I feel like this sentence is in every article for a reason. Thank goodness there are such obsessive people and here's a toast to those counter-factuals that never get mentioned.

  > I feel like this sentence is in every article for a reason.

I think we do a lot of disservice by dismissing the role of the dark horses. They are necessary. Like you suggest, there are many that fail, probably most do. But considering the impact, even just a small percentage succeeding warrants significant encouragement. Yet we often act in reverse, we discourage going against the grain. Often with reasons about fear of failure. In research, most things fail. But the only real failure is the ones you don't learn from (currently it is very hard to publish negative results. Resulting it not even being attempted. The system encourages "safe" research, which by its nature, can only be incremental. Fine, we want this, but it's ironic considering how many works get rejected due to "lack of novelty")

This is wrong. It's not inherent in the meaning of the word "breakthrough" that a breakthrough can occur only when someone has gone against the grain, and there are countless breakthroughs that have not gone against the grain. See: the four-minute mile; the Manhattan Project; the sequencing of the human genome; the decipherment of Linear B; research into protein folding. These breakthroughs have largely been the result of being first to find the solution to the problem or cross the theshold. That's it. That doesn't mean the people who managed to do that were working against the grain.

> Yet we often act in reverse, we discourage going against the grain. Often with reasons about fear of failure.

I don't know which "we" you're referring to, but just about everybody would agree with the statement that it's good to think creatively, experiment, and pursue either new lines of inquiry or old lines in new ways, so, again, your claim seems clearly wrong.

If you're discussing just scientific research, though, sure, there are plenty of incentives that encourage labs and PIs to make the safe choice rather than the bold or innovative choice.

Running the 4 minute mile, climbing everest - those are achievements rather than breakthroughs.

I'd also class the atomic bomb as an achievement - it was the expected/desired result of a massive investment program - though no doubt there were many breakthroughs required in order to achieve that result.

Even if we decide that breakthroughs require some kind of discontinuity, break, or, as the comment said, "paradigm shift," such discontinuity isn't necessarily "against the grain," as this would imply some kind of resistance to or rejection of "the grain."

Words can indeed mean multiple things. Their meanings aren't infinitely flexible. You wrote that the word (or concept?) "by definition" means some specific thing or must meet some specific requirement. You defined it. You didn't imply that you were relying on some specific meaning that happens to be relevant to your point. To the contrary, you wrote that the definition you provided is THE meaning -- the ONE meaning -- of the word.

That isn't the meaning of the word. It isn't any of the word's meanings. A breakthrough can "go against the grain." It isn't required to. So I didn't "turn it into something different." I read and responded to exactly what you wrote.

Your broader point, too, is, I think, clearly wrong. It paints a needlessly, inaccurately adversarial, even defensive and persecutory, picture of what you're calling "paradigm shifts" -- work that may not fit into existing lines of inquiry or research. I strongly disagree with you.

> If you understood what I meant then why turn it into something different unless you just want to argue?

Did I want to argue? Not particularly. You made a point. You stated it strongly. Why wouldn't I offer a counterpoint if I disagree? Or why shouldn't I? Moreover, I offered a refinement of your point: I said that your claims make more sense and are less objectionable if we apply them to contemporary scientific research -- research requiring grants and external funding. That's not disagreement. It's also decidedly not the point your comment makes; your post isn't about academic, scientific, or mathematical research. Your point is much, much broader. There's no evidence in your comment that "[you] meant" to make the narrower point that I made. It is literally not the point you made. It's the point I made. I had to supply it for you.

If that were it, I would agree.

But I don't agree. I think people who discourage going against the grain are more fearful of the loss of economic input. It's unproductive to do something you know will fail; it's very expensive to encourage that failure.

They are what Gladwell calls, in "David and Goliath", being unreasonable in the face of so-called "prevailing wisdom".

Personally I think governments should fund more moonshot solo or small team efforts because high risk / high reward pays off when you reduce the variance by spreading it out over so many people. But it looks like we’re going headstrong the other direction in terms of funding in the U.S. right now, so I’m not optimistic.

  > I want financial independence for the sole reason that I can work on interesting problems like this without any outside nagging or funding issues from anyone else

I don't have a great idea for it tbh. I'll pitch a bad one at least to put something out there. Maybe we can convince some billionaire to perform a post scarcity experiment? To buy large chunk of land, gather people who have both high passion and expertise in domains, and let them run wild. Less like Star Trek and more like Eureka[0]. I don't think it is a realistic expectation to get billionaire funding, but I think the idea of looking at something like Eureka (or this category of groups you see in many Sci-Fi stories) is worth drawing inspiration from: effectively post scarce (possible on small scale?), high levels of freedom (very doable), high levels of creativity and expressiveness (this is the experiment, to see if it is decreased/maintained/increased).

