‘There is a tendency in much contemporary literature to present material in a highly systematised fashion, in which an abstract definition will typically come before the list of examples that reveals the original motivation for that definition. Paedogogically, a disadvantage of this approach is that the student is not actually shown the genesis of concepts — how and why they evolved — and is thereby taught nothing about the mechanisms of creative thinking. Apart from lending the topic an often illusory impression of completedness, the method also has the drawback of inflating prerequisites to understanding. Category theory has more than once been referred to as ‘abstract nonsense’. In my experience, that reaction is the result of features that are not intrinsic to the subject itself, but are due merely to the style of some of its expositors. The approach I have taken here is to try to move always from the particular to the general, following through the steps of the abstraction process until the abstract concept emerges naturally. The starting points are elementary (in the ‘first principles’ sense), and at the finish it would be quite appropriate for the reader to feel that (s)he had just arrived at the subject, rather than reached the end of the story.’
Robert Goldblatt. Topoi -- The Categorial Analysis of Logic, 1979, page ix.
‘But anyone who has done mathematics knows what comes first—a problem … Sometimes a solution is a set of axioms! … In developing and understanding a subject, axioms come late. Then in formal presentations, they come early … The view that mathematics is in essence derivations from axioms is backward. In fact, it's wrong.’
Reuben Hersh. What is Mathematics, Really? Jonathan Cape, London, 1997, page 6.
Paul Lockhart's A Mathematician's Lament [my pdf copy]. Mark Taver's Why I am Not a Professor or The Decline and Fall of the British University [my pdf copy].