Mo Putera

Long-time lurker (c. 2013), recent poster. I also write on the EA Forum.

Wikitag Contributions

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Unbundling Tools for Thought is an essay by Fernando Borretti I found via Gwern's comment which immediately resonated with me (emphasis mine): 

I’ve written something like six or seven personal wikis over the past decade. It’s actually an incredibly advanced form of procrastination1. At this point I’ve tried every possible design choice. 

  1. Lifecycle: I’ve built a few compiler-style wikis: plain-text files in a git repo statically compiled to HTML. I’ve built a couple using live servers with server-side rendering. The latest one is an API server with a React frontend.

  2. Storage: I started with plain text files in a git repo, then moved to an SQLite database with a simple schema. The latest version is an avant-garde object-oriented hypermedia database with bidirectional links implemented on top of SQLite.

  3. Markup: I used Markdown here and there. Then I built my own TeX-inspired markup language. Then I tried XML, with mixed results. The latest version uses a WYSIWYG editor made with ProseMirror.

And yet I don’t use them. Why? Building them was fun, sure, but there must be utility to a personal database.

At first I thought the problem was friction: the higher the activation energy to using a tool, the less likely you are to use it. Even a small amount of friction can cause me to go, oh, who cares, can’t be bothered. So each version gets progressively more frictionless2. The latest version uses a WYSIWYG editor built on top of ProseMirror (it took a great deal for me to actually give in to WYSIWYG). It also has a link to the daily note page, to make journalling easier. The only friction is in clicking the bookmark to localhost:5000. It is literally two clicks to get to the daily note.

And yet I still don’t use it. Why? I’m a great deal more organized now than I was a few years ago. My filesystem is beautifully structured and everything is where it should be. I could fill out the contents of a personal wiki.

I’ve come to the conclusion that there’s no point: because everything I can do with a personal wiki I can do better with a specialized app, and the few remaining use cases are useless. Let’s break it down.

I've tried three different times to create a personal wiki, using the last one for a solid year and a half before finally giving up and just defaulting to a janky combination of Notion and Google Docs/Sheets, seduced by sites like Cosma Shalizi's and Gwern's long content philosophy (emphasis mine): 

... I have read blogs for many years and most blog posts are the triumph of the hare over the tortoise. They are meant to be read by a few people on a weekday in 2004 and never again, and are quickly abandoned—and perhaps as Assange says, not a moment too soon. (But isn’t that sad? Isn’t it a terrible ROI for one’s time?) On the other hand, the best blogs always seem to be building something: they are rough drafts—works in progress15. So I did not wish to write a blog. Then what? More than just “evergreen content”, what would constitute Long Content as opposed to the existing culture of Short Content? How does one live in a Long Now sort of way?16⁠

My answer is that one uses such a framework to work on projects that are too big to work on normally or too tedious. (Conscientiousness is often lacking online or in volunteer communities18 and many useful things go undone.) Knowing your site will survive for decades to come gives you the mental wherewithal to tackle long-term tasks like gathering information for years, and such persistence can be useful19—if one holds onto every glimmer of genius for years, then even the dullest person may look a bit like a genius himself20. (Even experienced professionals can only write at their peak for a few hours a day—usually first thing in the morning, it seems.) Half the challenge of fighting procrastination is the pain of starting⁠—I find when I actually get ⁠into the swing⁠ of working on even dull tasks, it’s not so bad. So this suggests a solution: never start. Merely have perpetual drafts, which one tweaks from time to time. And the rest takes care of itself.

Fernando unbundles the use cases of a tool for thought in his essay; I'll just quote the part that resonated with me: 

The following use cases are very naturally separable: ...

  • Learning: if you’re studying something, you can keep your notes in a TfT. This is one of the biggest use cases. But the problem is never note-taking, but reviewing notes. Over the years I’ve found that long-form lecture notes are all but useless, not just because you have to remember to review them on a schedule, but because spaced repetition can subsume every single lecture note. It takes practice and discipline to write good spaced repetition flashcards, but once you do, the long-form prose notes are themselves redundant.

(Tangentially, an interesting example of how comprehensively subsuming spaced repetition is is Michael Nielsen's Using spaced repetition systems to see through a piece of mathematics, in which he describes how he used "deep Ankification" to better understand the theorem that a complex normal matrix is always diagonalizable by a unitary matrix, as an illustration of a heuristic one could use to deepen one's understanding of a piece of mathematics in an open-ended way, inspired by Andrey Kolmogorov's essay on, of all things, the equals sign. I wish I read that while I was still studying physics in school.)

