Mathematics as a Science of Patterns is the definitive exposition of a system of ideas about the nature of mathematics which Michael Resnik has been elaborating for a number of years. In calling mathematics a science he implies that it has a factual subject-matter and that mathematical knowledge is on a par with other scientific knowledge; in calling it a science of patterns he expresses his commitment to a structuralist philosophy of mathematics. He links this to a defence (...) of realism about the metaphysics of mathematics--the view that mathematics is about things that really exist. Resnik's distinctive philosophy of mathematics is here presented in an accessible and systematic form: it will be of value not only to specialists in this area, but to philosophers, mathematicians, and logicians interested in the relationship between these three disciplines, or in truth, realism, and epistemology. (shrink)

This paper is a nontechnical exposition of the author's view that mathematics is a science of patterns and that mathematical objects are positions in patterns. the new elements in this paper are epistemological, i.e., first steps towards a postulational theory of the genesis of our knowledge of patterns.

This paper argues against Logical Realism, in particular against the view that there are facts of matters of logic that obtain independently of us, our linguistic conventions and inferential practices. The paper challenges logical realists to provide a non-intuition based epistemology, one which would be compatible with the empiricist and naturalist convictions motivating much recent anti-realist philosophy of mathematics.

By a logical theory I mean a formal system together with its semantics, meta-theory, and rules for translating ordinary language into its notation. Logical theories can be used descriptively (for example, to represent particular arguments or to depict the logical form of certain sentences). Here the logician uses the usual methods of empirical science to assess the correctness of his descriptions. However, the most important applications of logical theories are normative, and here, I argue, the epistemology is that of wide (...) reflective equilibrium. The result is that logic not only assesses our inferential practice but also changes it. I tie my discussion to Thagard's views concerning the relationship between psychology and logic, arguing against him that psychology has and should have only a peripheral role in normative (and most descriptive) applications of logic. (shrink)

In Science without numbers Hartry Field attempted to formulate a nominalist version of Newtonian physics?one free of ontic commitment to numbers, functions or sets?sufficiently strong to have the standard platonist version as a conservative extension. However, when uses for abstract entities kept popping up like hydra heads, Field enriched his logic to avoid them. This paper reviews some of Field's attempts to deflate his ontology by inflating his logic.

We have seen that despite Feferman's results Gödel's second theorem vitiates the use of Hilbert-type epistemological programs and consistency proofs as a response to mathematical skepticism. Thus consistency proofs fail to have the philosophical significance often attributed to them.This does not mean that consistency proofs are of no interest to philosophers. We know that a ‘non-pathological’ consistency proof for a system S will use methods which are not available in S. When S is as strong a system as we are (...) willing to entertain seriously then a consistency proof for it will yield no epistemological gain. But in other cases philosophers might argue that the proof uses methods which are merely different rather than stronger than those available in the system in question. This claim has been made, for example, in the case of the constructive consistency proofs for elementary number theory. Similar philosophical investigations can be made on relative consistency proofs, since these differ from each other in the principles they employ. For example, most relative consistency proofs can be carried out within elementary number theory, but without using the theory of real numbers, no one has been able to prove the consistency of Quine's ML relative to that of his NF.What about the consistency of all mathematics or of some strong system for set theory? How do we answer the skeptic? Since here a convincing proof is not possible, we have established that the skeptic demands too much. We cannot be certain that our axioms are free from contradiction and must treat them as hypotheses which may be abandoned or modified in the face of further mathematical experience. This attitude is taken by many foundational workers who also go on to voice opinions about thelikelihood that various systems are consistent. Since these opinions are variously supported by appeals to the clarity of the mathematical concept formalized, the existence or non-existence of ‘weird’ models for the system and actual empirical experience with the system, this is surely a fruitful area for philosophical research. (shrink)

Nothing has been more central to philosophy of mathematics than the distinction between mathematical and physical objects. Yet consideration of quantum particles shows the inadequacy of the popular spacetime and causal characterizations of the distinction. It also raises problems for an assumption used recently by Field, Hellman and Horgan, namely, that the mathematical realm is metaphysically independent of the physical one.

I present a theory of justification for mathematical beliefs that is both non‐foundationalist, in that it claims that some mathematics must be justified indirectly in terms of its consequences, and holistic, in that it maintains that no claim of theoretical science can be confirmed or refuted in isolation but only as a part of a system of hypotheses. Our evidence for mathematics is ultimately empirical because the mathematics that is part of theoretical science, is, in principle, revisable in light of (...) experience and confirmed by experience. Following Henry Kyburg, I develop this idea by claiming that in science we use combinations of mathematical and scientific principles to develop models that allow us to calculate values that we then compare with the data. ‘Separatists’ objections to holism revolve around the claim that holism fails to respect our intuitions about mathematics, e.g. that mathematics is clearly distinct from science and that mathematical evidence comes from proofs, rather than from experience. My response to these objections is that holism can accommodate these intuitions by appealing to pragmatic rationality that, on the one hand, underwrites the special role of mathematics and bids us to treat it as if it were a priori, and, on the other, justifies, on the grounds of simplicity and success, a local conception of evidence according to which data can confirm specific hypotheses. (shrink)

