Logique et Analyse
http://poj.peeters-leuven.be/content.php?url=journal&journal_code=LEA
Recent articlesPrefacepoj@peeters-leuven.behttp://dx.doi.org/10.2143/LEA.237.0.3197355
http://poj.peeters-leuven.be/content.php?url=article&id=3197355
Thu, 13 Apr 2017 15:19:07 GMT
Introduction
Mathematical Aims beyond Justificationpoj@peeters-leuven.behttp://dx.doi.org/10.2143/LEA.237.0.3197356
http://poj.peeters-leuven.be/content.php?url=article&id=3197356
Thu, 13 Apr 2017 15:19:30 GMT
The past decades, there has been an increased philosophical interest in mathematical practice. These philosophers intend to answer questions related to the activities of working mathematicians. One of these questions is what aims direct mathematical research. After all, mathematicians often express the desire for good or beautiful mathematics. These types of reflections indicate the need for an understanding of aims that go beyond justification. In this paper, we explore potential ways in which philosophy can clarify what these aims are. Furthermore, we stress the importance of a multidisciplinary approach to this problem.
Characterizing Properties and Explanation in Mathematicspoj@peeters-leuven.behttp://dx.doi.org/10.2143/LEA.237.0.3197357
http://poj.peeters-leuven.be/content.php?url=article&id=3197357
Thu, 13 Apr 2017 15:20:43 GMT
Mark Steiner proposes one of the earliest contemporary accounts of mathematical explanation, which appeals to characterizing properties of entities referred to in proofs. Unfortunately Steiner’s remarks are often quite vague, sometimes described as ‘very puzzling indeed’, and this lack of clarity has led to a lack of understanding and a tendency to reject Steiner’s account in the philosophical literature. I argue that Steiner’s account repays deeper analysis by providing a sympathetic reading that makes sense of his puzzling remarks and draws out some important questions. I focus on a simple mathematical example involving sums of number sequences and identify three key conditions that the proof must meet to count as explanatory for Steiner. I propose a suitable characterizing property and show that on my suggestion, the proof indeed fits Steiner’s account. Subsequently, I present a few potential problems relating to Steiner’s focus on the generalizability of proofs, and show how my reading of generalizability helps to avoid these worries. Finally, I show how (my extension of) Steiner’s proposal can account for what I take to be the primary epistemic function of an explanation, namely, to help us see why the fact to be explained is true.
A Quasi-Interventionist Theory of Mathematical Explanationpoj@peeters-leuven.behttp://dx.doi.org/10.2143/LEA.237.0.3197358
http://poj.peeters-leuven.be/content.php?url=article&id=3197358
Thu, 13 Apr 2017 15:21:50 GMT
Explanations in mathematics are not yet well understood. I discuss Steiner’s theory of mathematical explanation, then attempt to improve it by assimilating its core intuitions to Woodward’s counterfactual theory of explanation. The theory that results deals successfully with many cases, but it fails to handle certain types of explanatory asymmetry. To fix this, I draw on Woodward’s interventionist theory of causation and develop a quasi-interventionist theory of mathematical explanation. According to this theory, the asymmetry of mathematical explanations is subjective in the sense that it does not depend on the objective structure of mathematics itself; but I argue that this is not a problem, since the same subjectivity can be found in causal explanation.
Explanatory Proofs in Mathematicspoj@peeters-leuven.behttp://dx.doi.org/10.2143/LEA.237.0.3197359
http://poj.peeters-leuven.be/content.php?url=article&id=3197359
Thu, 13 Apr 2017 15:23:02 GMT
In recent times philosophers of mathematics have generated great interest in explanations in mathematics. They have focused their attention on mathematical practice and searched for special cases that seem to own some kind of explanatory power. Two main views can be identified, namely noneism and someism: the first is the view that no proof is explanatory, whereas the second is the view that some proofs are explanatory while others are not. The present paper aims to discuss the plausibility of the latter view. I first point out the main difficulties involved in this kind of research and I focus on a recent someist account, namely Frans and Weber’s mechanistic model. Their approach seems promising, but further research is needed before accepting this someist model as a proper one. I then outline a general assessment on someism which doesn’t turn out to be so convincing. I therefore suggest another view that I call allism, i.e. the view that all proofs are explanatory, at least in some sense, and I argue for its plausibility.
The Complementary Faces of Mathematical Beautypoj@peeters-leuven.behttp://dx.doi.org/10.2143/LEA.237.0.3197360
http://poj.peeters-leuven.be/content.php?url=article&id=3197360
Thu, 13 Apr 2017 15:24:42 GMT
This article focuses on the writings of Hardy, Poincaré, Birkhoff, and Whitehead, in order to substantiate the claim that mathematicians experience a mathematical proof as beautiful when it offers a maximum of insight while demanding a minimum of effort. In other words, it claims that the study of the aesthetic success of theorem-proofs can benefit from the analogy with the economic success of a business, which involves maximizing return on investment. On the other hand, the article also draws on Le Lionnais and Whitehead (again) in order to show that, whereas the kind of aesthetic delight offered by beautiful proofs is typical for well-established branches of mathematics, a romantic and adventurous spirit that goes beyond the search for classical aesthetic delights is needed when the exploration of new mathematics is at stake. The history of mathematics is not only a story of feelings of beauty invoked by perfect products, but also a survey of sublime periods of creative production. No account of mathematical beauty can be complete if it does not complement the classical product aesthetics with a romantic creation aesthetics.
Hilbert on Consistency as a Guide to Mathematical Realitypoj@peeters-leuven.behttp://dx.doi.org/10.2143/LEA.237.0.3197361
http://poj.peeters-leuven.be/content.php?url=article&id=3197361
Thu, 13 Apr 2017 15:26:22 GMT
In his early work Hilbert puts forward the principle that in mathematics consistency is enough for existence. Moriconi (2003) claims that the standard understanding of Hilbert’s contention is that he is assuming the completeness of his system. I look at the evidence for this interpretation and conclude that at the time he made this claim Hilbert had not yet developed a sophisticated conception of meta-mathematical concepts like consistency and completeness to allow him to formulate the completeness theorem. I then consider how we should understand Hilbert’s contention in light of this and suggest that, for Hilbert, consistency is <i>conceptually prior</i> to existence. On the basis of this I present a new reading of Hilbert’s Principle which recovers Hilbert’s true contention, and along with it the philosophical significance of Hilbert’s early work which – in particular – provides a new approach to questions of ontology in mathematics.