The view that a mathematical proof is a sketch of or recipe for a formal derivation requires the proof to function as an argument that there is a suitable derivation. This is a mathematical conclusion, and to avoid a regress we require some other account of how the proof can establish it.

It is conventionally understood that computers play a rather limited role in theoretical mathematics. While computation is indispensable in applied mathematics and the theory of computing and algorithms is rich and thriving, one does not, even today, expect to find computers in theoretical mathematics settings beyond the theory of computing. Where computers are used, by those studying combinatorics , algebra, number theory, or dynamical systems, the computer most often assumes the role of an automated and speedy theoretician, performing manipulations and (...) checking cases in a way assumed to be possible for human theoreticians, if only they had the time, the memory, and the precision. Automated proofs have become standard tools in mathematical logic, and it is often expected that proofs be published in a computer-checkable format. It is not surprising, then, that most philosophical work on computers in theoretical mathematics has been on computers' roles as supplementary mathematicians. Donald MacKenzie's 2001 book Mechanizing Proof demonstrates the rich potential for social and historical studies to complement the substantial analytic debate in this area of philosophy. But what of computers in theoretical mathematics behaving as computers, and not as mere mechanized mathematicians? Very little role is commonly assumed for computers working as supplements to mathematicians, rather than as supplementary mathematicians themselves. Accordingly, very little philosophy has attempted to grapple with theoretical mathematics in which computers play an essential but essentially non-theoretical role. My presentation will draw on work I conducted as a researcher in harmonic analysis on fractals at Cornell University. I will analyze the explicit and implicit conceptual apparatus employed in my and my fellow researchers' use of computers in the theoretical study of second order differential equations, such as those for sound and heat flow, on various fractal analogues of the Sierpinski gasket. Such gaskets are easy to visualize in very crude approximation in a low number of dimensions. As one increases the complexity of the gasket or the refinement of one's analysis, visualization and precise computation become impossible, and soon computers are unable to produce even approximate data to model differential equations in these situations. We thus had to carefully choose analytic approaches and methods to make our theoretical mathematics amenable to computer simulation. In my case, studying the transformation of the gaskets as they are expanded into increasingly high dimensions, computer simulation eventually required that the problem be reimagined entirely in terms of interlinked systems of parameters. This computer-approximation-driven theoretical orientation shaped my mathematical intuitions toward the problem and guided my fellow researchers and me in both theoretical and computational directions. We discovered both that computer approximation could be incredibly powerful as an aid to intuition, and that it can be incredibly difficult to transfer computer-oriented mathematics back into the purely theoretical standards of our area of specialty. I will address the philosophical implications of computer-driven theoretical mathematics, asking how computer experiments can shape both the content and standards of theoretical sciences. (shrink)

In recent years, philosophical work directly concerned with the practice of mathematics has intensified, giving rise to a movement known as the philosophy of mathematical practice . In this paper we offer a survey of this movement aimed at mathematics educators. We first describe the core questions philosophers of mathematical practice investigate as well as the philosophical methods they use to tackle them. We then provide a selective overview of work in the philosophy of mathematical practice covering topics including the (...) distinction between formal and informal proofs, visualization and artefacts, mathematical explanation and understanding, value judgments, and mathematical design. We conclude with some remarks on the potential connections between the philosophy of mathematical practice and mathematics education. (shrink)

