A quotation from A. Einstein
Math Int. 12 (1990) no. 2, p. 31
How does a normally talented research scientist come to concern himself with the theory of knowledge? Is there not more valuable work to be done in his field? I hear this from many of my professional colleagues; or rather, I sense in the case of many more of them that this is what they feel.
I cannot share this opinion. When I think of the ablest students whom I have encountered in teaching - i.e., those who have distinguished themselves by their independence and judgement and not only mere agility - I find that they have a lively concern for the theory of knowledge. They like to start discussions concerning the aims and methods of the sciences, and showed unequivocally by the obstinacy with which they defend their views that this subject seemed important to them.
This is not really astonishing. For when I turn to science not for some superficial reason such as money-making or ambition, and also not (or at least exclusively) for the pleasure of the sport, the delights of brain-athletics, then the following questions must burningly interest me as a disciple of science: What goal will be reached by the science to which I am dedicating myself? To what extent are its general results `true'? What is essential and what is based only on the accidents of development?... Concepts which have proved useful for ordering things easily assume so great an authority over us, that we forget their terrestrial origin and accept them as unalterable facts. They then become labelled as `conceptual necessities', `a priori situations', etc. The road of scientific progress is frequently blocked for long periods by such errors. It is therefore not just an idle game to exercise our ability to analyse familiar concepts, and to demonstrate the conditions on which their justification and usefulness depend, and the way in which these developed, little by little...
"The Einstein-Wertheimer Correspondence on Geometric Proofs and Mathematical Puzzles" (MI, 12(2)(1990), 35-43.)
December 12, 2014
Here is a link to a presentation in Galway, Dec 2, 2014, discussing some "Anomalies" in algebraic topology:
"A homotopical approach to algebraic topology via compositions of cubes ".
It is useful every so often to consider not what a subject does, but what it does not do, and in some sense should do.
Link to Ronnie Brown's home page
Link to articles on popularisation and teaching