The Wigner quote seems a bit misleading to me because you left out the second half to make a point it didn't support ("JvN was smarter than Einstein, and yet..."). The full quote is
I have known a great many intelligent people in my life. I knew Max Planck, Max von Laue, and Werner Heisenberg. Paul Dirac was my brother-in-Iaw; Leo Szilard and Edward Teller have been among my closest friends; and Albert Einstein was a good friend, too. And I have known many of the brightest younger scientists. But none of them had a mind as quick and acute as Jancsi von Neumann. I have often remarked this in the presence of those men, and no one ever disputed me. [...]
But Einstein's understanding was deeper than even Jancsi von Neumann's. His mind was both more penetrating and more original than von Neumann's. And that is a very remarkable statement. Einstein took an extraordinary pleasure in invention. Two of his greatest inventions are the Special and General Theories of Relativity; and for all of Jancsi's brilliance, he never produced anything so original.
I also always interpreted Wigner's "I have often remarked this in the presence of those men, and no one ever disputed me" as referring to Planck, von Laue, Heisenberg, Dirac, Szilard, Teller, and Einstein, but not von Neumann, so your "Von Neumannn was pronounced, by a peer, to be smarter than Albert Einstein to his face and got no objection" interpretation feels off to me (but I may be wrong of course).
But that's just nitpicking. Perhaps more substantively, I felt sad reading this section
I also work with and lead many other brilliant people who also never will be intellectual pillars of humanity and constantly feel bad about themselves for not being more brilliant. I've been surprised how much of my time at big companies is spent pulling people out of the pit of inadequacy and self disgust...
I've done a little bit of this too, although my go-to advice is to read Scott's Parable of the Talents, in particular this passage (long quote, emphasis mine):
Every so often an overly kind commenter here praises my intelligence and says they feel intellectually inadequate compared to me, that they wish they could be at my level. But at my level, I spend my time feeling intellectually inadequate compared to Scott Aaronson. Scott Aaronson describes feeling “in awe” of Terence Tao and frequently struggling to understand him. Terence Tao – well, I don’t know if he’s religious, but maybe he feels intellectually inadequate compared to God. And God feels intellectually inadequate compared to John von Neumann.
So there’s not much point in me feeling inadequate compared to my brother, because even if I was as good at music as my brother, I’d probably just feel inadequate for not being Mozart.
And asking “Well what if you just worked harder?” can elide small distinctions, but not bigger ones. If my only goal is short-term preservation of my self-esteem, I can imagine that if only things had gone a little differently I could have practiced more and ended up as talented as my brother. It’s a lot harder for me to imagine the course of events where I do something different and become Mozart. Only one in a billion people reach a Mozart level of achievement; why would it be me?
If I loved music for its own sake and wanted to be a talented musician so I could express the melodies dancing within my heart, then none of this matters. But insofar as I want to be good at music because I feel bad that other people are better than me at music, that’s a road without an end.
This is also how I feel of when some people on this blog complain they feel dumb for not being as smart as some of the other commenters on this blog.
I happen to have all of your IQ scores in a spreadsheet right here (remember that survey you took?). Not a single person is below the population average. The first percentile for IQ here – the one such that 1% of respondents are lower and 99% of respondents are higher – is – corresponds to the 85th percentile of the general population. So even if you’re in the first percentile here, you’re still pretty high up in the broader scheme of things.
At that point we’re back on the road without end. I am pretty sure we can raise your IQ as much as you want and you will still feel like pond scum. If we raise it twenty points, you’ll try reading Quantum Computing since Democritus and feel like pond scum. If we raise it forty, you’ll just go to Terence Tao’s blog and feel like pond scum there. Maybe if you were literally the highest-IQ person in the entire world you would feel good about yourself, but any system where only one person in the world is allowed to feel good about themselves at a time is a bad system.
