Tag Archives: Indispensability argument

Re: Paper and slides on indefiniteness of CH

Dear Sy,

In answer to your questions below, it seems to me that my work has philosophical significance in several ways. First, it shows that the reach of Quine’s (and perhaps Putnam’s) indispensability argument is extremely limited (for whatever that’s worth). Secondly, I believe it shows that one can’t sustain the view from Galileo to Tegmark that mathematics (and the continuum in particular) is somehow embedded in nature. Relatedly, it does not sustain the view that the success of analysis in natural science must be due to the independent reality of the real number system.

My results tell us nothing new about physics. And indeed, they do not tell us that physics is somehow conservative over PA. In fact it can’t because if Michael Beeson is right, quantum mechanics is inconsistent with general relativity; see his article, “Constructivity, computability, and the continuum”, in G. Sica (ed.) Essays on the Foundations of Mathematics and Logic, Volume 2 (2005), pp. 23-25. It just tells us that the mathematics used in the different parts of physics is conservative over PA.

Finally, to be “quite happy with ZFC” is not the same as saying that there is a good philosophical justification for it.

Best,
Sol

Re: Paper and slides on indefiniteness of CH

Dear Sol,

This message is not specifically about your rebuttal of Hilary’s claim, but about a more general issue which I hope that you can shed light on.

You write:

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?

My very general question is: What do we hope to gain by showing that something can be “captured by limited means” (in this case regarding what mathematics is needed for physical theory)? Does this tell us something new about what we have “captured”?

I am of course familiar with advantages of, for example, establishing that some computable function is in fact provably total in PA, as then one might extract useful and new information about the growth rate of such a function. In set theory is something analagous, which is if you can bring down the large cardinal strength enough, core model theory kicks in and you have a good chance of achieving a much better understanding. Or if one starts with a philosophical position, like predicativity, it is somehow gratifying to know that one can capture it precisely with formal means.

But frankly speaking, too often there is a connotation of “of, we don’t really need all of that bad set theory to do this”, or even more outdated: “what a relief, now we know that this is consistent because we captured it in a system conservative over PA!”. Surely in the 21st century we are not going to worry anymore about the consistency of ZFC.

Is the point that (as you say at the end of your message) that you think you have to invoke some kind of platonistic ontology if you are not using limited means, and for some reason this makes you feel uncomfortable (even though I presume you don’t have inconsistency worries)?

It is tempting to think that your result using your system W might tell us something new about physics. Does it? On the other hand you have not claimed that “physics is conservative over PA” exactly, but only that the math needed to do a certain amount of physics is conservative over PA.

Finally, how is it that you claim that “only a platonistic philosophy of mathematics provides justification” for impredicative 2nd order arithmetic? That just seems wrong, as there are plenty of non-platonists out there (I am one) who are quite happy with ZFC. But maybe I don’t understand how you are using the word “justification”.

Thanks in advance for your clarifications. And please understand, I am not suggesting that it is not valuable to “capture things by limited means”, I just want to have a better understanding of what you feel is gained by doing that.

All the best,
Sy

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.

Best,
Sol