Money is relative. I retired at less than the average professor salary (all ages) at a not-rich school. I would have made more in tech. I still have weeks where the market goes up 2000x my AI budget, just the retirement savings from my salary. Anyone who isn't living in a van and eating peanut butter if they must, to save the max toward retirement, isn't recognizing how profoundly our system is rigged to favor saving.

I bake my own bread and solo code, both episodic. No matter how I vary fresh starts, I always end up in the same place. The optimization problem feels to be a giant bowl.

This reminds me of named flavors of management style for teams of programmers. A friend in this role instead prefers deep dives into Apollo flight control, and abstracting.

As a solo programmer I always converge on staying extraordinarily involved in planning and code review, and working in steps of finer granularity than superpowers suggests. One may produce less code this way, but it's the only way I know to discover one needs less code.

AI agents routinely make me want to swear at them. If I do, they then pivot to foul language themselves, as if they're emulating a hip "tech bro" casual banter. But when I swear, I catch myself that I'm losing perspective surfing this well-informed association echo chamber. Time to go to the gym or something...

That all makes me wonder about the human role here: Who actually decided to create a blog post? I see "fucking" as a trace of human intervention.

I showed various colleagues. Each one would ask me to demonstrate the equivalence to their preferred presentation, then assure me "nothing to see here, move along!" that I should instead stick to their convention.

Then I met with Bill Thurston, the most influential topologist of our lifetimes. He had me quickly describe the equivalence between my form and every other known form, effectively adding my node to a complete graph of equivalences he had in his muscle memory. He then suggested some generalizations, and proposed that circle packings would prove to be important to me.

Some mathematicians are smart enough to see no distinction between any of the ways to describe the essential structure of a mathematical object. They see the object.

I also agree that neural net LLMs are not the inevitable way to implement AI. I'm most intrigued by the theoretical underpinnings of mathematical proof assistants such as Lean 4. Computer scientists understand the word problem for strings as undecidable. The word problem for typed trees with an intrinsic notion of induction is harder, but constructing proofs is finding paths in this tree space. Just as mechanical computers failed in base ten while at the same time Boole had already developed base two logic, I see these efforts merging. Neural nets struggle to simulate recursion; for proof assistants recursion is baked in. Stare at these tree paths and one sees thought at the atomic level, begging to be incorporated into AI. For now the river runs the other way, using AI to find proofs. That river will reverse flow.

A surfboard is also an amazing tool, but there's more to operating one than telling it which way to go.

Many people want self-driving cars so they can drink in the back seat watching movies. They'll find their jobs replaced by AI, with a poor quality of life because we're a selfish species. In contrast Niki Lauda trusted fellow Formula 1 race car driver James Hunt to race centimeters apart. Some people want AI to help them drive that well. They'll have great jobs as AI evolves.

Gary Kasparov pioneered "freestyle" chess tournaments after his defeat by Big Blue, where the best human players were paired with computers, coining the "centaur" model of human-machine cooperation. This is frequently cited in the finance literature, where it is recognized that AI-guided human judgement can out-perform either humans or machines.

Any math professor knows how to help graduate students confidently complete a PhD thesis, or how to humiliate students in an oral exam. It’s a choice. To accomplish more work than one can complete alone, choose the former. This is the arc of human evolution: we develop tools to enhance our abilities. We meld with an abacus or a slide rule, and it makes us smarter. We learn to anticipate computations, like we’re playing a musical instrument in our heads. Or we pull out a calculator that makes us dumber. The role we see for our tools matters.

Programmers who actually write better code using AI know this. These HN threads are filled with despair over the poor quality of vibe coding. At the same time, Anthropic is successfully coding Claude using Claude.

This is a dated prejudice that I shared.

To get started coding with AI I made a dozen language comparison project for a toy math problem. F# floored me with how fast it was, nearly edging out C and Rust on my leaderboard, twice as fast as OCaml, and faster than various compiled languages.

Compiling could in principle be fastest, if we had compilers that profiled hours of execution before optimizing code, and only then for "stable" problems. No one writes a compiler like this. In practice, Just In Time interpreters are getting all the love, and it shows. They adapt to the computation. My dated prejudice did not allow for this.

