Two types of mathematician

There's an obvious but cool meta-thing going on with the number 2 that might be useful to pick out. Some pieces of this thing:

All over the place, we speak in terms of dichotomies and not trichotomies or more. The reason is basically that each dichotomy corresponds to doing PCA and projecting space onto a single axis, and a one-dimensional line has two directions. This suggests that much of the interesting conversation about any given topic (i.e. axis) can be picked up by having exactly two people talk about it. Two people will always differ slightly on the axis. Adding any additional people to a conversation has rapidly diminishing returns: you may have more total disagreement, but rarely more total dimensionality in the disagreement.

Duo Talks is the idea that most productive conversations occur between two people. Even with Trio walks, the setup is for one person to stay on the sidelines and wait for a chance to rotate in. A single conversation really only happens along one dimension, and it requires only two distinct people to detect.

I wonder if monogamy is some kind of attractor state because interactions between two people are the most productive.

Applying the Solitaire Principle, for all the same reasons it's useful to have conversations between two people, it's most useful to draw dichotomies between two pieces of the mind instead of more. This is why we have Internal Double Crux instead of Triple or more. A conversation/internal conflict is always about something and should have a purpose, and that purpose projects the entire conversation onto the one relevant dimension, so it's really only necessary to divide into two sides along this axis. Thus we get S1/S2, Elephant/Rider, Episodic/Diachronic, etc.

Very pleased to see all of these dichotomies collected in one place. The natural question is whether these divides can be integrated to a useful picture with more pieces.

My take on the "Two Cultures" model of problem-solvers and theory-builders: theory-building fields of mathematics like algebraic topology (say) are those where the goal is to articulate grand meta-theorems that are bigger than any particular application. This was the work of a Grothendieck.

Meanwhile, concrete problem-solving fields of mathematics like combinatorics are those where the goal is to become the grand meta-theorem that contains more understanding than any particular theorem you can prove. This was the style of an Erdos. The inarticulate grand meta-theorems lived in his cognitive strategies so that the theorems he actually proved are individually only faint impressions thereof.

I eat corn like an an analyst, and I am an analyst. I also use vim over emacs, like Lisp, and find object-oriented programming weirdly distasteful.

However I don’t think analysis and algebra are usually lumped together and opposed to geometry; my understanding was that traditionally algebra, analysis, and geometry were the three main fields of math.

I tend to think of the distinctions within math as about how much we posit that we know about the objects we work with. The objects of study of mathematical logic are very general and thus can be very “perverse”; the objects of study of algebra and topology are also quite general; the objects of study of geometry are more pinned down because you have a metric; the objects of study of analysis are the ”best behaved” of all, because they have smoothness and integrability properties.

I find analysis much easier than algebra because I rely a lot on the concreteness of being able to measure, estimate, and (sometimes) visualize. People who are more algebra-oriented are more likely than me to become irritated by doing fiddly computations, but they have more ability to reason about very abstract objects.

Great! These are some of my favorite essays on mathematics and I'm excited to see what other rationalists think of them.

I think many mathematicians would object to lumping algebra and analysis together; I know a lot of people who are great at algebra and terrible at analysis. My clusters here are not terribly well-formed but my rough sense is that algebra, analysis, and geometry are three fairly different things.

Having been a geometer that migrated to computer science via formal logic, I can testify to this division - to some extent.

When I first learnt formal logic and then machine learning, I had the same plodding, 'algebra' approach. But now that I've grasped it better, I've started to develop an intuition in these areas, that can shortcut most of the plodding approach (and it's so much more fun).

I think the difference might be more in the way the ideas are communicated. You can communicate semi-rigorous geometric ideas in a (somewhat) intuitive way, and have other geometers grasp them, at least enough that they can re-create them rigorously if needed. But algebraic ideas have to be more explicit if you want anyone beyond your immediate circle to get them.

See for instance Bourbaki, where the internal discussions were filled with intuition and imagery, but where the written outputs were famously tedious and rigorous.

Ok, here’s a 2x2 that captures a lot of the variation in OP:

abstract/concrete x intuitive/methodical.

Intuitive vs. Methodical is what Atiyah, Klein, and Poincare are talking about. Abstract vs Concrete is what Gowers, Rota, and Dyson are talking about.

Abstract and intuitive is like Grothendieck.

Concrete and intuitive is like geometry or combinatorics.

Concrete and methodical is like analysis.

