Tazerenix

Tao is not the one who needs to worry, but it bears pointing out that Tao is probably one of the most famous archetypal "problem solving" mathematicians in history (though at that echelon all mathematicians are both problem solvers and theory builders, in the same way that Chess Super GMs are all universalists).

Tao only changed the heading of his blog to that quote in the last month, which is part of my point. Somehow despite this framework for viewing the field of mathematics having been beautifully described by one of our greatest leaders over 30 years ago, Terence Tao who has spent the last 5 years thinking about this has only just come around to it. Perhaps it has something psychologically to do with his greatest skills being those most under attack (though of course I'm not actually implying anything about Tao, he is obviously an honest and good-natured contributor to the community).

The people who need to read On Proof and Progress are the undergraduates and PhD students who have ended up down an academic pathway without looking up because they've always been good at proof and understanding mathematics, without ever pausing to ask why.

And already on page 2 Thurston cuts the the root of the issue, 30 years ago:

>On a more everyday level, it is common for people first starting to grapple with computers to make large-scale computations of things they might have done on a smaller scale by hand. They might print out a table of the first 10,000 primes, only to find that their printout isn’t something they really wanted after all. They discover by this kind of experience that what they really want is usually not some collection of “answers”—what they want is understanding.

The difference in reactions by the maths community should probably be spit up into those who have read and understood On Proof and Progress and actually thought deeply about why they do mathematics, and those who haven't.

There's a quote Atiyah has about spin geometry which he heard from his advisor Hodge, who had an office next to Dirac at Cambridge.

Only two people understand spinors: God, and Dirac. And Dirac's dead.

I think you're a little naive if you think this is because of gate-keeping by mathematicians rather than the essential complexity of mathematics. Mathematicians individually and as a group would love nothing more than a world which is capable of understanding their work more deeply.

In the exact same way that LLMs allow anyone to vibe code an app but do not replace real understanding of system design due to its essential complexity, non-mathematicians will quickly learn that asking an LLM to pump out advanced mathematical statements to you, even if they are correct (and even if you could verify them) does not constitute understanding, and that the human brain is the bottleneck either way.

It is only if the LLM is super-human at simplification and explaining that a difference will be noted. This would be excellent for mathematics but its not a foregone conclusion (and the argument of most mathematicians, such as Terence Tao, is that this distillation process is one of the key parts of doing mathematics, and that LLMs so far seem to be going in the opposite direction. I suspect its probably user error and leveraging the tools better will produce different outcomes, but mathematicians are only just starting the journey that software developers have been going through, so patience is needed).

Yeah I should have phrased that better. The traits themselves are not the effects, they're bits of code which can have side effects. Rust doesn't track those side effects in the type system yet, but !Forget especially is the essence of that idea. If you implement it for everything owned by a function, you can basically infer that the function does not have the leak effect (which would be an effect in the effect row of a Forget trait implementstjon).

If you try treat memory as an effect you gain the need for several polymorphic effect type functions drop, forget, etc which map a type to the effect row charged by its corresponding Drop, Forget impl. Rust doesn't have that type system obviously.

The traits are essentially effect handlers for effects like `drop<T>`, `move<T>`, `forget<T>` which are implicitly charged to the a function which owns a `T` and does drops, moves, or forgets it.

Inferring the capabilities of the function from the traits of the types of the arguments is similar to tracking effects. The function charges `drop<T>` when `x: T` goes out of scope, which is handled by the trait implementation. If Rust had a proper algebraic effects type system, you would be able to see this directly in the signature of the function (and even more, if the trait impls themselves had their effects tracked, you'd be able to see from the signature of the function the side effects of deallocation of its owned variables, like if `drop<File>` performs `io`).

More algebraic effects being retrofitted onto Rust.

Yep, no doubt there are definitely issues with inflation. My main point is that the size of black holes in the early universe is largely orthogonal to the whole big bang/inflation debate. Even if the model changes, the lack of IMBH evidence means that any universal history model must have some kind of time asymmetry.

The evidence for the big bang is generally not that if you look far enough back in a telescope, the universe looks younger, which is somewhat the layperson's confusion.

Evidence for the big bang is about measuring redshift of galaxies throughout universal history, homgeneity and thermal equilibrium of the universe and CMBR, which could only be explained by it all having been in a compressed location where it could reach thermal equilibrium at some point in the distant past.

None of that is challenged by the Webb observations about very young supermassive black holes.

In fact, the existence of supermassive black holes themselves has basically always been an unsolved problem even before Webb. The only known possible explanation (stellar collapse -> accretion -> supermassive black hole) could be ruled out even before Webb on theoretical and experimental grounds, we just have stronger evidence against it now. (To wit: if supermassive black holes form from stellar black holes by growing, you would expect to see lots of intermediate mass black holes. We see almost none. Furthermore, the process of accretion is extremely energetic, so IMBHs would be the most visible objects in the night sky. The fact we see none is doubly damning)

The mainstream position now will be big bang + some kind of primordial black hole formation during the very early stages of the universe. Work of Hawking/Penrose shows that black holes can form under generic conditions in solutions to the EFE equations. We have a general understanding of how they could come about from certain dense matter layouts in a standard GR cosmological model.

Willingness to look stupid and intellectual self-confidence are two sides of the same coin.

If you can find internal (rather than external) reasons to trust/believe in your own intelligence and capabilities, it makes it easier to be willing to look foolish. Also, a lack of knowledge/ability in a new area (or even a familiar area) is not a sign of a lack of capability. There's a difference between being a novice and being an idiot. So long as your source of intellectual self-confidence is strong enough (say, you have made great intellectual achievements in some other area of your life unrelated to the thing you're struggling with right now) its irrelevant if other people think you the fool: they're simply mistaken, and that's no skin off your back.

You need to differentiate between special and general relativity when making these statements.

It is absolutely true that someone else would have come up with special relativity very soon after Einstein. All that would be necessary is someone else to have the wherewithal to say "perhaps the aether does not need to exist" for the equations already known at the time by others before Einstein to lead to the general theory.

General relativity is different. Witten contends that it is entirely possible that without Einstein, we may have had to wait for the early string theorists of the 1960s to discover GR as a classical limit of the first string theories in their quest to understand the strong nuclear force.

As opposed to SR, GR is one of the most singular innovative intellectual achievements in human history. It's definitely "out of distribution" in some sense.