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Philosophy of Mathematics

Philosophy of Mathematics: Structure and Ontology by Stewart Shapiro (Oxford University Press) Do numbers, sets, and so forth, exist? What do mathematical statements mean? Are they literally true or false, or do they lack truth values altogether? Addressing questions that have attracted lively debate in recent years, Stewart Shapiro contends that standard realist and antirealist accounts of mathematics are both problematic.

As Benacerraf first noted, we are confronted with the following powerful dilemma. The desired continuity between mathematical and, say, scientific language suggests realism, but realism in this context suggests seemingly intractable epistemic problems. As a way out of this dilemma, Shapiro articulates a structuralist approach. On this view, the subject matter of arithmetic, for example, is not a fixed domain of numbers independent of each other, but rather is the natural number structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle. Using this framework, realism in mathematics can be preserved without troublesome epistemic consequences.

Shapiro concludes by showing how a structuralist approach can be applied to wider philosophical questions such as the nature of an "object" and the Quinean nature of ontological commitment. Clear, compelling, and tautly argued, Shapiro's work, noteworthy both in its attempt to develop a full-length structuralist approach to mathematics and to trace its emergence in the history of mathematics, will be of deep interest to both philosophers and mathematicians.

Mathematics and Its Philosophy
Object and Truth: A Realist Manifesto
Interlude on Antirealism
A Role for the External
Ontology: Object
Ontology: Structure
Theories of Structure
Mathematics: Structures, All the Way Down
Addendum: Function and Structure
Epistemology and Reference
Epistemic Preamble
Small Finite Structure: Abstraction and Pattern Recognition
Long Strings and Large Natural Numbers
To the Infinite: The Natural-number Structure
Indiscernibility, Identity, and Object
Ontological Interlude
Implicit Definition and Structure
Existence and Uniqueness: Coherence and
Conclusions: Language, Reference, and
How We Got Here
When Does Structuralism Begin?
Geometry, Space, Structure
A Tale of Two Debates
Dedekind and ante rem Structures
Nicholas Bourbaki
Practice: Construction, Modality, Logic
Dynamic Language
Idealization to the Max
Construction, Semantics, and Ontology
Construction, Logic, and Object
Dynamic Language and Structure
Assertion, Modality, and Truth
Practice, Logic, and Metaphysics
Modality, Structure, Ontology
Modal Fictionalism
Modal Structuralism
Other Bargains
What Is a Structuralist to Make of All This?
Life Outside Mathematics: Structure and Reality
Structure and Science-the Problem
Application and Structure
Maybe It Is Structures All the Way Down

This book has both an old topic and a relatively new one. The old topic is the ontological status of mathematical objects: do numbers, sets, and so on, exist? The relatively new topic is the semantical status of mathematical statements: what do mathematical statements mean? Are they literally true or false, are they vacuous, or do they lack truth-values altogether? The bulk of this book is devoted to providing and defending answers to these questions and tracing some implications of the answers, but the first order of business is to shed some light on the questions themselves. What is at stake when one either adopts or rejects answers?

Much contemporary philosophy of mathematics has its roots in Benacerraf [1973], which sketches an intriguing dilemma for our subject. One strong desideratum is that mathematical statements have the same semantics as ordinary statements, or at least respectable scientific statements. Because mathematics is a dignified and vitally important endeavor, one ought to try to take mathematical assertions literally, "at face value." This is just to hypothesize that mathematicians probably know what they are talking about, at least most of the time, and that they mean what they say. Another motivation for the desideratum comes from the fact that scientific language is thoroughly intertwined with mathematical language. It would be awkward and counterintuitive to provide separate semantic accounts for mathematical and scientific language, and yet another account of how various discourses interact.

Among philosophers, the prevailing semantic theory today is a truth-valued account, sometimes called "Tarskian." Model theory provides the framework. The desideratum, then, is that the model-theoretic scheme be applied to mathematical and ordinary (or scientific) language alike, or else the scheme be rejected for both discourses.

The prevailing model-theoretic semantics suggests realism in mathematics, in two senses. First, according to model-theoretic semantics, the singular terms of a mathematical language denote objects, and the variables range over a domain-of-discourse. Thus, mathematical objects-numbers, functions, sets, and the like-exist. This is what I call realism in ontology. A popular and closely related theme is the Quinean dictum that one's ontology consists of the range of the bound variables in properly regimented discourse. The slogan is "to be is to be the value of a variable." The second sense of realism suggested by the model-theoretic framework is that each wellformed, meaningful sentence has a determinate and nonvacuous truth-value, either truth or falsehood. This is realism in truth-value.

We now approach Benacerraf's dilemma. From the realism in ontology, we have the existence of mathematical objects. It would appear that these objects are abstract, in the sense that they are causally inert, not located in space and time, and so on. Moreover, from the realism in truth-value, it would appear that assertions like the twin­prime conjecture and the continuum hypothesis are either true or false, independently of the mind, language, or convention of the mathematician. Thus, we are led to a view much like traditional Platonism, and the notorious epistemological problems that come with it. If mathematical objects are outside the causal nexus, how can we know anything about them? How can we have any confidence in what the mathematicians say about mathematical objects? Again, I take it as "data" that most contemporary mathematics is correct. Thus, it is incumbent to show how it is possible for mathematicians to get it right most of the time. Under the suggested realism, this requires epistemic access to an acausal, eternal, and detached mathematical realm. This is the most serious problem for realism.

