Metaphysical Myths, Mathematical Practice: The Ontology and Epistemology of the Exact Sciences. – book reviews

Jesus Alcolea Banegas

This book can be read as an attempt to rehabilitate formalism, more generally nominalism, as a philosophy of mathematics. To be sure the author is only half willing to acknowledge that his theory is nominalist (“nominalism on the cheap” is just one of the labels he suggests for his view (p.213)). Another way to read the book is as a defense of linguistic realism, for the author insists that his theory is platonistic because it countenances both talk of mathematical objects and talk of mathematical truth (“platonism without puzzles” is that other label he suggests for this view (p. 213)). Nevertheless we feel that Azzouni’s view is basically nominalist: the ultimate stuff of mathematics is postulate based systems. Postulate based systems are slightly more abstract than formal systems as standardly defined in mathematical logic. Roughly speaking, a postulate based system is an equivalence class of formal systems – eg., all those formal variants of elementary arithmetic, or of group theory, or of set theory (p. 82). Only in an “algorithmic system” do proofs ever occur and, Azzouni assumes, the business of mathematicians is proving theorems (p. 3). So in his postulate based systems Azzouni already has enough stuff for all of mathematics.

Nevertheless, Azzouni introduces a still higher level of abstraction. His ultimate claim to platonism is his willingness to allow postulate based systems to be grouped according to what those systems apparently refer to and to count their theorems as true. For example, many different postulate based systems apparently talk about the number two and establish truths about it. This move is necessary both to account for the actual practice of mathematicians and for the applicability of mathematics to science. What keeps Azzouni a nominalist is his insistence that the coreferentiality of terms in different postulate based systems is entirely a matter of stipulation or convention (“platonism on the cheap”, one might say). His book is a well thought-out defense of this way of talking about things.

There is one other important ingredient that is added to Azzouni’s mix, and that is an account of the philosophy of mathematics according to Quine. To a large degree Azzouni’s philosophy is developed in contrast to Quine’s (“Quine” dominates the index). This is somewhat surprising since it is already apparent that Azzouni’s philosophy is profoundly anti-Quinean. The very idea of separating logic and mathematics from natural science by convention violates Quine’s restrictions on truth by convention as well as his repudiation of the analytic-synthetic distinction. We think that Azzouni’s nominalism is fundamentally a bold attempt to recast Quine’s apparent platonism in terms more appealing to the average mathematician. Indeed, a positive feature of the book is its clear, detailed expositions of a Quinean philosophy of mathematics. It is by modifying Quine’s notion of posit that Azzouni thinks he can get a nonpuzzling platonism.

Metaphysical Myths, Mathematical Practice is divided into three parts. The opening is devoted to a set of puzzles apparently posed by mathematical practice. Azzouni compares two sets of puzzles: the traditional puzzles about epistemic access, such as Benacerraf’s puzzle, and about referential access to mathematical objects, and what he calls the epistemic role puzzle about the apparent non-involvement of mathematical objects in mathematical practice. The latter is supplemented by “the various intuitions and practices regarding reference that seem to offer prima facie evidence against various positions for what the referential mechanism in mathematics could look like” (p. 61) The lesson of the puzzles is to first ask whether mathematical practice needs access to its objects in any substantial sense, and Azzouni answers that it does not.

The second part details Azzouni’s own theory of algorithmic systems. The algorithmic basis of his theory solves the epistemological questions about mathematics. The referential questions (along with applicability) are solved by his allowing mathematics only a very weak notion of posits – “ultrathin posits” he calls them – according to which mathematical objects, as well as mathematical truth, are metaphysically inert (we might call this a deflationary platonism”).

Finally, the third part of his book applies his theory to three issues: a priori truth, the normativity of mathematics and the success of applied mathematics. An Appendix at the end of the book shows the compatibility of the truth predicate with a set of logical systems. All in all, Metaphysical Myths and Mathematical Practice is an important contribution to the philosophy of mathematics if for no other reason than the author’s recognition and criticisms of Quine’s central place in contemporary philosophy of mathematics.

