fibonacci's wabbit

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[Daniel Litt @littmath]

In 1974 Deligne proved the last of the four Weil conjectures—an analogue of the Riemann hypothesis in a world where some prime number has been set to zero. While Deligne’s work was celebrated, his doctoral advisor, Grothendieck, was unhappy. Can solving a problem be bad? I think not, but I want to explain my understanding of the story, which in my view sheds some light on the present moment.

In his memoir, “Récoltes et Semailles” (Reaping and Sowing), Grothendieck explains his approach to problem solving and mathematics more generally: "The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration… the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it... yet it finally surrounds the resistant substance." He compared a precise mathematical question to a nut, which, rather than being cracked open by a chisel, yields at the perfect time to simple hand pressure once it has been soaked in the ocean of mathematical understanding. According to this account of mathematical progress, solutions, though important, are ancillary to the advance of the rising sea of mathematical understanding. At best they measure it to some extent, but the goal is to raise the water level.

Grothendieck had laid out a program to resolve the Weil conjectures: the standard conjectures, which are part of the philosophy of motives. The proof circumvented this program. Deligne characterized Grothendieck's reaction as "he did not care...the proof was a trick" but in fact Grothendieck excoriated Deligne in his memoir, claiming Deligne and others tried to "bury" the theory of motives, and Grothendieck himself. The standard conjectures remain open today.

Deligne's result relied on a lot of work that was not yet well-exposited in the literature, around the theory of étale cohomology. Deligne himself wrote an exposition of this in SGA 4.5, and Illusie contributed additional necessary material in SGA 5. Some have referred to this additional work as "thankless," but it was completed.

While Grothendieck is one of my heroes, and I find his view of mathematics very appealing, I think he was wrong here. Though the standard conjectures are still open, there is an immense amount of work on the theory of motives. People worked very hard to clarify the issues left unclear by Deligne's argument. And crucially, Deligne followed up on his work with a different proof, in "La conjecture de Weil II," that completely revolutionized arithmetic geometry, in many ways going beyond the philosophy of motives as it existed at that time.

I think Grothendieck's view that the first proof Deligne provided was insufficient was, in some sense, correct. But he was wrong about the fact that the theory he so desired, and the attendant quest for understanding, would be abandoned, and that his ideas would be buried. I don't think anything can stop us from trying to understand.

[David Bessis @davidbessis]

10 remarks on AI and math that can be true all at once:
1. The recent progress is stunning and exciting.
2. The Quasi-Riemann Hypothesis is arguably Fields-medal level.
3. Mathematics is being transformed and there is no way back.
4. OpenAI doesn't give a fuck about the future of mathematics.
5. Half-baked unreadable preprints claiming major results can effectively stall mathematical progress.
6. Some mathematicians are responding with naive, overly defensive comments.
7. The social-media wankers framing this as "OpenAI tramples gatekeeping" have no experience of mathematical research.
8. The product of mathematics is clarity and understanding. Not theorems, by themselves.
9. Mathematics is a critical infrastructure of human cognition. The human need for understanding isn't obsolete and it can't be offloaded to machines.
10. We haven't "solved" mathematics and we won't get there before the Sun burns out.

[Carles Sáez @csaez_math]

Fun fact: they were invented in a nazi prisoners' camp, where Leray (an analyist and expert in fluid mechanics) was pretending to be a topologist in order to avoid nazis to make him collaborate with them. He got so into the role that he ended up revolutionizing algebraic topology

[Daniel Litt @littmath]

The vibe on here re: AIxMath is kind of apocalyptic, and what’s true is that highly capable AI models have significant impacts on (non-mathematical!) questions around hiring, journals, etc. But if you just log off in fact the day to day of doing math research is really not so different, except that you can talk to your computer about it and sometimes it’s helpful.

What does this help look like? In practice high quality math research has never looked like “run a for loop through open conjectures and try to resolve them all,” and it still doesn’t. Most interesting work does not have the form of a problem set, but is rather an open-ended exploration of some phenomenon. There are lots of ways talking to one’s computer can help with this, but I don’t feel particularly worried that it will be automated. After all, it’s a process that is only done when I say it is!

