Metamathematics and the Philosophical Tradition

William Boos (1943-2014) was a mathematical logician whose work took him deep into the discipline of philosophy, as the philosophers do it, so to speak.  And, accordingly, this book, containing ten long essays on themes that one doesn’t really encounter in mathematical logic, is both idiosyncratic and singular.  It is certainly not for everyone in the sense that a mathematician with no background in philosophy would be out of his element and indeed out of his depth unless he’d be willing to read up on a number of subjects not generally part of his bailiwick.  Conversely, a philosopher should have a lot more comfort with mathematical logic than is the norm if he is to read this book with any profit. For example, Boos deals with such themes as Leibniz’ metaphysics, Kant’s ethics, and free will and determinism, which are by no means the stuff one finds in e.g. Shoenfield, Kleene, Moschovakis, or Enderton (pick your favorite book on introductory mathematical logic).

To illustrate the foregoing point, or suggestion, here are a couple of samples of what one encounters between the covers of the book under review.  On p. 33, in a section titled, “’Coherent idealism in the hieratic ideals of Leibniz and Berkeley,” we read the assertion that “Leibniz also accepted the ‘internal’ fatalism the newly devised dynamical determinism seemed to impose ‘within’ a given world.  But he also struggled to relativize ‘absolute’ notions of fate of the sort Spinoza had accepted, and broaden the regulative force of his metaphysical and metalogical constructions to include ‘ethical’ decisions in humanly (and humanely) acceptable terms.” Clearly, this is the sort of thing one in most likely to find in philosophy departments, not in mathematics departments.  On the other hand, on p. 263, in a chapter titled, “The Second-order Idealism of David Hume,” in the section suggestively labelled “Whatever is clearly conceived …”, Boos hits topics that are somewhat closer to a(n ideal?) mathematician’s heart: “Conceivability, of course, is not a standard metalogical notion. That one should not expect to find a single, uniform theoretical φ(v) for ‘v is conceivable’ already follows from Goedel’s diagonal lemma, which generates from φ a sentence ↔¬φ(), which provably asserts [its own] inconceivability.”  Yes, the more things change the more they stay the same.

So there it is, then, Boos’ Metamathematics and the Philosophical Tradition, certainly very aptly titled, is not everyone’s cup of tea, but those who like this flavor of tea will certainly enjoy much of what is being offered.  It strikes me that, if anything, the book leans more heavily in the direction of philosophy than in the direction of mathematics, but I guess a proper riposte would be that Boos’s objectives certainly include rebuilding bridges that were severed long and not so long ago, and maybe these separations did more damage than imagined, and it’s time to reconsider.  After all, Plato insisted on geometry as a prerequisite for admission into his academy, and Kurt Goedel presented his own version of the ontological argument. Boos has a point …

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.