Jargon often appears in lectures, and sometimes in print, as informal shorthand for rigorous arguments or precise ideas.
[Grothendieck] raised algebraic geometry to a new level of abstraction...if certain mathematicians could console themselves for a time with the hope that all these complicated structures were 'abstract nonsense'...the later papers of Grothendieck and others showed that classical problems...which had resisted efforts of several generations of talented mathematicians, could be solved in terms of...complicated concepts.There are two canonical proofs that are always used to show non-mathematicians what a mathematical proof is like: The beauty of a mathematical theory is independent of the aesthetic qualities...of the theory's rigorous expositions.
We acknowledge a theorem's beauty when we see how the theorem 'fits' in its place....We say that a proof is beautiful when such a proof finally gives away the secret of the theorem....Many of the results mentioned in this paper should be considered "folklore" in that they merely formally state ideas that are well-known to researchers in the area, but may not be obvious to beginners and to the best of my knowledge do not appear elsewhere in print.Since half a century we have seen arise a crowd of bizarre functions which seem to try to resemble as little as possible the honest functions which serve some purpose....Nay more, from the logical point of view, it is these strange functions which are the most general....to-day they are invented expressly to put at fault the reasonings of our fathers....[The Dirichlet function] took on an enormous importance...as giving an incentive for the creation of new types of function whose properties departed completely from what intuitively seemed admissible.
A celebrated example of such a so-called 'pathological' function...is the one provided by Weierstrass....This function is continuous but not differentiable.Although ultimately every mathematical argument must meet a high standard of precision, mathematicians use descriptive but informal statements to discuss recurring themes or concepts with unwieldy formal statements.
Norbert A'Campo of the University of Basel once asked Grothendieck about something related to the Platonic solids.