Long-time lurker (c. 2013), recent poster. I also write on the EA Forum.
Mo Putera
While Dyson’s birds and frogs archetypes of mathematicians is oft-mentioned, David Mumford’s tribes of mathematicians is underappreciated, and I find myself pointing to it often in discussions that devolve into “my preferred kind of math research is better than yours”-type aesthetic arguments:
… the subjective nature and attendant excitement during mathematical activity, including a sense of its beauty, varies greatly from mathematician to mathematician… I think one can make a case for dividing mathematicians into several tribes depending on what most strongly drives them into their esoteric world. I like to call these tribes explorers, alchemists, wrestlers and detectives. Of course, many mathematicians move between tribes and some results are not cleanly part the property of one tribe.
Explorers are people who ask—are there objects with such and such properties and if so, how many? They feel they are discovering what lies in some distant mathematical continent and, by dint of pure thought, shining a light and reporting back what lies out there. The most beautiful things for them are the wholly new objects that they discover (the phrase ‘bright shiny objects’ has been in vogue recently) and these are especially sought by a sub-tribe that I call Gem Collectors. Explorers have another sub-tribe that I call Mappers who want to describe these new continents by making some sort of map as opposed to a simple list of ‘sehenswürdigkeiten’.
Alchemists, on the other hand, are those whose greatest excitement comes from finding connections between two areas of math that no one had previously seen as having anything to do with each other. This is like pouring the contents of one flask into another and—something amazing occurs, like an explosion!
Wrestlers are those who are focussed on relative sizes and strengths of this or that object. They thrive not on equalities between numbers but on inequalities, what quantity can be estimated or bounded by what other quantity, and on asymptotic estimates of size or rate of growth. This tribe consists chiefly of analysts and integrals that measure the size of functions but people in every field get drawn in.
Finally Detectives are those who doggedly pursue the most difficult, deep questions, seeking clues here and there, sure there is a trail somewhere, often searching for years or decades. These too have a sub-tribe that I call Strip Miners: these mathematicians are convinced that underneath the visible superficial layer, there is a whole hidden layer and that the superficial layer must be stripped off to solve the problem. The hidden layer is typically more abstract, not unlike the ‘deep structure’ pursued by syntactical linguists. Another sub-tribe are the Baptizers, people who name something new, making explicit a key object that has often been implicit earlier but whose significance is clearly seen only when it is formally defined and given a name.
Mumford’s examples of each, both results and mathematicians:
Explorers:
Theaetetus (ncient Greek list of the five Platonic solids)
Ludwig Schläfli (extended the Greek list to regular polytopes in n dimensions)
Bill Thurston (“I never met anyone with anything close to his skill in visualization”)
the list of finite simple groups
Michael Artin (discovered non-commutative rings “lying in the middle ground between the almost commutative area and the truly huge free rings”)
Set theorists (“exploring that most peculiar, almost theological world of ‘higher infinities’”)
Mappers:
Mumford himself
arguably, the earliest mathematicians (the story told by cuneiform surveying tablets)
the Mandelbrot set
Ramanujan’s “integer expressible two ways as a sum of two cubes”
the Concinnitas project of Bob Feldman and Dan Rockmore of ten aquatints
Alchemists:
Abraham De Moivre
Oscar Zariski, Mumford’s PhD advisor (“his deepest work was showing how the tools of commutative algebra, that had been developed by straight algebraists, had major geometric meaning and could be used to solve some of the most vexing issues of the Italian school of algebraic geometry”)
the Riemann-Roch theorem (“it was from the beginning a link between complex analysis and the geometry of algebraic curves. It was extended by pure algebra to characteristic p, then generalized to higher dimensions by Fritz Hirzebruch using the latest tools of algebraic topology. Then Michael Atiyah and Isadore Singer linked it to general systems of elliptic partial differential equations, thus connecting analysis, topology and geometry at one fell swoop”)
Wrestlers:
Archimedes (“he loved estimating π and concocting gigantic numbers”)
Calculus (“stems from the work of Newton and Leibniz and in Leibniz’s approach depends on distinguishing the size of infinitesimals from the size of their squares which are infinitely smaller”)
Euler’s strange infinite series formulas
Stirling’s formula for the approximate size of n!
