I was reading Dominic Cummings’ blog post *On the referendum #33: high performance government, ‘cognitive technologies’, Michael Nielsen, Bret Victor, & ‘Seeing Rooms’*, and at one point he references a nice passage by Nielsen that immediately reminded me of Bill Thurston’s experience trying to communicate his ways of thinking in his seminal retrospective *On proof and progress in mathematics*, one of the highest wisdom-density (not just insight-density!) opinion pieces I have ever had the privilege of reading, so I thought I’d record it here. In doing so I went to Nielsen’s essay *Thought as a technology* where Cummings got the passage from, and discovered that Nielsen directly references Thurston *right then and there*. It is with supreme modesty that I henceforth claim that for one blindingly-inspired moment of reasoning-by-vague-association, I was *precisely* as bright as a man who in his mid-twenties co-wrote one of the ten most cited physics texts of *all time*.

(NOT! After all, a stopped clock is still right twice a day. But don’t you *dare* deny me my time in the sun.)

Cummings’ post itself is an interesting infodump. I am somehow turned on by infodense run-on sentences, because they’re the most reliable way to sound Stereotypically Ferociously Smart on paper. He certainly delivers there. But more so than that, he’s one of the very few people ‘in politics’ (broadly construed) who understands the need for quantitative literacy and complex systems intuition (in particular as it relates to high-performance grand-scale complex project management, like the Apollo mission). Also I’m biased, because he checks all the Right Names – Nielsen, Bret Victor, Alan Kay, Colonel ‘OODA loop’ Boyd, Tetlock etc – in other words, despite me having almost nothing else in common with him he is firmly One Of My People (albeit clearly better endowed in the grey matter department). His introduction to *On the referendum #33* certainly promises loads:

This blog looks at an intersection of decision-making, technology, high performance teams and government. It sketches some ideas of physicist Michael Nielsen about cognitive technologies and of computer visionary Bret Victor about the creation of dynamic tools to help understand complex systems and ‘argue with evidence’, such as ‘tools for authoring dynamic documents’, and ‘Seeing Rooms’ for decision-makers — i.e rooms designed to support decisions in complex environments. It compares normal Cabinet rooms, such as that used in summer 1914 or October 1962, with state-of-the-art Seeing Rooms. There is very powerful feedback between: a) creating dynamic tools to

seecomplex systems deeper (to see inside, see across time, and see across possibilities), thus making it easier to work with reliable knowledge and interactive quantitative models, semi-automating error-correction etc, and b) the potential for big improvements in the performance of political and government decision-making.It is relevant to Brexit and anybody thinking ‘how on earth do we escape this nightmare’ but 1) these ideas are

not at alldependent on whether you support or oppose Brexit, about which reasonable people disagree, and 2) they are generally applicable to how to improve decision-making — for example, they are relevant to problems like ‘how to make decisions during a fast moving nuclear crisis’ which I blogged about recently, or if you are a journalist ‘what future media could look liketo help improve debate of politics’. One of the tools Nielsen discusses isa tool to make memory a choice by embedding learning in long-term memoryrather than, as it is for almost all of us, an accident. I know from my days working on education reform in government that it’s almost impossible to exaggerate howlittlethose who work on education policy think about ‘how to improve learning’.Fields make huge progress when they move from

stories(e.g Icarus) andauthority(e.g ‘witch doctor’) toevidence/experiment(e.g physics, wind tunnels) andquantitative models(e.g design of modern aircraft). Political ‘debate’ and the processes of government are largely what they have always been —largely conflict over stories and authorities where almost nobody even tries to keep track of the facts/arguments/models they’re supposedly arguing about, or tries to learn from evidence, or tries to infer useful principles from examples of extreme success/failure. We can see much better than people could in the past how to shift towards processes of government being ‘partially rational discussion over facts and models and learning from the best examples of organisational success‘. But one of the most fundamental and striking aspects of government is that practically nobody involved in it has the faintest interest in or knowledge ofhow to create high performance teams to make decisions amid uncertainty and complexity. This blindness is connected to another fundamental fact:critical institutions (including the senior civil service and the parties) are programmed to fight to stay dysfunctional, they fight to stay closed and avoid learning about high performance, they fight to exclude the most able people.

