# A Role for Quasi-Empiricism in Mathematics Education

## 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.

## Keywords

Mathematical Knowledge Thought Experiment Mathematical Object Objective Knowledge Human Consciousness## References

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