In response to:
Is Mathematics for Real? from the August 13, 1981 issue
To the Editors:
Martin Gardner’s criticisms [NYR, August 13] of the mathematical conceptualism he found in Davis and Hersh’s The Mathematical Experience ring true. That kind of conceptualism is of a piece with the various “philosophies” mathematicians—and scientists—espouse not as a result of argument and reflection but as, one suspects, a means of deflecting them. One can appreciate that working mathematicians and scientists will typically find philosophical enquiry into their disciplines distracting while wishing that they didn’t dress up their irritation or mere lack of interest as a rival “philosophy.”
Conceptualism, though espoused in order to deflect problems, faces, as Gardner rightly pointed out, problems of its own. Most notable is conceptualism’s inability to give any convincing account of the role mathematics plays in successful science and technology. If mathematics is just a human conceptual or cultural creation, how is it that it stands so strikingly apart from other such human creations in being applicable to reality? Conceptualism has no answer; it presents us with a mystery. Gardner is surely right, too, to see part of conceptualism’s appeal as being to those with an abhorrence of “talk of anything that transcends human experience.”
Gardner attacks conceptualism in the name of mathematical realism, the view that “if all intelligent minds in the universe disappeared, the universe would still have a mathematical structure, and that in some sense even the theorems of pure mathematics would continue to be ‘true.”‘ The correctness or otherwise of such a view I won’t discuss here; one thing that is sure, though, is that mathematical realism is not so easily dismissed as conceptualists will have it. Gardner’s comments on Davis and Hersh indicate why.
Unfortunately, having done such a good job in realism’s behalf, Gardner then proceeds to undo it. Just when one takes him to have offered strong arguments for mathematical realism, one finds Gardner writing that his “unabashed” realism is held for mainly “emotional reasons;” his only other announced ground for being a realist is the “efficiency” of the language of realism. Gardner sees the dispute between realists and conceptualists as one to be settled not by evidence and argument, but by choice. He takes Rudolf Carnap as his philosophical guide, modeling his view of mathematics on the view he takes Carnap to have held about rival theories of physical reality: “Rudolf Carnap was able to show, in his Logical Structure of the World, that a phenomeno-logical language, never going beyond human experience, is capable of expressing the same empirical content as any realistic language….” Carnap showed no such thing; at most that was what he tried to show. Most later philosophers—Carnap’s later self included—have taken him to have failed in his attempt. (Gardner’s metaphysically tolerant attitude, by the way, fits rather more Carnap’s views of 1950, as expressed in the essay “Empiricism, Semantics and Ontology.”)
Gardner’s mathematical realism now looks rather abashed, having no more philosophic worth than someone else’s unabashed conceptualism, perhaps also held for “emotional reasons.”
Gardner’s realism takes another beating when he, later in his review, informs us that mathematics, in particular Euclidean geometry, is analytic; that is, that the theorems of Euclidean geometry are true solely in virtue of their logical structures and of the meanings of the terms in them. According to Gardner, Euclidean geometry can’t be wrong—though many have thought it so—because its theorems are “logical tautologies,” without content. But if geometrical—and, in general, mathematical—theorems are without content, what becomes of Gardner’s mathematical “realism”? A genuine realist about geometry will see it as being about points, lines, planes and such. If the realist wants geometry to be necessarily true, the necessity will have to come from elsewhere than analyticity. Gardner’s Carnapian tolerance of various geometries is but another symptom of a deeper antirealism, one which affects his view of all mathematics.
The question of whether mathematical realism is correct or not will only get settled if one is, first of all, clear as to what it is, and, second, clear as to what the criteria for settling such questions are. Anyone interested in the question would do well to ignore any “emotional reasons” one way or the other, and look to the work of those who have approached it with those two desiderata in mind; to them I recommend Hartry Field’s recent Science Without Numbers, an impressive—and rather technical—attempt to answer the question in the negative, though not in favour of conceptualism, but of a version of Hilbert’s formalism.
La Trovi University
Bundoona, Victoria, Australia
Martin Gardner replies:
Robert Farrell is right in chiding me for saying Carnap “was able to show.” I should have said “believed he could show.” But Farrell is wrong is suggesting that Carnap later gave up his “metaphysical tolerance” toward the rival languages of phenomenalism and realism, or toward the rival languages for talking about the foundations of mathematics.
Der logische Aufbau der Welt (The Logical Structure of the World, which I will henceforth call the Aufbau program) was Carnap’s first major work. Carnap himself considered it no more than a tentative sketch of a program. He early recognized its many faults, and became his own severest critic. The Aufbau’s major error, he declares in his 1961 preface to the second edition, was basing the program on a single primitive relation (similarity) instead of a multiplicity of relations. He remains convinced, however, that his thesis of the “reducibility of thing concepts to autopsychological concepts remains valid.”
Carnap’s Aufbau program was taken up by Nelson Goodman in his book The Structure of Appearance, and later vigorously championed in his contribution to The Philosophy of Rudolf Carnap (edited by P.A. Schilpp, 1963). Goodman argues that the incompleteness of a phenomenal language no more counts against it than the inability to trisect any angle counts against Euclidean geometry. Equally irrelevant is the charge that a phenomenal language is epistemologically false, because the language is not designed to say anything about an external world. Goodman concludes that Carnap’s errors were “serious, unoriginal, and worthwhile.”
Commenting favorably on Goodman’s paper, in the same volume, Carnap left no doubt that he considered the choice between a phenomenal language and the realistic language of physics to be based only on the “practical decision” as to which language is the most efficient. Phenomenalism is rejected because “it is an absolutely private language which can only be used for soliloquy, but not for common communication between two persons.”
If realism is taken as an ontological thesis, Carnap writes, he is not a realist. But “if ‘realism’ is understood as preference for the reistic language [Carnap’s term for a language about material, observable things] over the phenomenal language, then I am also a realist.” This metaphysical neutrality was never abandoned by Carnap, and I cannot comprehend why Farrell seems to think it applies only to the Carnap of 1950.
The problem of “realism” with respect to the entities of pure mathematics is an altogether different question, but here again Carnap never discarded his “principle of tolerance.” When Farrell says: “A genuine realist about geometry will see it as being about points, lines, planes and such,” I don’t know what he means. Euclidean geometry was formalized by Hilbert, and others, as an uninterpreted system. One interpretation is to take its symbols as representing abstract points, lines, planes and so on. Even so, one is still inside a formal system which says nothing about the world “out there.” To get to that world one must apply what Carnap called correspondence rules which link such ideal concepts as points and lines to observed physical structures.
Insofar as geometry applies to the outside world, it loses its certainty. By the same token, it is necessarily true only when its empirical meanings are abandoned. I am sure Farrell intends to say something important in his paragraph about this, but exactly what he wants to say eludes me. I am unfamiliar with the book Farrell recommends, so I cannot comment on it.