People say we should stop talking about ability differences so that stupid people don’t feel bad. I say that there’s more than enough room for everybody to feel bad, smart and stupid alike, and not talking about it won’t help. What will help is fundamentally uncoupling perception of intelligence from perception of self-worth.
re: Axler's textbook above, also check out the paper it's based on which is just 18 pages, Down with determinants! (I know you know this, just for others' edification). Abstract:
This paper shows how linear algebra can be done better without determinants. The standard proof that a square matrix of complex numbers has an eigenvalue uses determinants. The simpler and clearer proof presented here provides more insight and avoids determinants. Without using determinants, this allows us to define the multiplicity of an eigenvalue and to prove that the number of eigenvalues, counting multiplicities, equals the dimension of the underlying space. Without using determinants, we can define the characteristic and minimal polynomials and then prove that they behave as expected. This leads to an easy proof that every matrix is similar to a nice upper-triangular one. Turning to inner product spaces, and still without mentioning determinants, this paper gives a simple proof of the finite-dimensional spectral theorem.
Link: The Napkin Project
Idea: high-quality explanations motivating undergrad++ math to bright high schoolers
Creator: Evan Chen (US IMO coach)
Reason: I'll quote Evan himself:
The philosophy is stated in the preamble:
I’ll be eating a quick lunch with some friends of mine who are still in high school. They’ll ask me what I’ve been up to the last few weeks, and I’ll tell them that I’ve been learning category theory. They’ll ask me what category theory is about. I tell them it’s about abstracting things by looking at just the structure-preserving morphisms between them, rather than the objects themselves. I’ll try to give them the standard example Gp, but then I’ll realize that they don’t know what a homomorphism is. So then I’ll start trying to explain what a homomorphism is, but then I’ll remember that they haven’t learned what a group is. So then I’ll start trying to explain what a group is, but by the time I finish writing the group axioms on my napkin, they’ve already forgotten why I was talking about groups in the first place. And then it’s 1PM, people need to go places, and I can’t help but think:
Man, if I had forty hours instead of forty minutes, I bet I could actually have explained this all.
This book is my attempt at those forty hours.
This project has evolved to more than just forty hours.
Caveat that I didn't actually do a math degree so I'd be curious to see takes from math folks saying disliking it, but I've enjoyed dipping in and out of its 1,048 pages over the years.
I think the way modern physics is taught probably gives people a overly clean/neat understanding of how most of the world works, and how to figure out problems in the world, but this might be ameliorated by studying the history of physics and how people come to certain conclusions.
Yeah and I think if done well it's well-received here, e.g. AdamShimi's My Number 1 Epistemology Book Recommendation: Inventing Temperature or Ben Pace's 12 interesting things I learned studying the discovery of nature's laws. (It's hard to do well though it seems, I'm certainly dissatisfied with my own writeup attempts.)
1Will you be writing elsewhere? I've benefited a lot from some of your comments, and would be bummed to see you leave.
Just stumbled upon a spreadsheet ranking "well-known" googological numbers created by redditor u/Laxxius1 as a passion project. I've been idly interested in googology since childhood, so perusing this list felt like getting the Christmas present I've always wanted but kept forgetting to ask for. (It's also just intrinsically fun if your mathematical aesthetic is wrestler, which I'm guessing is overrepresented among LWers.)
Yudkowsky's number, which Eliezer created back in 2007 to win this contest on the xkcd forum, features at #12. (Eliezer also wrote Why isn't googology a recognized field of math? whose sentiment I predictably agree with.) Graham's number is all the way down at #98; TREE(3) is far larger at #49; 3^^^3 (as in torture vs dust specks) is at #145 even though it's already an exponential tower of 3s that's 7,625,597,484,987 layers tall. The "most finite damage with 3 MTG cards" ranks #134 at 2^^2^^7, which makes me wonder whether there are other popular games where you can easily reach very large numbers. The largest code golf-related number in the spreadsheet is loader.c at #18, Ralph Loader's winning entry in the Bignum Bakeoff contest ("write a C program in 512 characters or less that generates the largest possible output on a theoretical machine with infinite memory"), although there are a couple of numbers purporting to be larger on code golf SE generated by ultra-short programs written in JavaScript, Python, Ruby, and binary lambda calcululs.