That passage is tempting to dive further into, but I am trying to discipline my blogging/associative note-taking output by ‘modularizing’ it to better build upon it in the future (a long-term personal knowledge management project of mine), so I’ll save that for a later post.

Back to the topic at hand with a relevant quote:

Language and writing were

cognitive technologiescreated thousands of years ago which enabled us to think previously unthinkable thoughts. Mathematical notation did the same over the past 1,000 years. [A math problem that took al-Khawarizmi a convoluted paragraph to express can now be written as a quadratic equation an inch long.]Michael Nielsen uses a similar analogy. Descartes and Fermat demonstrated that equations can be represented on a diagram and a diagram can be represented as an equation. This was a

new cognitive technology, a new way of seeing and thinking: algebraic geometry. Changes to the ‘user interface’ of mathematics were critical to its evolution and allowed us to think unthinkable thoughts.Similarly in the 18th Century, there was the

creation of data graphicsto demonstrate trade figures. Before this, people could only read huge tables. …

Segue to Nielsen’s essay elaborating on the above, then noting that visual thinking is another great example of a cognitive technology:

Language is an example of a

cognitive technology: an external artifact, designed by humans, which can be internalized, and used as a substrate for cognition. That technology is made up of many individual pieces – words and phrases, in the case of language – which become basicelements of cognition. These elements of cognition are things we can think with.Language isn’t the only cognitive technology we internalize.

Consider visual thinking. If, like me, you sometimes think visually, it’s tempting to suppose your mind’s eye is a raster display, capable of conceiving any image. But while tempting, this is wrong. In fact, our visual thinking is done using visual cognitive technologies we’ve previously internalized.

For instance, one of the world’s best-known art teachers, Betty Edwards, explains that the visual thinking of most non-artist adults is limited to what she refers to as a simple “symbol system”, and that this constrains both what they see and what they can visually conceive:

[A]dult students beginning in art generally do not really see what is in front of their eyes — that is, they do not perceive in the way required for drawing. They take note of what’s there, and quickly translate the perception into words and symbols mainly based on the symbol system developed throughout childhood and on what they know about the perceived object.It requires extraordinary imagination to conceive new forms of visual meaning – i.e., new visual cognitive technologies. Many of our best-known artists and visual explorers are famous in part because they discovered such forms. When exposed to that work, other people can internalize those new cognitive technologies, and so expand the range of their own visual thinking.

For example, cubist artists such as Picasso developed the technique of using multiple points of view in a single painting. Once you’ve learnt to see cubist art, it can give you a richer sense of the structure of what’s being shown…

Another example is the work of Doc Edgerton, a pioneer of high-speed photography, whose photographs revealed previously unsuspected structure in the world. If you study such photographs, you begin to build new mental models of everyday phenomena, enlarging your range of visual thought…

Another class of examples comes from the many cartographers who’ve developed ways to visually depict geography. Consider, for example, the 1933 map of the London Underground, developed by Harry Beck. In the early 1930s, Beck noticed that the official map of the Underground was growing too complex for readers to understand. He simplified the map by abandoning exact geographic fidelity, as was commonly used on most maps up to that point. He concentrated instead on showing the

topologicalstructure of the network of stations, i.e., what connects to what…Images such as these are not natural or obvious. No-one would ever have these visual thoughts without the cognitive technologies developed by Picasso, Edgerton, Beck, and many other pioneers. Of course, only a small fraction of people really internalize these ways of visual thinking. But in principle, once the technologies have been invented, most of us can learn to think in these new ways.

