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Humboldt-Universität zu Berlin - Philosophy, Science and the Sciences

Benjamin Wilck

Thesis on Mathematical Definition and Dialectical Refutation. Aristotle’s Criticism of the Foundations of Mathematics

In my dissertation, I reconstruct the ways in which ancient philosophers criticise principles of mathematical proof, especially definitions. In order to explore the interrelations between the philosophical theory and the actual practice of mathematical definition, I also examine to what extent these criticisms are reflected in ancient mathematical works. The focus of my dissertation lies in Aristotle’s theory of definition in the Organon and Euclid’s Elements. In the Topics, Aristotle launches a major attack on the foundations of ancient mathematics, insofar as these foundations concern definitions. He provides a deductive method to test whether a given pair of terms is a definition or not and illustrates this procedure with contemporary mathematical examples. Book VI of the Topics contains a series of dialectical arguments to refute putative definitions of fundamental geometrical items such as point, line, plane, solid, and finite straight-line, as well as of the arithmetical terms odd and even.

Many of the ancient mathematical definitions do not in fact meet Aristotle’s requirements for adequate definition. The question arises whether, or to what extent, the tensions between Aristotle’s theory and the ancient practice of defining can be harmonised. Moreover, some of the definitions that Aristotle rejects in the Topics have been expressly restated and deductively employed as premisses in mathematical proofs in Euclid’s Elements. This gives rise to the deeper question concerning the objective and outcome of overthrowing an established definition, especially if it appears to serve its purpose as an explanatory principle in scientific demonstration. Why should a mathematician care about dialectical arguments against mathematical principles? How seriously should we take Aristotle’s tests and rules in the Topics? Should they actually make us worried about the foundations of mathematics?


Benjamin Wilck is a doctoral candidate in the RTG Philosophy, Science and the Sciences of the Graduate Program in Ancient Philosophy of Humboldt University Berlin. His doctoral supervisors are Professor Jonathan Beere (Berlin / Princeton), Professor Stephen Menn (Montréal), and Professor Benjamin Morison (Princeton). After a successful career in high-performance swimming, Benjamin studied Philosophy, Cultural Studies, and Aesthetics in Berlin and Paris. In March 2014, he received a Magister Artium with honours from Humboldt University Berlin. In his M.A. dissertation, he enquired into the explanatory status of metaphysical arguments by discussing the question whether Aristotelian metaphysics is a demonstrative science. His research interests encompass Ancient Philosophy (especially Ancient Logic) and Ancient Mathematics as well as Philosophical Logic, History of Logic, and Metaphysics.

Honours & grants


Princeton University (NJ, USA), VSCR at the Faculty of Philosophy, funded by the Deutsche Forschungsgemeinschaft (DFG)


Doctoral fellowship, Deutsche Forschungsgemeinschaft (DFG)


Award from the Carl & Max Schneider Foundation for the M.A. thesis "Is Aristotelian Metaphysics a Science?"


Université Paris-1 Panthéon-Sorbonne, Erasmus student at the Faculty of Philosophy, funded by the European Union


Prix Apollinaire for Extraordinary Achievements in French Literacy from the Robert Bosch Foundation


Summer 2012

B.A. Seminar Introduction to the Theory of Forms in Plato and Aristotle (with Jan Matthijs Menting) at the Department of Philosophy of University of Potsdam (Germany)



M.A. Seminar Aristotle’s Posterior Analytics (with Colin G. King) at the Department of Philosophy of Humboldt University Berlin (Germany)

Summer 2011

B.A. Tutorial Introduction to Formal Logic at the Department of Philosophy of Humboldt University Berlin (Germany)

Academic talks (selection)

  • "What’s odd about Euclid’s definitions of odd and even?" at Avtonomi Akadimia, October 8th 2015, Athens (Greece)
  • Comments on Marilù Papandreou’s "Artefacts and Substances" at the 1st Berlin-Munich Workshop at LMU München, November 27th-28th 2015, Munich (Germany)
  • "The Refutation of Mathematical Definitions in Aristotle’s Topics" at the 2nd Berlin-Munich Workshop at LMU München, February 19th-20th 2016, Munich (Germany)
  • "Aristotelian Dialectic and the Ancient Greek Practice of Mathematical Definition" at the Vortragskolloquium of TOPOI & Berlin Graduate School of Ancient Studies, June 2nd 2016, Berlin (Germany)
  • "Mathematical Babbling. Aristotle on Definitions with Repetitions" at the Annual Meeting of Postgraduates in Ancient Literature (AMPAL) on Repetition in Ancient Literature at University of Oxford, September 20th-21st 2016, Oxford (UK)
  • "How Many Noses Did Socrates Have? Aristotle’s distinction between two kinds of repetition and the solution to the problem of babbling in the Topics" at the 20th Oxford Philosophy Graduate Conference at University of Oxford, November 11th-13th 2016, Oxford (UK)


Conference organization and co-organization


Knowledge and Demonstrative Science: Aristotle’s Posterior Analytics, April 14th-18th 2012, at Humboldt University Berlin (Germany)

Further academic activities


Research assistant to Jonathan Beere, Chair for Ancient Philosophy and History of Knowledge at the Department of Philosophy of Humboldt University Berlin


Administrative assistant to the President of Humboldt University Berlin, Professor Jan-Hendrik Olbertz


Event management & public relations for the 200th anniversary of Humboldt University Berlin

Contact: bwilck@princeton.edu