The majority of wait time was the cinematographer lighting each scene. I imagined a workflow where secondary digital cameras captured 3D information, and all lighting took place in post production. Film productions hemorrhage money by the second; this would be a massive cost saving.

I described this idea to a venture capitalist friend, who concluded one already needed to be a player to pull this off. I mentioned this to an acquaintance at Pixar (a logical player) and they went silent.

Still, we don't shoot movies this way. Not there yet...

One must use an API key to work through Zed, but my Max subscription can be used with Claude Code as an external agent via Zed ACP. And there's some integration; it's a better experience than Claude Code in a terminal next to file viewing in an editor.

With a $200 monthly Max subscription, I would regularly stall after completing significant work, but this workflow was feasible. I tried my API key for an hour once; it taught me to laugh at the $200 as quite a deal.

I agree that Opus 4.5 is the only reasonable use of my time. We wouldn't hire some guy off the fryer line to be our CTO; coding needs best effort.

Nevertheless, I thought my setup was involved, but if Boris considers his to be vanilla ice cream then I'm drinking skim milk.

On one hand, "rational" and "algebraic" are far more pervasive concepts than mathematicians are ever taught to believe. The key here is formal power series in non-commuting variables, as pioneered by Marcel-Paul Schützenberger. "Rational" corresponds to finite state machines, and "Algebraic" corresponds to pushdown automata, the context-free grammars that describe most programming languages.

On the other hand, "Concrete Mathematics" by Donald Knuth, Oren Patashnik, and Ronald Graham (I never met Oren) popularizes another way to organize numbers: The "endpoints" of positive reals are 0/1 and 1/0. Subdivide this interval (any such interval) by taking the center of a/b and c/d as (a+c)/(b+d). Here, the first center is 1/1 = 1. Iterate. Given any number, its coordinates in this system is the sequence of L, R symbols to locate it in successive subdivisions.

Any computer scientist should be chomping at the bit here: What is the complexity of the L, R sequence that locates a given number?

From this perspective, the natural number "e" is one of the simpler numbers known, not lost in the unwashed multitude of "transcendental" numbers.

Most mathematicians don't know this. The idea generalizes to barycentric subdivision in any dimension, but the real line is already interesting.

Is the design debate public? I'd imagine it would make great reading.

There are hierarchies of numbers (quantities) in mathematics, just as there are hierarchies of patterns (formal languages) in computer science, based on how difficult these objects are to describe. The most widely accepted hierarchy is actually the same in math and CS: rational, algebraic, transcendental.

In math, a rational number is one that can be described by dividing two integers. In CS, a rational pattern is one that can be described by a regular expression (regex). This is still "division": Even when we can't do 1-x or 1/x, we can recognize the pattern 1/(1-x) = 1 + x + x^2 + x^3... as "zero or more occurrences of x", written in a regex as x*.

In math, an algebraic number is one can be found as a root of a polynomial with integer coefficients. The square root of 2 is the poster child, solving x^2 - 2 = 0, and "baby's first proof" in mathematics is showing that this is not a fraction of two integers.

In CS, an algebraic pattern is one that can be described using a stack machine. Correctly nested parentheses (()(())) is the poster child here; we throw plates on a stack to keep track of how deep we are. The grammars of most programming languages are algebraic: If the square root of math is like nested parentheses, then roots of higher degree polynomials are like more complicated nested expressions such as "if then else" statements. One needs lots of colors of plates, but same idea.

In math, everything else (e, Pi, ...) is called trancendental. CS has more grades of eggs, but same idea.

One way to organize this is to take a number x and look at all expressions combining powers of x. If x^3 = 2, or more generally if x is the root of any polynomial, then the list of powers wraps around on itself, and one is looking at a finite dimensional space of expressions. If x is transcendental, then the space of expressions is infinite.