Why, you know who was good at visual thinking, perhaps the best geometric thinker in the *history* of mathematics (a tall claim!)? Bill Thurston. In Thinking and explaining, one of the most upvoted MathOverflow questions of all time, Thurston asked:

How big a gap is there between how you think about mathematics and what you say to others? Do you say what you’re thinking? Please give either personal examples of how your thoughts and words differ, or describe how they are connected for you.

I’ve been fascinated by the phenomenon the question addresses for a long time. We have complex minds evolved over many millions of years, with many modules always at work. A lot we don’t habitually verbalize, and some of it is very challenging to verbalize or to communicate in any medium. Whether for this or other reasons, I’m under the impression that mathematicians often have unspoken thought processes guiding their work which may be difficult to explain, or they feel too inhibited to try.

One prototypical situation is this: there’s a mathematical object that’s obviously (to you) invariant under a certain transformation. For instant, a linear map might conserve volume for an ‘obvious’ reason. But you don’t have good language to explain your reason—so instead of explaining, or perhaps after trying to explain and failing, you fall back on computation. You turn the crank and without undue effort, demonstrate that the object is indeed invariant.

Here’s a specific example. Once I mentioned this phenomenon to Andy Gleason; he immediately responded that when he taught algebra courses, if he was discussing cyclic subgroups of a group, he had a mental image of group elements breaking into a formation organized into circular groups. He said that ‘we’ never would say anything like that to the students. His words made a vivid picture in my head, because it fit with how I thought about groups. I was reminded of my long struggle as a student, trying to attach

meaningto ‘group’, rather than just a collection of symbols, words, definitions, theorems and proofs that I read in a textbook.

There’s a reason this question matters so much to Thurston, and it’s *exactly *what this post of mine is building up to. But I like my foreplay, so I shall prolong it just a tad further with more related-ish quotes. (Also, “Connect Everything!” can be a hard impulse to resist.)

It turns out that Nielsen’s essay *Thought as a technology* also quotes part of the Thurston passage above (before Tao), just without referencing the MO question. (I knew it anyway because I’d seen it before. Gosh, yet *another* instance where my thoughts rise to Nielsen’s level! Where do I collect my MacArthur Fellowship??) He paraphrases Thurston like so:

… mathematicians often don’t think about mathematical objects using the conventional representations found in books. Rather, they rely heavily on what we might call

hiddenrepresentations, such as the mental imagery Thurston describes, of groups breaking into formations of circular groups. Such hidden representations help them reason more easily than the conventional representations, and occasionally provide them with what may seem to others like magical levels of insight.

Nielsen notes that “the use of hidden representations occurs in many fields”. In electrical engineering, for instance, Gerald Sussman has the following quote about analyzing electrical circuits:

I was teaching my first classes in electrical engineering at MIT, in circuit theory… and I observed that what we taught the students wasn’t at all what the students were actually expected to learn. That is, what an expert person did when presented with a circuit… was quite different from what we tell [the students] to write down – the node equations… and then you’re supposed to grind these equations together somehow and solve them, to find out what’s going on. Well, you know, that’s not what a really good engineer does. …

Nielsen himself is a theoretical physicist, so he can draw from personal experience:

The energy surface prototype is based on the kind of hidden representation described by Thurston and Sussman. In particular, it’s based on the way I often visualize one-dimensional motion, in my work as a theoretical physicist. The visuals are not original to me: when I’ve shown the prototype to other physicists, several have told me “Oh, I think about one-dimensional motion like this”. But while this way of understanding may be common among physicists, they rarely talk about it. For instance, it’s not the kind of thing one would use in teaching a class on one-dimensional motion. At most, you might make a few ancillary sketches along these lines for the students. Certainly, you would not put this way of thinking front and center, or expect students to answer homework or exam questions using energy surfaces. Nor would you use such a representation in a research paper.

The situation is strange. A powerful way of thinking about one-dimensional motion is largely absent from our shared conversations. The reason is that traditional media are poorly adapted to working with such representations.

Okay, why not just share those representations? The answer is that they do, but it’s hard nonetheless, and there can be reasons why they don’t. Nielsen:

To answer that question, suppose you think hard about a subject for several years – say, cyclic subgroups of a group, to use Thurston’s example. Eventually you push up against the limits of existing representations. If you’re strongly motivated – perhaps by the desire to solve a research problem – you may begin inventing new representations, to provide insights difficult through conventional means. You are effectively acting as your own interface designer. But the new representations you develop may be held entirely in your mind, and so are not constrained by traditional static media forms. Or even if based on static media, they may break social norms about what is an “acceptable” argument. Whatever the reason, they may be difficult to communicate using traditional media. And so they remain private, or are only discussed informally with expert colleagues.

*This* is the passage I alluded to at the very beginning. It’s precisely what Thurston ran up against, and the answer to it will be the climax of this post.

But one last interlude – Terry Tao has also expressed the same sentiments, which is to be expected given the highly collaborative nature of his research style. He describes these in his answer to Thurston’s MO question, which is *fantastic* in its sheer range:

I find there is a world of difference between explaining things to a colleague, and explaining things to a close collaborator. With the latter, one really can communicate at the intuitive level, because one already has a reasonable idea of what the other person’s mental model of the problem is. In some ways, I find that throwing out things to a collaborator is closer to the mathematical thought process than just thinking about maths on one’s own, if that makes any sense.

One specific mental image that I can communicate easily with collaborators, but not always to more general audiences, is to think of quantifiers in game theoretic terms. Do we need to show that for every epsilon there exists a delta? Then imagine that you have a bag of deltas in your hand, but you can wait until your opponent (or some malicious force of nature) produces an epsilon to bother you, at which point you can reach into your bag and find the right delta to deal with the problem. Somehow, anthropomorphising the “enemy” (as well as one’s “allies”) can focus one’s thoughts quite well. This intuition also combines well with probabilistic methods, in which case in addition to you and the adversary, there is also a Random player who spits out mathematical quantities in a way that is neither maximally helpful nor maximally adverse to your cause, but just some randomly chosen quantity in between. The trick is then to harness this randomness to let you evade and confuse your adversary.

Is there a quantity in one’s PDE or dynamical system that one can bound, but not otherwise estimate very well? Then imagine that it is controlled by an adversary or by Murphy’s law, and will always push things in the most unfavorable direction for whatever you are trying to accomplish. Sometimes this will make that term “win” the game, in which case one either gives up (or starts hunting for negative results), or looks for additional ways to “tame” or “constrain” that troublesome term, for instance by exploiting some conservation law structure of the PDE.

For evolutionary PDEs in particular, I find there is a rich zoo of colourful physical analogies that one can use to get a grip on a problem. I’ve used the metaphor of an egg yolk frying in a pool of oil, or a jetski riding ocean waves, to understand the behaviour of a fine-scaled or high-frequency component of a wave when under the influence of a lower frequency field, and how it exchanges mass, energy, or momentum with its environment. In one extreme case, I ended up rolling around on the floor with my eyes closed in order to understand the effect of a gauge transformation that was based on this type of interaction between different frequencies. (Incidentally, that particular gauge transformation won me a Bocher prize, once I understood how it worked.) I guess this last example is one that I would have difficulty communicating to even my closest collaborators. Needless to say, none of these analogies show up in my published papers, although I did try to convey some of them in my PDE book eventually.

ADDED LATER: I think one reason why one cannot communicate most of one’s internal mathematical thoughts is that one’s internal mathematical model is very much a function of one’s mathematical upbringing. For instance, my background is in harmonic analysis, and so I try to visualise as much as possible in terms of things like interactions between frequencies, or contests between different quantitative bounds. This is probably quite a different perspective from someone brought up from, say, an algebraic, geometric, or logical background. I can appreciate these other perspectives, but still tend to revert to the ones I am most personally comfortable with when I am thinking about these things on my own.