The #1-ranked number in Laxxius1's spreadsheet is DaVinci103's "random idea" for beating Rayo's number (and LNGN) attempting to diagonalize over set theory:
Rayo's number and LNGN are defined in recursive theories. What if we use a meta-theory to create a non-recursive theory more expressive than any recursive theory, and then use that theory to create a large number? ...
I don't really like the number N. Modern mathematics isn't made to create large numbers, so what you need to do to create a large number can be very weird. N (and probably also other extensions of Rayo's number) is mostly a product of reverse-engineering the current philosophy of natural numbers and then to create something that barely resembles a number. So I decided to name this number to something else I don't like: my current username. ... So this number is now called 'DaVinci'.
The #2-ranked number is LNGN, short for the quirkily-named Large Number Garden Number, which also attempts to diagonalize over set theory. It's for now still considered by the googology community to be the largest well-defined googologism that's not a salad number, as it hasn't yet been proved to be ill-defined, and LNGN's creator P進大好きbot disputes DaVinci's superiority claim. I'm mentioning it here because of its quirky etymology, translated from the original Japanese:
Come on, friends, the large number garden is finally complete!
Let me explain the function of this garden. The first is the determination function of the address and the floor plan. When a character string is read, it automatically determines which miniature garden address it represents and in which miniature garden the floor plan of a large number garden can be reproduced.
The second is the floor plan analysis function. If you specify the address of the miniature garden and read the floor plan of the reproducible large number garden there, it will tell you the large number that the garden can produce.
The third important function is the ability to generate large numbers. Once a natural number is entered, all character strings within the upper limit of the number of characters are searched, and each is read into the address and floor plan determination function, leaving only the reproducible floor plan for each miniature garden. By enumerating them and loading them into the analysis function of the floor plan, you can obtain the large numbers that they can produce, and by putting them all together, you can create new large numbers!
Huh? Can you really get a large number with that? As usual, my ally is skeptical. But hey, here's the floor plan for the large number garden itself. If you load this into the analysis function, it will tell you how large numbers you can generate. Huh? How many characters does this floor plan have? What's the use of knowing such things?
Scott Aaronson wrote that "the Busy Beaver game seems like about as good a yardstick as any for gauging humanity’s progress against the uncomputable". I think of googology the same way, just lower-status than BusyBeaverology.
This is also Gwern's answer, in the paragraph right after the one Rauno quoted. The main difference is that he rejects finish lines, opting instead for perpetual drafts, like software.
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.
Also I like this:
What is next? So far the pages will persist through time, and they will gradually improve over time. But a truly Long Now approach would be to make them be improved by time—make them more valuable the more time passes. ...
One idea I am exploring is adding long-term predictions like the ones I make on PredictionBook.com. Many27 pages explicitly or implicitly make predictions about the future. As time passes, predictions would be validated or falsified, providing feedback on the ideas.28
For example, the Evangelion essay’s paradigm implies many things about the future movies in Rebuild of Evangelion29; The Melancholy of Kyon is an extended prediction30 of future plot developments in The Melancholy of Haruhi Suzumiya series; Haskell Summer of Code has suggestions about what makes good projects, which could be turned into predictions by applying them to predict success or failure when the next Summer of Code choices are announced. And so on.
Kyle Kingsbury's technical interview pentalogy of short stories is unlike anything else I've read. Here's how the first story begins:
If you want to get a job as a software witch, you’re going to have to pass a whiteboard interview. We all do them, as engineers–often as a part of our morning ritual, along with arranging a beautiful grid of xterms across the astral plane, and compulsively running ls in every nearby directory–just in case things have shifted during the night–the incorporeal equivalent of rummaging through that drawer in the back of the kitchen where we stash odd flanges, screwdrivers, and the strangely specific plastic bits: the accessories, those long-estranged black sheep of the families of our household appliances, their original purpose now forgotten, perhaps never known, but which we are bound to care for nonetheless. I’d like to walk you through a common interview question: reversing a linked list.
