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.

>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.

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

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).

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.

To study a combinatorial conjecture all your life but fail to produce a proof of anything is almost definitionally a failure in that field, because if you failed to prove anything, no new techniques were developed which could be leveraged to solve other similar or different combinatorial problems.

The same is not true of other, more theoretical areas of mathematics, where the balance between problem solving and theory building is more on the theory side. In such areas, a lot of valuable work can come out of pursuing conjectures without achieving the ultimate glory of proving them.

In such parts of mathematics, the most famous conjectures are usually upheld because of the volume of circumstantial or theoretical evidence in their favour. Most of the time a disproof of such a conjecture is far more likely to be "for trivial reasons" rather than revealing some fundamental failure of the theory. Take Yang-Mills existence and mass gap: what would it even mean to mathematics for that famous conjecture to be "disproved"? Well, our universe exists and appears to be mathematical, so (unless the disproof was really remarkable) it cannot mean that no such theory can exist, it just means that the axioms declared need to be tinkered with.

It is still absolutely the case that a disproof which reveals something fundamental or interesting would be held up as a remarkable result. But there are many areas of mathematics where the deck has been stacked so favourably for the positive case (because the conjecture is in essence "obviously true" but in practice we do not know the right statement of the conjecture, and constructing the statement requires developing theory, this is the Grothendieck nut-crack quote at the conjecture-forming level) that a non-trivial disproof is almost impossible to imagine. The Clay problems which do not have a significant reward for a disproof are all of this form.

A disproof of the Jacobian conjecture which is just an oracle giving us a polynomial which is not injective is not such a fundamental or interesting result. An oracle giving us a non-trivial zero of the Riemann zeta function not on the critical line would be similar, and so on.

In other words, for many parts of mathematics, there is a big difference between conjectures being false for trivial reasons [1] or false for non-trivial reasons.

[1] https://www.sciencedirect.com/science/article/pii/0040938369...

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.

Suppose kinetic energy was E = m|v| instead, linearly dependent on speed |v|. What does that mean for the universe?

The traditional Lagrangian is L = 1/2 mv^2 - V(x). This kinetic energy gives a different formula:

L = m|v|ln|v|-V(x).

Deriving the corresponding equations of motion, you get:

p = m(1+ln|v|)sgn(v)

ma = |v|F

A few things we can note from these formulas:

1. They are not boost invariant: Galilean relativity is violated. That means there is necessarily a privileged reference frame (i.e. an aether) in which the universe is at rest, and all dynamics must be understood relative to this reference frame.

2. Newton's first law has a pathological interpretation in regards to the above reference frame: If ma = |v|F and |v| = 0 (i.e. you are at rest relative to the aether), then a = 0 no matter what F is. That is, for objects which are stationary with respect to the aether, no motion is possible regardless of what force is applied.

It is still true that objects in motion (relative to the aether) remain in motion unless acted upon by an outside force, and Newton's third law is still true, but such a universe basically makes no sense.

You could essentially argue from the anthropic principle that such a universe would have such pathological dynamics that it could not permit life, and therefore we cannot observe it.

This is the contrapositive of the argument presented on stackexchange. There they say "given Galilean relativity, you get the quadratic scaling law". This argument says "if you don't have the quadratic scaling law, you don't have relativity".

The point of the counterfactual is a bit like Richard Feynman's "why" argument [1]. There is no fundamental reason why this kind of dynamics couldn't exist. We can only ever reduce our explanation to a more fundamental intuition we have about the same universe we live in (i.e. from kinetic energy scaling laws to Galilean relativity). But without a mathematical proof of the incoherence even in principle of the alternative, its perfectly valid to imagine an alternative universe with different dynamics. It's just not our universe.

[1] https://www.youtube.com/watch?v=36GT2zI8lVA

https://github.com/npm/cli/pull/8965

https://github.com/npm/cli/issues/8994

Its good that that they finally got there but....

I would be avoiding npm itself on principle in the JS ecosystem. Use a package manager that has a history of actually caring about these issues in a timely manner.

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.

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.

- Albert Einstein

most mathematicians act as though they are Platonists, even though, if pressed to defend the position carefully, they may retreat to formalism https://en.wikipedia.org/wiki/Mathematical_Platonism

Mathematicians begrudgingly retreat to formalism and foundations when pressed because its easier to defend, but the day-to-day of contemporary mathematics is much more an explorative process of a "real" mathematical landscape. They aren't concerned with foundations because it "feels" self-evident that the mathematics they are discovering is true (because their means of discovery, rigour and proof, "guarantee" it to be so).

A lot of the comments here are making false assumptions like "but surely mathematicians all know that their field is ultimately justified as a symbol-pushing game from some axiomatic system right?" in the same way one might say "surely all computer scientists know that every language ultimately compiles down to 1s and 0s processed by a CPU" but that is not at all how most mathematicians think about doing mathematics.

Here's Gromov, one of the greatest geometers of the last 50 years, discussing his viewpoint on this.

The conclusion is that with a sufficiently large number of actors in the market all seeking profits by trying to find misevaluation of stock prices, the excess profits of any individual actor will (assuming they all have access to the same information) converge to zero.

Its less a paradox and more a matter of game theory. Every investment firm which gives up trying to look for alpha (believing it is fruitless) means the remaining firms have more opportunities to find stocks with available information not reflected in the price. There's no paradox here: each individual actor is incentivized to participate in order to not miss out on that potential for excess profits, and the net effect is the EMH.