I defend a new position in philosophy of mathematics that I call mathematical inferentialism. It holds that a mathematical sentence can perform the function of facilitating deductive inferences from some concrete sentences to other concrete sentences, that a mathematical sentence is true if and only if all of its concrete consequences are true, that the abstract world does not exist, and that we acquire mathematical knowledge by confirming concrete sentences. Mathematical inferentialism has several advantages over mathematical realism and fictionalism.

State of the art paper on the topic realism/anti-realism. The first part of the paper elucidates the notions of existence and independence of the metaphysical characterization of the realism/anti-realism dispute. The second part of the paper presents a critical taxonomy of the most important positions and doctrines in the contemporary literature on the domains of science and mathematics: scientific realism, scientific anti-realism, constructive empiricism, structural realism, mathematical Platonism, mathematical indispensability, mathematical empiricism, intuitionism, mathematical fictionalism and second philosophy.

The philosophy of mathematics plays an important role in analytic philosophy, both as a subject of inquiry in its own right, and as an important landmark in the broader philosophical landscape. Mathematical knowledge has long been regarded as a paradigm of human knowledge with truths that are both necessary and certain, so giving an account of mathematical knowledge is an important part of epistemology. Mathematical objects like numbers and sets are archetypical examples of abstracta, since we treat such objects in (...) our discourse as though they are independent of time and space; finding a place for such objects in a broader framework of thought is a central task of ontology, or metaphysics. The rigor and precision of mathematical language depends on the fact that it is based on a limited vocabulary and very structured grammar, and semantic accounts of mathematical discourse often serve as a starting point for the philosophy of language. Although mathematical thought has exhibited a strong degree of stability through history, the practice has also evolved over time, and some developments have evoked controversy and debate; clarifying the basic goals of the practice and the methods that are appropriate to it is therefore an important foundational and methodological task, locating the philosophy of mathematics within the broader philosophy of science. (shrink)

Most science requires applied mathematics. This truism underlies the Quine-Putnam indispensability argument: we cannot be mathematical nominalists without rejecting whole swaths of good science that are seamlessly linked with mathematics. One style of response accepts the challenge head-on and attempts to show how to do science without mathematics. There is some consensus that the response fails because the nominalistic apparatus deployed either is not extendible to all of mathematical physics or is merely a deft reconstrual equivalent to standard mathematics. A (...) second style of response denies that indispensability entails realism: when we mathematize a physical problem we treat its physical content as if it were the mathematical representation; provided the two are sufficiently similar, we can use the mathematics to draw conclusions about the physics; even if we cannot represent physical facts without mathematical tools, as-if-fictionalism is reasonable. In this paper I argue that uses of mathematics in science reach deeper than is appreciated by this second response and, indeed, in the more general literature. More specifically, our confidence that we can use the mathematics to draw conclusions about the physics itself depends on mathematics. If the mathematical premises we employ in concluding that a certain application is trustworthy are false, we may lack a justification for supposing that the application will reliably lead us from correct input to correct output. For example, solutions to many physical problems require the determination of a function satisfying a differential equation. Sometimes the existence of a solution for initial value problems can be established directly; where direct methods fail, the existence of a solution must be established indirectly, generally by constructing a sequence of functions that converges to a limit function that satisfies the initial value problem. Moreover, the solution often cannot be evaluated by analytic methods, and scientists must rely on finite element numerical methods to approximate the solution. Mathematical analysis of errors provides further useful information governing the choice of approximation method and of the step size and number of elements needed for the approximation to reach a desired precision. Mathematical physicists rely on the background mathematical theories presupposed in proving the existence of the solutions and approximating them. It is difficult to see how they could do this while adding the fictionalist disclaimer, “But, you know, I don’t believe any of the mathematics I’m using”. It is difficult to see how a fictionalist pursuing the second strategy can account for the soundness of mathematical reasoning in mathematical physics and elsewhere in the sciences. The paper will fill out this argument by appeal to examples and attempt to make clear how mathematics is indispensable to understanding – and thus underwriting our confidence in – applications that would otherwise be shaky approximations and idealizations and how this role is difficult to square with fictionalism. (shrink)

The contemporary Platonists in the philosophy of mathematics argue that mathematical objects exist. One of the arguments by which they support this standpoint is the so-called Enhanced Indispensability Argument (EIA). This paper aims at pointing out the difficulties inherent to the EIA. The first is contained in the vague formulation of the Argument, which is the reason why not even an approximate scope of the set objects whose existence is stated by the Argument can be established. The second problem is (...) reflected in the vagueness of the very term indispensability, which is essential to the Argument. The paper will remind of a recent definition of the concept of indispensability of a mathematical object, reveal its deficiency and propose an improvement of this definition. Following this, we will deal with one of the consequences of the arbitrary employment of the concept of indispensability of a mathematical theory. We will propose a definition of this concept as well, in accordance with the common intuition about it. Eventually, on the basis of these two definitions, the paper will describe the relation between these two concepts, in the attempt to clarify the conceptual apparatus of the EIA. (shrink)