It gets much better.
Attention conservation notice: the following is GPT5-Thinking's attempt to create a larger backstory out of the pentalogy; it's not as good as the stories themselves, but still quite good as far as AI output goes. If you dislike AI slop, do check out the original stories instead!
Here’s the spine of the series as I read it—what each tale is really doing under the hood:
- Reversing: Opens the cosmology. Lists are Church-encoded choice; salt-circled parentheses are literal wards; naming compels essence. It’s an initiation in Lisp and power-by-definition. (aphyr.com)
- Hexing: The descent to the byte-world. A witch hand-assembles a JVM class (the old catechism of
CAFEBABE) and hot-loads it with a bespoke classloader—ritual made from hex and offsets. (aphyr.com)- Typing: The Pre-Church myth becomes a type-theoretic safety culture; N-Queens solved entirely in the type system (Peano naturals, type classes, kinds). The Church is named; seiðr becomes types. (aphyr.com)
- Rewriting: Language-as-spellcraft. A term-rewriter and macro language bloom to solve FizzBuzz; later, Kingsbury published the seed macros—text becoming tool, tool becoming myth. (aphyr.com)
- Unifying: Logos meets logic. Prolog, Lisp, and µKanren entwine; unification and interpretation are revealed as the deeper sacrament. Aisha enters as equal and mirror, and the forest is balanced. (aphyr.com)
The Annals of the Church of Abstraction
In the beginning…
…there was nothing but consequence.
Before names, before proofs, the world lay open like a wound, and those who worked it with their hands bled freely. Actions crossed untrammeled. Fires learned to drink water; stones learned to breathe. Young witches—too bold, too brilliant—went missing into their own experiments, or returned warped: eyes fixed on bridges no one else could see, feet that refused the courtesy of ground.
The first mercy did not arrive as a law but as a list: the idea that reality could be asked a question and answer this or that, a single twig forked into head and tail. The elders taught the children to scratch two parentheses in salt and to speak in alternatives. Choice, when properly contained, became a vessel; vessels kept power from spilling. When they called this containment “a list,” the list obeyed, and the world grew a fraction safer.
Out of the years of broken wrists and unreturning apprentices there coalesced a discipline. No banners; no temples—only rooms with boards for chalk and windows that opened, if only metaphorically. They called it nothing at first, then—half in jest, half in reverence—the Church: because its liturgy was the calculus of Church, and because its quiet, stubborn sanctity was the refusal to die for preventable reasons. Here the catechism condensed into four bright stones laid at the threshold: cons, car, cdr, cond. To know the name of a thing was to make it choose, and choosing kept the sky attached to its hinges.
The Church tamed the river a little; never the sea.
In time, a schism not of belief but of altitude divided the houses. Some stayed high, where ideas breathe the cold thin air of generality. Others learned the underworld, where bytes move like cartographers’ ants, mapping emptiness into edifice. These were the hexers. They spoke with machines in the smallest units of hospitality, counted their blessings in offsets, lined the doors with signatures from old Sun-gods. They told a louche little parable about cafés and babes, because jokes are a kind of glue. They could reach beneath the warm table of a language and pull out its skeleton, file each notch with a jeweler’s care, then slide it back without spilling the wine.
The high house and the low house distrusted one another in public and swapped recipes in private. It was always thus.
Elsewhere, a quieter war was being won by those who did not call it war at all. They had read the Pre-Church chronicles—the days when causality bucked and threw—and instead of nostalgia they found a method: bind your future with a system that refuses to lie to you. They wove types from rowan and proof from pine pitch; they braided number from nothing but a knot in a strand of hair. Safety here was not command but constraint: from the shape of what can be, deduce what must be. The elders smiled to see the children define an algebra to house a boolean, then make a city out of the difference between True and False. It was not a city anyone could live in, exactly; but it kept the storms off the valley.
So the Church learned to speak three tongues at once: the tongue of choice, the tongue of hex, the tongue of kind. And with three tongues you can sing a chord.