ADDED (MUCH) LATER: Another mode of thought that I and many others use routinely, but which I realised only recently was not as ubiquitious as I believed, is to use an “economic” mindset to prove inequalities such as X≤Y or X≤CY for various positive quantities X,Y, interpreting them in the form “If I can afford Y, can I therefore afford X?” or “If I can afford lots of Y, can I therefore afford X?” respectively. This frame of reference starts one thinking about what types of quantities are “cheap” and what are “expensive”, and whether the use of various standard inequalities constitutes a “good deal” or not. It also helps one understand the role of weights, which make things more expensive when the weight is large, and cheaper when the weight is small.

ADDED (MUCH, MUCH) LATER: One visualisation technique that I have found very helpful is to incorporate the ambient symmetries of the problem (a la Klein) as little “wobbles” to the objects being visualised. This is most familiarly done in topology (“rubber sheet mathematics”), where every object considered is a bit “rubbery” and thus deforming all the time by infinitesimal homeomorphisms. But geometric objects in a scale-invariant problem could be thought of as being viewed through a camera with a slightly wobbly zoom lens, so that one’s mental image of these objects is always varying a little in size. Similarly, if one is in a translation-invariant setting, one’s mental camera should be sliding back and forth just a little to remind you of this, if one is working in a Euclidean space then the camera might be jiggling through all the rigid motions, and so forth. A more advanced example: if the problem is invariant under tensor products, as per the tensor product trick, then one’s low dimensional objects should have a tiny bit of shadowing (or perhaps look like one of these 3D images when one doesn’t have the polarised glasses, with the slightly separated red and blue components) that suggest that they are projections of a higher dimensional Cartesian product.

One reason why one wants to do this is that it helps suggest useful normalisations. If one is viewing a situation with a wobbly zoom lens and there is some length that appears all over one’s analysis, one is reminded that one can spend the scale invariance of the problem to zoom up or down as appropriate to normalise this scale to equal 1. Similarly for other ambient symmetries.This sort of wobbling of symmetries is also available in less geometric settings. … In analysis, one often only cares about the order of magnitude of some very large or very small quantity X, rather than its exact value; so one should view this quantity as being a bit squishy in size, growing or shrinking by a factor of two or so every time one looks at the problem. If there is some probability theory in one’s problem, and some of your objects are random variables rather than deterministic variables, then you can imagine that every so often the “game resets”, with the random variables jumping around to different values in their range (and any quantities depending on these variables changing accordingly), whereas the deterministic variables stay fixed. Similarly if one has generic points in a variety, or nonstandard objects in a space (with the point being that if something bad happens if, say, your generic point is trapped in a subvariety, you can “reset the game” in which the generic point is now outside the subvariety; similarly one can “reset” an unbounded nonstandard number to be larger than any given standard number, etc.).

Hot *damn*, Terry!

I have digressed enough. Here’s Thurston’s reminiscences, the answer to the prompt at the beginning of this post.

First, foliations, where Thurston “did wrong”:

First I will discuss briefly the theory of foliations, which was my first subject, starting when I was a graduate student. (It doesn’t matter here whether you know what foliations are.)

At that time, foliations had become a big center of attention among geometric topologists, dynamical systems people, and differential geometers. I fairly rapidly proved some dramatic theorems. I proved a classification theorem for foliations, giving a necessary and sufficient condition for a manifold to admit a foliation. I proved a number of other significant theorems. I wrote respectable papers and published at least the most important theorems. It was hard to find the time to write to keep up with what I could prove, and I built up a backlog.