Augustin-Louis Cauchy (“his eponymous inequality remains the single most important inequality in math”)
Sergei Sobolev
Shing-Tung Yau
Detectives:
Andrew Wiles is probably the archetypal example
Roger Penrose (“”My own way of thinking is to ponder long and, I hope, deeply on problems and for a long time … and I never really let them go.”)
Strip Miners:
Alexander Grothendieck (“he greatest contemporary practitioner of this philosophy in the 20th century… Of all the mathematicians that I have met, he was the one whom I would unreservedly call a “genius”. … He considered that the real work in solving a mathematical problem was to find le niveau juste in which one finds the right statement of the problem at its proper level of generality. And indeed, his radical abstractions of schemes, functors, K-groups, etc. proved their worth by solving a raft of old problems and transforming the whole face of algebraic geometry)
Leonard Euler from Switzerland and Carl Fredrich Gauss (“both showed how two dimensional geometry lay behind the algebra of complex numbers”)
Eudoxus and his spiritual successor Archimedes (“he level they reached was essentially that of a rigorous theory of real numbers with which they are able to calculate many specific integrals. Book V in Euclid’s Elements and Archimedes The Method of Mechanical Theorems testify to how deeply they dug”)
Aryabhata
Some miscellaneous humorous quotes:
When I was teaching algebraic geometry at Harvard, we used to think of the NYU Courant Institute analysts as the macho guys on the scene, all wrestlers. I have heard that conversely they used the phrase ‘French pastry’ to describe the abstract approach that had leapt the Atlantic from Paris to Harvard.
Besides the Courant crowd, Shing-Tung Yau is the most amazing wrestler I have talked to. At one time, he showed me a quick derivation of inequalities I had sweated blood over and has told me that mastering this skill was one of the big steps in his graduate education. Its crucial to realize that outside pure math, inequalities are central in economics, computer science, statistics, game theory, and operations research. Perhaps the obsession with equalities is an aberration unique to pure math while most of the real world runs on inequalities.
In many ways [the Detective approach to mathematical research exemplified by e.g. Andrew Wiles] is the public’s standard idea of what a mathematician does: seek clues, pursue a trail, often hitting dead ends, all in pursuit of a proof of the big theorem. But I think it’s more correct to say this is one way of doing math, one style. Many are leery of getting trapped in a quest that they may never fulfill.
Mingyuan has written Cryonics signup guide #1: Overview.
Can you clarify the calculation behind this estimate?
See here.
I kind of envy that you figured this out yourself — I learned the parallelipiped hypervolume interpretation of the determinant from browsing forums (probably this MSE question’s responses). Also, please do write that blog article.
And if I keep doing that, hypothetically speaking, some of those discoveries might even be original.
Yeah, I hope you will! I’m reminded of what Scott Aaronson said recently:
When I was a kid, I too started by rediscovering things (like the integral for the length of a curve) that were centuries old, then rediscovering things (like an efficient algorithm for isotonic regression) that were decades old, then rediscovering things (like BQP⊆PP) that were about a year old … until I finally started discovering things (like the collision lower bound) that were zero years old. This is the way.
What were you outputting over a million words in a week for?
And given that there are 7 x 16 x 60 = 6,720 minutes in a week of 16-hour days, you’d need to output 150 wpm at minimum over the entire duration to hit a million words, which doesn’t seem humanly possible. How did you do it?
I suspect you’ve probably seen Scott’s Varieties Of Argumentative Experience, so this is mostly meant for others. He says of Graham’s hierarchy:
Graham’s hierarchy is useful for its intended purpose, but it isn’t really a hierarchy of disagreements. It’s a hierarchy of types of response, within a disagreement. Sometimes things are refutations of other people’s points, but the points should never have been made at all, and refuting them doesn’t help. Sometimes it’s unclear how the argument even connects to the sorts of things that in principle could be proven or refuted.
