Martin Davis’s Realism in Mathematics
In “Pragmatic Platonism”, Davis points out that an ancestor of the integral calculus, “the method of indivisibles” was used by Torricelli in the seventeenth century to obtain results—one of the most surprising at the time being the existence of a solid (the “Torricelli trumpet”) with infinite surface area and finite volume. These results could be checked (but not discovered) by other methods, and the method of indivisibles (and other methods that lacked rigorous justification until the nineteenth century, including the use of complex numbers in calculating the real roots of an equation) became part of the mathematician’s repertoire. Nor does the story stop in the nineteenth century; the Axiom of choice was introduced by Zermelo in 1904, but cannot be justified from the other axioms of Zermelo-Frankel set theory.” Yet, as Davis remarks (op. cit. p.9), “The obligation to always point out a use of the axiom of choice is a thing of the past.” And he adds, “I haven’t heard of anyone calling the proof of Fermat’s Last Theorem into question because of the large infinities implicit in Grothendieck universes.” (In “What is Mathematical Truth”, I similarly pointed out that since Descartes the isomorphism of the geometrical line with the continuum of real numbers has become fundamental to virtually all of analysis without a “proof” from other axioms.) Davis boldly concludes, “What can we say about Torricelli’s methodology? He was certainly not seeking to obtain results by ‘cogent proofs from the definitions’ or ‘in ontological terms, from the essences of things’. He was experimenting with a mathematical technique that he had learned, and was attempting to see whether it would work in an uncharted realm. In the process, something new about the infinite was discovered. I insist that this was induction from a body of mathematical experience.”
Although the remarks I have just quoted are primarily epistemological, both the use of the term “discovered” and the title “Pragmatic Platonism” (emphasis added) indicate that Davis believes that mathematical knowledge is objective, and, in fact, he goes on to say so explicitly. (I shall quote the place in a moment.) But Davis (like myself) cannot go along with Godel’s view that mathematical objects exist in a Platonic realm that (parts of which) the mind is somehow capable of perceiving. I now quote two paragraphs from Davis’s essay that seem to me to capture the essence of what I am calling his “realism”.
If the objects of mathematics are not in nature and not in a “second plane of reality,” then where are they? Perhaps we can learn something from the physicists. Consider for example, the discussion of the “Anthropic Principle” . The advocates of this principle note that the values of certain critical constants are finely tuned to our very existence. Given even minor deviations, the consequence would be: no human race. It is not relevant here whether this principle is regarded as profound or merely tautological. What I find interesting in this discussion of alternate universes whose properties exclude the existence of us, is that no one worries about their ontology. There is simply a blithe confidence that the same reasoning faculty that serves physicists so well in studying the world that we actually do inhabit, will work just as well in deducing the properties of a somewhat different hypothetical world.
A more mundane example is the ubiquitous use of idealization. When Newton calculated the motions of the planets assuming that each of the heavenly bodies is a perfect sphere of uniform density or even a mass particle, no one complained that the ontology of his idealized worlds was obscure. The evidence that our minds are up to the challenge of discovering the properties of alternative worlds is simply that we have successfully done so. Induction indeed! This reassurance is not at all absolute. Like all empirical knowledge it comes without a guarantee that it is certain.
My claim is that what mathematicians do is very much the same. We explore simple austere worlds that differ from the one we inhabit both by their stark simplicity and by their openness to the infinite. It is simply an empirical fact that we are able to obtain apparently reliable and objective information about such worlds. And, because of this, any illusion that this knowledge is certain must be abandoned.
The key notions in these paragraphs are “hypothetical worlds”, “idealized worlds”, and “objective information”. For Davis, mathematics is not about worlds that actually exist in some hyper-cosmology (unlike David Lewis’s “possible worlds”9), but about what would be the case if certain idealized worlds existed, worlds that would contain infinities (in some cases “large infinities”) if they really did exist. If a reader were to ask me whether the difference between Davis’s hypothetical worlds, which I find it reasonable to talk about, and Lewis’s possible worlds, which I don’t, isn’t “rather thin”, I would reply that it is like the difference between saying that a solid gold mountain actually exists somewhere, only not in this world, which was Lewis’s view (“real” is just what we call our world; all possible worlds are equally real to their inhabitants) and saying that physically possibly a solid gold mountain could exist, which is the case according to present day physics. I interpret Davis’s term “hypothetical” to mean that he, like me, conceives of mathematical structures as ones that, in some sense of “could”, could exist. If I have him right, they exist hypothetically, but not actually, not even in a Platonic heaven. I don’t find the difference between saying that certain worlds or structures are possible and saying that they exist “thin” at all. We reason about such hypothetical worlds by using our human abilities to imagine and to idealize and to deduce from given assumptions. Because it is obtained in this way, mathematical knowledge is fallible (pace Godel, there is nothing “perceptual” about it), but the consilience of the results justifies our taking the results to be objective information. There is a fact of the matter about what would be the case if those “hypothetical worlds” were real.
-  The method of indivisibles was invented by Bonaventura Cavalieri in 1637.
-  For a detailed account, see Kanamori, Akihiro (2004), “Zermelo and set theory”, The Bulletin ofSymbolic Logic 10 (4): 487-553, doi:10.2178/bsl/1102083759, ISSN 1079-8986, MR 2136635.
-  Here I am going by Davis’s reference to the use of Grothendieck’s infinity topoi by Wiles andTaylor in their proof of Fermat’s “Last Theorem”.
-  By “consilience” I mean that the results are not only consistent, but that they extend one another,often in unexpected directions.