The fact that mathematics is ordinarily practised as an autonomous science with its own, peculiar type of evidence constituted mainly by deductive reasoning is often taken as evidence that mathematics and science have specifically different evidential supports and specifically different subject matters. I argue against this conclusion by first analysing deductive proofs, and the type of evidence that is usually required for axioms, and claiming that most of the evidence for the most elementary and fundamental parts of mathematics is empirical. (...) I then appeal to the role of computation to argue that non‐deductive inference from empirical premises is part of the contemporary methodology of mathematics, and so some of our proofs turn out not to be purely logical deductions. Finally, I discuss the relation between mathematics and logic and argue against logical realism by denying that statements attributing logical properties or relations are true independently of our holding them to be true, our psychology, our linguistic and inferential conventions, or other facts about human beings. In the end, both mathematics and logic turn out to be a priori only in the sense that some mathematical and logical truths are obtained through deductive proofs, and for pragmatic reasons, are insulated from experience; but neither mathematics nor logic are a priori in the sense of being immune to empirical revision. (shrink)

It is commonly believed that the epistemology of mathematics must be different from the epistemology of science because their objects are different in kind, i.e. metaphysically different. In this chapter, I want to suggest that some careful work must be done before we can take the distinction between physical and mathematical objects for granted. This distinction has traditionally been drawn by making reference to location, causal powers, detectability in principle, and change in properties. By analysing the ontology of theoretical physics, (...) I wish to show that some physical objects, like quantum particles as they are described by D. Bohm, are as much mathematical as physical, for they do not seem to be located in space and time, and they are as incomplete as mathematical objects. The upshot of this discussion is to address those who are realists about physical objects and anti‐realists about mathematical objects, and, more importantly, to suggest that mathematical realists should not assume that a realist epistemology should be radically different from ordinary scientific epistemology. (shrink)

The chapter presents a largely sympathetic account of W. V. O. Quine’s account of mathematics and logic. The themes of naturalism, confirmational holism, and ontological relativity are discussed in detail, along with the indispensability argument for the truth of mathematical theories and the existence of mathematical objects.

It is usual to regard mathematical objects as entities that can be identified, characterized, and known in isolation. In this chapter, I propose a contrasting view according to which mathematical entities are structureless points or positions in structures that are not distinguishable or identifiable outside the structure. By analysing the various relations that can hold between patterns, like congruence, equivalence, mutual occurrence, I also account for the incompleteness of mathematical objects, for mathematics turns out to be a conglomeration of theories, (...) each dealing with its own structure and each forgoing identities leading outside its structure. My suggestion that mathematical objects are positions in structures or patterns is not intended as an ontological reduction, but rather as a way of viewing mathematical objects and theories that put the phenomena of multiple reductions and ontological and referential relativity in a clearer light. (shrink)

I explore the relation between structuralism and other theses that I have presented in the rest of the book, in particular, my holism, realism about mathematical objects, and the disquotational account of truth. In developing my theory, I have claimed that there is no fact of the matter as to whether the patterns that the various mathematical theories describe are themselves mathematical objects, so I first try to explain what the locution ‘there is no fact of the matter’ means. Next, (...) I discuss the relativity of key structuralist concepts like sub‐pattern and pattern equivalence, and then explore the possibility of formulating structuralist versions of mathematical theories. Even if my holism precludes me from drawing a sharp distinction between philosophy and science I do not regard my structuralism as a mathematical theory but rather as a philosophical account of mathematics that tries to achieve a deeper understanding of the epistemology and ontology of mathematics. My structuralism is epistemic rather than ontic because it is not an ontological reduction or a foundation for mathematics but a philosophical view about the nature of its objects. So when it is combined with realism concerning mathematical objects, it need not commit one to the existence of structures. (shrink)

The application of mathematics to science and the enormous success that derives from it is, perhaps, the strongest evidence in favour of mathematical realism. Quine and Putnam have taken the indispensability of mathematics in doing science as the main premise in an argument for both the truth of mathematics and the existence of mathematical objects. This argument has been criticized, among other things, for presupposing a realist position with regard to science. In this chapter, I propose a new argument, the (...) pragmatic indispensability argument that avoids the problem by failing to presuppose that our best scientific theories are true. I argue that the justification for doing science also justifies our accepting as true, the mathematics that science uses. (shrink)

This paper is concerned with the genesis of mathematical knowledge. While some philosophers might argue that mathematics has no real subject matter and thus is not a body of knowledge, I will not try to dissuade them directly. I shall not attempt such a refutation because it seems clear to me that mathematicians do know such things as the Mean Value Theorem, The Fundamental Theorem of Arithmetic, Godel's Theorems, etc. Moreover, this is much more evident to me than any philosophical (...) view of mathematics I know of — including my own. So I am going to take mathematics as my starting point. (shrink)

I present a hypothetical account of how the ancients might have come to introduce mathematical objects in order to describe patterns, and I explain how working with patterns can generate information about the mathematical realm. The ancients might have started using what I call templates, i.e. concrete devices, like blueprints or drawings, to represent how things are shaped or structured, and this could have evolved into representing the abstract patterns that concrete things might fit. In this way, they might have (...) come to believe that written constructions and computations could provide information about the mathematical realm, for by their very nature, patterns should be structurally analogous to their templates and in positing that they are, one simply projects onto structures and features already attributed to templates. By reflecting on systems of dots representing cardinalities, the ancients could generate a body of results that then evolved into a systematic theory of numbers, but the approach fails when there is no direct connection between the computations and the patterns they are supposed to concern, e.g. those concerning trigonometric functions or transfinite ordinal numbers. In these cases, we forge a connection between proofs and patterns by positing that the premises of the proofs state uncontroversial features of the patterns, i.e. the premises constitute implicit definitions of the patterns. I explain how this position does not entail that mathematics is analytic or a priori. (shrink)