligence system that imitates genuine mathematical insight. There seem to be no promising plans on the drawing board.

[sciborg2]

Put up some stuff from Massimo's critique of reductionism and mechanism in the relevant thread, but wanted to make note of his plug for Platonism  :

One last parting shot, about a topic that the astute reader may have noticed I have bypassed so far: if every thing is gone and we only have mathematical structures and relations, what is the ontological status of mathematical objects themselves? Here are the only relevant quotes from Ladyman and Ross that I could find:

 "   OSR as we develop it is in principle friendly to a naturalized version of Platonism. ... One distinct, and very interesting, possibility is that as we become truly used to thinking of the stuff of the physical universe as being patterns rather than little things, the traditional gulf between Platonistic realism about mathematics and naturalistic realism about physics will shrink or even vanish. ... [Bertrand Russell] was first and foremost a Platonist. But as we pointed out there are versions of Platonism that are compatible with naturalism; and Russell’s Platonism was motivated by facts about mathematics and its relationship to science, so was PNC [Principle of Naturalistic Closure] -compatible."

Wild stuff, no? Now I don’t feel too badly about having written in sympathetic terms about mathematical Platonism...