If we were to classify disagreements themselves – talk about what people are doing when they’re even having an argument – I think it would look something like this:
Most people are either meta-debating – debating whether some parties in the debate are violating norms – or they’re just shaming, trying to push one side of the debate outside the bounds of respectability.
If you can get past that level, you end up discussing facts (blue column on the left) and/or philosophizing about how the argument has to fit together before one side is “right” or “wrong” (red column on the right). Either of these can be anywhere from throwing out a one-line claim and adding “Checkmate, atheists” at the end of it, to cooperating with the other person to try to figure out exactly what considerations are relevant and which sources best resolve them.
If you can get past that level, you run into really high-level disagreements about overall moral systems, or which goods are more valuable than others, or what “freedom” means, or stuff like that. These are basically unresolvable with anything less than a lifetime of philosophical work, but they usually allow mutual understanding and respect.
Scott’s take on the relative futility of resolving high-level generators of disagreement (which seems to be beyond Level 7? Not sure) within reasonable timeframes is kind of depressing.
A bit more on the high-level generators:
High-level generators of disagreement are what remains when everyone understands exactly what’s being argued, and agrees on what all the evidence says, but have vague and hard-to-define reasons for disagreeing anyway. In retrospect, these are probably why the disagreement arose in the first place, with a lot of the more specific points being downstream of them and kind of made-up justifications. These are almost impossible to resolve even in principle.
“I feel like a populace that owns guns is free and has some level of control over its own destiny, but that if they take away our guns we’re pretty much just subjects and have to hope the government treats us well.”
“Yes, there are some arguments for why this war might be just, and how it might liberate people who are suffering terribly. But I feel like we always hear this kind of thing and it never pans out. And every time we declare war, that reinforces a culture where things can be solved by force. I think we need to take an unconditional stance against aggressive war, always and forever.”
“Even though I can’t tell you how this regulation would go wrong, in past experience a lot of well-intentioned regulations have ended up backfiring horribly. I just think we should have a bias against solving all problems by regulating them.”
“Capital punishment might decrease crime, but I draw the line at intentionally killing people. I don’t want to live in a society that does that, no matter what its reasons.”
Some of these involve what social signal an action might send; for example, even a just war might have the subtle effect of legitimizing war in people’s minds. Others involve cases where we expect our information to be biased or our analysis to be inaccurate; for example, if past regulations that seemed good have gone wrong, we might expect the next one to go wrong even if we can’t think of arguments against it. Others involve differences in very vague and long-term predictions, like whether it’s reasonable to worry about the government descending into tyranny or anarchy. Others involve fundamentally different moral systems, like if it’s okay to kill someone for a greater good. And the most frustrating involve chaotic and uncomputable situations that have to be solved by metis or phronesis or similar-sounding Greek words, where different people’s Greek words give them different opinions.
You can always try debating these points further. But these sorts of high-level generators are usually formed from hundreds of different cases and can’t easily be simplified or disproven. Maybe the best you can do is share the situations that led to you having the generators you do. Sometimes good art can help.
The high-level generators of disagreement can sound a lot like really bad and stupid arguments from previous levels. “We just have fundamentally different values” can sound a lot like “You’re just an evil person”. “I’ve got a heuristic here based on a lot of other cases I’ve seen” can sound a lot like “I prefer anecdotal evidence to facts”. And “I don’t think we can trust explicit reasoning in an area as fraught as this” can sound a lot like “I hate logic and am going to do whatever my biases say”. If there’s a difference, I think it comes from having gone through all the previous steps – having confirmed that the other person knows as much as you might be intellectual equals who are both equally concerned about doing the moral thing – and realizing that both of you alike are controlled by high-level generators. High-level generators aren’t biases in the sense of mistakes. They’re the strategies everyone uses to guide themselves in uncertain situations.
