Online research in philosophy

According to standard mathematical platonism, mathematical entities (numbers, sets, etc.) are abstract entities. As such, they lack causal powers and spatio-temporal location. Platonists owe us an account of how we acquire knowledge of this inaccessible mathematical realm. Some recent versions of mathematical platonism postulate a plenitude of mathematical entities, and Mark Balaguer has argued that, given the existence of such a plenitude, the attainment of mathematical knowledge is rendered non-problematic. I assess his epistemology for such a profligate platonism and find it unsatisfactory because it lacks an adequate semantics, in particular, an adequate account of reference.

In this article, I will discuss the relationship between mathematical intuition and mathematical visualization. I will argue that in order to investigate this relationship, it is necessary to consider mathematical activity as a complex phenomenon, which involves many different cognitive resources. I will focus on two kinds of danger in recurring to visualization and I will show that they are not a good reason to conclude that visualization is not reliable, if we consider its use in mathematical practice. Then, I will give an example of mathematical reasoning with a figure, and show that both visualization and intuition are involved. I claim that mathematical intuition depends on background knowledge and expertise, and that it allows to see the generality of the conclusions obtained by means of visualization.

This paper attempts to show that mathematical knowledge does not grow by a simple process of accumulation and that it is possible to provide a quasi-empirical (in Lakatos's sense) account of mathematical theories. Arguments supporting the first thesis are based on the study of the changes occurred within Eudidean geometry from the time of Euclid to that of Hilbert; whereas those in favour of the second arise from reflections on the criteria for refutation of mathematical theories.

Suggests that the recent emphasis on Benacerraf's access problem locates the peculiarity of mathematical knowledge in the wrong place. Instead we should focus on the sense in which mathematical concepts are or might be "armchair concepts" – concepts about which non-trivial knowledge is obtainable a priori.

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).

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..

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.

Penelope Maddy’s original solution to the dilemma posed by Benacerraf in his (1973) ‘Mathematical Truth’ was to reconcile mathematical empiricism with mathematical realism by arguing that we can perceive realistically construed sets. Though her hypothesis has attracted considerable critical attention, much of it, in my view, misses the point. In this paper I vigorously defend Maddy’s (1990) account against published criticisms, not because I think it is true, but because these criticisms have functioned to obscure a more fundamental issue that is well worth addressing: in general – and not only in the mathematical domain – empiricism and realism simply cannot be reconciled by means of an account of perception anything like Maddy’s. But because Maddy’s account of perception is so plausible, this conclusion raises the specter of the broader incompatibility of realism and empiricism, which contemporary philosophers are frequently at pains to forget.

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.

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.