Putting pure math in its proper place

Epistemic status: playing around with ideas here, let me know what you think.

Here’s something I have more clarity on now – good thinking is not just about being rigorous. There was a very long while where I thought of rigorous thinking – the kind involved in writing math proofs – as the most elevated kind of thinking. I believed, for example, that a reliable way to become a more “brilliant” person, to refine your intellectual capacity, would be to do more and more hard math. Why did I think this? There were two misconceptions at the root of it.

One misconception was the idea that math itself is a purer or more elevated kind of truth. Of course, math is indeed pure; “correctness” in math is clearer than all other disciplines. But that is because it is devoid of meaning!¹ You can call math the study of necessary truth; equivalently you could call math the study of the mechanics of meaningless symbols that follow axiomatic rules. Math is a powerful tool but it's really only one way of trying to grapple with the world.²

The other misconception was just that because math is so hard – it’s probably uncontroversial to say that pure math and physics are the hardest intellectual disciplines you could try to master – that this hardness also elevated it in some way. The fact that a field is “more challenging to understand” than other fields does say something, of course. But what exactly does it say? Does the fact that it’s more challenging lead you to actually get better at all other fields, at other kinds of thinking?

What does it mean to be intellectually capable? There are many different ways of putting this. How many novel ideas have you put out there that have changed people’s thinking? How many discoveries have you made at the frontier of human knowledge? How many great essays/lectures have you produced that were viewed as insightful by other people who have produced insightful essays/lectures? How good are you at asking questions? How good at you at identifying ideas that are likely to be consequential in the future? How good are you at spotting intellectual grift?

If we loosely bucket all of these skills as one vague concept of “thinking clearly,”³ what I want to say is that simply doing more math does not lead to clearer thinking. How do we make this case? One empirical observation is that mathematicians are not regarded as being uniquely good at everything else. It’s not true that you see PhDs in math going on to establish a disproportionate number of groundbreaking discoveries in all other fields.⁴

That’s an empirical argument, now here’s a more psychological point: the fact that math is fundamentally about following logical rules, means you don’t get to do as much “flexing your intuition muscles”; thinking in math is a way of thinking very rigidly.⁵

Creativity and insight require the opposite of mental rigidity – they require mental relaxation. Insight is fundamentally a kind of analogy-making – forming previously unseen connections between disparate ideas.⁶ How do you make connections between disparate ideas? There is some kind of role here for relaxation. I’m not exactly sure why, but relaxing seems to allow for unexpected connections to be made. Consider the fact that our best ideas often come to us when we’re not thinking hard about them; consider Ramanujan's dreams, or the role of LSD in Francis Crick’s discovery of the DNA helix or Karry Mullis’s invention of the polymerase chain reaction (PCR), and the vast literature on how psychedelics increase plasticity in the brain.

There are also arguments that mental relaxation is crucial for psychological change—so mental flexibility is important not just for intellectual insights but also for emotional ones.

I think to foster more insight you need to dance. Math (or, precise and rigorous thinking more broadly) is like a weight-lifting of intellectual activity – you are building up pure muscle mass. But intellectual insight also requires dexterity – which you get more from dance and stretching. You want to marinate in your intuitions, appreciate art, relax, dream, doodle, journal, roll around on the floor, listen to music, and (quite literally) dance and stretch.

There’s an ironic historical note here, pointed out by Julian Gough, which is that math was originally meant to be a “humble language,” delimiting science to a narrow domain (that which can be described by math) and leaving all the other “important stuff” in the hands of the Catholic Church. This view has now been turned upside down: many people think of math as the only language of truth, or as the purest form of truth. Gough calls science’s insistence on mathematical language as a “vow of poverty.” At least for me, it was revelatory that I had never even contemplated this view – that math might be a deeply limited language rather than some kind of superior language. Math is a language of quantities devoid of qualities. It’s a language of form with no substance. It’s syntax without semantics.