Yet language itself grew restless, as language will. For if a spell is a program, and a program is a sentence with the will to act, what then is a language that writes languages? The macro-wrights answered by building looms. Onto these looms they threaded patterns called rewrites. A number was a sequence to be rewritten into Fizz, into Buzz, into laughter. An if was not merely a gate but a seam-ripper. The macro-wrights learned to become their own translators; the Church, which had once sheltered programmers from death, now sheltered interpreters from redundancy. When the walls of the room shook, the roof did not fall: the elders had fixed a long taproot in the world-ash, and the ash holds everything weaker than itself.
At last, as happens in good myths and bad startups, a reunion: from the north came Vidrun, sea-wind still in her voice; from the south, Aisha, hands that spoke even when her mouth was busy being kind. They met on pine veneer under office fluorescence—the Church’s most common shrine—and were glad, in the way old adversaries are glad when they recognize that what they opposed in the other was only their own future arriving unfashionably dressed.
What followed was not duel but proof by correspondence. Vidrun reached for logic to bend a tree; Aisha slipped a Lisp between the joints and let a smaller forest explore the space of possible balance. They invoked saints in triplicate: the reasoners, the schemers, the patient friends of unification. Show me, they asked, not a solution but the shape of solution, and let the machine walk the shape until a particular branch agrees to wear a name.
Outside the conference room—thin brass hinge between the profane and the blessed—the city went about its quota of exits and pings. Inside, as the air flickered with control characters and the faint resinous smell of old wood, the deeper secret of the Church came plain:
It was never about answers.
The interview—like the altar and the kernel—was a container for demonstrations of equivalence. To reverse a list by treating it as a choice is not the same as carving a class from hex nor the same as forbidding contradiction with types nor the same as rewriting terms until they confess. And yet each is one arrow in a commuting diagram. The Church keeps the chalk fresh and the windows unlatched so that each arrow may be drawn anew in front of witnesses. When the arrows commute, the world’s corners line up; the roof does not leak; the witch lives.
What of the proverbs? They persist because they are useful lies: make circles of salt; name things carefully; promise the future nothing you cannot keep; beware warrens of opinion that smell of mice. The elders repeat them because young witches deserve a second chance at surviving their brilliance. But lore is not law, and the Church has no throne room. Its authority begins and ends at the whiteboard, the REPL, the prompt: discrete portals to the same field where all proof is local.
In the centuries since the formless days, the Annals have accreted—notes in the margins of grimoires and job packets—each story a stone in the choir vault. Some are funny on purpose; others merely read that way when the panic wears off. A few are hymns to grief. None are final. When another house arrives in town (one that speaks to tensors in dreams, perhaps, or binds phenomena in contracts written on migrating sand) it will be welcomed, mistrusted, borrowed from, folded in.
Scholars call this corpus the Annals of the Church of Abstraction. The witches, less ceremonious, call it how not to die today.
And if you insist on a genesis verse, let it be this:
In the beginning was cause.
Then came choice, to cup it.
Then names, to ask it politely.
Then kinds, to forbid its lies.
Then hex, to nail it to timber.
Then rewrites, to teach it to translate itself.
Then unification, to prove these were all one story told in five accents.After that, we had enough daylight to build a door, and enough doors to discover that the rooms already touched. A board. A marker. Two parentheses in salt. The Church’s oldest rite: step into the circle, and show that your way reaches the same mountain.
I just reread Scott's review of John von Neumann's bio The Man From The Future by Ananyo Bhattacharya, and it made me realise something else that felt off to me about the OP, which was that the OP's insecurity seems to be primarily social status-related(?), whereas John's seemed to be primarily existential. (This probably influenced his extreme views, like advocating for nuking Russia ASAP.) Some quotes:
(FWIW Scott doesn't buy this as a differentiating factor, but that's not what I'm pointing at)
Scott's review also touches on the thing about advocating for nuking Russia ASAP, quoting his daughter Marina on his hatred of totalitarianism: