We've all got a chicken-duck-woman thing waiting for us

One thing that's come up talking to @perdvivly and @bubbliterally about the differences between our lists, both this one and the physicist one from a while ago, is that I tend to take a narrower view of the subject in question than they do. One of my big criteria is "what's the chance a modern mathematician encounters this person's work on any given day," which creates a bias towards certain subjects and eras. That's why Turing and Shannon aren't on this list, for instance; what they did was very impressive and had massive implications for society writ large, but it lives somewhat in its own sphere that's peripheral to the broader mathematical landscape. Someone working in differential geometry, algebraic number theory, stable homotopy theory, etc can probably go their whole career without thinking about computability or information entropy.

Euclid's a similar case but with a different texture: we just don't really do synthetic geometry anymore. That's nothing against him or his significance to the history of math in a broader sense, but it makes it hard for me to put him on the list. I wouldn't know how to place him fairly given my standards. (Viv has argued that he deserves recognition for essentially inventing the modern concept of a proof, and if that's true it might make me reconsider, but I'm not sure how much I believe it.) It's analogous to me leaving people like Copernicus, Galileo and Kepler off my physicists list. There are reasonable arguments that the field wouldn't exist without them, but Newton is just such a dividing line in its history that any work done before him at most only loosely resembles what we'd call "physics" today. (Except maybe Huygens, guy was a beast.)

That's why Al-Khwarizmi's different from Euclid in my eyes. The things he was primarily concerned with—solving linear and quadratic equations, trig identities—are still familiar to and fundamental for any modern mathematician (or hell, any modern high school student). He doesn't lie behind that dividing line because, like Newton, he is the line. I credit him with being the first to develop a system of techniques for working with numbers that's sufficiently powerful to make it believable that they could form the basis for not just mathematics but basically every field of technical inquiry. Put anther way, he is (along with lots of people like Descartes later on) a big part of why the mathematics of Euclid eventually became obsolete.

1. The Euclid thing:

To be clear, the claim isn’t that Euclid invented the modern notion of proof. The claim is that Euclid’s Elements is the earliest known work that makes use of the axiomatic system for proof. I think ~proof~ in a more general mathematical sense goes back earlier than Euclid, at least to Thales and probably earlier than that (but ancient history is hard and we don’t have a ton of sources. Thales is generally considered to be the first natural philosopher and that’s good enough for me to make an offhand (but not too serious) claim about his priority <- I know this is Aristotelian propaganda but it’s also the mainstream view as of the 18th century.)

The remarkable thing about Euclid is that his axiomatic system was early and popular enough to survive. We don’t have great accounts of people using the axiomatic method in the same way for a long time afterwards. And we know that when the modern notion of proof was being developed by Hilbert et al. it was Euclid they were modelling their systems on. This isn’t to say that Euclid single-handedly shaped the axiomatic method into what it is today (Zermelo did that) it’s just that he was doing something epistemically remarkable for his time and that survived to be widely read at around the time modern mathematics was being born.

2. The cs is a subfield of mathematics thing:

You have elsewhere described a “dense core” of mathematics that involves analysis, algebra, topology, and algebraic geometry. And I think your claim is that these subjects are inescapable for any modern mathematician whereas other fields are specialties. A modern mathematician may specialise in whatever field but they will still be familiar with the more ubiquitous fields and it’s those that should be understood to have the biggest influence.

And I don’t fully hate that idea! I think it’s vaguely borne out by this graph (though perhaps you are over valuing analysis and undervaluing probability, combinatorics and number theory)

Which was made by mapping mathematical research on ArXiv.

But as a non-mathematician that’s not how I see it. I’m not spending my time reading mathematical research papers. I’ll occasionally read a textbook but that feels like a different sort of type of engagement with the topic.

I think I have a view of the value of mathematics that is more about the ways of thinking it can afford. And geometric and algebraic ways of thinking are very powerful! But so is the information theoretic view. And as I become gradually more and more Shannon-pilled, I’m starting to think that that view is actually more important for navigating the world than any other. *Zac Oyama voice* but enough about that.

I think if I did relegate the value of the fields of mathematics to how well they integrate with other fields, the entire top 10 list would just be foundations people, which seems absurd to me. Maybe more working mathematicians know more about algebraic geometry, but the field of algebraic geometry, topology, whatever, all rest more deeply on foundations than they do any other. But enough about that.

my daughter cannot, through action or inaction, harm a human or allow a human to come to harm

a daughter at rest or in constant motion remains at rest or in constant motion unless acted upon by another force

daughters are never created or destroyed, only transformed

always treat every daughter as loaded, even if you know she isn't

Euler

Turing

Al-Khwarizmi

Gauss

Leibniz

Napier

Euclid

Zermelo

Shannon

1. Euler

2. Al-Khwarizmi

3. Gauss

4. Turing

5. Shannon

6. Leibniz

7. Zermelo

8. Hilbert

9. Riemann

Euclid

Alexander Grothendieck

Henri Poincaré

Isaac Newton

Al-Khwarizmi

Alan Turing

Emmy Noether

Alfred Tarski

Euler

Turing

Al-Khwarizmi

Gauss

Leibniz

Napier

Euclid

Zermelo

Shannon

1. Euler

2. Al-Khwarizmi

3. Gauss

4. Turing

5. Shannon

6. Leibniz

7. Zermelo

8. Hilbert

9. Riemann

Asking a yes/ no question that narrows the pool from 11 options to 1 gained you 3.459 bits of information. Which means if the answer had been no you would have narrowed the pool from 11 options to 10 and gained 0.138 bits of information.

The expected information gain from the question is the probability weighted average of the yes/ no responses, which works out to 0.439 bits.

You could have asked a question with an expected information gain of 0.994 bits by asking something that would split remaining options into 5 vs 6.