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.


Empirical science only part of knowledge



Can what we know be reduced to the descriptions and explanations of the empirical sciences?

Another way to think of this:  Is all human knowledge empirical?  Of course, everything we can know about derives from contact with external reality.  But knowledge only occurs in the brain’s emergent ordering of experience as ‘mind.’

Consider the argument:

All human knowledge is empirical, [because] it derives from contact with empirical reality.
All knowledge derives from contact with empirical reality;
what derives from contact with empirical reality is itself empirical;
All knowledge is empirical.

The first problem is terminological; the “empirical” describing reality is not the same “empirical” describing knowledge.  Reality, just as such, is not ’empirical’ (or anything else), it is just what is.

The proper phrase here is “empirical contact with reality” (where “empirical” means ‘sensory experience’ of reality). But once we rephrase this, the problem becomes clear – whatever could we mean by “all knowledge is ‘sensory experience'”?  Obviously what we want to say is that “all knowledge is derived from sensory experience,” but then the argument is a tautology:

All knowledge derives from sensory experience; whatever derives from sensory experience is (derived from) sensory experience;
therefore all knowledge is (derived from) sensory experience.

One way to see this problem is that the “empirical” of “all knowledge is empirical” is part of an assertion of empiricist epistemology, while the “empirical” of “contact-with-reality” is a common language description of experience with reality.  So what we really want to say is, “all knowledge derives (originates from, is developed out of) sensory experience.”  After this we can then make a case for empiricism; but that won’t get us a case that ‘all knowledge reduces to what can be described in the empirical sciences.’

Just as I can grant that mathematics originated from counting, I can grant that all knowledge originates from (in the sense that the questions developing knowledge begin with questions about)  sensory experience.  But as I’ve just made clear, this is not at all the same as saying that “all knowledge is [sensory experience].”  That’s just silly.  That knowledge originates in sensory experience (via questions about it) makes no claim at all on its later development.

Consider the origin of number, as derived from counting, e.g. “1+1=2.”  I know some philosophers do go back to that – but as a problem.  The problem is, how do we get from there to complex deductive mathematics?  That answer is not at all clear.   Algebra is the problem (let alone, for now, calculus, and higher order mathematics dealing with large numbers, or say, n-dimensional geometry), not simple addition. *

Talking about adding apples as a basis of higher math is like discussing Newton’s law of gravity in terms of an apple falling on his head.

Newton himself said that he deduced the law of gravity.  And while the physical issue involved actual masses, e.g., the moon and the earth; and while, had the movements of these spheres not tallied with the law, the hypothesis would have been falsified; the mathematical deduction itself involves abstract conceptual entities, not empirical or physical spheres:  “Every point mass attracts every single other point mass by a force pointing along the line intersecting both points.”

Now, physics seems to me to be a complex interweaving of mathematical deductions and empirical testing; that’s exactly one of the things that makes epistemology of science, and the philosophy of science in general, so fascinating.  Some want deduction, empirical testing, knowledge about abstract entities, knowledge about physical entities, and the sometimes difficult trail that somehow connects them all, to reduce to the same basic stuff.    I think they would need, at the very least, engage in the epistemological problems directly before making such a claim.

No science can survive productively on the basis of a primitive empiricism (which is really little different from naive Realism trashed by various philosophers during the Middle Ages).

Such labels are important as markers for positions in a field of competing claims.  They help us understand what claims relate to others, and what their issues are.  John Stuart Mill once claimed that all mathematics were reducible to counting (he was quite a sophisticated empiricist, but he didn’t understand mathematics).  And Hegel also claimed that all knowledge was essentially acquired one way, through dialectics (through which he once deduced that there could be only seven planets in the solar system).  Basically the claim that all knowledge reduces to the empirical is a variant (and reductive) Millsian position totalizing knowledge in a manner similar to Hegelian claims for the dialectic. I don’t think Mill would appreciate that (and I know Hegel wouldn’t).

It is sometimes claimed that our knowledge of reality is unified through the sciences, a seamless whole.  What’s odd about this is that it is rarely argued, simply asserted or implied, in variant ways.

Mathematics is primarily a deductive reasoning.  It is all in the mind.  Even geometry:  Two dimensional forms cannot (as far as we know) exist in the 3+1 dimensional reality in which we actually live.  Their reality is entirely of human mental construction.  Their measurements, and the various maths we use to describe these measurements, are also a matter of mental construction.  That these mental constructions often overlap physical reality is fortuitous – but not necessary: either as point of origin or as final culmination. They are simply useful tools for understanding the world in which we live; they are not that world.

Because mathematicians deal with formulas (derived or posited deductively), and physicists also work with theoretical postulates (not axioms in the strict mathematical sense, BTW), in mathematical formulas, derived from previous hypothesis and empirical research, does not at all mean that mathematicians and physicists are doing the same things – it means precisely that they’re not.

So basically, in the claim that all knowledge reduces to description and explanation in the empirical sciences, we have an incomplete philosophical argument for a primitive empiricism offered as a totalization of knowledge based on shaky premises and lacking historical accountability or epistemological insight.

This is not convincing, I’m sorry.


* BTW, here’s another problem equating number with counting, and it is simpler, and more definitive, than the problem of higher mathematics: zero.  Zero doesn’t count anything, it doesn’t even count ‘nothing.’  It can not be derived from sensory experience. [In fact it was derived from Hindu/Jainist metaphysics.]  Yet it’s function in mathematics is undeniable.  Problems such as this have to be accounted for, not “1+1=2” – in whatever universe we like to imagine.