"The footpath is blocked by that dog. If I kick the dog out of the way, Ganyu will be upset, so Master will be upset. But if I don't kick the dog out of the way, I will have to slow down, which is inefficient. But if I kick the dog out of the way... 🤔"
—Shenhe, at least once a week
Shenhe reaching the conclusion that she can subtly nudge the dog a few steps to the right but it does take quite a few tries to get the strength right... so she ends up kicking a few dogs (accidentally) into the river
Ganyu wonders why one or two dogs look particularly afraid of Shenhe while the others are fine...
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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 Feynman’s challenge: by a series of clear, unsurprising steps, one arrives at an unexpected conclusion.
surprising rep-theory consequences of Young diagrams, Ferrers sequences, and so on (you could say the strangeness of integer partitions is really to blame here…)
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 Coxeter’s 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, I’ve found similarly flush with shockng content—quite 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 — 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 “looks” — 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 it’s clearer than elsewhere whether you understand something or not—compare to economic theory for example), there is a clear delination of what’s obvious and what’s 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 Teichmüller stuff and mapping class groups because it’s the confluence of three rivers.
Sometimes mathematical explanations require fortitude (Gromov’s "energy") and sometimes a shift in perspective (Gromov’s (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 — like mathematicians do — is the best kind of thought.
(twitter.com/replicakill, the author of twitter.com/logicians, ragged on David Lewis by saying “What do mathematicians like?” “What do mathematicians think?” —— 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 generalisation’s sake, as did V. I. Arnol’d. And it seems that the people a Harvard mathematician spends her time with make reasonable demands of a mathematical idea as well. It shouldn’t just contain previous theories; it should surprise. In Buliga’s Blake/Reynolds dispute, Blake wins hands down.
Bivector transformation, and reciprocal frame for column vectors of a transformation
[Click here for a PDF version of this and previous two posts]
The author of a book draft I am reading pointed out if a vector transforms as
\begin{equation}\label{eqn:adjoint:760}
\Bv \rightarrow M \Bv,
\end{equation}
then cross products must transform as
\begin{equation}\label{eqn:adjoint:780}
\Ba \cross \Bb \rightarrow \lr{ \textrm{adj}\, M }^\T \lr{ \Ba \cross \Bb }.
\end{equation}
Bivectors…
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[Click here for a PDF version of this post]
I started reviewing a book draft that mentions the adjoint in passing, but I’ve forgotten what I knew about the adjoint (not counting self-adjoint operators, which is different.) I do recall that adjoint matrices were covered in high school linear algebra (now 30+ years ago!), but never really used after that.
It appears that the basic property of the…