•  182
    Review of C. Mortensen, Inconsistent Geometry (review)
    Philosophia Mathematica 20 (3): 365-372. 2012.
  •  133
    Ross' paradox is an impossible super-task
    British Journal for the Philosophy of Science 45 (2): 743-748. 1994.
  •  88
    Zeno's paradoxes and the tile argument
    Philosophy of Science 54 (2): 295-302. 1987.
    A solution of the zeno paradoxes in terms of a discrete space is usually rejected on the basis of an argument formulated by hermann weyl, The so-Called tile argument. This note shows that, Given a set of reasonable assumptions for a discrete geometry, The weyl argument does not apply. The crucial step is to stress the importance of the nonzero width of a line. The pythagorean theorem is shown to hold for arbitrary right triangles
  •  86
    Mathematical arguments in context
    with Bart Van Kerkhove
    Foundations of Science 14 (1-2): 45-57. 2009.
    Except in very poor mathematical contexts, mathematical arguments do not stand in isolation of other mathematical arguments. Rather, they form trains of formal and informal arguments, adding up to interconnected theorems, theories and eventually entire fields. This paper critically comments on some common views on the relation between formal and informal mathematical arguments, most particularly applications of Toulmin’s argumentation model, and launches a number of alternative ideas of presenta…Read more
  •  73
    A Defense of Strict Finitism
    Constructivist Foundations 7 (2): 141-149. 2012.
    Context: Strict finitism is usually not taken seriously as a possible view on what mathematics is and how it functions. This is due mainly to unfamiliarity with the topic. Problem: First, it is necessary to present a “decent” history of strict finitism (which is now lacking) and, secondly, to show that common counterarguments against strict finitism can be properly addressed and refuted. Method: For the historical part, the historical material is situated in a broader context, and for the argume…Read more
  •  63
  •  55
    A Defense of Strict Finitism
    Constructivist Foundations 7 (2): 141-149. 2012.
    Context: Strict finitism is usually not taken seriously as a possible view on what mathematics is and how it functions. This is due mainly to unfamiliarity with the topic. Problem: First, it is necessary to present a “decent” history of strict finitism and, secondly, to show that common counterarguments against strict finitism can be properly addressed and refuted. Method: For the historical part, the historical material is situated in a broader context, and for the argumentative part, an evalua…Read more
  •  53
    What is philosophy of mathematics and what is it about? The most popular answer, I suppose, to this question would be that philosophers should provide a justification for our presently most cherished mathematical theories and for the most important tool to develop such theories, namely logico-mathematical proof. In fact, it does cover a large part of the activity of philosophers that think about mathematics. Discussions about the merits and faults of classical logic versus one or other ‘deviant’…Read more
  •  48
    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 epistemi…Read more
  •  47
    Pi on Earth, or Mathematics in the Real World
    with Bart Van Kerkhove
    Erkenntnis 68 (3): 421-435. 2008.
    We explore aspects of an experimental approach to mathematical proof, most notably number crunching, or the verification of subsequent particular cases of universal propositions. Since the rise of the computer age, this technique has indeed conquered practice, although it implies the abandonment of the ideal of absolute certainty. It seems that also in mathematical research, the qualitative criterion of effectiveness, i.e. to reach one’s goals, gets increasingly balanced against the quantitative…Read more
  •  46
    The Collatz conjecture. A case study in mathematical problem solving
    Logic and Logical Philosophy 14 (1): 7-23. 2005.
    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 …Read more
  •  45
    Introductory Note
    Philosophica 42. 1988.
  •  42
    No one will dispute, looking at the history of mathematics, that there are plenty of moments where mathematics is “in trouble”, when paradoxes and inconsistencies crop up and anomalies multiply. This need not lead, however, to the view that mathematics is intrinsically inconsistent, as it is compatible with the view that these are just transient moments. Once the problems are resolved, consistency (in some sense or other) is restored. Even when one accepts this view, what remains is the question…Read more
  •  40
    Proofs and arguments: The special case of mathematics
    Poznan Studies in the Philosophy of the Sciences and the Humanities 84 (1): 157-169. 2005.
    Most philosophers still tend to believe that mathematics is basically about producing formal proofs. A consequence of this view is that some aspects of mathematical practice are entirely lost from view. My contention is that it is precisely in those aspects that similarities can be found between practices in the exact sciences and in mathematics. Hence, if we are looking for a (more) unified treatment of science and mathematics it is necessary to incorporate these elements into our view of what …Read more
  •  30
    Schoonheid in de wiskunde: Birkhoff revisited
    Tijdschrift Voor Filosofie 60 (1): 106-130. 1998.
    Everyone is familiar with the measure of beauty that has been proposed by Birkhoff, the famous formula M = O/C. Although I show that the formula in its original form cannot be maintained, I present a reinterpretation that adapts the formula for measuring the beauty of mathematical proofs. However, this type of measure is not the only aesthetic element in mathematics. There exists a 'romantic' side as well, to use the term introduced by François Le Lionnais. Thus, a more complex proposal of mathe…Read more
  •  29
    Non-Formal Properties of Real Mathematical Proofs
    PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1988 249-254. 1988.
    The heuristics and strategies presented in Lakatos' Proofs and Refutations are well-known. However they hardly present the whole story as many authors have shown. In this paper a recent, rather spectacular, event in the history of mathematics is examined to gather evidence for two new strategies. The first heuristic concerns the expectations mathematicians have that a statement will be proved using given methods. The second heuristic tries to make sense of the mathematicians' notion of the quali…Read more
  •  27
    Significs and mathematics: Creative and other subjects
    Semiotica 2013 (196): 307-323. 2013.
    Journal Name: Semiotica - Journal of the International Association for Semiotic Studies / Revue de l'Association Internationale de Sémiotique Volume: 2013 Issue: 196 Pages: 307-323
  •  25
    Dialogue Logic and Problem-Solving
    Philosophica 35. 1985.
  •  25
    Een metalogische referentietheorie
    Tijdschrift Voor Filosofie 56 (2): 350-354. 1994.
  •  24
    The Unreasonable Richness of Mathematics
    with Bart Van Kerkhove
    Journal of Cognition and Culture 4 (3-4): 525-549. 2004.
  •  24
    It is a rather safe statement to claim that the social dimensions of the scientific process are accepted in a fair share of studies in the philosophy of science. It is a somewhat safe statement to claim that the social dimensions are now seen as an essential element in the understanding of what human cognition is and how it functions. But it would be a rather unsafe statement to claim that the social is fully accepted in the philosophy of mathematics. And we are not quite sure what kind of state…Read more
  •  24