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.

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Thinking Nominalism, Living Pragmatism

Nobody really wants the sloppy, childlike relativism that some self-proclaimed ‘post-Modernists’ espouse – even they don’t want it, since it would make their proclamations and espousals nonsensical. But relativism is not all one thing, it’s available in various types and to varying degrees. Dealing with any relativism in a useful manner requires considerable thought, caution, and care.

It is one of the most difficult concepts to get our minds around, that the world we know is only known through the concepts our minds generate (or that are communicated to us by others). Since these concepts are generally constructed via some linguistic or otherwise systematized communication processes, it follows that our ‘knowledge’ of the world is really largely a knowledge of what we say about the world. Even if I kick a rock (ala Sam Johnson), this experience will only make sense through my signifying response to it in a given context. Even expressions like ‘ow!’ or ‘ouch!’ can be seen to be some responsive effort to make sense of the experience; i.e., announcement that a painful event/sensation has occurred.

We’ve all had the experience of feeling some tiny sting on our arms; we slap at it reflexively. What is it? I pull my hand away, and there on the palm is a flattened body with broken wings, and I say, ‘oh, a bug.’ But if I pull my hand away and there is no flattened body on it, there still arises some thought in mind, such as ‘oh, probably a bug.’ And it is probably a bug, but that doesn’t matter – more important is recognizing that whatever it was, I have made sense of it by interpreting it and expressing this interpretation. And if it never happens again, and I never find any further evidence that it was a bug, yet a bug it will be in my memory.

I confess that I am something of a classical (i.e., traditional or Medieval) Nominalist – I’m sometimes unsure that we know anything ‘out there’ at all, except that it exists (but I’m also something of a Pragmatist, so this doesn’t really cause me any loss of sleep). But one doesn’t have to go so far as Nominalism to see that any claim we can make of the world beyond ourselves is thoroughly mediated by the system of the language by which we make the claim, and thoroughly dependent on context – not only the context of the particular world in which we speak, but the the context of the language we speak itself, and all the social reality that requires we admit.

Nominalism is a position taken regarding the problematic relationship between universals and particulars. This relationship can only be worked through in language.

It should be noted that there are certainly signifying practices other than language; but there can be no experience with reality that does not engage – and hence is not mediated by – signifying practices. (An infant reaching for the mother’s breast is signifying something, and reaching for what signifies to it.) Whether infants have ‘concepts’ seems irrelevant, or badly phrased. That an infant responds to the world reliant on persistence of objects hardly means that it has a concept of persistence of objects. This seems to beggar the very concept of a concept.

One of the questions inadvertently raised here is whether knowledge is to be equated with the hoary Positivist standard of Justified True Belief; because an infant certainly has no belief to be justified. – the truth of the breast is the immediate presence of the breast, and the justification of that is satisfaction of hunger. But the infant surely does not ‘believe’ this in any way  he or she can articulate, but merely reaches for the breast. Yet infants surely know, in a meaningful way, the breast – and the success or failure to get satisfaction from it – and intimately.

I’m not sure that the notion of knowledge being reducible to Justified True Belief, makes any sense outside of language, since analysis of a ‘justified true belief’ requires formulation into claims in a language system.

I noted parenthetically that my Nominalist position (concerning universals) did not cause me loss of sleep because I am also something of a Pragmatist. In pragmatism, knowledge need not be equitable to JTB. Reliability, as ground for responding to the world, often seems to have a stronger claim.

I earlier used the term “signifying” exactly to avoid getting into a technical distinctions between signifying systems. But I will introduce one technical term which may be of use here, which is that of Charles Sanders Peirce: interpretant. The interpretant to a sign is primarily composed of responses to the sign, which may be conceptualization or may be some form of action or speech-act, or some inner sensation. If we think in terms of signification and how various organisms respond to signs, we can avoid the dangers of ascribing language to an infant, and still have a means of addressing how they interact with their environment and each other in significant ways. And we can also avoid the trap of conceiving of our entire existence as somehow fundamentally linguistic. We are the language speaking animal, but we have other non-linguistic significant interactions with each other and the environment.

Pragmatism is a post-Idealist philosophy (Peirce was taught to recite Kant’s First Critique – in German! – at an early age; Dewey was an avowed Hegelian until WWI). Idealism makes a claim, actually similar to that of Logical Positivism, that knowledge is primarily or wholly the result of theory construction, and thus must be articulated linguistically. * Pragmatism begins with the recognition that this cannot be the case.

So the question may come down to whether what we know needs be communicated in language, or whether some other form of signification can be rich enough to inform our responses to the world.

But that does not mean we can be free of signification all together. The sting on the arm is a sign; what I say of it is an attempt to understand its significance, as response to it. If (assuming the scenario that I cannot see or find the bug or bug-parts) I come down with symptoms (signs) of malaria, that will enrich the signification of my response, and will also point to (sign) the species of bug that stung me. None of this need be predicated on the understanding that there is an inherent ‘bugness’ (some universal bug-hood) in the bug, the theory of which I must be familiar with before I form a proposition concerning it. And that is what I see as the real issue here.

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* This falls into the Nominalist trap: if all knowledge is theoretical, and all theories concern universals, and all existent entities are individuals, then the most we can say we know is our own theories, since individuals are not universals, but universals need to be constructed to account for them.. Unless, that is, we allow that knowledge is not all one thing and that there is not only one way of knowing. I’m glad that my doctor has a theory of malaria that can be relied on should I come down with it, so I can get properly treated. But I know I was stung, and what that felt like, without any theory to account for it. The interpretation of it is, however inevitable, as making sense of the matter, and certainly necessary if I become sick and need to articulate to a doctor what I think happened.