This doesn’t mean everyone is equally right and okay. You’ve reached this level when you agree that the situation is complicated enough that a reasonable person with reasonable high-level generators could disagree with you. If 100% of the evidence supports your side, and there’s no reasonable way that any set of sane heuristics or caveats could make someone disagree, then (unless you’re missing something) your opponent might just be an idiot.
Depends on the app. Tinder for instance has a section called “What are you looking for?” that everyone else can see, whose selectable options include “New friends”, “Still figuring it out”, “Short-term fun”, “Long-term partner”, and a mix of the last two. People in my area use a pretty even mix of these, and their signaling is usually honest.
I should’ve been clearer, my bad.
For most dating app users, I’m genuinely uncertain how representative both assumptions are, and I’d be curious to see more data regarding both (Aella’s surveys maybe?)
For me, neither assumption holds; I suspect this makes me un-representative of most users:
I decouple dating from sex, and do use these apps to find platonic acquaintances
I swipe right mostly if I predict the person is interesting to meet up with, and swipe left on the majority of “lust at first sight” profiles
Not sure how representative your guess is of most dating app users. Certainly isn’t the case for me.
This Nature post looks into theories of why GPL-1 drugs seem to help with essentially everything.
There’s also Scott’s Why Does Ozempic Cure All Diseases? from awhile back. The Nature article takes a more straightforward scientific journalism approach and largely focuses on immediate biological mechanisms, while Scott is Scott.
I enjoyed these passages from Henrik Karlsson’s essay Cultivating a state of mind where new ideas are born on the introspections of Alexander Grothendieck, arguably the deepest mathematical thinker of the 20th century.
In June 1983, Alexander Grothendieck sits down to write the preface to a mathematical manuscript called Pursuing Stacks. He is concerned by what he sees as a tacit disdain for the more “feminine side” of mathematics (which is related to what I’m calling the solitary creative state) in favor of the “hammer and chisel” of the finished theorem. By elevating the finished theorems, he feels that mathematics has been flattened: people only learn how to do the mechanical work of hammering out proofs, they do not know how to enter the dreamlike states where truly original mathematics arises. To counteract this, Grothendieck in the 1980s has decided to write in a new way, detailing how the “work is carried day after day [. . .] including all the mistakes and mess-ups, the frequent look-backs as well as the sudden leaps forward”, as well as “the early steps [. . .] while still on the lookout for [. . .] initial ideas and intuitions—the latter of which often prove to be elusive and escaping the meshes of language.”
This was how he had written Pursuing Stacks, the manuscript at hand, and it was the method he meant to employ in the preface as well. Except here he would be probing not a theorem but his psychology and the very nature of the creative act. He would sit with his mind, observing it as he wrote, until he had been able to put in words what he meant to say. It took him 29 months.
When the preface, known as Récoltes et Semailles, was finished, in October 1986, it numbered, in some accounts, more than 2000 pages. It is in an unnerving piece of writing, seething with pain, curling with insanity at the edges—Grothendieck is convinced that the mathematical community is morally degraded and intent on burying his work, and aligns himself with a series of saints (and the mathematician Riemann) whom he calls les mutants. One of his colleagues, who received a copy over mail, noticed that Grothendieck had written with such force that the letters at times punched holes through the pages. Despite this unhinged quality, or rather because of it, Récoltes et Semailles is a profound portrait of the creative act and the conditions that enable our ability to reach out toward the unknown. (Extracts from it can be read in unauthorized English translations, here and here.)
On the capacity to be alone as necessary prerequisite to doing groundbreaking work:
An important part of the notes has Grothendieck meditating on how he first established contact with the cognitive space needed to do groundbreaking work. This happened in his late teens. It was, he writes, this profound contact with himself which he established between 17 and 20 that later set him apart—he was not as strong a mathematician as his peers when he came to Paris at 20, in 1947. That wasn’t the key to his ability to do great work.
I admired the facility with which [my fellow students] picked up, as if at play, new ideas, juggling them as if familiar with them from the cradle—while for myself I felt clumsy, even oafish, wandering painfully up an arduous track, like a dumb ox faced with an amorphous mountain of things that I had to learn (so I was assured), things I felt incapable of understanding[.] …
In fact, most of these comrades who I gauged to be more brilliant than I have gone on to become distinguished mathematicians. Still, from the perspective of 30 or 35 years, I can state that their imprint upon the mathematics of our time has not been very profound. They’ve all done things, often beautiful things, in a context that was already set out before them, which they had no inclination to disturb. Without being aware of it, they’ve remained prisoners of those invisible and despotic circles which delimit the universe of a certain milieu in a given era. To have broken these bounds they would have had to rediscover in themselves that capability which was their birth-right, as it was mine: the capacity to be alone.
The capacity to be alone. This was what Grothendieck had developed. In the camp during the war, a fellow prisoner named Maria had taught him that a circle can be defined as all points that are equally far from a point. This clear abstraction attracted him immensely. After the war, having only a limited understanding of high school mathematics, Grothendieck ended up at the University of Montpellier, which was not an important center for mathematics. The teachers disappointed him, as did the textbooks: they couldn’t even provide a decent definition of what they meant when they said length! Instead of attending lectures, he spent the years from 17 to 20 catching up on high school mathematics and working out proper definitions of concepts like arc length and volume. Had he been in a good mathematical institution, he would have known that the problems he was working on had already been solved 30 years earlier. Being isolated from mentors he instead painstakingly reinvent parts of what is known as measurement theory and the Lebesgue integral.
A few years after I finally established contact with the world of mathematics at Paris, I learned, among other things, that the work I’d done in my little niche [. . . had] been long known to the whole world [. . .]. In the eyes of my mentors, to whom I’d described this work, and even showed them the manuscript, I’d simply “wasted my time”, merely doing over again something that was “already known”. But I don’t recall feeling any sense of disappointment. [. . .]
(I think that last sentence resonates with me in a way that I don’t think it does for most science & math folks I know, for whom discovery (as opposed to rediscovery) takes precedent emotionally.)
This experience is common in the childhoods of people who go on to do great work, as I have written elsewhere. Nearly everyone who does great work has some episode of early solitary work. As the philosopher Bertrand Russell remarked, the development of gifted and creative individuals, such as Newton or Whitehead, seems to require a period in which there is little or no pressure for conformity, a time in which they can develop and pursue their interests no matter how unusual or bizarre. In so doing, there is often an element of reinventing the already known. Einstein reinvented parts of statistical physics. Pascal, self-teaching mathematics because his father did not approve, rederived several Euclidean proofs. There is also a lot of confusion and pursuit of dead ends. Newton looking for numerical patterns in the Bible, for instance. This might look wasteful if you think what they are doing is research. But it is not if you realize that they are building up their ability to perceive the evolution of their own thought, their capacity for attention.
On the willingness to linger in confusion, and the primacy of good question generation over answering them:
One thing that sets these intensely creative individuals apart, as far as I can tell, is that when sitting with their thoughts they are uncommonly willing to linger in confusion. To be curious about that which confuses. Not too rapidly seeking the safety of knowing or the safety of a legible question, but waiting for a more powerful and subtle question to arise from loose and open attention. This patience with confusion makes them good at surfacing new questions. It is this capacity to surface questions that set Grothendieck apart, more so than his capacity to answer them. When he writes that his peers were more brilliant than him, he is referring to their ability to answer questions1. It was just that their questions were unoriginal. As Paul Graham observes:
People show much more originality in solving problems than in deciding which problems to solve. Even the smartest can be surprisingly conservative when deciding what to work on. People who’d never dream of being fashionable in any other way get sucked into working on fashionable problems.
Grothendieck had a talent to notice (and admit!) that he was subtly bewildered and intrigued by things that for others seemed self-evident (what is length?) or already settled (the Lebesgue integral) or downright bizarre (as were many of his meditations on God and dreams). From this arose some truly astonishing questions, surfacing powerful ideas, such as topoi, schemes, and K-theory.