Fernando, emphasis mine:

So I often wonder: what do other people use their personal knowledge bases for? And I look up blog and forum posts where Obsidian and Roam power users explain their setup. And most of what I see is junk. It’s never the Zettelkasten of the next Vannevar Bush, it’s always a setup with tens of plugins, a daily note three pages long that is subdivided into fifty subpages recording all the inane minutiae of life. This is a recipe for burnout.

People have this aspirational idea of building a vast, oppressively colossal, deeply interlinked knowledge graph to the point that it almost mirrors every discrete concept and memory in their brain. And I get the appeal of maximalism. But they’re counting on the wrong side of the ledger. Every node in your knowledge graph is a debt. Every link doubly so. The more you have, the more in the red you are. Every node that has utility—an interesting excerpt from a book, a pithy quote, a poem, a fiction fragment, a few sentences that are the seed of a future essay, a list of links that are the launching-off point of a project—is drowned in an ocean of banality. Most of our thoughts appear and pass away instantly, for good reason.

I enjoyed Brian Potter's Energy infrastructure cheat sheet tables over at Construction Physics, it's a great fact post. Here are some of Brian's tables — if they whet your appetite, do check out his full essay.

Energy quantities:


Units and quantities
Kilowatt-hoursMegawatt-hoursGigawatt-hours
1 British Thermal Unit (BTU)0.000293  
iPhone 14 battery0.012700  
1 pound of a Tesla battery pack0.1  
1 cubic foot of natural gas0.3  
2000 calories of food2.3  
1 pound of coal2.95  
1 gallon of milk (calorie value)3.0  
1 gallon of gas33.7  
Tesla Model 3 standard battery pack57.5  
Typical ICE car gas tank (15 gallons)506  
1 ton of TNT1,162  
1 barrel of oil1,700  
1 ton of oil11,62912 
Tanker truck full of gasoline (9300 gallons)313,410313 
LNG carrier (180,000 cubic meters)1,125,214,7401,125,2151,125
1 million tons of TNT (1 megaton)1,162,223,1521,162,2231,162
Oil supertanker (2 million barrels)3,400,000,0003,400,0003,400

It's amazing that a Tesla Model 3's standard battery pack has an OOM less energy capacity than a typical 15-gallon ICE car gas tank, and is probably heavier too, yet a Model 3 isn't too far behind in range and is far more performant. It's also amazing that an oil supertanker carries ~3 megatons(!) of TNT worth of energy.

Energy of various activities:

ActivityKilowatt-hours
Fired 9mm bullet0.0001389
Making 1 pound of steel in an electric arc furnace0.238
Driving a mile in a Tesla Model 30.240
Making 1 pound of cement0.478
Driving a mile in a 2025 ICE Toyota Corolla0.950
Boiling a gallon of room temperature water2.7
Synthesizing 1 kilogram of ammonia (NH3) via Haber-Bosch11.4
Making 1 pound of aluminum via Hall-Heroult process7.0
Average US household monthly electricity use899.0
Moving a shipping container from Shanghai to Los Angeles2,000.0
Average US household monthly gasoline use2,010.8
Heating and cooling a 2500 ft2 home in California for a year4,615.9
Heating and cooling a 2500 ft2 home in New York for a year23,445.8
Average annual US energy consumption per capita81,900.0

Power output:


Activity or infrastructure
KilowattsMegawattsGigawatts
Sustainable daily output of a laborer0.08  
Output from 1 square meter of typical solar panels (21% efficiency)0.21  
Tesla wall connector11.50  
Tesla supercharger250  
Large on-shore wind turbine6,1006 
Typical electrical distribution line (15 kV)8,0008 
Large off-shore wind turbine14,70015 
Typical US gas pump20,22020 
Typical daily production of an oil well (500 barrels)35,41735 
Typical transmission line (150 kV)150,000150 
Large gas station (20 pumps)404,400404 
Large gas turbine500,000500 
Output from 1 square mile of typical solar panels543,900544 
Electrical output of a large nuclear power reactor1,000,0001,0001
Single LNG carrier crossing the Atlantic (18 day trip time)2,604,6642,6053
Nord Stream Gas pipeline33,582,50033,58334
Trans Alaska pipeline151,300,000151,300151
US electrical generation capacity1,189,000,0001,189,0001,189

This observation by Brian is remarkable:

A typical US gas pump operates at 10 gallons per minute (600 gallons an hour). At 33.7 kilowatt-hours per gallon of gas, that’s a power output of over 20 megawatts, greater than the power output of an 800-foot tall offshore wind turbine. The Trans-Alaska pipeline, a 4-foot diameter pipe, can move as much energy as 1,000 medium-sized transmission lines, and 8 such pipelines would move more energy than provided by every US electrical power plant combined.