An interesting phenomenon occurred. Within a couple of years, a dramatic evacuation of the field started to take place. I heard from a number of mathematicians that they were giving or receiving advice not to go into foliations—they were saying that Thurston was cleaning it out. People told me (not as a complaint, but as a compliment) that I was killing the field. Graduate students stopped studying foliations, and fairly soon, I turned to other interests as well.

I do not think that the evacuation occurred because the territory was intellectually exhausted—there were (and still are) many interesting questions that remain and that are probably approachable. Since those years, there have been interesting developments carried out by the few people who stayed in the field or who entered the field, and there have also been important developments in neighboring areas that I think would have been much accelerated had mathematicians continued to pursue foliation theory vigorously.

Today, I think there are few mathematicians who understand anything approaching the state of the art of foliations as it lived at that time, although there are some parts of the theory of foliations, including developments since that time, that are still thriving.

What happened?

I believe that two ecological effects were much more important in putting a damper on the subject than any exhaustion of intellectual resources that occurred.

First, the results I proved (as well as some important results of other people) were documented in a conventional, formidable mathematician’s style. They depended heavily on readers who shared certain background and certain insights. The theory of foliations was a young, opportunistic subfield, and the background was not standardized. I did not hesitate to draw on any of the mathematics I had learned from others. The papers I wrote did not (and could not) spend much time explaining the background culture. They documented top-level reasoning and conclusions that I often had achieved after much reflection and effort. I also threw out prize cryptic tidbits of insight, such as “the Godbillon-Vey invariant measures the helical wobble of a foliation”, that remained mysterious to most mathematicans who read them. This created a high entry barrier: I think many graduate students and mathematicians were discouraged that it was hard to learn and understand the proofs of key theorems.

Second is the issue of what is in it for other people in the subfield. When I started working on foliations, I had the conception that what people wanted was to know the answers. I thought that what they sought was a collection of powerful proven theorems that might be applied to answer further mathematical questions. But that’s only one part of the story. More than the knowledge, people want

personal understanding. And in our credit-driven system, they also want and needtheorem-credits.

And secondly, 3-manifolds and hyperbolic geometry, where Thurston “did right”:

I’ll skip ahead a few years, to the subject that Jaffe and Quinn alluded to, when I began studying 3-dimensional manifolds and their relationship to hyperbolic geometry. (Again, it matters little if you know what this is about.) I gradually built up over a number of years a certain intuition for hyperbolic three-manifolds, with a repertoire of constructions, examples and proofs. (This process actually started when I was an undergraduate, and was strongly bolstered by applications to foliations.) After a while, I conjectured or speculated that all three-manifolds have a certain geometric structure; this conjecture eventually became known as the geometrization conjecture. About two or three years later, I proved the geometrization theorem for Haken manifolds. It was a hard theorem, and I spent a tremendous amount of effort thinking about it. When I completed the proof, I spent a lot more effort checking the proof, searching for difficulties and testing it against independent information.

I’d like to spell out more what I mean when I say I proved this theorem. It meant that I had a clear and complete flow of ideas, including details, that withstood a great deal of scrutiny by myself and by others. Mathematicians have many different styles of thought. My style is not one of making broad sweeping but careless generalities, which are merely hints or inspirations: I make clear mental models, and I think things through. My proofs have turned out to be quite reliable. I have not had trouble backing up claims or producing details for things I have proven. I am good in detecting flaws in my own reasoning as well as in the reasoning of others.

However, there is sometimes a huge expansion factor in translating from the encoding in my own thinking to something that can be conveyed to someone else. My mathematical education was rather independent and idiosyncratic, where for a number of years I learned things on my own, developing personal mental models for how to think about mathematics. This has often been a big advantage for me in thinking about mathematics, because it’s easy to pick up later the standard mental models shared by groups of mathematicians. This means that some concepts that I use freely and naturally in my personal thinking are foreign to most mathematicians I talk to. My personal mental models and structures are similar in character to the kinds of models groups of mathematicians share—but they are often different models. At the time of the formulation of the geometrization conjecture, my understanding of hyperbolic geometry was a good example. A random continuing example is an understanding of finite topological spaces, an oddball topic that can lend good insight to a variety of questions but that is generally not worth developing in any one case because there are standard circumlocutions that avoid it.

