## Abstract

Although there are quite a few directions in modern philosophy of mathematics that invoke some essential role for (quasi-)empirical material, this chapter will be devoted exclusively to what may be considered the seminal tradition. This enabled me to present the subject as one coherent whole and to forestall the discussion getting scattered in a diversity of directions without doing justice to any one of them.

Quasi-empiricism in this tradition is the view that the logic of mathematical inquiry is based, like the logic of scientific discovery, on the bottom-up retransmission of falsity (by means of counterexamples) rather than the top-down transmission of axiomatic truth. Imre Lakatos, who originally introduced the distinction between Euclidean and quasi-empirical theories, construed mathematics as a quasi-empirical science following in essence Popper’s method of conjectures and refutations. His essay on *Proofs and Refutations* was an attempt to apply, and thereby to test, Popper’s critical method and fallibilist theory of knowledge in a field – mathematics – to which it had not primarily been intended to apply. While Popper’s proposal stood up gloriously to this test, the new application gave rise also to new insights, in particular the construal of proofs as thought experiments with a role similar to that of testing and corroborating experiments in science.

After examination of a number of examples of mathematical thought-experiments from this perspective, the doctrine of the relative autonomy of objective knowledge is focussed upon, a doctrine which is part and parcel of the dialectic of proofs and refutations and hence of the quasi-empiricist programme.

Some remarks on the educational implications of quasi-empiricism conclude this chapter. These implications have no direct bearing on the actual practice of teaching mathematics in the classroom, but are mainly, if not exclusively, concerned with the *image* of mathematics that is conveyed in education.

Throughout this chapter, extensive use has been made of previous articles mentioned in the references.

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### Suggested Further Reading

Brown, J.R. (1999)

*Philosophy of Mathematics: an introduction to the world of proofs and pictures*, RoutledgeCorfield, D. (2003)

*Towards a Philosophy of Real Mathematics*, Cambridge University PressErnest, P. (1997) ‘The Legacy of Lakatos: Reconceptualizing the Philosophy of Mathematics’,

*Philosophia Mathematica***5**, p. 116–134François, K. & Van Bendegem, J.P. (eds.) (2007)

*Philosophical Dimensions in Mathematics Education*, SpringerGavroglu, K., Goudraroulis, Y. & Nicolacopoulos, P. (eds.) (1989)

*Imre Lakatos and Theories of Scientific Change*, KluwerHersh, R. (2006)

*18 Unconventional Essays on the Nature of Mathematics*, SpringerKampis, G., Kvasz, L. & Stöltzner, M. (eds.) (2002)

*Appraising Lakatos: Mathematics, Methodology and the Man*, KluwerLarvor, B. (1998)

*Lakatos: An Introduction*, RoutledgeOliveri, G. (1997) ‘Criticism and the Growth of Mathematical Knowledge’,

*Philosophia Mathematica***5**, p. 228–259Oliveri, G. (2006) ‘Mathematics as a Quasi-Empirical Science’,

*Foundations of Science***11**, p. 41–79Tymoczko, T. (ed.) (1986)

*New Directions in the Philosophy of Mathematics*, BirkhäuserVan Kerkhove, B. (ed.) (2009)

*Mathematical Practices: Essays in Philosophy and History of Mathematics*, World ScientificVan Kerkhove, B. & Van Bendegem, J.P. (eds.) (2007)

*Perspectives on Mathematical Practices: Bringing together Philosophy of Mathematics, Sociology of Mathematics, and Mathematics Education*, SpringerZheng, Y. (1990) ‘From the Logic of Mathematical Discovery to the Methodology of Scientific Research Programmes’,

*British Journal for the Philosophy of Science***41**, p. 377–399

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Glas, E. (2014). A Role for Quasi-Empiricism in Mathematics Education. In: Matthews, M. (eds) International Handbook of Research in History, Philosophy and Science Teaching. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7654-8_23

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