Mathematical Platonism: A Comedy

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.

Reality has no name

We find it so easy to use words as the ‘names’ of things that we don’t recognize when these ‘names’ over-extend their ‘nameness’ or naming function into generalities that cannot possibly be considered as actual entities.

It seems to make sense, and it is even commonplace, to refer to that tall woody thing that dumps leaves on my lawn every autumn a ‘tree.’ further extending that to other entities like it, collectively, as ‘trees’ is simply grammatical pluralization, again making sense according to the rules of both classical logic and modern biology. But – what is a thing called ‘forest’? where is such an entity? Is it just more ‘trees’? A spot on a map? Something that isn’t a desert? What’s a desert?

Most people are not aware that the Antarctic is considered a desert by scientists. ( http://www.answers.com/Q/Is_Antarctica_a_desert ) I think most of us believe that ‘desert’ signifies a geographical space that is hot and sandy. And because most of us use the word in this way and refer to images, and experiences, and narratives including these qualities, we would be right. The Antarctic is something else. But only in so far as we imagine it, and experience it (if we are fortunate enough). Otherwise – well, scientists have agreed to call it a desert. Fine; who would argue with science? But it’s no ‘desert’ as most of us who use the term would imagine it to be.

The fact is, language is not about naming objects; it is not a mirror of nature, it does not validate ‘justified true beliefs’.

I see no difficulty in saying that forests don’t exist in nature.

We walk among these tall, rough-shelled things with irregular poles sticking out of them, all covered with thin green things. We’ve come to call these things ‘trees,’ with their ‘branches’ growing ‘leaves.’ To the extent we can differentiate a large number of these ‘trees’ from areas without much of them, or any of them, we have come to call the area with the ‘trees’ a ‘forest,’ so we have a location in space to which to refer for the sake of directions.

There are a number of ways we’ve developed to address these phenomena, including careful dissection and analyses of activities and events that permit the objects under study continued existence. But do ‘forests’ exist? Do even ‘trees’ exist except within the discourse of humans needing to refer to ‘those things there’? I confess I doubt this.

What we call tomatoes are fruit, and watermelons are vegetables – until we get to the table; then it is neither wrong nor nonsense to toss the tomato into the vegetable salad, and afterwards, enjoy the fruity flavor of the watermelon.

Individuals cannot define words arbitrarily. But the meaning of words is found in their usage, not in the existence of objects to which they are used to refer. That gives us some collective power over how we can or should use words. So distinctions between words and their references are very important.

Do forests exist ‘in reality?’ I’m not sure I recognize that as a legitimate question.