[0] https://www.imdb.com/title/tt0796264/

It's amazing to me how much thought and work has gone into the seemingly trivial things we encounter on a daily basis.

I think of this every time I see a blue LED. Or a rice cooker!! So easy to take for granted.

Is the nifty part about the rice cooker the temperature cutoff at 105C? Induction? That my cats turn on my zojirushi three times a day and open the lid and it doesn't harm it because it knows there's nothing in the pot?

One of these days I need to track down who actually got the patent for using IR LED in a ring around a camera lens to see in the dark.

If it was at&t I am gunna be pissed.

  > Just let them be

Just make sure "let them be" applies in both directions

Hans Asperger could only save his Austic children from nazi death camps by convincing the nazis that they had value to produce rockets and bombs.

It’s quite remarkable that the USA is so advanced given how deep and ruthless our anti-intellectualism goes.

This is a wildly inaccurate picture of the average person. I don't even think this is true of 10% of people.

But 90% is just too much to be feasible.

My point was more that its doesn't take a large chunk of a population to mistreat others to make mistreatment almost inescapable.

For child sexual abuse, which is studied more than "anti-nerd bullying" offenders are just a few percent of the population, if that (I can't actually find good sources for population incidence of being an abuser) but the offended against are 1/6 to 1/3 of the population.

Another interesting point is that some nerdy folks might not even notice the bullying that much - I once had a friend from high school say that they always felt bad for how much bullying I experienced, and I said "what? what do you mean?" I felt that many of the folks in high school were terribly evil people, but I didn't get that upset by the bullying itself. Just determined to move as soon as I could. Not like as a strategy to over come the bullying, just to get among people I can respect.

He has been taught to love others since he was born, and the Path of Love has borne fruit for all those around all four of us.

All the people who say it can't be done have never tried consciously evolving with Divine help.

{Complete lyrics} --Sinead O'Connor in Massive Attack's "What Your Soul Sings"

The average person is probably uncomfortable around autistic people because they don't know how to deal with them and, when you see someone like that, it's usually best to avoid interacting with them. Not because autistic people are dangerous but people acting out of the ordinary sometimes are.

The lumping in of nerdy with autistic is ridiculous too. The average person just doesn't care about your interests unless they are in common. Nerdy typically just means having a niche interest or hobby.

Your parent reeks of a persecution complex.

The average zoomer isn't your average person. People are sometimes cruel. The average person is not. The average person didn't vote for Trump.

I also took most issue with grouping a autism with "nerdy". Being a nerd has never been easier.

> It's not a persecution complex to identify the ways that hate gets used, deployed, and repackaged for a new generation.

Okay, but that's not what happened. Cool strawman though.

I imagine it's a lot like my experience of having ADHD and being trapped between wanting leeway and support but also wanting to be held accountable and considered capable of improving and watching the Tiktokification of my disorder force me to argue that actually people with ADHD can do things and no having it doesn't fully excuse you from ever meeting commitments or doing the fucking dishes, especially if you're rejecting all and any treatment or strategy.

your name is reference to Stirner is it not? Hegel's realm of ideality is just that, a realm of ideality. Hegel's totality as well, a spiritual abstraction upon the world, but a good one to deny the Abrahamics and other mental rots of the west. The irony of totality is also, that it resolves back to the self, the process of becoming, the unique.

Dialectical analysis is a never ending, self-reflected, and interpenetrated spiral toward an asymptotic singularity.... but that is exactly what being is. Some X posited/constructed condition is always appearing, but X is not only X, the synthesized revelation of these arise in a new more real truth, along with a final truth that this process leads back, to some condition is always appearing.

I think the difference is^W was that USA celebrated it in a Homer Simpson kind of "Ha ha! NERDS!" way, while meth-Hitler was like "let's sterilize them but try to extract math from them to... (whatever batshit goal)"

Anti-intellectualism seems to be a thing when the intellectual/moderately-competent people have already brought success. (Until then it's more like anti-witchcraft, or whatever...)

That sort of positive support was one of the elements I really liked in working on combinatorial problems. People like Persi Diaconis and D.J.A. Welsh were so nice it makes the whole field seem more inviting.

"Positive vibration, yeah." --Bob Marley

Suppose I'm interested in representing a Group as matrices over the complex numbers. There are usually many ways of doing this. Each one of them has a so-called character, which is like fingerprint of such a representation.

Along another line, it has been known that all groups contain large subgroup having an order which is a power of a prime--call it P. This group in turn has a normalizer in which P is normal--call it N(P).

The surprising thing is that the number of characters of G and of N(P)--which is is only a small part of G--is equal.

*technical note in both cases we exclude representation the degree of which is a multiple of p.

