Chapter 7 uses pictures of objects with visible holes, like this obwarzanek (a Polish soft pretzel)
to illustrate (co-)homology theory, which is in turn introduced to motivate ∞-categories. At one point it is mentioned that
there is more than one way to define ∞-categories, so that they can all be seen as avatars — the word “incarnation” is actually used in the literature — but when one asks “avatars of what?” the only sensible answer is that they are avatars of the theory of ∞-categories mathematicians need.
Mark Goresky, who is here in Montreal, pointed out that the introduction to his book Stratified Morse Theory, with Robert MacPherson, describes the genesis of homology theory in a similar way:
The century-long story of the taming of homology theory is one of the greatest in mathematical history, and has not yet been adequately recorded by historians.
(That sentence was written before the publication of the History of Topology, edited by I. M. James, with contributions by historians as well as mathematicians. Much more remains to be recorded.)
Two avatars were proposed in the 1920s by Morse and Lefschetz, but, according to Goresky and MacPherson,
If mathematical journals in 1924 had the same standards of rigor that they have today, neither Morse theory nor Lefschetz theory could have been published. Morse and Lefschetz both attributed their success to their use of intuitive homology theory without insisting on adequate foundations.
They quote Morse, who wrote (after the dust settled):
Mathematicians of today are perhaps too exuberant in their desire to build new logical foundations for everything. Forever the foundation and never the cathedral.
Goresky and MacPherson add in parentheses: “We feel a kinship with this sentiment” and continue, in a philosophical vein one doesn’t often encounter in a book of this importance:
true geometers often feel its language misses the essential geometric ideas. Language is not well adapted to describing geometry, as the facilities for language and geometry live on opposite sides of the human brain.
Many more passages in this three-page segment of the book are worth quoting, but for now, three thoughts come to mind:
The “true geometers” of Goresky and MacPherson could equally well be realists (platonists) or constructivists; and the passage is neutral on whether the “essential geometric ideas” reside in the external world or in the perceiving brain;
No one who knows Goresky and MacPherson would ever accuse either of them of elitism; but the expression “true geometers” brings to mind the claim by David Pimm and Nathalie Sinclair (quoted in Chapter 2) that an “elitist view of mathematics and mathematical aesthetics … dominates today;” they read Poincaré’s writings on mathematical aesthetics as suggesting that “only mathematicians are privy to the aesthetic sensibilities that enable” the decision of “what is worth studying.” This is certainly not what the authors have in mind; but what words can we use to convey the sense of authentic knowledge of a subject without leaving ourselves open to accusations of elitism?
And one wonders: will (mechanical) geometers, without the inconvenience of brains, have as much trouble expressing their ideas in language?