Online research in philosophy

Proof and Knowledge in Mathematics tackles the main problem that arises when considering an epistemology for mathematics, the nature and sources of mathematical justification. Focusing both on particular and general issues, these essays from leading philosophers of mathematics raise important issues for our current understanding of mathematics. Is mathematical justification a priori or a posteriori? What role, if any, does logic play in mathematical reasoning or inference? And how epistemologically important is the formalizability of proof? Michael Detlefsen has brought together an outstanding collection of essays in a volume which will be essential for philosophers and historians of mathematics who are interested in the nature of reasoning and justification. A companion volume, Proof, Knowledge and Formalization is also available from Routledge.

Do mathematical objects exist in some realm inaccessible to our senses? It may be tempting to deny this. For how we could come to know mathematical truths, if such knowledge must arise from causal interaction with non-empirical objects? Among current positions, literalists argue that mathematical objects simply exist in the empirical world, and fictionalists hold that, strictly speaking, mathematical objects do not exist but are rather harmless fictions. Both positions have been ascribed to Aristotle. The ascription of literalism to Aristotle, however, commits Aristotle to the unattractive view that mathematics studies but a small fragment of the physical world; and there is evidence that Aristotle would deny the literalist position that mathematical objects are perceivable. The ascription of fictionalism also faces a difficult challenge: there is evidence that Aristotle would deny the fictionalist position that mathematics is false. I will argue that, in Aristotle’s view, the fiction of mathematics is not to treat what does not exist as if existing but to treat mathematical objects with an ontological status they lack. This form of fictionalism is consistent with holding that mathematics is true.

This paper compares the statement ‘Mathematics is the study of structure’ with the actual practice of mathematics. We present two examples from contemporary mathematical practice where the notion of structure plays different roles. In the first case a structure is defined over a certain set. It is argued firstly that this set may not be regarded as a structure and secondly that what is important to mathematical practice is the relation that exists between the structure and the set. In the second case, from algebraic topology, one point is that an object can be a place in different structures. Which structure one chooses to place the object in depends on what one wishes to do with it. Overall the paper argues that mathematics certainly deals with structures, but that structures may not be all there is to mathematics.

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.

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.

The philosophy of mathematics has long been concerned with determining the means that are appropriate for justifying claims of mathematical knowledge, and the metaphysical considerations that render them so. But, as of late, many philosophers have called attention to the fact that a much broader range of normative judgments arise in ordinary mathematical practice; for example, questions can be interesting, theorems important, proofs explanatory, concepts powerful, and so on. The associated values are often loosely classified as aspects of “mathematical understanding.” Meanwhile, in a branch of computer science known as “formal verification,” the practice of interactive theorem proving has given rise to software tools and systems designed to support the development of complex formal axiomatic proofs. Such efforts require one to develop models of mathematical language and inference that are more robust than the the simple foundational models of the last century. This essay explores some of the insights that emerge from this work, and some of the ways that these insights can inform, and be informed by, philosophical theories of mathematical understanding.

In this paper, I introduce and examine the notion of “mathematical engineering” and its impact on mathematical change. Mathematical engineering is an important part of contemporary mathematics and it roughly consists of the “construction” and development of various machines, probes and instruments used in numerous mathematical fields. As an example of such constructions, I briefly present the basic steps and properties of homology theory. I then try to show that this aspect of contemporary mathematics has important consequences on our conception of mathematical knowledge, in particular mathematical growth.

Platonism about mathematics (or mathematical platonism as I will mostly call it) is typically defined as the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices. Just as electrons and planets exist independently of us, so do numbers and sets. And just as statements about electrons and planets are made true or false by the objects with which they are concerned and these objects’ perfectly objective properties, so are statements about numbers and sets. If true, mathematical platonism would be of great philosophical significance: it would be a counterexample to common physicalist views, and it would put great pressure on the epistemology of mathematics. The view would also be of significance for mathematical practice.

The philosophy of mathematics of the last few decades is usually distinguished into mainstream and maverick.1 The mainstream philosophy of mathematics considers mathematics as a static body of knowledge; it is mainly concerned with the question of the justification of mathematical knowledge; it holds that there is an absolutely certain, or at least fairly reliable, foundation for mathematics; it considers mathematical logic as a canon for the philosophy of mathematics; it assumes that a detailed account of mathematical practice would be desirable but not really essential; it generally sets itself within the framework of analytic philosophy. The maverick philosophy of mathematics considers mathematics as a dynamic body of knowledge; it is mainly concerned with the question of the growth of mathematical knowledge, including the dynamics of mathematical discovery; it holds that there is no absolutely certain foundation for mathematics; it considers mathematical logic very useful to show the limitations of the mainstream philosophy of mathematics by means of the limitative results, but inadequate to deal with the question of the growth of mathematical knowledge; it assumes that only a detailed analysis of mathematical practice could lead to a philosophy of mathematics worth its name; it generally sets itself outside the framework of analytic philosophy. The mainstream philosophy of mathematics consists of the three big foundational schools of the first few decades of the twentieth century, namely logicism (Frege, Russell), formalism (Hilbert), intuitionism (Brouwer, Heyting), and the positions which ensued from them in the second half of the twentieth..

If mathematics is regarded as a science, then the philosophy of mathematics can be regarded as a branch of the philosophy of science, next to disciplines such as the philosophy of physics and the philosophy of biology. However, because of its subject matter, the philosophy of mathematics occupies a special place in the philosophy of science. Whereas the natural sciences investigate entities that are located in space and time, it is not at all obvious that this is also the case with respect to the objects that are studied in mathematics. In addition to that, the methods of investigation of mathematics differ markedly from the methods of investigation in the natural sciences. Whereas the latter acquire general knowledge using inductive methods, mathematical knowledge appears to be acquired in a different way, namely, by deduction from basic principles. The status of mathematical knowledge also appears to differ from the status of knowledge in the natural sciences. The theories of the natural sciences appear to be less certain and more open to revision than mathematical theories. For these reasons mathematics poses problems of a quite distinctive kind for philosophy. Therefore philosophers have accorded special attention to ontological and epistemological questions concerning mathematics.