Anyway, it’s a great time to be a congenitally calm mathematician.

[fibonacci's wabbit @FibbinWabbit]

The only takeaway I'm getting from this whole AI Math discourse is that a *lot* of people have felt needlessly intimidated by mathematics and a fair number of those folks have harbored a secret humiliation because of it.

[fibonacci's wabbit @FibbinWabbit]

@RevivalCare 🫂 😞

[Daniel Rosiak @DanRosiak]

I keep seeing things like this, and it's important to dismantle. I don't care about this particular post, but am using it as an example.

There is an extremely important distinction between delight in an enlargement of mathematical understanding and delight in the imagined humiliation of mathematicians and what you think they represent.

There is another difference between welcoming a discovery that happens to be humbling and welcoming it because someone is being humbled.

If you've done math for decades, you're used to being humbled and feeling dumb. Being wrong, spending months or years on something that turns out to be nothing like you thought, this is ordinary life when you do math for a living.

The widespread responses to all this ai math that say things like "cope", "get f**ked", "they're just mad because something is better than them" have an uncanny resemblance to classic revenge against distinction, as though someone else's mathematical achievement and passion had been an injury, and its supposed devaluation were a form of redress.

Notice what's nearly always missing from these sorts of responses: any interest in the math. What is interesting about the result? What do we understand now that we didn't before? How, concretely, does this show that now more people can participate? Instead, the first and loudest response is glee at somebody else's perceived loss of standing. This is what "cope", "permanent underclass", and all those sorts of snarky responses mean. Not "now more of us can do math, awesome!", but "now your math does not make you anything special."

This should feel familiar, it's pretty textbook. Nietzsche described this very clearly: someone "who does not possess enough intellect to enjoy it, and just enough education to know this--bored, disgusted, contemptuous of himself." What does this sort of person need? Revenge. This sort of perverse delight in the perceived humiliation of somebody else is a way for them to feel superior to those whose abilities they resent.

Finally, they have "found a standard before which even the intellectually gifted are no better than oneself." Equality before the machine.

Let me be clear, because this is much much larger than math and this battle will keep recurring as ai appears to eat more and more of human life. At its limit, this sort of response is a verdict on humanity itself: collective self-contempt becomes their final source of pride.

This is a war between the will to humanity's self-belittlement and love of humanity and the conviction that such love is one of the foundations of rationality (not its enemy).

If your picture of the world comes from trash romantic accounts of the enlightenment, or their residue absorbed by osmosis, you have probably learned to equate rationality and objectivity with showing that humanity is nothing special. In that fairy-tale, each successive humiliation is seen as another victory for reason.

Get off x and out of your bubble and live a real life. Forge bonds with real, complex human beings. Realize how much you and everything you fight for, everything you desire, your dreams, are bound up with other human beings and our fate. You are living through one of the most interesting times in human history. WAKE UP. It's not cool to be cynical anymore.

Spinoza is a nice antidote to the idiotic romantic fairy-tale. He showed why we do not have to choose between an unsparing understanding of nature as it really is (where we are not exceptional) and an affirmative attitude toward human life and our right to take special interest in it.

Humiliation, especially of humanity itself, does not arise from reason. If you correctly understand something more powerful than ourselves, it helps you understand ourselves more clearly, and assists our power of acting. It even helps us love ourselves more, which is a requirement of reason. It doesn't lead to reveling in our own lack of power.

Understanding our limits is not the same thing as learning to despise ourselves, and there is nothing rational or objective about delight in perceived humiliation of people. And contempt for human beings is not a qualification for deciding what should happen to them.

This is about power, in the end. Do we still have any power to decide how this is all going to turn out? Of course we do.

If you are feeling defeated by all this stuff, try to remember what an extraordinary gift so much of this can be. There's so much you can now do with all of this to enhance your powers and share those powers with others. For every perceived demonstration of our limitations, 50 other limitations have been lifted. Wake up with a burning desire to explore those. Discovering that we are less powerful than we supposed (at least in certain areas) does not have to make us less capable than we were.

