There are no theorems in category theory.
Emily Riehl, Category Theory In Context
Mathematicians often tell her this; hence the book.
If I had to summarise her views in one sentence, it would be:
Everything is an adjunction.
I also like the division these mathematicians are making to her: essentially, a theorem is anything that solves Feynmanâs challenge: by a series of clear, unsurprising steps, one arrives at an unexpected conclusion.
Examples for me include:
17 possible tessellations
6 ways to foliate a surface
27 lines on a cubic
1, 1, 1, 1, 1, 1, 28, 2, 8, 6, 992, 1, 3, 2, 16256, 2, 16, 16, 523264, 24, 8, 4 ways to link any-dimensional spheres.
the existence of sporadic groups
surprising rep-theory consequences of Young diagrams, Ferrers sequences, and so on (you could say the strangeness of integer partitions is really to blame hereâŚ)
59 icosahedra
8 geometric layouts
Books which are bristling with mathematical ideas of this kind include Montesinos on tessellations, Geometry and the Imagination (the original one), and Coxeterâs book on polyhedra (start with Baez on A-D-E if you want to follow my path). Moonshine and anything by Thurston or his students, Iâve found similarly flush with shockng contentâquite different to what I thought mathematics would be like. (I had pictured something more like a formal logic book: row by row of symbols. But instead, the deeper I got into mathematics, the fewer the symbols and the more the surnames thanking the person who came up with some good idea.)
Note that a theorem is different here to some geometry â as in The Geometry of Schemes. The word geometry used in that sense, I feel, is to have a comprehensive enough vision of a subject to say how it âlooksâ â but the word theorem means the result is surprising or unintuitive.
This definition of a theorem, to me, presents a useful challenge to annoying pop-psychology that today lurks under the headings of Bayesianism, cognitive _______, behavioural econ/finance, and so on.
Following Buliga and Thurston to understand the nature of mathematical progress, within mathematics at least (where itâs clearer than elsewhere whether you understand something or notâcompare to economic theory for example), there is a clear delination of whatâs obvious and whatâs not.
What is definitely not the case in mathematics, is that every logical or computable consequence of a set of definitions is computed and known immediately when the definitions are stated! You can look at a (particularly a good) mathematical exposition as walking you through the steps of which shifts in perspective you need to take to understand a conclusion. For example start with some group, then consider it as a topological object with a cohomology to get the centraliser. Or in Fourier analysis: re-present line-elements on a series of widening circles. Use hyperbolic geometry to learn about integers. Use stable commutator length (geometry) to learn about groups. Or read about TeichmĂźller stuff and mapping class groups because itâs the confluence of three rivers.
Sometimes mathematical explanations require fortitude (Gromovâs "energy") and sometimes a shift in perspective (Gromovâs (neg)"entropy").
This view of theorems should be contrasted to the disease of generalisation in mathematical culture. Citing two real-life grad students and a tenured professor in logic (one philosophical, one mathematical, the professor in computer science):
I like your distinction between hemi-toposes, demi-toposes, and semi-toposes
I care about hyper-reals, sur-reals, para-consistency, and so on
Abstract thought â like mathematicians do â is the best kind of thought.
(twitter.com/replicakill, the author of twitter.com/logicians, ragged on David Lewis by saying âWhat do mathematicians like?â âWhat do mathematicians think?â ââ And Corey Mohler has done a wonderful job of mocking Platonism, which is how I guess the thirst for over-generalisation reaches non-mathematicians.)
Paul Halmos knew that cool examples beat generalisations for generalisationâs sake, as did V. I. Arnolâd. And it seems that the people a Harvard mathematician spends her time with make reasonable demands of a mathematical idea as well. It shouldnât just contain previous theories; it should surprise. In Buligaâs Blake/Reynolds dispute, Blake wins hands down.
