Ontology and the Foundations of Mathematics

This Element looks at the problem of inter-translation between mathematical realism and anti-realism and argues that so far as realism is inter-translatable with anti-realism, there is a burden on the realist to show how her posited reality differs from that of the anti-realist. It also argues that an effective defence of just such a difference needs a commitment to the independence of mathematical reality, which in turn involves a commitment to the ontological access problem – the problem of how knowable mathematical truths are identifiable with a reality independent of us as knowers. Specifically, if the only access problem acknowledged is the epistemological problem – i.e. the problem of how we come to know mathematical truths – then nothing is gained by the realist notion of an independent reality and in effect, nothing distinguishes realism from anti-realism in mathematics.

Social constructivism in mathematics? The promise and shortcomings of Julian Cole’s institutional account

The core idea of social constructivism in mathematics is that mathematical entities are social constructs that exist in virtue of social practices, similar to more familiar social entities like institutions and money. Julian C. Cole has presented an institutional version of social constructivism about mathematics based on John Searle’s theory of the construction of the social reality. In this paper, I consider what merits social constructivism has and examine how well Cole’s institutional account meets the challenge of accounting for the characteristic features of mathematics, especially objectivity and applicability. I propose that in general social constructivism shows promise as an ontology of mathematics, because the view can agree with mathematical practice and it offers a way of understanding how mathematical entities can be real without conflicting with a scientific picture of reality. However, I argue that Cole’s specific theory does not provide an adequate social constructivist account of mathematics. His institutional account fails to sufficiently explain the objectivity and applicability of mathematics, because the explanations are weakened and limited by the three-level theoretical model underlying Cole’s account of the construction of mathematical reality and by the use of the Searlean institutional framework. The shortcomings of Cole’s theory give reason to suspect that the Searlean framework is not an optimal way to defend the view that mathematical reality is socially constructed.

Computers as Fresh Mathematical Reality. I. Personal account

In the last decades there was much ado about computer proofs, computer aided proofs, computer verified proofs, etc. It is obvious that the advent and proliferation of computers have drastically changed applications of mathematics. What one discusses much less, however, is how computers changed mathematics itself, and mathematicians’ stance in regard of mathematical reality, both as far as the possibilities to immediately observe it, and the apprehension of what we can hope to prove. I am recounting my personal experience of using computers as a mathematical tool, and the experience of such similar use in the works of my colleagues that I could observe at close range. This experience has radically changed my perception of many aspects of mathematics, how it functions, and especially, how it should be taught. This first introductory part consists mostly of reminiscences and some philosophical observations. Further parts describe several specific important advances in algebra and number theory, that would had been impossible without computers.

Computers as novel mathematical reality. III. Mersenne numbers and divisor sums

Nowhere in mathematics is the progress resulting from the advent of computers is as apparent, as in the additive number theory. In this part, we describe the role of computers in the investigation of the oldest function studied in mathematics, the divisor sum. The disciples of Pythagoras started to systematically explore its behaviour more that 2500 years ago. A description of the trajectories of this function — perfect numbers, amicable numbers, sociable numbers, and the like — constitute the contents of several problems stated over 2500 years ago, which still seem completely inaccessible. A theorem due to Euclid and Euler reduces classification of even perfect numbers to Mersenne primes. After 1914 not a single new Mersenne prime was ever produced manually, since 1952 all of them have been discovered by computers. Using computers, now we construct hundreds or thousands times more new amicable pairs daily, than what was constructed by humans over several millenia. At the end of the paper, we discuss yet another problem posed by Catalan and Dickson