Benacerraf [1973] argues that antirealist philosophies of mathematics have a more tractable line on epistemology, but then the semantic desideratum is in danger. Here is our dilemma: the desired continuity between mathematical language and everyday and scientific language suggests realism, and this leaves us with seemingly intractable epistemic problems. We must either solve the problems with realism, give up the continuity between mathematical and everyday discourse, or give up the prevailing semantical accounts of ordinary and scientific language.

Most contemporary work in philosophy of mathematics begins here. Realists grab one horn of the trilemma, antirealists grab one of the others. The straightforward, but daunting strategy for realists is to develop an epistemology for mathematics while maintaining the ontological and semantic commitments. A more modest strategy is to argue that even if we are clueless concerning the epistemic problems with mathematics, these problems are close analogues of (presumably unsolved) epistemic problems with ordinary or scientific discourse. Clearly, we do have scientific and ordinary knowledge, even if we do not know how it works. The strategy is to link mathematical knowledge to scientific knowledge. The ploy would not solve the epistemic problems with mathematics, of course, but it would suggest that the problems are no more troublesome than those of scientific or ordinary discourse. The modest strategy conforms nicely to the seamless interplay between mathematical and ordinary or scientific discourse. On this view, everyday or scientific knowledge just is, in part, mathematical knowledge.

For a realist, however, the modest strategy exacerbates the dichotomy between the abstract mathematical realm and the ordinary physical realm, bringing the problem of applicability to the fore. The realist needs an account of the relationship between the eternal, acausal, detached mathematical universe and the subject matter of

science and everyday language-the material world. How it is that an abstract, eternal, acausal realm manages to get entangled with the ordinary, physical world around us, so much so that mathematical knowledge is essential for scientific knowledge?

Antirealist programs, on the other hand, try to account for mathematics without assuming the independent existence of mathematical objects, or that mathematical statements have objective truth-values. On the antirealist programs, the semantic desideratum is not fulfilled, unless one goes on to give an antirealist semantics for ordinary or scientific language. Benacerraf's observation is that some antirealist programs have promising beginnings, but one burden of this book is to show that the promise is not delivered. If attention is restricted to those antirealist programs that accept and account for the bulk of contemporary mathematics, without demanding major revisions in mathematics, then the epistemic (and semantic) problems are just as troublesome as those of realism. In a sense, the problems are equivalent. For example, a common maneuver today is to introduce a "primitive," such as a modal operator, in order to reduce ontology. The proposal is to trade ontology for ideology. However, in the context at hand-mathematics-the ideology introduces epistemic problems quite in line with the problems with realism. The epistemic difficulties with realism are generated by the richness of mathematics itself.

In an earlier paper, Benacerraf [1965] raises another problem for realism in ontology (see also Kitcher [ 1983, chapter 6]). It is well known that virtually every field of mathematics can be reduced to, or modeled in, set theory. Matters of economy suggest that there be a single type of object for all of mathematics-sets. Why have numbers, points, functions, functionals, and sets when sets alone will do? However, there are several reductions of arithmetic to set theory. If natural numbers are mathematical objects, as the realist contends, and if all mathematical objects are sets, then there is a fact concerning which sets the natural numbers are. According to one account, due to von Neumann, the natural numbers are finite ordinals. Thus, 2 is (0, { 0 ) ), 4 is { 0,1 0 ), { o,1 0 } }, { ~, ( }, { 0,1 0 ) ] ) ) , and so 2 E 4. According to Zermelo's account, 2 is 110) ), 4 is { { { { ~) ]) }, and so 2 0 4. Moreover, there seems to be no principled way to decide between the reductions. Each serves whatever purpose a reduction is supposed to serve. So we are left without an answer to the question of whether 2 is really a member of 4 or not. Will the real 2 please stand up? What, after all, are the natural numbers? Are they finite von Neumann ordinals, Zermelo numerals, or other sets? From these observations and questions, Benacerraf and Kitcher conclude that numbers are not objects, against realism in ontology. This conclusion, I believe, is not warranted. It all depends on what it is to 6e an object, a matter that is presently under discussion. Benacerraf sand Kitcher's conclusion depends on what sorts of questions can legitimately be asked about objects and what sorts of questions have determinate answers waiting to be discovered.

The philosophy of mathematics to be articulated in this book goes by the name "structuralism," and its slogan is "mathematics is the science of structure." The subject matter of arithmetic is the natural-number structure, the pattern common to any system of objects that has a distinguished initial object and a successor relation that satisfies the induction principle. Roughly speaking, the essence of a natural number is the relations it has with other natural numbers. There is no more to being the natural number 2 than being the successor of the successor of 0, the predecessor of 3, the first prime, and so on. The natural-number structure is exemplified by the von Neumann finite ordinals, the Zermelo numerals, the arabic numerals, a sequence of distinct moments of time, and so forth. The structure is common to all of the reductions of arithmetic. Similarly, Euclidean geometry is about Euclidean-space structure, topology about topological structures, and so on. As articulated here, structuralism is a variety of realism.

A natural number, then, is a place in the natural-number structure. From this perspective, the issue raised by Benacerraf and Kitcher concerns the extent to which a place in a structure is an object. This depends on how structures and their places are construed. Thus, in addition to providing a line on solving the traditional problems in philosophy of mathematics, structuralism has something to say about what a mathematical object is. With this, the ordinary notion of "object" is illuminated as well.

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