Abstract and methodical — I don’t know what goes in this space.

Promoted to frontpage.

(You mention being unsure about whether it's a good fit for the frontpage. We've been having a debate about how intuition works lately on LW, and I think a post that brings in a bunch of solid data from a healthy field like mathematics is absolutely appropriate, and really appreciated.)

This post is great, and is probably one of my favorite things on LessWrong 2.0 so far. Thank you a lot for writing this, and I am looking forward to reading all the essays in full when I find the time for that.

There are two similar clusters/tensions in arts:

• visual art: on one hand you have to design the "big picture", with all its equilibia, balances and tensions, on the other hand you have to design the local and fine details, wich is something less imaginative and more formal and technical, with strict rules (for anatomy, shadows,...)
• creative writing: your story need to have emotional tensions on the scale of the general plot, but need also to be realistic and credible on the scale of more detailed single events and interactions, wich must respect some stricter constrains
• music composition: you need to design a general theme and mood and then you have to articulate the detailed development of the melodies and rithms which need to observe stricter rules in order to work appropriately

In the chakra system, the 5th (throat) is associated with algebra, while 6th (third eye) is associated with geometry. (Yet another place where this distinction has been noted.) I could probably go off on an intuition-based rant about what this might imply, which should tell you which one I am.

I'd also predict birds prefer games with some variance in them (card games), and frogs prefer deterministic games (chess, go).

Ka-kaw!

I've referenced the Grothendieck quote in this post many times since it came out, and the quote itself seems important enough to be worth curating it. 

I've also referenced this post a few times in a broader context around different mathematical practices, though definitely much less frequently than I've referenced the Grothendieck quote. 

Late to commenting on this post, but where would Grigori Perelman, the prover of the Poincare conjecture, fall then? I remember this quote from his biography:

Golovanov, who studied and occasionally competed alongside Perelman for more than ten years, tagged him as an unambiguous geometer: Perelman had a geometry problem solved in the time it took Golovanov to grasp the question. This was because Golovanov was an algebraist. Sudakov, who spent about six years studying and occasionally competing with Perelman, claimed Perelman reduced every problem to a formula. This, it appears, was because Sudakov was a geometer: his favorite proof of the classic theorem above was an entirely graphical one, requiring no formulas and no language to demonstrate. In other words, each of them was convinced Perelman’s mind was profoundly different from his own. Neither had any hard evidence. Perelman did his thinking almost entirely inside his head, neither writing nor sketching on scrap paper. He did a lot of other things—he hummed, moaned, threw a Ping-Pong ball against the desk, rocked back and forth, knocked out a rhythm on the desk with his pen, rubbed his thighs until his pant legs shone, and then rubbed his hands together—a sign that the solution would now be written down, fully formed. For the rest of his career, even after he chose to work with shapes, he never dazzled colleagues with his geometric imagination, but he almost never failed to impress them with the single-minded precision with which he plowed through problems. His brain seemed to be a universal math compactor, capable of compressing problems to their essence. Club mates eventually dubbed whatever it was he had inside his head the “Perelman stick”—a very large imaginary instrument with which he sat quietly before striking an always-fatal blow.

from Perfect Rigor.

These essays had a pretty large impact on how I go about learning mathematics, I always had an easier time when formulas or arguments could be mapped onto visual structure. In-fact, before writing this comment (and in general when constructing arguments) I imagined a mind map containing all the relevant ideas and relations I wanted to portray. I am now (somewhat poorly) attempting to translate my 3-D visual argument into a linear verbal one.

Something else to be noted is visual reasoning and complementary cognitive artifacts seem to go hand in hand. Consider that learning to use an abacus can allow someone to simulate an abacus in their mind and produce the outputs of an abacus without needing to actually have one. A similar thing can be done with a slide rule. This practice can also produce other, positive effects on certain parts of cognition*.

I would be surprised if the skill of being able to construct complementary cognitive artifacts wasn't potentially helpful in many domains. I don't know how one would go about learning this, but it seems like something to consider as having positive value if investigated.

*Those papers are the first things that came up with a google search. So I reserve the right to be wrong about the exact consequences.

This is a very valuable effort in outlining a hypothesis, and using the author’s wide-ranging taste and knowledge to pull loads of sources together. Definitely helped me a bit think about mathematics and thought, and some of my friends too. I've especially thought about that Grothendieck quote a lot.