The n-Category Café

In the Preface to the marvelous book

• I.M. Gelfand, M.M. Kapranov, A.V. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants, Birkhäuser, 1994

great tribute is paid to the work of Arthur Cayley:

“…Moreover, in the process of writing we discovered a beautiful area which had nearly been forgotten so that our work can be regarded as a natural continuation of the classical developments in algebra during the 19th century.

We found that Cayley and other mathematicans of the period understood many of the concepts which today are commonly thought of as modern and quite recent. Thus, in an 1848 note on the resultant, Cayley in fact laid out the foundations of modern homological algebra. We were happy to enter into spiritual contact with this great mathematician.”

The 1848 article is reproduced in an appendix. They also credit Cayley with discovering hyperdeterminants (the multidimensional determinants in their title).

On reading Cantor’s work I was surprised by how much of it was metaphysical (and application orientated) rather than (in the modern sense) mathematical, and then thereby encouraged to add to it and to criticise it on metaphysical grounds.

Once, as a student, I was looking at an old textbook on theoretical physics, from the beginning of the 20th century.

Apparently, it was from before the time when linear algebra had been cleaned up to a point that a physics textbook could freely play around with vector spaces.

Instead, that book didn’t know the concept of a vector – or almost knew it, but not quite. Instead of using vectors, the text was full of direction cosines.

This way, many very simple statements, from the present perspective, appeared in a comparatively opaque fashion.

Remarkably, at one point there was a discussion of the classical gravitational field outside of a solid ball.

In a footnote to the rather simple computation, the author remarked that, allegedly, Newton’s publication of his Principia was held back for many years because he wasn’t able to prove his conjecture that this field is the same as that of a point source of the same mass.

The footnote concluded with an emphasis of how important it is, in theoretical physics and in math, to use the right symbolic language. If only Newton had bothered to invent something better than his fluxion notation for doing calculus, he would have easily been able to do this calculation in two minutes, as every student today does.

I found that interesting. An author who could see of every linear equation only a faint shadow of direction cosines was marvelling at the power of notational elegance.

The older I get, the more I get interested in the history of math and physics. I hope it’s not just because my past is getting longer than my future. I think it’s because I keep learning new stuff from the past.

One big lesson is that: what seems obvious now was not obvious before… so what seems nonobvious now may seem obvious later!

For example, when Newton’s Principia came out, it caused a big stir, but very few people could follow it — classical mechanics and calculus were comprehensible only to a few top experts. Now they’re routinely taught in good high schools.

(The Principia is still very hard to read.)

This gives me hope that differential geometry, topology, category theory and so on will eventually be taught in good high schools. Which is important, because otherwise people will eventually be quite old by the time they reach the frontiers of knowledge.

(Of course, life spans are increasing, too…)

Another big lesson is: what smart people once did often seems strange or silly now — so what smart people are doing now may seem strange or silly in the future.

This lesson makes me feel less compelled to follow every trend that comes along.

There are also lots of specific technical facts one can learn from studying the history of science — and I could talk about these for hours — but in a way these are less important.

Thanks everyone for these comments.

To these I can add that on my old blog ‘dt’ wrote:

It’s worth pointing out that, from time to time, modern mathematicians find it extremely fruitful to read the classics from eras with inferior epistemic resources. Geometric invariant theory certainly owes a lot to Mumford reading Hilbert. Intersection homology owes a lot to MacPherson reading Poincare. Manjul Bhargava of course read Gauss.

I also mention there that Gian-Carlo Rota once described reading nineteenth century mathematical texts as like entering a hothouse full of exotic plants whose existence you had never suspected.

Robert Hermann wrote (see end of this)

In reading Lie’s work in preparation for my commentary on these translations, I was overwhelmed by the richness and beauty of the geometric ideas flowing from Lie’s work. Only a small part of this has been absorbed into mainstream mathematics. He thought and wrote in grandiose terms, in a style that has now gone out of fashion, and that would be censored by our scientific journals! The papers translated here and in the succeeding volumes of our translations present Lie in his wildest and greatest form.

I seem to recall he also says somewhere that he periodically returns to Darboux’s works, finding he can understand more each time.

I wonder if Pierre Cartier read Euler to write Mathemagics (A Tribute to L. Euler and R. Feynman)

I suppose one might still argue that there’s an essential difference between mathematics and philosophy along the following lines. One can practice mathematics without returning to earlier texts, even though many good mathematicians have profited greatly by doing so, whereas one should not practice philosophy without having read Plato and Aristotle.

I’m not so sure this points to an essential difference.