Neither the geometrization conjecture nor its proof for Haken manifolds was in the path of any group of mathematicians at the time—it went against the trends in topology for the preceding 30 years, and it took people by surprise. To most topologists at the time, hyperbolic geometry was an arcane side branch of mathematics, although there were other groups of mathematicians such as differential geometers who did understand it from certain points of view. It took topologists a while just to understand what the geometrization conjecture meant, what it was good for, and why it was relevant.

*This* is the answer! Continuing:

At the same time, I started writing notes on the geometry and topology of 3-manifolds, in conjunction with the graduate course I was teaching. I distributed them to a few people, and before long many others from around the world were writing for copies. The mailing list grew to about 1200 people to whom I was sending notes every couple of months. I tried to communicate my real thoughts in these notes. People ran many seminars based on my notes, and I got lots of feedback. Overwhelmingly, the feedback ran something like “Your notes are really inspiring and beautiful, but I have to tell you that we spent 3 weeks in our seminar working out the details of §n.n. More explanation would sure help.”

I also gave many presentations to groups of mathematicians about the ideas of studying 3-manifolds from the point of view of geometry, and about the proof of the geometrization conjecture for Haken manifolds. At the beginning, this subject was foreign to almost everyone. It was hard to communicate—the infrastructure was in my head, not in the mathematical community. There were several mathematical theories that fed into the cluster of ideas: three-manifold topology, Kleinian groups, dynamical systems, geometric topology, discrete subgroups of Lie groups, foliations, Teichmuller spaces, pseudo-Anosov diffeomorphisms, geometric group theory, as well as hyperbolic geometry.

We held an AMS summer workshop at Bowdoin in 1980, where many mathematicans in the subfields of low-dimensional topology, dynamical systems and Kleinian groups came. It was an interesting experience exchanging cultures.

It became dramatically clear how much proofs depend on the audience. We prove things in a social context and address them to a certain audience. Parts of this proof I could communicate in two minutes to the topologists, but the analysts would need an hour lecture before they would begin to understand it. Similarly, there were some things that could be said in two minutes to the analysts that would take an hour before the topologists would begin to get it. And there were many other parts of the proof which should take two minutes in the abstract, but that none of the audience at the time had the mental infrastructure to get in less than an hour.

At that time, there was practically no infrastructure and practically no context for this theorem, so the expansion from how an idea was keyed in my head to what I had to say to get it across, not to mention how much energy the audience had to devote to understand it, was very dramatic.

In reaction to my experience with foliations and in response to social pressures, I concentrated most of my attention on developing and presenting the infrastructure in what I wrote and in what I talked to people about. I explained the details to the few people who were “up” for it. I wrote some papers giving the substantive parts of the proof of the geometrization theorem for Haken manifolds—for these papers, I got almost no feedback. Similarly, few people actually worked through the harder and deeper sections of my notes until much later.

The result has been that now quite a number of mathematicians have what was dramatically lacking in the beginning: a working understanding of the concepts and the infrastructure that are natural for this subject. There has been and there continues to be a great deal of thriving mathematical activity. By concentrating on building the infrastructure and explaining and publishing definitions and ways of thinking but being slow in stating or in publishing proofs of all the “theorems” I knew how to prove, I left room for many other people to pick up credit. There has been room for people to discover and publish other proofs of the geometrization theorem. These proofs helped develop mathematical concepts which are quite interesting in themselves, and lead to further mathematics.

What mathematicians most wanted and needed from me was to learn my ways of thinking, and not in fact to learn my proof of the geometrization conjecture for Haken manifolds. It is unlikely that the proof of the general geometrization conjecture will consist of pushing the same proof further.