Mathematics as a Science of Patterns

Abstract

Abstract Mathematics is regarded as our most developed science, and yet philosophical troubles surface as soon as we inquire about its subject matter partly because mathematics itself says nothing about the metaphysical nature of its objects. Taking mathematics at face value seems to favour the Platonist view according to which mathematics concerns causally inert objects existing outside space‐time, but this view seems to preclude any account of how we acquire mathematical knowledge without using some mysterious intellectual intuition. In this book, I defend a version of mathematical realism, motivated by the indispensability of mathematics in science, according to which (1) mathematical objects exist independently of us and our constructions, (2) much of contemporary mathematics is true, and (3) mathematical truths obtain independently of our beliefs, theories, and proofs. The ontological component of my realism is a form of structuralism according to which mathematical objects are featureless, abstract positions in structures, or patterns, and like geometric points, their identities are fixed only through their relationships to each other. Structuralism is also part of my epistemology in that material objects ‘fit’ simple patterns, and in doing so, they ‘fill’ the positions of simple mathematical structures. We may perceive the arrangements of objects but we cannot perceive their positions i.e. the abstract, non‐spatiotemporal mathematical objects, and the problem then consists in explaining how we can form beliefs about them. Answering this question introduces a central notion of my epistemology, that of a posit: by representing and designing patterned objects our ancestors posited geometric objects as sui generis and started describing them by describing the patterns in which they are positions. Since positing mathematical objects, like positing new scientific entities, is an activity similar to making up a story, one might wonder how such an activity can lead to mathematical knowledge and truth, but I believe that our ancestors were justified in introducing mathematical objects and we are justified in retaining them, by pragmatic and global considerations: mathematics has proved immensely fruitful for science, technology, and practical life, and doing without it is now virtually impossible. This account of justification introduces a further problem: if our justification for believing in mathematical truths is global and pragmatic, then it might turn out that one is not justified in accepting a mathematical claim unless it is accepted by science, and this is clearly at odds with the practice of mathematics where we hardly ever invoke such global considerations in order to justify a mathematical claim. In mathematics, we usually employ a local conception of evidence made up mainly of a priori proofs. However, arguing from the perspective of a Quinean epistemic holism, I claim that this feature of the practice should not make us conclude that mathematics is an a priori science, disconnected evidentially from both observation and natural science, for observation is relevant to mathematics, and technological and scientific success forms a vital part of our justification for believing in the truth of mathematics.