US energy flows Sankey diagram by LLNL (a "quad" is short for “a quadrillion British Thermal Units,” or 293 terawatt-hours):

I had a vague inkling that a lot of energy is lost on the way to useful consumption, but I was surprised by the two-thirds fraction; the 61.5 quads of rejected energy is more than every other country in the world consumes except China. I also wrongly thought that the largest source of inefficiency was in transmission losses. Brian explains:

The biggest source of losses is probably heat engine inefficiencies. In our hydrocarbon-based energy economy, we often need to transform energy by burning fuel and converting the heat into useful work. There are limits to how efficiently we can transform heat into mechanical work (for more about how heat engines work, see my essay about gas turbines).

The thermal efficiency of an engine is the fraction of heat energy it can transform into useful work. Coal power plant typically operates at around 30 to 40% thermal efficiency. A combined cycle gas turbine will hit closer to 60% thermal efficiency. A gas-powered car, on the other hand, operates at around 25% thermal efficiency. The large fraction of energy lost by heat engines is why some thermal electricity generation plants list their capacity in MWe, the power output in megawatts of electricity.

Most other losses aren’t so egregious, but they show up at every step of the energy transportation chain. Moving electricity along transmission and distribution lines results in losses as some electrical energy gets converted into heat. Electrical transformers, which minimize these losses by transforming electrical energy into high-voltage, low-current before transmission, operate at around 98% efficiency or more.

I also didn't realise that biomass is so much larger than solar in the US (I expect this of developing countries), although likely not for long given the ~25% annual growth rate.

Energy conversion efficiency:


Energy equipment or infrastructure
Conversion efficiency
Tesla Model 3 electric motor97%
Electrical transformer97-99%
Transmission lines96-98%
Hydroelectric dam90%
Lithium-ion battery86-99+%
Natural gas furnace80-95%
Max multi-layer solar cell efficiency on earth68.70%
Max theoretical wind turbine efficiency (Betz limit)59%
Combined cycle natural gas plant55-60%
Typical wind turbine50%
Gas water heater50-60%
Typical US coal power plant33%
Max theoretical single-layer solar cell efficiency33.16%
Heat pump300-400%
Typical solar panel21%
Typical ICE car16-25%

Finally, (US) storage:


Type
Quads of capacity
Grid electrical storage0.002
Gas station underground tanks0.26
Petroleum refineries3.58
Other crude oil3.79
Strategic petroleum reserve4.14
Natural gas fields5.18
Bulk petroleum terminals5.64
Total22.59

I vaguely knew grid energy storage was much less than hydrocarbon, but I didn't realise it was 10,000 times less!

Thanks for compiling these quotes :) seemed too important not to make them at least somewhat actionable for myself, hence the checklist. 

Much appreciated, thanks!

Lex Fridman spent five hours talking AI and other things with Dylan Patel of SemiAnalysis. This is probably worthwhile for me and at least some of you, but man that’s a lot of hours.

Are we already at the point where AI, or some app, can summarize podcasts accurately and extract key takeaways with relatively technical interviewees like Dylan, so we don't need 5 hours (or even 2.5h at 2x)?

I currently work in policy research, which feels very different from my intrinsic aesthetic inclination, in a way that I think Tanner Greer captures well in The Silicon Valley Canon: On the Paıdeía of the American Tech Elite:

I often draw a distinction between the political elites of Washington DC and the industrial elites of Silicon Valley with a joke: in San Francisco reading books, and talking about what you have read, is a matter of high prestige. Not so in Washington DC. In Washington people never read books—they just write them.

To write a book, of course, one must read a good few. But the distinction I drive at is quite real. In Washington, the man of ideas is a wonk. The wonk is not a generalist. The ideal wonk knows more about his or her chosen topic than you ever will. She can comment on every line of a select arms limitation treaty, recite all Chinese human rights violations that occurred in the year 2023, or explain to you the exact implications of the new residential clean energy tax credit—but never all at once. ...

Washington intellectuals are masters of small mountains. Some of their peaks are more difficult to summit than others. Many smaller slopes are nonetheless jagged and foreboding; climbing these is a mark of true intellectual achievement. But whether the way is smoothly paved or roughly made, the destinations are the same: small heights, little occupied. Those who reach these heights can rest secure. Out of humanity’s many billions there are only a handful of individuals who know their chosen domain as well as they do. They have mastered their mountain: they know its every crag, they have walked its every gully. But it is a small mountain. At its summit their field of view is limited to the narrow range of their own expertise.

In Washington that is no insult: both legislators and regulators call on the man of deep but narrow learning. Yet I trust you now see why a city full of such men has so little love for books. One must read many books, laws, and reports to fully master one’s small mountain, but these are books, laws, and reports that the men of other mountains do not care about. One is strongly encouraged to write books (or reports, which are simply books made less sexy by having an “executive summary” tacked up front) but again, the books one writes will be read only by the elect few climbing your mountain.

The social function of such a book is entirely unrelated to its erudition, elegance, or analytical clarity. It is only partially related to the actual ideas or policy recommendations inside it. In this world of small mountains, books and reports are a sort of proof, a sign of achievement that can be seen by climbers of other peaks. An author has mastered her mountain. The wonk thirsts for authority: once she has written a book, other wonks will give it to her.

While I don't work in Washington, this description rings true to my experience, and I find it aesthetically undesirable. Greer contrasts this with the Silicon Valley aesthetic, which is far more like the communities I'm familiar with:

The technologists of Silicon Valley do not believe in authority. They merrily ignore credentials, discount expertise, and rebel against everything settled and staid.  There is a charming arrogance to their attitude. This arrogance is not entirely unfounded. The heroes of this industry are men who understood in their youth that some pillar of the global economy might be completely overturned by an emerging technology. These industries were helmed by men with decades of experience; they spent millions—in some cases, billions—of dollars on strategic planning and market analysis. They employed thousands of economists and business strategists, all with impeccable credentials. Arrayed against these forces were a gaggle of nerds not yet thirty. They were armed with nothing but some seed funding, insight, and an indomitable urge to conquer.

And so they conquered.

This is the story the old men of the Valley tell; it is the dream that the young men of the Valley strive for. For our purposes it shapes the mindset of Silicon Valley in two powerful ways. The first is a distrust of established expertise. The technologist knows he is smart—and in terms of raw intelligence, he is in fact often smarter than any random small-mountain subject expert he might encounter. But intelligence is only one of the two altars worshiped in Silicon Valley. The other is action. The founders of the Valley invariably think of themselves as men of action: they code, they build, disrupt, they invent, they conquer. This is a culture where insight, intelligence, and knowledge are treasured—but treasured as tools of action, not goods in and of themselves.

This silicon union of intellect and action creates a culture fond of big ideas. The expectation that anyone sufficiently intelligent can grasp, and perhaps master, any conceivable subject incentivizes technologists to become conversant in as many subjects as possible. The technologist is thus attracted to general, sweeping ideas with application across many fields. To a remarkable extent conversations at San Fransisco dinner parties morph into passionate discussions of philosophy, literature, psychology, and natural science. If the Washington intellectual aims for authority and expertise, the Silicon Valley intellectual seeks novel or counter-intuitive insights. He claims to judge ideas on their utility; in practice I find he cares mostly for how interesting an idea seems at first glance. He likes concepts that force him to puzzle and ponder.  

This is fertile soil for the dabbler, the heretic, and the philosopher from first principles.  It is also a good breeding ground for books. Not for writing books—being men of action, most Silicon Valley sorts do not have time to write books. But they make time to read books—or barring that, time to read the number of book reviews or podcast interviews needed to fool other people into thinking they have read a book (As an aside:  I suspect this accounts somewhat for the popularity of this blog among the technologists. I am an able dealer in second-hand ideas).

Out of curiosity, I asked Claude Sonnet 3.5 to create a checklist-style version of The Way to "serve as a daily reminder and also guide to practical daily action and thinking", with the understanding that (quoting Zvi) "The Way that can be specified is not The Way". Seems decent. (All bullet lists are meant to be checkboxes, except the last list of bullets.)

