 There is an important pedagogic point to all this, too. The dialectic of proofs and refutations can generate, in the ways explained above, quite complicated definitions of mathematical concepts, definitions that can only really be understood by considering the process that gave rise to them. But mathematics teaching is not historical, or even quasi-historical. (One sense in which Lakatos’s theory is dialectical: it represents a process as rational even though the terms of the debate are not clearly defined.) But students nowadays are presented with the latest definitions at the outset, and required to learn them and apply them, without ever really understanding them.