Mathematical Platonism holds that mathematical forms – equations, geometric forms, measurable relationships – are somehow embedded into the fabric of the universe, and are ‘discovered’ rather than invented by human minds.
From my perspective, humans respond to challenges of experience. However, within a given condition of experience, the range of possible responses is limited. In differing cultures, where similar conditions of experience apply, the resulting responses can also be expected to be similar. The precise responses and their precise consequences generate new conditions to be responded to – but again only within a range. So while the developments we find in differing cultures can oft end up being very different, they can also end up being very similar, and the trajectories of these developments can be traced backward, revealing their histories. These histories produce the truths we find in these cultures, and the facts that have been agreed upon within them. As these facts and the truths concerning them prove reliable, they are sustained until they don’t, at which point each culture will generate new responses that prove more reliable.
Since, again, the range of these responses within any given set of conditions is actually limited by the history of their development, we can expect differing cultures with similar sets of conditions to recognize a similar set of facts and truths in each other when they at last make contact. That’s when history really gets interesting, as the cultures attempt to come into concordance, or instead come into conflict – but, interestingly, in either case, partly what follows is that the two cultures begin borrowing from each other facts, truths, and possible responses to given challenges. ‘Universal’ truths, are simply those that all cultures have found equally reliable over time.
This is true about mathematical forms as well, the most resilient truths we develop in response to our experiences. I don’t mean that maths are reducible to the empirical; our experiences include reading, social interatction, professional demands, etc., many of which will require continued development of previous inventions. However, there’s no doubt that a great deal of practical mathematics have proven considerably reliable over the years. Whereas, on the contrary, I find useless Platonic assertions that two-dimensional triangles or the formula ‘A = Π * r * r’ simply float around in space, waiting to be discovered.
So, in considering this issue, I came up with a little dialogue, concerning two friends trying to find – that is, discover – the mathematical rules for chess (since the Platonic position is that these rules, as they involve measurable trajectories, effectively comprise a mathematical form, and hence were discovered rather than invented).
Bob: Tom, I need some help here; I’m trying to find something, but it will require two participants.
Tom: Sure, what are we looking for.
B.: Well, it’s a kind of game. It has pieces named after court positions in a medieval castle.
T.: How do you know this?
B.: I reasoned it through, using the dialectic process as demonstrated in Plato’s dialogues. I asked myself, what is the good to be found in playing a game? And it occurred to me, that the good was best realized in the Middle Ages. Therefore, the game would need to be a miniaturization of Medieval courts and the contests held in them.
T.: Okay, fine, then let’s start with research into the history of the Middle Ages –
B.: No, no, history has nothing to do with this. That would mean that humans brought forth such a game through trial and error. We’re looking for the game as it existed prior to any human involvement.
T.: Well, why would there be anything like a game unless humans were involved in it?
B.: Because its a form; as a form, it is pure and inviolate by human interest.
T.: Then what’s the point in finding this game? Aren’t we interested in playing it?
B.: No, I want to find the form! Playing the game is irrelevant.
T.: I don’t see it, but where do you want to start.
B.: In the Middle Ages, they thought the world was flat; we’ll start with a flat surface.
T.: Fine, how about this skillet.
B.: But it must be such that pieces can move across it in an orderly fashion.
T.: All right, let’s try a highway; but not the 490 at rush hour….
B. But these orderly moves must follow a perpendicular or diagonal pattern; or they can jump part way forward and then to the side.
T.: You’re just making this up as you go along.
B.: No! The eternally true game must have pieces moving in a perpendicular, a diagonal, or a jump forward and laterally.
T.: Why not a circle?
B.: Circles are dangerous; they almost look like vaginas. We’re looking for the morally perfect game to play.
T.: Then maybe it’s some sort of building with an elevator that goes both up and sideways.
B.: No, it’s flat, I tell you… aha! a board is flat!
T.: So is a pancake.
B.: But a rectangular board allows perpendicular moves, straight linear moves, diagonal moves, and even jumping moves –
T.: It also allows circular moves.
B.: Shut your dirty mouth! At least now we know what we’re looking for. Come on, help me find it. (begins rummaging through a trash can.) Here it is, I’ve discovered it!
T.: What, that old box marked “chess?”
B.: It’s inside. It’s always inside, if you look for it.
T.: My kid brother threw that out yesterday. He invented a new game called ‘shmess’ which he says is far more interesting. Pieces can move in circles in that one!.
B,: (Pause.) I don’t want to play this game anymore. Can you help me discover the Higgs Boson?
T.: Is that anywhere near the bathroom? I gotta go….
Bob wants a “Truth” and Tom wants to play a game. Why is there any game unless humans wish to play it?
A mathematical form comes into use in one culture, and then years later again in a completely other culture; assuming the form true, did it become true twice through invention? Yes. This is one of the unfortunate truths about truth: it can be invented multiple times. That is precisely what history tells us.
So, Bob wants to validate certain ideas from history, while rejecting the history of those ideas. You can’t have it both ways. Either there is a history of ideas, in which humans participated to the extent of invention, or history is irrelevant, and you lose even “discovery.” The Higgs Boson, on the other hand, gets ‘discovered,’ because there is an hypothesis based on theory which is itself based on previous observations and validated theory, experimentation, observation, etc. In other words, a history of adapting thought to experience. (No one doubts that there is a certain particle that seems to function in a certain way. But there is no Higgs Boson without a history of research in our effort to conceptualize a universe in which such is possible, and to bump into it, so to speak, using our invented instrumentation, and to name it, all to our own purposes.)
Plato was wrong, largely because he had no sense of history. Beyond the poetry of his dialogues (which has undoubted force), what was most interesting in his philosophy had to be corrected and systematized by Aristotle, who understood history; the practical value of education; the differences between cultures; and the weight of differing opinions. Perhaps we should call philosophy “Footnotes to Aristotle.”
But I will leave it to the readers here whether they are willing to grapple with a history of human invention in response to the challenges of experiences, however difficult that may seem; or whether they prefer chasing immaterial objects for which we can find no evidence beyond the ideas we ourselves produce.