196 karma · joined September 1, 2023
https://northdakotastate-ndus.nbsstore.net/mathematics-genea...
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.
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`).
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.
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.
Mathematicians actually do the same thing as scientists: hypothesis building by extensive investigation of examples. Looking for examples which catch the boundary of established knowledge and try to break existing assumptions, etc. The difference comes after that in the nature of the concluding argument. A scientist performs experiments to validate or refute the hypothesis, establishing scientific proof (a kind of conditional or statistical truth required only to hold up to certain conditions, those upon which the claim was tested). A mathematician finds and writes a proof or creates a counter example.
The failure of logical positivism and the rise of Popperian philosophy is obviously correct that we can't approach that end process in the natural sciences the way we do for maths, but the practical distinction between the subjects is not so clear.
This is all without mention the much tighter coupling between the two modes of investigation at the boundary between maths and science in subjects like theoretical physics. There the line blurs almost completely and a major tool used by genuine physicists is literally purusiing mathematical consistency in their theories. This has been used to tremendous success (GR, Yang-Mills, the weak force) and with some difficulties (string theory).
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Einstein understood all this:
> If, then, it is true that the axiomatic basis of theoretical physics cannot be extracted from experience but must be freely invented, can we ever hope to find the right way? Nay, more, has this right way any existence outside our illusions? Can we hope to be guided safely by experience at all when there exist theories (such as classical mechanics) which to a large extent do justice to experience, without getting to the root of the matter? I answer without hesitation that there is, in my opinion, a right way, and that we are capable of finding it. Our experience hitherto justifies us in believing that nature is the realisation of the simplest conceivable mathematical ideas. I am convinced that we can discover by means of purely mathematical constructions the concepts and the laws connecting them with each other, which furnish the key to the understanding of natural phenomena. Experience may suggest the appropriate mathematical concepts, but they most certainly cannot be deduced from it. Experience remains, of course, the sole criterion of the physical utility of a mathematical construction. But the creative principle resides in mathematics. In a certain sense, therefore, I hold it true that pure thought can grasp reality, as the ancients dreamed. - Albert Einstein
Pure mathematics is regarded as an abstract science, which it is by definition. Arnol'd argued vehemently and much more convincingly for the viewpoint that all mathematics is (and must be) linked to the natural sciences.
>On forums such as Stack Exchange, trained mathematicians may sneer at newcomers who ask for intuitive explanations of mathematical constructs.
Mathematicians use intuition routinely at all levels of investigation. This is captured for example by Tao's famous stages of rigour (https://terrytao.wordpress.com/career-advice/theres-more-to-...). Mathematicians require that their intuition is useful for mathematics: if intuition disagrees with rigour, the intuition must be discarded or modified so that it becomes a sharper, more useful razor. If intuition leads one to believe and pursue false mathematical statements, then it isn't (mathematical) intuition after all. Most beginners in mathematics do not have the knowledge to discern the difference (because mathematics is very subtle) and many experts lack the patience required to help navigate beginners through building (and appreciating the importance of) that intuition.
The next paragraph about how mathematics was closely coupled to reality for most of history and only recently with our understanding of infinite sets became too abstract is not really at all accurate of the history of mathematics. Euclid's Elements is 2300 years old and is presented in a completely abstract way.
The mainstream view in mathematics is that infinite sets, especially ones as pedestrian as the naturals or the reals, are not particularly weird after all. Once one develops the aforementioned mathematical intuition (that is, once one discards the naive, human-centric notion that our intuition about finite things should be the "correct" lens through which to understand infinite things, and instead allows our rigorous understanding of infinite sets to inform our intuition for what to expect) the confusion fades away like a mirage. That process occurs for all abstract parts of mathematics as one comes to appreciate them (expect, possibly, for things like spectral sequences).
As Tao puts it, the value of intuition becomes much higher in the post-rigorous stage once you have sufficiently developed your technical skills.
https://terrytao.wordpress.com/career-advice/theres-more-to-...
Unfortunately a chip (even a legitimately earned one) on ones shoulder about the bad parts of academia doesn't save you from being criticized for being crank-y.
Sabine, like many contrarians, takes advantage of the fact that there are smart and convincing criticisms of many mainstream ideas, and she does her best to rely on those criticisms. However like all contrarians she presents a biased and exaggerated view of things in order to stoke engagement, and unless you are an expert it can be difficult/impossible to determine whether the view she is giving is balanced.
This is a classic issue with string theory critics, because string theory has many legitimate problems with it, but many of the critics are intellectually dishonest and you probably shouldn't listen to their criticisms on principle (but even I must admit it's quite hard to find good quality intellectually honest criticism of string theory which is digestible, so these contrarians tend to be the only loud voice).
In Sabine's case it is not so bad, because it is clear from some of her other positions that she is basically a crank. MOND and superdeterminism are basically crank physics at this point but she supports them purely because she is a contrarian. On this evidence alone you should not trust anything she says on any other subject, otherwise you're falling for a kind of Gell-Mann amnesia.
That is, there are models of Peano arithmetic which contain all of the natural numbers we know and love, and some other ones on top of that and there are some Goodstein sequences using those extra "non-standard" natural numbers which do not terminate at zero.
https://en.wikipedia.org/wiki/Non-standard_model_of_arithmet...
Either what we know about black hole formation is basically complete (it goes gas -> star -> black hole -> accretion + collisions) but the environment in the early universe was sufficiently different/dense that parameters which rule out the formation of supermassive black holes now were different. Maybe there were many intermediate black holes just in the millions of years after the big bang and things were still close enough together that accretion could happen and collisions were "likely" at the rate needed to form SMBHs after just a billion years. If that is true we might expect to see many many active galactic nuclei as we get better telescopes and look further back, depending on how quickly such black holes formed.
The other option is there is a mechanism of black hole formation that bypasses the above chain which we understand. People talk about supermassive stars, gas clouds collapsing directly into black holes, or primordial black holes that existed due to essentially random distributions of density moments after the big bang causing some regions of space to collapse into massive black holes which then persisted. Such things are far more difficult to observe, but could be inferred if we don't see many many active nuclei as we get better telescopes but all other indications of the accuracy of the big bang + inflationary theory hold true.