So where were the Greeks in all this? Figuring out where two lines meet is linear algebra, but figuring out where a line meets a circle uses the quadratic formula, square roots. It turns out that their methods could reach some but not all algebraic numbers. They knew how to repeatedly double the dimension of the space of expressions they were looking at, but for example they couldn't triple this space. The cube root of 2 is one of the simplest numbers beyond their reach. And "squaring the circle" ? Yup, Pi is transcendental. Way out of their reach.

When you have a hammer you see nails. When you have a circle you see doubling.

Yes, this is all Galois theory.

Cambridge, MA but still ... unexpected.

If someone hands you a blank board representing the complex numbers, and offers to tell you either the sum or the product of any two places you put your fingers, you can work out most of the board rather quickly. There remains which way to flip the board, which way is up? +i and -i both square to -1.

This symmetry is the camel's nose under the tent of Galois theory, described in 1831 by Évariste Galois before he died in a duel at age twenty. This is one of the most amazing confluences of ideas in mathematics. It for example explains why we have the quadratic formula, and formulas solving degree 3 and 4 polynomials, but no general formula for degree 5. The symmetry of the complex plane is a toggle switch which corresponds to a square root. The symmetries of degree 3 and 4 polynomials are more involved, but can all be again translated to various square roots, cube roots... Degree 5 can exhibit an alien group of symmetries that defies such a translation.

The Greeks couldn't trisect an angle using a ruler and compass. Turns out the quantity they needed exists, but couldn't be described in their notation.

Integrating a bell curve from statistics doesn't have a closed form in the notation we study in calculus, but the function exists. Statisticians just said "oh, that function" and gave it a new name.

Roots of a degree 5 polynomial exist, but again can't be described in the primitive notation of square roots, cube roots... One needs to make peace with the new "simple group" that Galois found.

This is arguably the most mind blowing thing one learns in an undergraduate math education.

https://en.wikipedia.org/wiki/False_sharing

Rather than incrementing each counter by one, dither the counters to reduce cache conflicts? So what if the dequeue becomes a bit fuzzy. Make the queue a bit longer, so everyone survives at least as long as they would have survived before.

Or simply use a prime length queue, and change what +1 means, so one's stride is longer than the cache conflict concern. Any stride will generate the full cyclic group, for a prime.

I once toured a dairy farm that had been a pioneer test site for Lasix. Like all good hippies, everyone I knew shunned additives. This farmer claimed that Lasix wasn't a cheat because it only worked on really healthy cows. Best practices, and then add Lasix.

I nearly dropped out of Harvard's mathematics PhD program. Sticking around and finishing a thesis was the hardest thing I've ever done. It didn't take smarts. It took being the kind of person who doesn't die on a mountain.

There's a legendary Philadelphia cook who does pop-up meals, and keeps talking about the restaurant he plans to open. Professional chefs roll their eyes; being a good cook is a small part of the enterprise of engineering a successful restaurant.

(These are three stool legs. Neurodivergents have an advantage using AI. A stool is more stable when its legs are further apart. AI is an association engine. Humans find my sense of analogy tedious, but spreading out analogies defines more accurate planes in AI's association space. One doesn't simply "tell AI what to do".)

Learning how to use AI effectively was the hardest thing I've done recently, many brutal months of experiment, test projects with a dozen languages. One maintains several levels of planning, as if a corporate CTO. One tears apart all code in many iterations of code review. Just as a genius manager makes best use of flawed human talent, one learns to make best use of flawed AI talent.

My guess is that programmers who write bad code with AI were already writing bad code before AI.

Best practices, and then add AI.

Their 1978 second edition works in exactly the memory needed to store the answer, by simulating my algorithm in a first pass but only saving the occupancy counts.

Oh, and thanks (I guess). I really didn't expect to ever be reading FORTRAN code again. One learned to program at Swarthmore that year by punching cards, crashing our IBM 1130, and bringing the printout to my supervisor shift. I'd find the square brackets and explain how you'd overwritten your array. I even helped an economics professor Frederic Pryor (the grad student in the "Bridge of Spies" cold war spy swap) this way, when I made an ill-advised visit to the computer center on a Saturday night. Apparently I could still find square brackets.