The question remaining here is: what role should math play, in a serious intellectual engagement with life? I do think that having done many proof-based math courses (and also having been a programmer for over a decade), I have learned useful things. Continuing the acrobatics analogy, I've built the necessary muscle mass for certain things that will just be hard to have any intuition about if you've never done hard math or programming. Examples: loops, recursion, self-reference/strange loops, proof-by-contradiction, uncountable numbers, diagonalization, dimensionality, orthogonality, continuity and differentiation, the idea of “invariance under stretching and bending” (topology), and “invariance under rotation and translation” (geometry), infinitesimals. Debatably, you could get a feel for all these things without ever having done the actual math or programming; I'm not sure. But having done it certainly helps.

I think a reasonable approach to math is to learn it to the extent that you need to, for the questions you want to ask, and not feel any obligation to go further than that. For example, I am somewhat convinced that understanding the physics of fields could potentially be relevant for consciousness (at least, the QRI folks claim that consciousness could be explained by electromagnetic field topologies) and so that's something I’m interested in. They’ve also piqued my interest in group theory and symmetries, since they think of valence of conscious experiences as having to do with symmetry.

There is an unanswered meta-question here, which is “how far will formal methods take us?” The big shift for me the past five years has been losing my unquestioned faith in formal methods as “the correct way to think about ultimate truth.”⁷ I am less convinced that the way to understand the world is to make formal mathematical models of it and prove theorems about those models. The way I could be wrong is if further study of pure math led to some groundbreaking discovery that had tangible implications for our understanding of the world. Some people believe, for example, the universe is itself a mathematical object, and in that case, there is a very strong case for doing as much math as possible because you are studying the fundamental structure of the universe itself.

My friend Will recently made the following observation about AI models: “it's interesting that 1.5 billion parameters is all you need to crush math competitions, but you need like 15 trillion to make the model be funny. maybe humor is the right measure of true intelligence.” I think it was said in jest, but I wonder if there’s some deeper insight there. An insight that sounds like something like: that which is formalizable is not necessarily that which is most powerful, or most important to know.

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Thanks to Santi for discussion on a draft.

1. Let me try to explain what I mean here. Math does not study physical entities; it does not point to anything in the actual world. Math is the study of abstractions, whose existence is controversial to say the least. When mathematicians do math they are unconcerned about any practical facts about the world. Math can certainly be applied to the real world, but is not constrained by it; all the objects that mathematicians study are idealizations (e.g. spheres) which are never found in the world itself. (You could argue that this is all a mistaken understanding of what math is – see David Bessis’s book Mathematica which I haven’t read yet.)

2. Objection: isn’t math unreasonably effective? Isn’t it the most powerful tool we’ve found for understanding the world? I think we can get perspective on this question but inverting it: what are all the fields where math has not proven fundamental to understanding? I’d argue there are many: psychology, (parts of) biology, history, literature, (much of) philosophy, anthropology, art, etc. Even in more STEM-y fields like machine learning, you could argue that pure math has not been particularly helpful – the math behind the last decade of AI advances is pretty straightforward, and progress has been driven more by engineering than by mathematical discoveries.

3. Some might call this “intelligence”? I haven’t thought much about how I’d distinguish my notion of “thinking clearly” from “intelligence.” But e.g. I think Einstein could be described as a “clearer thinker” than von Neumann, while von Neumann could be described as “more intelligent,” see Erik Hoel.

4. I’m not sure if there are systematic studies on this but this is my general impression. An interesting counterpoint that a mathematician friend of mine made is that while math PhDs aren’t disproportionately successful in other fields, math undergrads do seem to make outstanding contributions beyond just math (e.g. Jim Simons, Sergey Brin, Ed Thorp).

5. I don’t mean here that mathematical thinking involves no intuition at all; see e.g. Terry Tao’s post about the pre-rigorous, rigorous, and post-rigorous stages of the mathematician’s journey. But math clearly demands much more rigor than other disciplines and prizes it like no other field.

6. This is very much inspired by Hofstadter’s work on analogy as the “core engine” of thought. See his lecture, and his book Surfaces and Essences, which I wrote a tweet thread about.

7. “Truth” itself is tricky, so I might rephrase this as saying that I went from thinking that “formal methods are the correct way to think about what’s most important”, to no longer believing that.