On working with others without losing yourself:
After his three years of solitary work, Grothendieck did integrate into the world of mathematics. He learned the tools of the trade, he got up to date on the latest mathematical findings, he found mentors and collaborators—but he was doing that from within his framework. His peers, who had been raised within the system, had not developed this feel for themselves and so were more susceptible to the influence of others. Grothendieck knew what he found interesting and productively confusing because he had spent three years observing his thought and tracing where it wanted to go. He was not at the mercy of the social world he entered; rather, he “used” it to “further his aims.” (I put things in quotation marks here because what he’s doing isn’t exactly this deliberate.) He picked mentors that were aligned with his goals, and peers that unblock his particular genius.
I do not remember a single occasion when I was treated with condescension by one of these men, nor an occasion when my thirst for knowledge, and later, anew, my joy of discovery, was rejected by complacency or by disdain. Had it not been so, I would not have “become a mathematician” as they say—I would have chosen another profession, where I could give my whole strength without having to face scorn. [My emphasis.]
He could interface with the mathematical community with integrity because he had a deep familiarity with his inner space. If he had not known the shape of his interests and aims, he would have been more vulnerable to the standards and norms of the community—at least he seems to think so.
You’re right, I mis-paraphrased. Thanks for the correction Said.
I’ve read most of your stories over at Narrative Ark and wanted to remark that The Gentle Romance did feel more concrete than usual, which was nice. Given how much effort it took for you however, I suppose I shouldn’t expect future stories at Narrative Ark to be similarly concrete?
I’m surprised to see this take so disagree-voted, given how sensible the policy of adopting a vibes-invariant strategy is. Anyone who disagree-voted care to explain?
Found an annotated version of Vernor Vinge’s A Fire Upon The Deep.
Mo Putera’s Shortform
The part of Ajeya’s comment that stood out to me was this:
On a meta level I now defer heavily to Ryan and people in his reference class (METR and Redwood engineers) on AI timelines, because they have a similarly deep understanding of the conceptual arguments I consider most important while having much more hands-on experience with the frontier of useful AI capabilities (I still don’t use AI systems regularly in my work).
I’d also look at Eli Lifland’s forecasts as well:
I don’t think you need that footnoted caveat, simply because there isn’t $150M/year worth of room for more funding in all of AMF, Malaria Consortium’s SMC program, HKI’s vitamin A supplementation program, and New Incentives’ cash incentives for routine vaccination program all combined; these comprise the full list of GiveWell’s top charities.
Another point is that the benefits of eradication keep adding up long after you’ve stopped paying for the costs, because the counterfactual that people keep suffering and dying of the disease is no longer happening. That’s how smallpox eradication’s cost-effectiveness can plausibly be less than a dollar per DALY averted so far and dropping (Guesstimate model, analysis). Quoting that analysis:
3.10.) For how many years should you consider benefits?
It is not clear for how long we should continue to consider benefits, since the benefits of vaccines would potentially continue indefinitely for hundreds of years. Perhaps these benefits would eventually be offset by some other future technology, and we could try to model that. Or perhaps we should consider a discount rate into the future, though we don’t find that idea appealing.
Instead, we decided to cap at an arbitrary fixed amount of years set to 20 by default, though adjustable as a variable in our spreadsheet model (or by copying and modifying our Guesstimate models). We picked 20 because it felt like a significant enough amount of time for technology and other dynamics to shift.
It’s important to think through what cap makes the most sense, though, as it can have a large effect on the final model, as seen in this table where we explore the ramifications of smallpox eradication with different benefit thresholds:
Smallpox Eradication Cost-effectiveness
I thought it’d be useful for others to link to your longer writings on this:
Just reread Scott Aaronson’s We Are the God of the Gaps (a little poem) from 2022:
Feels poignant.
Philosophy bear’s response to Scott is worth reading too.