Online research in philosophy

What is the nature of mathematical knowledge? Is it anything like scientific knowledge or is it sui generis? How do we acquire it? Should we believe what mathematicians themselves tell us about it? Are mathematical concepts innate or acquired? Eight new essays offer answers to these and many other questions. Written by some of the world's leading philosophers of mathematics, psychologists, and mathematicians, Mathematical Knowledge gives a lively sense of the current state of debate in this fascinating field. Contents 1. Mary Leng: Introduction 2. Michael Potter: What is the problem of mathematical knowledge? 3. Tim Gowers: Mathematics, memory, and mental arithmetic 4. Alan Baker: Is there a problem of induction for mathematics? 5. Marinella Cappelletti and Valeria Giardino: The cognitive basis of mathematical knowledge 6. Mary Leng: What's there to know? A fictionalist account of mathematical knowledge 7. Mark Colyvan: Mathematical recreation versus mathematical knowledge 8. Alexander Paseau: Scientific platonism 9. Crispin Wright: On quantifying into predicate position: Steps towards a (new)tralist position.

It is often assumed that empiricism in the philosophy of mathematics was laid to rest by Frege’s stinging attack on Mill. I will argue that empiricism is alive and well and able to deal with almost everything that’s thrown at it. In particular, I will show how the brand of empiricism I subscribe to is able to give a satisfying account of mathematical knowledge. This brand of mathematical empiricism has a rather curious feature though: some parts of mathematics (e.g., analysis, modern algebra, ZFC set theory) are taken to be theories about which we have genuine mathematical knowledge, while others (e.g., set theory with large cardinal axioms) are (following Quine) treated as “mathematical recreation”. I will defend this demarcation against some recent criticisms from Mary Leng.

I discuss a puzzle that shows there is a need to develop a new metaphysical interpretation of mathematical theories, because all well-known interpretations conflict with important aspects of mathematical activities. The new interpretation, I argue, must authenticate the ontological commitments of mathematical theories without curtailing mathematicians’ freedom and authority to creatively introduce mathematical ontology during mathematical problem-solving. Further, I argue that these two constraints are best met by a metaphysical interpretation of mathematics that takes mathematical entities to be constitutively constructed by human activity in a manner similar to the constitutive construction of the US Supreme Court by certain legal and political activities. Finally, I outline some of the philosophical merits of metaphysical interpretations of mathematical theories of this type.

In historical claims for nativism, mathematics is a paradigmatic example of innate knowledge. Claims by contemporary developmental psychologists of elementary mathematical skills in human infants are a legacy of this. However, the connection between these skills and more formal mathematical concepts and methods remains unclear. This paper assesses the current debates surrounding nativism and mathematical knowledge by teasing them apart into two distinct claims. First, in what way does the experimental evidence from infants, nonhuman animals and neuropsychology support the nativist hypothesis? Second, granting that infants have some elementary mathematical skills, does this mean that such skills play an important role in the development of mathematical knowledge?

This paper considers the nature and role of axioms from the point of view of the current debates about the status of category theory and, in particular, in relation to the “algebraic” approach to mathematical structuralism. My aim is to show that category theory has as much to say about an algebraic consideration of meta-mathematical analyses of logical structure as it does about mathematical analyses of mathematical structure, without either requiring an assertory mathematical or meta-mathematical background theory as a “foundation”, or turning meta-mathematical analyses of logical concepts into “philosophical” ones. Thus, we can use category theory to frame an interpretation of mathematics according to which we can be structuralists all the way down.

Fictionalism in the philosophy of mathematics is the view that mathematical statements, such as ‘8+5=13’ and ‘π is irrational’, are to be interpreted at face value and, thus interpreted, are false. Fictionalists are typically driven to reject the truth of such mathematical statements because these statements imply the existence of mathematical entities, and according to fictionalists there are no such entities. Fictionalism is a nominalist (or anti-realist) account of mathematics in that it denies the existence of a realm of abstract mathematical entities. It should be contrasted with mathematical realism (or Platonism) where mathematical statements are taken to be true, and, moreover, are taken to be truths about mathematical entities. Fictionalism should also be contrasted with other nominalist philosophical accounts of mathematics that propose a reinterpretation of mathematical statements, according to which the statements in question are true but no longer about mathematical entities. Fictionalism is thus an error theory of mathematical discourse: at face value mathematical discourse commits us to mathematical entities and although we normally take many of the statements of this discourse to be true, in doing so we are in error (cf. error theories in ethics).

Current versions of nominalism in the philosophy of mathematics face a significant problem to understand mathematical knowledge. They are unable to characterize mathematical knowledge as knowledge of the objects mathematical theories are taken to be about. Oswaldo Chateaubriand’s insightful reformulation of Platonism (Chateaubriand 2005) avoids this problem by advancing a broader conception of knowledge as justified truth beyond a reasonable doubt, and by introducing a suitable characterization of logical form in which the relevant mathematical facts play an important role in the truth of the corresponding mathematical propositions. In this paper, I contrast Chateaubriand’s proposal with an agnostic form of nominalism that is able to accommodate mathematical knowledge without the commitment to mathematical facts.

(I) MATHEMATICAL LECTURES. LECTURE I. Of the Name and general Division of the Mathematical Sciences. BEING about to treat upon the Mathematical Sciences, ...

Although all mathematical truths are necessary, mathematicians take certain combinations of mathematical truths to be ‘coincidental’, ‘accidental’, or ‘fortuitous’. The notion of a ‘mathematical coincidence’ has so far failed to receive sufficient attention from philosophers. I argue that a mathematical coincidence is not merely an unforeseen or surprising mathematical result, and that being a misleading combination of mathematical facts is neither necessary nor sufficient for qualifying as a mathematical coincidence. I argue that although the components of a mathematical coincidence may possess a common explainer, they have no common explanation; that two mathematical facts have a unified explanation makes their truth non-coincidental. I suggest that any motivation we may have for thinking that there are mathematical coincidences should also motivate us to think that there are mathematical explanations, since the notion of a mathematical coincidence can be understood only in terms of the notion of a mathematical explanation. I also argue that the notion of a mathematical coincidence plays an important role in scientific explanation. When two phenomenological laws of nature are similar, despite concerning physically distinct processes, it may be that any correct scientific explanation of their similarity proceeds by revealing their similarity to be no mathematical coincidence.

Baker (2005) claims to provide an example of mathematical explanation of an empirical phenomenon which leads to ontological commitment to mathematical objects. This is meant to show that the positing of mathematical entities is necessary for satisfactory scientific explanations and thus that the application of mathematics to science can be used, at least in some cases, to support mathematical realism. In this paper I show that the example of explanation Baker considers can actually be given without postulating mathematical objects and thus cannot be used by the mathematical realist. I also show that, despite this, mathematics keeps playing an important methodological role in the explanation and does not reduce to a merely computational or descriptive framework.