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)

Michael Dummett argued that Frege was a realist while Hans Sluga countered that he was an objective idealist in the rationalist tradition of Kant and Lotze. Sluga ties Frege's idealism to the context principle which he argues Frege never gave up. It is argued that Sluga has correctly interpreted the pre?1891 Frege while Dummett is correct concerning the later period. It is also claimed that the context principle was dropped prior to 1891 to be replaced by the doctrine of (...) unsaturated entities. (shrink)

CONTENTS: 1 Introductory Remark; 2 Formalism of Empirical Theories; 3 Semantics of Formalized Languages; 4 Interpretation of Empirical Theories; 5 Interpretation of Observational Terms; 6 Interpretation of Theoretical Terms; 7 Main Types of Meaning Postulates for Theoretical Terms; 8 Some Other Kinds of Meaning Postulates for Theoretical Terms; 9 Main Types of Statements in an Empirical Theory; 10 Towards a More Realistic Account; 11 Concluding Remarks; 12 Bibliographical Note.

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.

This book expounds 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 defense of realism about the metaphysics of mathematics--the view (...) that mathematics is about things that really exist. (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.

One of the strongest motivations for being an anti‐realist with regard to mathematics is the difficulty in formulating a plausible realist epistemology, given that there seems to be a lack of ties between the mathematical apparatus and observation. In this chapter, I discuss a few puzzles that the mathematical realist has to solve in order to formulate an acceptable epistemology, and I hint at the direction in which one might hope to find the solution to these puzzles. One of the (...) puzzles, that was first clearly formulated by Paul Benacerraf, is that since mathematical objects are supposed to be causally inert entities existing outside of space and time, it is hard to see how we can ever get in contact with them, i.e. how we can ever gain knowledge of them or refer to them. A second puzzle concerns the incompleteness of mathematical entities that has it, that we have no conceivable evidence and so no answers to questions of whether the objects that one mathematical theory discusses are identical to those that another theory treats. Realists can reject the presuppositions of both these puzzles. The first presupposes that there must be some sort of direct or indirect interaction between us and the objects of our knowledge. The second assumes that there is always a fact of the matter as to whether the objects of one theory are the same or distinct, as those of another. In later chapters, I will try to undermine both of these presuppositions. (shrink)

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)

Mathematics has often been described as the ‘Queen of Sciences’, yet philosophical problems arise as soon as one tries to define its subject matter. Anti‐realism concerning mathematical objects has proven to be problematic, but, on the other hand, realism gives rise to well‐known epistemological problems. In this book, I offer a type of mathematical realism according to which mathematical objects exist independently of our constructions, and mathematical truths are obtained independently of our beliefs. Central to my account of mathematical objects (...) is a sort of structuralism, according to which mathematical objects are featureless, abstract positions in structures. On the epistemological level, I reject the apriority of mathematical knowledge, and claim that our justification for believing in the truth of mathematics is derived from pragmatic and holistic considerations concerning science. (shrink)

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 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)

If, as I grant, mathematical objects are abstract entities existing outside of space and time, and if the idea of supernaturally grasping abstract entities is scientifically unacceptable, then we need to explain how we can attain mathematical knowledge using our ordinary faculties. I try to meet this challenge through a postulational account of the genesis of our mathematical knowledge, according to which our ancestors introduced mathematical objects by first positing geometric ideals and then postulating abstract mathematical entities. Since positing involves (...) simply introducing a discourse about objects and affirming their existence, positing mathematical objects involves nothing more serious than writing fiction. For this reason, postulational approaches seem better suited for conventionalists; so in the second part of this chapter, I explain how positing in mathematics is different from positing in fiction, and how we can gain knowledge from the former. Finally, I try to make sense of the idea that mathematical postulates are about an independent mathematical reality and that we can refer to that reality through them, by giving an immanent and disquotational account of reference and contrasting it with a transcendent/causal account. (shrink)

The indispensability thesis maintains both that using mathematical terms and assertions is an indispensable part of scientific practice and that this practice commits science to mathematical objects and truths. Anti‐realists have used several methods for attacking this thesis: Hartry Field has tried to show how science can do without mathematics by showing that it is possible to replace analytic mathematical scientific theories with synthetic versions that make no reference to mathematical objects. Phillip Kitcher and Charles Chichara have tried, instead, to (...) maintain the mathematical formalism in science without being committed to mathematical realism, by giving a non‐realist account of mathematical objects. Finally, Geoffrey Hellman devised a modal structuralism that uses modal operators to translate standard mathematical language into a ‘structuralist’ language. In this chapter I discuss these positions and claim that, on the one hand, they have failed to show that science can do without mathematical objects, and on the other, that these approaches to mathematics do not represent an ontic and epistemic gain over standard realism. (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)

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 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)

Talk of truth plays a major role in formulating realism, to the point that realist theories are often criticized by attacking the correspondence theory of truth that they are presumed to defend. In this chapter, I claim that there is an alternative theory of truth, which is both non‐epistemic and not based on correspondence relation that suffices to support mathematical realism. I describe the theory as a logical conception of truth because the truth predicate will turn out to be simply (...) a logical operator. The theory has two main features: it is disquotational, and immanent, in that it specifies the extension of the term ‘true’ only for the language in which it applies. (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)

Penelope Maddy and Elliott Sober recently attacked the confirmational indispensability argument for mathematical realism. We cannot count on science to provide evidence for the truth of mathematics, they say, because either scientific testing fails to confirm mathematics (Sober) or too much mathematics occurs in false scientific theories (Maddy). I present a pragmatic indispensability argument immune to these objections, and show that this argument supports mathematical realism independently of scientific realism. Mathematical realism, it turns out, may be even more firmly established (...) than scientific realism. (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.