Mathematical Philosophy

Mathematical Philosophy

In 1919 Bertrand Russell published a book titled Introduction to Mathematical Philosophy, written while he was in jail; but the title is somewhat misleading, because the book isn’t about doing philosophy mathematically but about the philosophy of mathematics. Still, Russell did do a fair amount of mathematical philosophy: the theory of descriptions, existence as a propositional function, mathematical constructions of the external world, etc. So did Frege, Wittgenstein, Carnap, Quine, Tarski, Montague, Kripke, Davidson, and others. It was a movement, a meta-philosophy. We might call it formal philosophy or symbolic logic philosophy or technical philosophy or scientific philosophy—as opposed to informal philosophy or phenomenological philosophy or commonsense philosophy or ordinary language philosophy or humanistic philosophy. Its model is mathematical science not vernacular literature—arithmetic not art. It aligns itself with the mathematical member of the two cultures. I think we can add that analytic philosophy is by nature mathematical in that it employs conceptual analysis, which is a breakdown of the whole into its parts, isolated and enumerated. Concepts are composites of added parts. Such analysis results in equations—e.g., knowledge = true justified belief; it seeks conceptual equations. It also employs rigorous deduction like mathematics, hence the emphasis on logic: syllogisms, axioms and theorems, formalized arguments with premises and conclusions, logical variables. Thus, we have today formal epistemology, formal politics, formal ethics, formal semantics. The subject is steeped in formal methods and conceits. Twentieth century philosophy was awash in mathematics, for good or ill. The dispute between analytical philosophy and so-called continental philosophy might well be seen as a disagreement about the role of mathematical methods in philosophy—what do you prefer, aesthetics or arithmetic?

But I think this story has history skewed: philosophy has been mathematical for much longer—indeed, it has never really been non-mathematical. I think philosophy is, and always at heart has been, a mathematical science, more or less successfully. That is what it aspires to be, though we must be liberal about the meaning of “mathematical” and “science”. It is about abstract and abstruse questions, aiming for precision and exactitude, obsessed with number, methodologically a priori, in love with logic, proudly technical and symbolic. It is a kind of geometry or geography of concepts or facts or realities. One of its most fundamental questions is numerical: is the world One or Many—and if many, how many? Hence, monism, dualism, and pluralism. Is the world finite or infinite, continuous or discrete, qualitative or quantitative? What kind of mathematics best captures its nature? Philosophy is all about distinctions, divisions, differences, relations. Identity is always a central concept. For example, Plato’s problem of the one universal and the many particulars: this is a quintessentially philosophical problem, and it is overtly numerical. We have the abstract form and the concrete particulars instantiating it: the form is a unity, but the particulars are a plurality. These are essentially different kinds of entity, yet they are intimately connected: many things are white, say. How can the One be Many, and how can Many be One? What is the relationship between them? It seems puzzling. More generally, is the whole world a unity or is it a plurality? How many types of entity are there? Is a person two things in one (mind plus body)? Are there values and facts or just facts? Is appearance the same as reality? How many kinds of good are there?  Are pleasures all of one kind? How many types of truth are there? Is necessity unitary? Are all words names? Polytheism or monism?  And so on. We want to know how many; we want to count our ontological commitments. We want to mathematically analyze the world. There are odd numbers and even numbers—are there universals and particulars, minds and bodies, values and facts? We look for unity but accept multiplicity. In arithmetic it all comes down to addition (subtraction is just the opposite of addition)—is that the way it is with the world generally? Metaphysics is a kind of mathematics of reality in its broadest outlines. We are applying mathematical conceptions, albeit elementary ones (but are any mathematical conceptions truly elementary?). This is not all we are doing, to be sure, but it is a large part of it; we are employing a diverse range of knowledge in conjunction with our mathematical knowledge.

So, the mathematical paradigm did not suddenly emerge in the twentieth century; it was alive and well for two thousand years or more. But it is so natural to us that we don’t notice it; we mathematize instinctively. Philosophy could hardly exist without it—or any science. It’s how we think scientifically, rigorously, systematically. The only question is how much we insist on it or extend it. I am not myself enamored of a sharp distinction between science and literature (I think literature is a type of science, mainly psychological). I think the human mind is essentially mathematical, here as elsewhere. It’s just a matter of admitting it, making it explicit as part of one’s meta-philosophy. We don’t have to mathematize philosophy because it is already mathematical. Even Sartre’s existentialism is mathematical, because it insists on the duality of the in-itself and the for-itself: Sartre is an ontological dualist—two things not one. The philosophy of language as it now exists is clearly a branch of mathematics—what with model theory and computational linguistics. Mathematical thinking is so entrenched in philosophy that we risk accusations of triviality just by drawing attention to it. It is an interesting question in the pre-history of ideas whether philosophy evolved in the human mind as an offshoot of an antecedent mathematical faculty. We are natural accountants, calculators, and bean counters; we brought these talents to bear on the most general questions of the cosmos. Mathematicians often have philosophical predilections. Mathematics and philosophy belong to the same intellectual natural kind.[1]

[1] Pretty much all science is mathematical to one degree or another; it would be strange if philosophical science were the exception. Inside each philosopher is a little mathematician struggling to strut his stuff.