In the rest of this essay, we want to focus on a few of the many interesting issues raised by the book. It is crucial to Azzouni’s philosophy that he examine an apparently deep problem about reference to mathematical objects. First order axiomatizations can’t fix reference by themselves as the well-known limitation theorems show (p. 9). Moreover, second order logic fares no better as Azzouni clearly demonstrates (section 3). (A simple argument is that second order logic presupposes reference to, e.g., the natural numbers in very much the way that specifying that a first order predicate Nx be interpreted as true only of the natural numbers presupposes reference to the numbers.)

But there remains a possibility for fixing reference that we felt Azzouni gave short shrift to, namely, the possibility that reference is fixed by the informal practice of mathematicians, as Kitcher has suggested (1984, The Nature of mathematical Knowledge. Oxford: Oxford University Press). Azzouni discusses this (p. 29 ff.) but we’re unconvinced by his negative answer. In fact, his arguments change from the detailed, developed criticisms that he gave of first order axiom systems and second order logic to anecdotes and opinions about just when was the first successful reference to [square root]-1. Kitcher’s notion of mathematical practice might leave something to be desired, but Azzouni’s arguments do not convince us that mathematical practice does not fix reference.

This issue of mathematical practice figures heavily in the new philosophical puzzle about mathematics that Azzouni calls “the Epistemic Role Puzzle”. In a nutshell, his claim is that the actual objects of mathematics (numbers, functions, spaces, etc.) are pragmatically, epistemologically, metaphysically, inert. The actual practice of mathematics could go on as it has even if none of these mathematical objects existed! This is in marked contrast to the robust posits of natural science that are involved in interactions with scientists. Quine’s notion of posits might apply also to mathematics, but in this case it seems to be posits only in a very thin sense. According to Azzouni some posits (in the sense of Quine) are involved in causal interactions with human beings, these are thick posits. Other posits are required in order to systematize our experience, these are thin posits. But the new puzzle is meant to show that the objects of mathematics are not even thin posits: they are ultrathin, irrelevant to mathematical practice and only a concession to our tendency to be linguistic realists.

Azzouni’s new puzzle is at the heart of his book, but we find some difficulties with it. Our basic worry is that he simply assumes, without much argument, that it is possible to describe mathematical practice without reference to mathematical objects. Given this assumption, it is easy to generate the puzzle that mathematical practice could continue even if all mathematical objects disappeared from the universe. This strategy can be applied outside of mathematics. If astronomical practice could be described without reference to actual astronomical bodies, but only with reference to the observations and calculations of astronomers, then exactly the same new puzzle that Azzouni proposes for mathematics would arise for astronomy. But when should we admit that a practice can be described without reference to the apparent referents of that practice? As we’ve already indicated, it seems that part of Azzouni’s motivation in allowing mathematical reference at all, even that purely conventional reference that identifies terms in different postulate based systems, is to make sense of mathematical practice, and we wonder if this motivation doesn’t conflict with his contrast between thin and ultrathin posits.

Moreover, it appears to us that Azzouni fails to characterize mathematical practice without reference to mathematical objects. Among mathematical objects are finite algorithms. Algorithms (Turing Machines, Ideal Computer Programs) are mathematical objects just as integers are (in fact, algorithms are prima facie slightly more complex mathematical objects than integers, even though they can be mathematically coded by integers). Now Azzouni insists that mathematical practice is just the manipulation of mathematical algorithms in certain ways; this is the primary thesis of the second part of the book. So if he is right, then mathematical practice cannot be described without reference to some mathematical objects, hence his claim that mathematical objects are epistemologically inert is just wrong. (This is the standard objection to formalist philosophies of mathematics: formal theories are mathematical objects just like integers are. Formalism/nominalism seems less to eliminate mathematical objects than to repeat Kronecker’s claim that mathematics reduces to the mathematics of natural numbers.) But we hasten to add that at one point Azzouni tries to distinguish versions of metamathematics in a way that might be relevant to our objection (pp. 171-4).