Although there exist today a variety of non-deductive reliable processes able to determine the truth of certain mathematical propositions, proof remains the only form of justification accepted in mathematical practice. Some philosophers and mathematicians have contested this commonly accepted epistemic superiority of proof on the ground that mathematicians are fallible: when the deductive method is carried out by a fallible agent, then it comes with its own level of reliability, and so might happen to be equally or even less reliable (...) than existing non-deductive reliable processes—I will refer to this as the reliability argument. The aim of this article is to examine whether the reliability argument forces us to reconsider the commonly accepted epistemic superiority of the deductive method over non-deductive reliable processes. I will argue that the reliability argument is fundamentally correct, but that there is another epistemic property differentiating the deductive method from non-deductive reliable processes. This property is based on the observation that although mathematicians are fallible agents, they are also self-correcting agents. This means that when a proof is produced that only contains repairable mistakes, given enough time and energy, a mathematician or a group thereof should be able to converge towards a correct proof through a finite number of verification and correction rounds, thus providing a guarantee that the considered proposition is true, something that non-deductive reliable processes will never be able to produce. From this perspective, the standard of justification adopted in mathematical practice should be read in a diachronic way: the demand is not that any proof that is ever produced be correct—which would amount to requiring that mathematicians are infallible—but rather that, over time, proofs that contain repairable mistakes be corrected, and proofs that cannot be repaired be rejected. (shrink)

In this contribution, I comment on Raffaella Campaner’s defense of explanatory pluralism in psychiatry (in this volume). In her paper, Campaner focuses primarily on explanatory pluralism in contrast to explanatory reductionism. Furthermore, she distinguishes between pluralists who consider pluralism to be a temporary state on the one hand and pluralists who consider it to be a persisting state on the other hand. I suggest that it would be helpful to distinguish more than those two versions of pluralism – different understandings (...) of explanatory pluralism both within philosophy of science and psychiatry – namely moderate/temporary pluralism, anything goes pluralism, isolationist pluralism, integrative pluralism and interactive pluralism. Next, I discuss the pros and cons of these different understandings of explanatory pluralism. Finally, I raise the question of how to implement or operationalize explanatory pluralism in scientific practice; how to structure the “genuine dialogue” or shape “the pluralistic attitude” Campaner is referring to. As tentative answers, I explore a question-based framework for explanatory pluralism as well as social-epistemological procedures for interaction among competing approaches and explanations. (shrink)

Philosophical analysis of mathematical knowledge are commonly conducted within the realist/antirealist dichotomy. Nevertheless, philosophers working within this dichotomy pay little attention to the way in which mathematics evolves and structures itself. Focusing on mathematical practice, I propose a weak notion of objectivity of mathematical knowledge that preserves the intersubjective character of mathematical knowledge but does not bear on a view of mathematics as a body of mind-independent necessary truths. Furthermore, I show how that the successful application of mathematics in science (...) is an important trigger for the objectivity of mathematical knowledge. (shrink)

This paper assumes the success of arguments against the view that informal mathematical proofs secure rational conviction in virtue of their relations with corresponding formal derivations. This assumption entails a need for an alternative account of the logic of informal mathematical proofs. Following examination of case studies by Manders, De Toffoli and Giardino, Leitgeb, Feferman and others, this paper proposes a framework for analysing those informal proofs that appeal to the perception or modification of diagrams or to the inspection or (...) imaginative manipulation of mental models of mathematical phenomena. Proofs relying on diagrams can be rigorous if it is easy to draw a diagram that shares or otherwise indicates the structure of the mathematical object, the information thus displayed is not metrical and it is possible to put the inferences into systematic mathematical relation with other mathematical inferential practices. Proofs that appeal to mental models can be rigorous if the mental models can be externalised as diagrammatic practice that satisfies these three conditions. (shrink)

This paper assumes the success of arguments against the view that informal mathematical proofs secure rational conviction in virtue of their relations with corresponding formal derivations. This assumption entails a need for an alternative account of the logic of informal mathematical proofs. Following examination of case studies by Manders, De Toffoli and Giardino, Leitgeb, Feferman and others, this paper proposes a framework for analysing those informal proofs that appeal to the perception or modification of diagrams or to the inspection or (...) imaginative manipulation of mental models of mathematical phenomena. Proofs relying on diagrams can be rigorous if it is easy to draw a diagram that shares or otherwise indicates the structure of the mathematical object, the information thus displayed is not metrical and it is possible to put the inferences into systematic mathematical relation with other mathematical inferential practices. Proofs that appeal to mental models can be rigorous if the mental models can be externalised as diagrammatic practice that satisfies these three conditions. (shrink)