But the article says that mathematicians are now searching for a deeper “structural reason” why the conjecture holds. Now that the result is known to be true, it’s giving more mathematicians the permission to attack it seriously.

For example, article says there are 50 groups of order 72 (chatGPT says there are 50 non-Abelian, 5 Abelian), this seems to be an important insight but into what?

As someone "junior programmer" , I think I should just keep this information . Many times I have tried to not use chatgpt. but its just so lucrative.

I really need better control of myself. Going to block chatgpt at a dns level.

Slightly trickier is that every group is a set of symmetry operations for something. So groups exactly capture the idea of symmetry. To a mathematician, "group" and "symmetries" are synonymous.

Finite groups can be interesting as the symmetries of e.g. molecules (e.g. rotating atoms around onto each other), which can tell you something about molecular structure, energy levels, spectra, bonding potential, etc. Infinite groups appear in physics (e.g. the laws of physics are the same when you rotate or translate your coordinates by arbitrary amounts). Symmetry also comes up as a way to study other mathematical objects, and mathematicians might just want to know what all possible groups look like.

Nobody bothers explaining why the primes are spaced like they are, rather people explain other phenomena by pointing out that certain things about it are prime or not.

For instance, there's this thing about having a nice neat formula for factoring second degree polynomials (the quadratic formula). One also exists for cubics and quartics (though they don't usually have you memorize these) but none exists for quintics. It took mathematicians a while to prove that such a thing doesn't exist (how to prove a negative?) but they managed it by arguing that all such things have an underlying finite group smaller than a certain size and look, we've listed them here, and there's no such group corresponding to a quintic formula, therefore there is no quintic formula.

So they're useful as a sort of primordial complexity that can be referenced without extensive explanation since properties about the small ones can be checked by hand. And as it turns out, quite a lot of things form groups if you bother you look at them that way.

I take it that means you also haven't taken (m)any number theory classes then ;-). Because people wildly care about that.

This is in a sense the background of another well-known conjecture, the Riemann conjecture...

I hope that their relationship deals well with the new reality, now that their principal goal has been achieved.

Unrelatedly, I'd be curious what this couple thinks about using AI tools in formal math problems. Did they use any AI tools in the past 2 years while working on this problem?

Mind you, my math is enough for BSEE. I do have a copy of one of my university professor's go-to work books: The Algebraic Eigenvalue Problem and consult it occasionally and briefly.

Yup, I think stubbornness/perseverance is the most useful transferrable skill I got from doing research math. It's a double-edged sword though as I often just can't give up working on something when I really should in my tech job.

You're really standing on the shoulders of giants when you rely on the classification of finite simple groups.

Giants whose work has (dirty little secret) never truly been verified. The proof totals about 10,000 pages. At the end of the effort to prove it there were lots of very long papers, with a shrinking pool of experts reviewing them. There have been efforts to reprove it with a more easily verified proof, but they've gone nowhere.

Hopefully, the growing ease of formalization will lead to a verification some day. But even optimistically that is still a few years out.

My understanding was that the so called "second generation proof" of the classification of finite simple groups led by Gorenstein, Lyons, Solomon has been progressing slowly but steadily, and only the quasithin case had a significant (but now fixed) hole. Are there other significant gaps that aren't as well known?

Still they have a couple more books of proof left, and I have to wonder how carefully it will be reviewed. This will still be a massive improvement, but I'd be a lot happier if the entire proof could be formalized.

Plus there is still a possibility that there proves to be another significant hole.

If any theorem needs to be formalized, this is the one. No other theorem is this big, this hard to prove, and this important to get right.

I believe this is apocryphal. Watson likely said this because he stole Rosalind Franklin's research.

Mind you he was helped by a nice coincidence. Franklin knew all 230 space symmetry groups, and so had to sort through them. Watson only really knew one - his PhD thesis was on a protein with the same group as DNA.

This is probably something I need to research more, because the questions scientists are dealing with are too deep to ignore the one big question, and I wonder who struggled with it.

It's happened several times that I struggled with a bug for hours, then suddenly came up with a key insight during the commute home (or while taking a shower or whatnot), while not actively thinking about the problem. I can't explain this any other way than some kind of subconscious "brainstorming" taking place.

As a side note, this doesn't mean the hours of conciously struggling with the problem were a waste. I bet that this period of focus on the problem is what allows for the later insights to happen. Whether it's data gathering that alows the insights to happen, or even just giving importance to the problem by focusing on it. Most likely it's both.

For me, the subconscious is the wellspring of sudden insights.

(n.b. historically speaking in the literature, "subconscious" was the word used to describe Jung's "woo-woo" ideas about a collective subconscious shared across populations, and unconscious was the word used for ideas about a single brain a la Freud which is what we are talking about here)