Traditional geometry concerns itself with planimetric and stereometric considerations, which are at the root of the division between plane and solid geometry. To raise the issue of the relation between these two areas brings with it a host of different problems that pertain to mathematical practice, epistemology, semantics, ontology, methodology, and logic. In addition, issues of psychology and pedagogy are also important here. To our knowledge there is no single contribution that studies in detail even one of the aforementioned areas.

There is an urgent need in philosophy of mathematics for new approaches which pay closer attention to mathematical practice. This book will blaze the trail: it offers philosophical analyses of important characteristics of contemporary mathematics and of many aspects of mathematical activity which escape purely formal logical treatment.

In his new foreword to this edition, Hilary Putnam forcefully rejects these nativist claims.

This book argues against the view that mathematical knowledge is a priori,contending that mathematics is an empirical science and develops historically,just as ...

Many explanations in science make use of mathematics. But are there cases where the mathematical component of a scientific explanation is explanatory in its own right? This issue of mathematical explanations in science has been for the most part neglected. I argue that there are genuine mathematical explanations in science, and present in some detail an example of such an explanation, taken from evolutionary biology, involving periodical cicadas. I also indicate how the answer to my title question impacts on broader (...) issues in the philosophy of mathematics; in particular it may help platonists respond to a recent challenge by Joseph Melia concerning the force of the Indispensability Argument. (shrink)

Unlike explanation in science, explanation in mathematics has received relatively scant attention from philosophers. Whereas there are canonical examples of scientific explanations, there are few examples that have become widely accepted as exhibiting the distinction between mathematical proofs that explain why some mathematical theorem holds and proofs that merely prove that the theorem holds without revealing the reason why it holds. This essay offers some examples of proofs that mathematicians have considered explanatory, and it argues that these examples suggest a (...) particular account of explanation in mathematics. The essay compares its account to Steiner's and Kitcher's. Among the topics that arise are proofs that exploit symmetries, mathematical coincidences, brute-force proofs, simplicity in mathematics, merely clever proofs, and proofs that unify what other proofs treat as separate cases. (shrink)

In this ambitious study, David Corfield attacks the widely held view that it is the nature of mathematical knowledge which has shaped the way in which mathematics is treated philosophically and claims that contingent factors have brought us to the present thematically limited discipline. Illustrating his discussion with a wealth of examples, he sets out a variety of approaches to new thinking about the philosophy of mathematics, ranging from an exploration of whether computers producing mathematical proofs or conjectures are doing (...) real mathematics, to the use of analogy, the prospects for a Bayesian confirmation theory, the notion of a mathematical research programme and the ways in which new concepts are justified. His inspiring book challenges both philosophers and mathematicians to develop the broadest and richest philosophical resources for work in their disciplines and points clearly to the ways in which this can be done. (shrink)

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. (shrink)

v. 1. Nominalism and realism.--v. 2. A theory of universals.

In 1908, Henri Poincar? claimed that: ...the mathematical facts worthy of being studied are those which, by their analogy with other facts, are capable of leading us to the knowledge of a mathematical law, just as experimental facts lead us to the knowledge of a physical law. They are those which reveal to us unsuspected kinship between other facts, long known, but wrongly believed to be strangers to one another. Towards the end of the twentieth century, with many more mathematical (...) facts since discovered, several mathematicians proposed overarching schemes to organise the facts they considered most significant. In this paper, I shall briefly discuss three of these schemes (those of Arnold, Atiyah, and Baez and Dolan), before drawing some philosophical consequences from their attempts. Rather than the kind of claim made by Frege that with the entry of imaginary numbers we reach the ?natural end of the domain of numbers?, we are dealing here with a more open-ended sense of conceptual growth. I shall illustrate this theme by discussing the elaboration of algebraic structures designed to measure symmetry. What emerges is that at any one time mathematicians are operating with a notion very similar to that discussed in the philosophy of science under the heading of ?natural kinds?. We find ?quasi-causal? talk of properties being ?responsible? for a phenomenon, projectability, the transfer of robust mechanisms between domains, and reference to entities not yet fully determined. Debates in philosophy of science prompt further questioning as to the ?naturalness? of these mathematical kinds, whether one should expect them to be dependent on varying human interests, whether there is a distinction between artefactual and real kinds, and whether there is convergence of kinds. I believe that these questions present a wonderful opportunity for a philosophy of mathematics to treat ?real? mathematics, while making powerful points of contact with philosophy of science, and philosophy in general. (shrink)

Certain scientific explanations of physical facts have recently been characterized as distinctively mathematical –that is, as mathematical in a different way from ordinary explanations that employ mathematics. This article identifies what it is that makes some scientific explanations distinctively mathematical and how such explanations work. These explanations are non-causal, but this does not mean that they fail to cite the explanandum’s causes, that they abstract away from detailed causal histories, or that they cite no natural laws. Rather, in these explanations, (...) the facts doing the explaining are modally stronger than ordinary causal laws or are understood in the why question’s context to be constitutive of the physical arrangement at issue. A distinctively mathematical explanation works by showing the explanandum to be more necessary than ordinary causal laws could render it. Distinctively mathematical explanations thus supply a kind of understanding that causal explanations cannot. 1 Introduction2 Some Distinctively Mathematical Scientific Explanations3 Are Distinctively Mathematical Explanations Set Apart by their Failure to Cite Causes? 4 Distinctively Mathematical Explanations do not Exploit Causal Powers5 How these Distinctively Mathematical Explanations Work6 Conclusion. (shrink)

v. 1. Language.--v. 2. Mythical thought.--v. 3. The phenomenology of knowledge.--v. 4. The metaphysics of symbolic forms.

In the spirit of James and Dewey, I ask what one might want from a theory of knowledge. Much Anglophone epistemology is centered on questions that were once highly pertinent, but are no longer central to broader human and scientific concerns. The first sense in which epistemology without history is blind lies in the tendency of philosophers to ignore the history of philosophical problems. A second sense consists in the perennial attraction of approaches to knowledge that divorce knowing subjects from (...) their societies and from the tradition of socially assembling a body of transmitted knowledge. When epistemology fails to use the history of inquiry as a laboratory in which methodological claims can be tested, there is a third way in which it becomes blind. Finally, lack of attention to the growth of knowledge in various domains leaves us with puzzles about the character of the knowledge we have. I illustrate this last theme by showing how reflections on the history of mathematics can expand our options for understanding mathematical knowledge. (shrink)

In this ambitious study, David Corfield attacks the widely held view that it is the nature of mathematical knowledge which has shaped the way in which mathematics is treated philosophically, and claims that contingent factors have brought us to the present thematically limited discipline. Illustrating his discussion with a wealth of examples, he sets out a variety of new ways to think philosophically about mathematics, ranging from an exploration of whether computers producing mathematical proofs or conjectures are doing real mathematics, (...) to the use of analogy, the prospects for a Bayesian confirmation theory, the notion of a mathematical research programme, and the ways in which new concepts are justified. His highly original book challenges both philosophers and mathematicians to develop the broadest and richest philosophical resources for work in their disciplines, and points clearly to the ways in which this can be done. (shrink)