Finally, we note that Azzouni’s idea that no causal or other direct theory of reference can be used to explain how we refer to mathematical objects is defended by a distinction between primary A-mishaps and primary ‘A’-mishaps (in fact the distinctions are more complicated). In the former case, speakers confuse the referent of the term with another object, in the latter case, speakers confuse the term used and what it refers to. Azzouni’s claim, very roughly, is that primary A-mishaps can occur only in science, primary ‘A’-mishaps occur in science and mathematics. Of course, the very project of distinguishing between these two types of mishaps is a version of the analyticsynthetic distinction and so would be resisted by Quine. More to the point, we wonder if the distinction makes the cut he wants it to, between mathematics and science. Are we able to make primary A-mishaps about electrons and quarks? Perhaps we can no more confuse this electron with that electron than we can confuse this number with that number on Azzouni’s account (the issue isn’t concreteness; it’s isolating individuals in the context of a theory).

Our kind of criticism could be applied to many points of Azzouni’s interesting book, but we can generalize by returning to an earlier claim: this book is a well thought-out defense of this way of talking about things. The general worry is whether Azzouni is presenting a genuinely new philosophy of mathematics or just a different way of talking about Quine’s philosophy of mathematics. We mention two points.

Azzouni develops his account of mathematical truth and applied mathematics essentially in contrast to Quine. Just after his treatment of applied mathematics he claims that one “objection to Quine’s views that arises regularly in the literature is that Quine is forced to treat unapplied mathematics in a way that differs from applied mathematics, in terms of both how it is justified and how it is interpreted (p. 104). Now just what is the objection to Quine’s view? Our interpretation is that Quine’s view aims to justify the applied aspects of mathematics, both theorems and theories. What’s left open is the philosophical significance of pure mathematics, what we say about those theorems and theories of mathematics that are as yet “unapplied”, e.g., issues, theorems and practices about the distribution of primes, about large cardinals, etc. These latter topics are central to mathematical practice.

However this is not the issue that Azzouni addresses when considering the truth of applied mathematics. His is the very general philosophical question that given any algorithmic system sufficiently large and coherent to service all of natural science (e.g., ZFC), what should we say about algorithmic systems that conflict with this one (e.g., intuitionistic mathematics or set theory without foundations)? In fact intuitionistic mathematics and groundless set theory can rightly claim to be alternatives to standard mathematics. Indeed, in all the natural sciences, physics included, the standard practice is to develop more than one theory and to keep active a multiplicity of theories. There’s no philosophical worry about the conflicting truth demands, for example, of the gradualist or punctuated equilibrium theories of evolution. They disagree.

We thought that the standard objection to Quine was his inability to account for pure mathematics – the idea that mathematics develops primarily without regard to use. Azzouni simply ignores this issue. On his account, almost all of so called pure mathematics is really applied, that is, representable in ZFC.

Now consider an even deeper contrast: Quine’s unusual view that mathematics is continuous with natural science and Azzouni’s defense of the common sense distinction between them. “What made classical physics physics was that it could be (and was) overthrown on the basis of experimental evidence. Were it a mathematical subject, this would be irrelevant. Furthermore, there is an ongoing study that looks just like classical physics but is a mathematical subject. In it, for example, one studies classical point masses. And the results in that subject have not been overthrown by anything because they are simply not about the physical world. Classical point masses are mathematical objects in precisely the same sense that Turing machines are.” (p. 108)

This way of talking differs from Quine’s. Quine (like Kant) assimilates calculus and Newtonian physics. Azzouni separates them by creating a doppelganger of Newtonian physics called “mathematical Newtonian physics”. This latter mathematical theory has not only not been overthrown by Einstein, but it continues to be taught in physics departments under the misleading label “classical physics” and it is frequently applied! So mathematics is different from physics according to Azzouni, but it follows from his distinction that the disciplines of mathematics and physics are confused about their subject matters! (As one of the authors has argued, Quine’s holism no more reduces mathematics to science than it reduces science to mathematics: Tymoczko: “Mathematics, Science and Ontology”, Synthese, 88 (1991), n. 2, pp. 201-28.)

Philosophers in general would be well rewarded by studying Metaphysical Myths, Mathematical Practice. It discusses many important questions in the philosophy of mathematics. Mathematicians, on the other hand, might be slightly disappointed. They might not recognize their own practice in Azzouni’s discussion and decide that the mathematical practice referred to in the title is just one more metaphysical myth.

COPYRIGHT 1996 Oxford University Press

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