Foreword to Special Issue on Mathematical Depth
2015-09-16 10:20:43 AM
On the Depth of Szemeredi's Theorem
Arana, A., 2015-09-16 10:20:43 AM
Many mathematicians have cited depth as an important value in their research. However, there is no single widely accepted account of mathematical depth. This article is an attempt to bridge this gap. The strategy is to begin with a discussion of Szemerédi's theorem, which says that each subset of the natural numbers that is sufficiently dense contains an arithmetical progression of arbitrary length. This theorem has been judged deep by many mathematicians, and so makes for a good case on which to focus in analyzing mathematical depth. After introducing the theorem, four accounts of mathematical depth will be considered.Depth -- A Gaussian Tradition in Mathematics
Gray, J., 2015-09-16 10:20:43 AM
Mathematicians use the word ‘deep’ to convey a high appreciation of a concept, theorem, or proof. This paper investigates the extent to which the term can be said to have an objective character by examining its first use in mathematics. It was a consequence of Gauss's work on number theory and the agreement among his successors that specific parts of Gauss's work were deep, on grounds that indicate that depth was a structural feature of mathematics for them. In contrast, French mathematicians had a less structural, more problem-oriented approach to mathematics and did not speak of depth so readily.Depth and Explanation in Mathematics
Lange, M., 2015-09-16 10:20:43 AM
This paper argues that in at least some cases, one proof of a given theorem is deeper than another by virtue of supplying a deeper explanation of the theorem — that is, a deeper account of why the theorem holds. There are cases of scientific depth that also involve a common abstract structure explaining a similarity between two otherwise unrelated phenomena, making their similarity no coincidence and purchasing depth by answering why questions that separate, dissimilar explanations of the two phenomena cannot correctly answer. The connections between explanation, depth, unification, power, and coincidence in mathematics and science are compared.What Does 'Depth' Mean in Mathematics?
Stillwell, J., 2015-09-16 10:20:43 AM
This paper explores different interpretations of the word ‘deep’ as it is used by mathematicians, with a large number of examples illustrating various criteria for depth. Most of the examples are theorems with ‘historical depth’, in the sense that many generations of mathematicians contributed to their proof. Some also have ‘foundational depth’, in the sense that they support large mathematical theories. Finally, concepts from mathematical logic suggest that it may be possible to order certain theorems or problems according to ‘logical depth’.Mathematical Depth
Urquhart, A., 2015-09-16 10:20:43 AM
The first part of the paper is devoted to surveying the remarks that philosophers and mathematicians such as Maddy, Hardy, Gowers, and Zeilberger have made about mathematical depth. The second part is devoted to the question of whether we can make the notion precise by a more formal proof-theoretical approach. The idea of measuring depth by the depth and bushiness of the proof is considered, and compared to the related notion of the depth of a chess combination.Afterword to Special Issue on Mathematical Depth
2015-09-16 10:20:43 AM
Depth and Clarity * Felix Muhlholzer. Braucht die Mathematik eine Grundlegung? Eine Kommentar des Teils III von Wittgensteins Bemerkungen uber die Grundlagen der Mathematik [Does Mathematics need a Foundation? A Commentary on Part III of Wittgenstein's Remarks on the Foundations of Mathematics]. Frankfurt: Vittorio Klostermann, 2010. ISBN: 978-3-465-03667-8. Pp. xiv + 602
Floyd, J., 2015-09-16 10:20:43 AM
Andrew Aberdein and Ian J. Dove, eds. The Argument of Mathematics. Logic, Epistemology, and the Unity of Science; 30. Dordrecht: Springer, 2013. ISBN: 978-94-007-6533-7 (hbk); 978-94-007-6534-4 (ebook). Pp. x + 393
DeVidi, D., 2015-09-16 10:20:43 AM
James Franklin. An Aristotelian Realist Philosophy of Mathematics. New York: Palgrave Macmillan, 2014. ISBN: 978-1-137-40072-7 (hbk); 978-1-137-40073-4 (pdf); 978-1-137-40074-1 (e-book). Pp. x + 308
Jones, M., 2015-09-16 10:20:43 AM
Richard G. Heck Jr. Reading Frege's Grundgesetze. Oxford: Oxford University Press, 2012. ISBN: 978-0-19-923370-0 (hbk); 978-0-19-874437-5 (pbk); 978-0-19-165535-7 (pdf). Pp. xvii + 296
Ebert, P. A., 2015-09-16 10:20:43 AM
Dagfinn Follesdal, guest ed. Mathematical Evidence. Special issue of Inquiry; An Interdisciplinary Journal of Philosophy 58 (2015) 1-98
2015-09-16 10:20:43 AM
Wednesday, September 16, 2015
Philosophia Mathematica
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