And if people at the top of their field, who have lived and breathed math for their entire lives, who feel connected to it in a way you probably do not fathom and who have accomplished extraordinary things, are attempting to express a serious misalignment between a new technology and their practice, it makes sense to listen, even if it's not always expressed in your preferred way.

If you are one of the "cope" people or think this is just about humbling some people and it's your job to point out what you take to be their humiliation, know that that is about YOU. And take comfort in the following: whatever bulldozer you think you see coming, it's coming for you too; and, if you had your way, for all things human. But YOU knew that already, you want that, and you may even believe that shepherding it in helps you signal your rationality. But is is just the will to self-destruction, which is always irrational.

Nobody serious judges a mathematical result by whether it flatter humanity. But whether certain developments are desirable for human beings, this cannot be settled by showing how it punctures a flattering belief.

If you are a mathematician, you need to be very careful right now to not internalize someone else's resentment into our own self-contempt, and let it diminish our powers. Seize the parts of those tools that genuinely enlarge what we can understand, create, and share and go all in on that. Ignore the resentment and extinctionist rhetoric. The goal of all that is to make you feel powerless, and there is no ultimate goal of those people other than to bring about the end of humanity.

Rather than rage against the people who take glee in pointing to these new powers as evidence that you have none, ignore them, and rage to use these newfound powers to enhance our powers and humanity.

One last thing: if you hate humanity, if the first thing you see in all the debate these new discoveries creates is a chance to take delight in humbling somebody else or a group of people, if you are compelled to say things like “cope” when someone expresses concern about where some developments seem to be going, if yu are a nepobaby who has never experienced anything real of life or suffering or the depth and stunning expressiveness of human beings, KEEP YOUR HANDS AWAY FROM ALIGNMENT and any discussions of what we're supposed to do with all this.

[David Bessis @davidbessis]

On top of the old arguments, here's a new one: mathematical training makes you a much stronger LLM user, even when the topic has no math in it.

[Greg Egan @gregeganSF]

Breaking: a team of human mathematicians working in secret over the last 3 years, and at least two frontier AI labs with 7-figure budgets, are believed to be neck-and-neck in a race to determine whether the Birch-Swinnerton-Dyer conjecture was posed by three people, or just two.

[fibonacci's wabbit @FibbinWabbit]

people that are going around promoting that ai has solved mathematics are perfect examples of the saying - it is better that people suspect that you are stupid then opening your mouth and proving it

[fibonacci's wabbit @FibbinWabbit]

@it_is_fareed tim ferris implemented

[fibonacci's wabbit @FibbinWabbit]

and we worry about agents cheating

[Mr Family Office @MrFamilyOffice]

An Excel horror show

[Isaac King 🔎 @IsaacKing314]

I recently came across an excellent dive into Qantas Flight 32 and the investigation into what caused the accident.

* An engine exploded
* Because a drive arm broke, allowing a turbine disk to overspeed
* Because a fire burned through the drive arm
* Because an oil pipe cracked, releasing hot oil that caught fire
* Because the wall of the oil pipe was too thin
* Because it had been misaligned by half a millimeter in the machine that manufactured it
* Because the manufacturing process required a realignment of certain clamps partway through the machining, and the part could shift during this process
* Because the correct part position had not been specified in the manufacturing documents
* Because the original manufacturing design had prevented this problem, and thus didn't need to specify a required measurement, and then the process was tweaked without catching this
* Because the product inspections were performed against the manufacturing drawings rather than the design drawings, thus no discrepancy was noticed
* Because there was "widespread procedural non-compliance" among the employees at the facility
* Because the company did not act on an internal review from 2 years earlier which had found that the facility lacked a “strong focus on quality within the business".

This is the level of investigation into root causes that you get when the people involved are competent and making a serious attempt to keep the public safe.

And the original accident didn't hurt a single person!

https://t.co/ySfqtUwd20

[Negro Libre @NegroLibre1]

This is also why I've been so persistent in saying anti-institutionalism is the biggest theat to American politics today, bc it transcend left vs right, socialism vs capitalism and any other ideological conflict, since if there's no trust, what difference does it make?

