What is dialectical philosophy of mathematics? (Q2765253)

The author considers Imre Lakatos' hope to found a school of dialectical philosophy of mathematics. He attempts to answer the question what a dialectical philosophy of mathematics could be, and illustrates his results with three examples from recent research in the philosophy of mathematics in the dialectical spirit.NEWLINENEWLINE If a deductivist view on mathematics is chosen the mathematician is seen as investigating the deductive closure of a freely chosen set of axioms under the only constraint of consistency. The question how the premises for proofs are chosen remains unanswered. The author opposes the opinion that this question has no philosophical significance. The dialectical philosopher of mathematics is open for such questions. He takes an ``inside-phenomenological stance'' (p. 214) insisting ``that changes in the body of mathematics usually take place for mathematical reasons'' (p.\ 215), influenced also by historical and environmental factors. The dialectical philosopher, thus, does not share the ``logical positivist and Popperian horror of `psychologism'.'' He understands that ``human minds, however, fallible, are the only available vehicles for the greater rationality of science'' (ibid.). According to the dialectical philosopher, the historical development of mathematics is best explained by an analysis of the concepts governing that development. He also deals with the mathematicians' choices leading to this development, not only of the axioms, but also of problems, techniques and proof-strategies, choices which are not simply subjective, but time dependent.NEWLINENEWLINE The cases discussed are \textit{Y. Rav's} [``Why do we prove theorems''. Philos. Math.\ (3) 7, 5--41 (1999; Zbl 0941.03003)] in which the epistemic elements of the proofs are stressed which are not passed on to the theorems. The second example is Mary Leng's participant-observer study on G. Elliot's 1998 Toronto course on the structure of \(C^*\)-algebras, developing a classification theorem for some suitable well behaved class of these objects [\textit{M. Leng}, ``Phenomenology and mathematical Practice'', Philos.\ Math.\ (3) 10, 3--25 (2002; Zbl 1011.00011)]. Finally David Corfield's studies on applications of Lakatos' ideas to more recent mathematics is discussed. Corfield formulates a research programme of his own, starting with an identification of mathematical ends, then investigating the means by which these ends are achieved (e.g., [\textit{D. Corfield}, ``Beyond the methodology of mathematics research programmes,'' Philos.\ Math. (3) 6, 272--301 (1998; Zbl 1062.00501)]).