Tag Archives: Predicativity

Re: Paper and slides on indefiniteness of CH

Dear Hilary,

Thank you for bringing my attention to Ch. 9 of your book, Philosophy in an Age of Science, and especially to its Appendix where you say something about my work on predicative foundations of applicable analysis. I appreciate your clarification in Ch. 9 of the relation of your arguments re indispensability to those of Quine; I’m afraid that I am one of those who has not carefully distinguished the two. In any case, what I addressed in my 1992 PSA article, “Why a little bit goes a long way. Logical foundations of scientifically applicable mathematics”, reprinted with some minor corrections and additions as Ch. 14 in my book, In the Light of Logic, was that if one accepts the indispensability arguments, there still remain two critical questions, namely:

Q1. Just which mathematical entities are indispensable to current scientific theories?, and

Q2. Just what principles concerning those entities are need for the required mathematics?

I provide one answer to these questions via a formal system W (‘W’ in honor of Hermann Weyl) that has variables ranging over a universe of individuals containing numbers, sets, functions, functionals, etc., and closed under pairing, together with variables ranging over classes of individuals. (Sets are those classes that have characteristic functions.) While thus conceptually rich, W is proof-theoretically weak. The main metatheorem, due to joint work with Gerhard Jäger, is that W is a conservative extension of Peano Arithmetic, PA. Nevertheless, a considerable part of modern analysis can be developed in W. In W we have the class (not a set) R of real numbers, the class of arbitrary functions from R to R, the class of functionals on such to R, and so on. I showed in detail in extensive unpublished notes from around 1980 how to develop all of 19th c. classical analysis and much of 20th c. functional analysis up to the spectral theorem for bounded self-adjoint operators. These notes have now been scanned in full and are available with an up to date introduction on my home page under the title, “How a little bit goes a long way. Predicative foundations of analysis.” The same methodology used there can no doubt be pushed much farther into modern analysis. (I also discuss in the introduction to those notes the relationship of my work to that of work on analysis by Friedman, Simpson, and others in the Reverse Mathematics program.)

Now it is a mistake in your appendix to Ch. 9 to say that I can’t quantify over all real numbers; given that we have the class R of “all” real numbers in W, we can express various propositions containing both universal and existential quantification over R. Of course, we do not have any physical language itself in W, so we can’t express directly that “there is a [physical] point corresponding to every triple of real numbers.” But we can formulate mathematical models of physical reality using triples of real numbers to represent the assumed continuum of physical space, and quadruples to represent that of physical space-time, and so on; moreover, we can quantify over “nice” kinds of regions of space and space-time as represented in these terms. So your criticism cannot be an objection to what is provided by the system W and the development of analysis in it.

As to the philosophical significance of this work, the conservation theorem shows that W is justified on predicative grounds, though it has a direct impredicative interpretation as well. When you say you disagree with my philosophical views, you seem to suggest that I am a predicativist; others also have mistakenly identified me in those terms. I am an avowed anti-platonist, but, as I wrote in the Preface to In the Light of Logic, p. ix, “[i]t should not be concluded from … the fact that I have spent many years working on different aspects of predicativity, that I consider it the be-all and end-all in nonplatonistic foundations. Rather, it should be looked upon as the philosophy of how we get off the ground and sustain flight mathematically without assuming more than the structure of natural numbers to begin with. There are less clear-cut conceptions that can lead us higher into the mathematical stratosphere, for example that of various kinds of sets generated by infinitary closure conditions. That such conceptions are less clear-cut than the natural number system is no reason not to use them, but one should look to see where it is necessary to use them and what we can say about what it is we know when we do use them.” As witness for these views, see my considerable work on theories of transfinitely iterated inductive definitions and systems of (what I call) explicit mathematics that have a constructive character in a generalized sense of the word. However, the philosophy of mathematics that I call “conceptual structuralism” and that has been referred to earlier in the discussion in this series is not to be identified with the acceptance or rejection of any one formal system, though I do reject full impredicative second-order arithmetic and its extensions in set theory on the grounds that only a platonistic philosophy of mathematics provides justification for it.