[Negro Libre @NegroLibre1]

7. It's also why you have to look at Covid, the Affordability Crisis and every other thing that drives Americans nuts about their standard of living as continuations of the GFC that we've never gotten over, on either side of aisle. And that happened nearly 20 yrs ago.

[fibonacci's wabbit @FibbinWabbit]

This outcome is not about ideology but about concentration of power and the rigidity that ensues.

[Izabella Kaminska @izakaminska]

The scam economy I keep going on about is no joke. It risks bringing down Western Civilisation.

Next time you have to complain to a company rep about anything, just watch how they react. Just watch how they have been trained to overreact to any minor expression of emotion.

Any complaint about core service is immediately twisted by them as representing a personal attack on them. It's insane. I am FED UP with having to say to people: “I appreciate you don’t make the policy, and I appreciate it’s not up to you personally, but at some point, somewhere, someone has to take responsibility for this.”

Because nobody you actually speak to appears to have any authority. What is the point of employing armies of people on the customer-service frontline if none of them is permitted to exercise judgment about anything? Every decision requires a manager. The manager, invariably, requires another manager. That manager must presumably consult a committee, which will refer the matter to the appropriate directorate, where somebody will discover that, regrettably, the relevant official is at lunch.

At the rate we’re going, @AndyBurnham himself will have to be summoned to adjudicate whether Mrs Miggins is entitled to a refund on her tram ticket.

What this represents is a complete collapse of corporate efficiency. In fact, what it reminds me of is.... COMMUNIST POLAND.

This is exactly how the world worked back then. I know, because I was there.

There was no meaningful avenue for an ordinary person to appeal anything. The system was unfair, arbitrary, and routinely out to fuck you. And, time and again, the fucking was justified by solemn reference to some abstract rulebook.

“Oh, I’m terribly sorry, madam, I can’t possibly stamp this. You’ve written two millimetres over the line.”

“Can I have another form?”

“No. We’ve run out.”

“When will you have more?”

“Tuesday.”

“It is Tuesday.”

“Yes, but now it’s lunch. Next!”

In fact, life soon became a giant game of how to hack and outsmart the system. They threw the rulebook at you; you threw the rulebook back at them. It forced ordinary people to develop the sort of command of administrative law normally associated with tax attorneys. “Towarzysz, if you consult Section 45, paragraph 4, subparagraph 8(b), you’ll find that as the registered occupant of a two-room dwelling with one dependent, I am in fact entitled to a second saucepan.”

They wielded the power of their monopolies; we found ways around them.

Their response was to change the rulebook and close the loophole. Ours was to find another loophole. They patched the system; we hacked their amendment. They built another wall; we found another way around it. Life became a permanent contest between “Them” and “Us.”

“Them” were the drones who enabled the system’s tyrannical assault on common sense by faithfully administering its absurdities in return for an increasingly lousy — but reassuringly secure — paycheck.

“Us” were the wily normies who just wanted to get shit done.

Eventually we discovered the obvious solution: if the official economy couldn’t reliably provide goods and services, we’d create an unofficial one that could. On the sly, naturally.

This is how black markets were born.

AND THIS IS WHERE WE ARE NOW.

[fibonacci's wabbit @FibbinWabbit]

@CollinWilkins7 @ashpreetbedi This is to be expected when extracting value from AI is viewed as an individual edge. Also, the tech strongly promotes siloing and idiosyncratic building. I've learned the hard way *not* to share what I've learned.

[Daniel Litt @littmath]

One thing I find obnoxious about “hitting a ceiling in math” discourse is that it’s mostly based on people’s self-reports of having found some topic too hard. But my experience teaching math is that people are often wrong when they make such claims!

[fibonacci's wabbit @FibbinWabbit]

@halvarflake @fchollet Thank you! I was looking for exactly this - the Grothendieck way

[Zara Zhang @zarazhangrui]

Don't use AI for writing until you develop your own taste and voice

Using AI to write isn't inherently bad. The danger is using AI to write before you've developed your taste for what is good content. If the AI produces slop, you won't even recognize it as slop

Read a lot to figure out what good looks like. Write a lot to know what your voice sounds like. Only then, use AI to help you write, but make sure it actually sounds like you