Logic and Mathematical Method

In the comments to my post Follow-up on “A Probability Puzzle” Jonathan of jd2718 posed another probability question that I answered incorrectly and somewhat pretentiously (note to self: beware the phrase “deep epistemological and logical question” unless you’re really sure you’re on the right track).  In my defense, I quickly saw the force behind the correct answer and began hedging my defense of the incorrect answer.  In my further defense, I was wrong in an interesting way that shed light on the relationship between the puzzle Jonathan posed and my own.

One thing that Jonathan impressed upon me was to “keep it simple stupid”, though he was polite enough not to put it that way.  List the outcomes, he kept insisting, and count.  The emphasis on basic methods of counting and charting relates nicely to some of Wittgenstein’s criticisms of logicism, which I have been reading about lately.  Especially in his later writings, but also to some extent in his early work, Wittgenstein attacked the logicist project as overly reductive.  He emphasized the dependence of mathematical understanding on basic rule-governed practices like counting, which rules are conceptually prior to and independent of logic (for detail see this reconstruction of Wittgenstein’s argument for this priority claim).

Wittgenstein also placed emphasis on the inventiveness involved in mathematical reasoning.  Neither rules nor invention, it was maintained, could be reduced to logic; hence, the Frege-Russell project could only give an at-best-incomplete account of the epistemology of mathematics.  (I should menition that the logicist line of defense would likely be a charge of psychologism, but it is not my purpose in this post to pursue that point).    Juliet Floyd, in her essay “Wittgenstein’s Philosophy of Logic and Mathematics”, summarizes the point about invention thusly:

Wittgenstein’s emphasis on the image of the mathematician as inventor or fashioner of models, pictures, and concepts was, in the main, directed at the philosophical talk of those, like Hardy and Russell, who insisted on speaking of mathematical reality in a freestanding way, picturing the logician or mathematician as a zoologist embarked on an expedition to new, hitherto unseen land, analogous to an empirical scientist…  For Wittgenstein, the mathematician is an inventor, not in the sense of making up truth willy-nilly as he or she goes along, as the pure conventionalist would suppose, but in the sense of engaging in the activities of fashioning proofs, diagrams, notations, routines, or algorithms that allow us to see and accept (understand, apply) results as answering to what does and does not make sense to us.

I think that this point about invention and its relationship to logicism, or indeed to any reductive/foundationalist mathematical epistemology, can be brought out by reflection on two proofs of the Pythagorean theorem.  First, consider this pictorially motivated proof by the (utterly brilliant) 12th century Indian mathematician Bhaskara:

The proof has the virtue of immediate surveyability and visual intuitiveness.  This is in contrast with Euclid’s proof, which while proceeding in simple and intuitive steps, takes a bit of effort to take in as a whole.  I’m struck by the inventiveness of Bhaskara’s proof.  It could not have been, in any straightforward way, been uncovered by just running through sequents with a computer program.  Given the axioms, Euclid’s proof, it seems to me, could have been discovered by a computer.  This is not to say that Euclid lacked inventiveness, of course, since stating his axioms and realizing that they could form a foundation for geometric study was an incredible insight.

I’ll let Floyd, on behalf of Wittgenstein, have the last word for now, but needless to say I think that there is much, much more to say about the relationship between “invention” and foundations in mathematics.  (When Floyd employs the term “metamathematics” you may think of set theory as well as Frege and Russell’s approach to foundations in higher-order logics):

The trouble with metamathematics, for Wittgenstein, is that it tends to mislead philosophers into thinking that the metamathematical language gives us a single way of surveying the core, or interpreting the meaning, of apparently fundamental mathematical and logical notions.   But ascent to the metalanguage is just another perspective on practices that gain their character within language from their working applications in human life.  Such ascent may change our perspective on our own language, but it grows from our current practices, and it is parasitic upon them: it cannot make them more epistemologicall certain.

An analogy: Bhaskara’s proof is to Euclid’s, perhaps, as a familiar arithmetic calculation is to a set-theoretical or logicist derivation.  It must be granted that if “surveying the core” or “interpreting the meaning” is our interest then the more familiar, the more picturable, and the more simply and totally grasped are to be prefered. . .  but that is only the last word for now.