The Collatz conjecture. A case study in mathematical problem solving

In previous papers (see Van Bendegem [1993], [1996], [1998], [2000], [2004], [2005], and jointly with Van Kerkhove [2005]) we have proposed the idea that, if we look at what mathematicians do in their daily work, one will find that conceiving and writing down proofs does not fully capture their activity. In other words, it is of course true that mathematicians spend lots of time proving theorems, but at the same time they also spend lots of time preparing the ground, if you like, to construct a proof

Why the naïve Derivation Recipe model cannot explain how mathematicians’ proofs secure mathematical knowledge

This is a pre-copyedited, author-produced PDF of an article accepted for publication in Philosophia Mathematica following peer review. Under embargo. Embargo end date: 7 July 2018 The version of record [Lavor, B., 'Why the Naive Derivation Recipe Model Cannot Explain How Mathematician's Proofs Secure Mathematical Knowledge', Philosophia Mathematica (2016) 24(3): 401-404, is available online at: https://doi.org/10.1093/philmat/nkw012. © The Author [2016]. Published by Oxford University Press. All rights reserved.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.Peer reviewedFinal Accepted Versio

Classical arithmetic is quite unnatural

It is a generally accepted idea that strict finitism is a rather marginal view within the community of philosophers of mathematics. If one therefore wants to defend such a position (as the present author does), then it is useful to search for as many different arguments as possible in support of strict finitism. Sometimes, as will be the case in this paper, the argument consists of, what one might call, a “rearrangement” of known materials. The novelty lies precisely in the rearrangement, hence on the formal-axiomatic level most of the results presented here are not new. In fact, the basic results are inspired by and based on Mycielski (1981)

Inconsistent models of artihmetic Part II : The general case

The paper establishes the general structure of the inconsistent models of arithmetic of [7]. It is shown that such models are constituted by a sequence of nuclei. The nuclei fall into three segments: the first contains improper nuclei: the second contains proper nuclei with linear chromosomes: the third contains proper nuclei with cyclical chromosomes. The nuclei have periods which are inherited up the ordering. It is also shown that the improper nuclei can have the order type of any ordinal. of the nationals. or of any other order type that can be embedded in the rationals in a certain way

Epistemic Injustice in Mathematics

We investigate how epistemic injustice can manifest itself in mathematical practices. We do this as both a social epistemological and virtue-theoretic investigation of mathematical practices. We delineate the concept both positively – we show that a certain type of folk theorem can be a source of epistemic injustice in mathematics – and negatively by exploring cases where the obstacles to participation in a mathematical practice do not amount to epistemic injustice. Having explored what epistemic injustice in mathematics can amount to, we use the concept to highlight a potential danger of intellectual enculturation

Epistemic injustice in mathematics

We investigate how epistemic injustice can manifest itself in mathematical practices. We do this as both a social epistemological and virtue-theoretic investigation of mathematical practices. We delineate the concept both positively – we show that a certain type of folk theorem can be a source of epistemic injustice in mathematics – and negatively by exploring cases where the obstacles to participation in a mathematical practice do not amount to epistemic injustice. Having explored what epistemic injustice in mathematics can amount to, we use the concept to highlight a potential danger of intellectual enculturation

Mathematical Practice and Naturalist Epistemology: Structures with Potential for Interaction

In current philosophical research, there is a rather one-sided focus on the foundations of proof. A full picture of mathematical practice should however additionally involve considerations about various methodological aspects. A number of these is identified, from large-scale to small-scale ones. After that, naturalism, a philosophical school concerned with scientific practice, is looked at, as far as the translations of its epistemic principles to mathematics is concerned. Finally, we call for intensifying the interaction between both dimensions of practice and epistemology

Can discrete time make continuous space look discrete?

Van Bendegem has recently offered an argument to the effect that, if time is discrete, then there should exist a correspondence between the motions of massive bodies and a discrete geometry. On this basis, he concludes that, even if space is continuous, it should nonetheless appear discrete. This paper examines the two possible ways of making sense of that correspondence, and shows that in neither case van Bendegem's conclusion logically follows

Introduction: From Practice to Results in Mathematics and Logic

1 Mathematical practice: a short overview This volume is a collection of essays that discuss the relationships between the practices deployed by logicians and mathematicians, either as individuals or as members of research communities, and the results from their research. We are interested in exploring the concept of 'practices' in the formal sciences. Though common in the history, philosophy and sociology of science, this concept has surprisingly thus far been little reflected upon in logic ..