The Way: A Living Checklist

Note: The Way that can be specified is not The Way. This is an incomplete approximation, meant to guide rather than constrain.

Core Principles

Truth-Seeking

  •  Have I written down my actual beliefs clearly and publicly?
  • Am I ready to be proven wrong and update accordingly?
  • Have I avoided fooling myself, especially about things I want to be true?
  • Am I reasoning things out explicitly, step by step?
  • Have I shown my work so others can check my reasoning?

Action & Impact

  •  Am I actually Doing The Thing, rather than just talking about it?
  • Have I found ways to create concrete improvements today, rather than waiting for perfect solutions?
  • Am I focusing on real outcomes rather than appearances or process?
  • Do I have meaningful skin in the game?
  • Am I using my comparative advantage effectively?

Decision Making

  •  Have I considered the actual price/tradeoffs involved?
  • Am I making decisions under uncertainty rather than waiting for perfect information?
  • Have I avoided false dichotomies and found the nuanced path?
  • Am I being appropriately careful with irreversible decisions?
  • Have I maintained enough slack in my systems and decisions?

Learning & Growth

  •  Am I willing to look stupid to become less wrong?
  • Have I learned from my mistakes and updated my models?
  • Am I experimenting and iterating to find better approaches?
  • Have I sought out worthy opponents who can challenge my thinking?
  • Am I building deep understanding rather than surface knowledge?

Character & Conduct

  •  Have I been honest, even when it's costly?
  • Am I following through on my commitments?
  • Have I avoided needless cruelty or control?
  • Am I using power and influence responsibly?
  • Have I maintained my integrity while pursuing my goals?

Balance & Wisdom

  •  Have I found room for joy and fun without compromising effectiveness?
  • Am I building lasting value rather than chasing short-term gains?
  • Have I avoided both reckless abandon and paralyzing caution?
  • Am I considering both practical utility and deeper principles?
  • Have I remained adaptable as circumstances change?

Remember

  • The Way is hard
  • The Way is not for everyone
  • The Way changes as reality changes
  • Violence is not The Way
  • The perfect need not be the enemy of the good
  • Having skin in the game focuses the mind
  • Mundane utility matters
  • The Way includes both effectiveness and joy

This checklist is intentionally incomplete. The Way that matters is the one you find through doing the work.

Just reread Scott Aaronson's We Are the God of the Gaps (a little poem) from 2022:

When the machines outperform us on every goal for which performance can be quantified,

When the machines outpredict us on all events whose probabilities are meaningful,

When they not only prove better theorems and build better bridges, but write better Shakespeare than Shakespeare and better Beatles than the Beatles,

All that will be left to us is the ill-defined and unquantifiable,

The interstices of Knightian uncertainty in the world,

The utility functions that no one has yet written down,

The arbitrary invention of new genres, new goals, new games,

None of which will be any “better” than what the machines could invent, but will be ours,

And which we can call “better,” since we won’t have told the machines the standards beforehand.

We can be totally unfair to the machines that way.

And for all that the machines will have over us,

We’ll still have this over them:

That we can’t be copied, backed up, reset, run again and again on the same data—

All the tragic limits of wet meat brains and sodium-ion channels buffeted by microscopic chaos,

Which we’ll strategically redefine as our last strengths.

On one task, I assure you, you’ll beat the machines forever:

That of calculating what you, in particular, would do or say.

There, even if deep networks someday boast 95% accuracy, you’ll have 100%.

But if the “insights” on which you pride yourself are impersonal, generalizable,

Then fear obsolescence as would a nineteenth-century coachman or seamstress.

From earliest childhood, those of us born good at math and such told ourselves a lie:

That while the tall, the beautiful, the strong, the socially adept might beat us in the external world of appearances,

Nevertheless, we beat them in the inner sanctum of truth, where it counts.

Turns out that anyplace you can beat or be beaten wasn’t the inner sanctum at all, but just another antechamber,

And the rising tide of the learning machines will flood them all,

Poker to poetry, physics to programming, painting to plumbing, which first and which last merely a technical puzzle,

One whose answers upturn and mock all our hierarchies.

And when the flood is over, the machines will outrank us in all the ways we can be ranked,

Leaving only the ways we can’t be.

Feels poignant. 

Philosophy bear's response to Scott is worth reading too. 

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.

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