Antirealism in the philosophy of mathematics

Realism in the philosophy of mathematics is the position that takes mathematics at face value. According to realists, mathematics is the science of mathematical objects (numbers, sets, lines and so on); mathematicians, to use the old metaphor, are discoverers, not inventors. Moreover, just as there may be truths about physical reality which we can never know, so too, realists say, there may be truths about mathematical reality which we can never know. It is this claim in particular which antirealists find unacceptable. Equating what can be known in mathematics with what can be proved, they insist that only what can be proved is true. (Only what can be proved: different accounts of what this ‘can’ means, facing different difficulties, generate different positions.) This leads antirealists to recoil not only from realism but also from the practice of mathematicians themselves. For the orthodox assumption that every mathematical statement is either true or false would be invalidated, on the antirealist view, by a statement that was neither provable nor disprovable. Not that antirealists themselves can see it in these terms. For if a statement were neither provable nor disprovable, that would itself be an unprovable truth about mathematical reality. Antirealists must learn how to be circumspect even in defence of their own circumspection.

Intuitionistic logic and antirealism

The law of excluded middle (LEM) says that every sentence of the form A∨¬A (‘A or not A’) is logically true. This law is accepted in classical logic, but not in intuitionistic logic. The reason for this difference over logical validity is a deeper difference about truth and meaning. In classical logic, the meanings of the logical connectives are explained by means of the truth tables, and these explanations justify LEM. However, the truth table explanations involve acceptance of the principle of bivalence, that is, the principle that every sentence is either true or false. The intuitionist does not accept bivalence, at least not in mathematics. The reason is the view that mathematical sentences are made true and false by proofs which mathematicians construct. On this view, bivalence can be assumed only if we have a guarantee that for each mathematical sentence, either there is a proof of the truth of the sentence, or a proof of its falsity. But we have no such guarantee. Therefore bivalence is not intuitionistically acceptable, and then neither is LEM. A realist about mathematics thinks that if a mathematical sentence is true, then it is rendered true by the obtaining of some particular state of affairs, whether or not we can know about it, and if that state of affairs does not obtain, then the sentence is false. The realist further thinks that mathematical reality is fully determinate, in that every mathematical state of affairs determinately either obtains or does not obtain. As a result, the principle of bivalence is taken to hold for mathematical sentences. The intuitionist is usually an antirealist about mathematics, rejecting the idea of a fully determinate, mind-independent mathematical reality. The intuitionist’s view about the truth-conditions of mathematical sentences is not obviously incompatible with realism about mathematical states of affairs. According to Michael Dummett, however, the view about truth-conditions implies antirealism. In Dummett’s view, a conflict over realism is fundamentally a conflict about what makes sentences true, and therefore about semantics, for there is no further question about, for example, the existence of a mathematical reality than as a truth ground for mathematical sentences. In this vein Dummett has proposed to take acceptance of bivalence as actually defining a realist position. If this is right, then both the choice between classical and intuitionistic logic and questions of realism are fundamentally questions of semantics, for whether or not bivalence holds depends on the proper semantics. The question of the proper semantics, in turn, belongs to the theory of meaning. Within the theory of meaning Dummett has laid down general principles, from which he argues that meaning cannot in general consist in bivalent truth-conditions. The principles concern the need for, and the possibility of, manifesting one’s knowledge of meaning to other speakers, and the nature of such manifestations. If Dummett’s argument is sound, then bivalence cannot be justified directly from semantics, and may not be justifiable at all.

Mathematical Enargeia: The Rhetoric of Early Modern Mathematical Notation

This article proposes and explicates a rhetorical model for the function of notational writing in sixteenth- and seventeenth-century European mathematics. Drawing on enargeia's requirement that both author and reader contribute to the full realization of a text, mathematical enargeia enables the transformation of images of mathematical imagination resulting from an encounter with mathematical writing into further written acts of mathematical creation. Mathematical enargeia provides readers with an ability to understand a text as if they created it themselves. Within the period's dominant reading of classical geometry as a synthetic presentation that suppressed, hid, or obscured analytic mathematical reality, notational mathematics found favor as a rhetorically unmediated expression of mathematical truth. Consequently, mathematical enargeia creates an operational and presentational link between mathematics' past and its future.