joigus

Ok. I finally really understood!! Somehow the proof I'm providing seems ugly to me. I'm sure there must be a simpler, more beautiful one. Let me re-state the answer with some formal embellishments. I'm sorry that I'm such a stickler for formalism: I hope I didn't make any silly mistake with the LaTeX. The possibility that p=2 is a silly one, because the theorem must be valid for all n. One can use many ideas to rule that one out.

More than just interesting. I think the point, is a fair one. The learning curve of new theories is generally steeper than the one corresponding to old theories. In the case of physics, foundational or not, students generally have to master sophisticated calculational tools in order to tackle the simplest problems of the most modern theories.

Agreed. There's some prevailing wind acting there. Social status is obviously a factor too. I don't expect the next Einstein to emerge from among the homeless either. What's peculiar is that people working in the IAS, or PIRSA, etc, working under a common umbrella, with all the facilitation that money can buy to fertilise each other's minds (never mind gender or race) seem to have been unable to parallel the high standards of creativity that lone individuals reached in past decades. Noether was one of these geniuses too, she was a woman, and didn't have it easy either. One would think that putting together an Einstein and a Noether, and throwing in a Fermi, all teleconferencing each other at the touch of a button, would have achieved much more much more quickly. It reminds me of the splitting of mayonnaise: If you put too much oil too soon, it fails to emulsify.

That was your first mistake. ChatGPT, as far as I can see, can only mimic human thinking based on previous patterns of human thinking that it's been fed. No wonder it led you to parallel dimensions, which is not a new idea, was tried for more than gravitation, and AFAIK belongs in the junkyard of discarded ideas. Thought-generating patterns must be fed something external. That something is experiment.

After all, an actuary for an insurance company is bound to have a more promising future (from an economic POV) than someone that spends time thinking about the collapse of the wave function. People who think about such things are usually in well-funded institutions, and it would conceivably be difficult to step out of line and initiate a whole new way of thinking. Well, yes I assume. It's not like you're gonna make this big discovery just when the Inquisition officials are pounding on your door. Otherwise you might end up like Archimedes in Syracuse. I'm sure the feel of urgency gets to you much earlier than the actual soldiers pounding on your door! Evariste Galois is another good example of what I'm saying. I'm well aware of the possibility of a cognitive bias. There are counterexamples, as @studiot pointed out.

The story of my life when it comes to women. No, you're right, of course. It could be the victim of a cognitive bias. I'm aware of that. That's why I was appealing to a good sounding board.

Oh, I'm not very good at number theory. I would have to review some related stuff, like the Euclidean algorithm, divisibility criteria, etc, and see if some of the central ideas illuminate me. But let me try to trim the statement a little bit. Is it: Find a polynomial f(n) with 0<deg(f)<infinity, such that f(n) is prime for all n in the integers? Of course, it'd be the integers. Sorry for mentioning the rationals. Something doesn't seem right, unless you set a bound for n.

Yeah, something like that. It's the sense of urgency that troubled times potentially give the right kind of people. I'm not suggesting that a third world war is necessary, of course. Yes, that's true. This is compatible with what I say, rather than contradictory. Of your other argument I'm not so convinced, as it may well be that you need another indipendent condition to fuel the causal drive. Example: Maybe those human groups did not have the proper cultural seeds planted. Let's say at that particular time they were planting other cultural seeds.

With all due respect, Galileo, Newton and Einstein are soooo in a different league! IMO, theory became orders of magnitude harder after WWII, so it's perhaps unfair to compare him... I sometimes think of Feynman as similar to Newton (developing of a new calculus, mathematical tools of the highest order), but he didn't get lucky in formulating new laws. The closest he got (according to his own admission) is the V-A Lagrangian for weak interactions.

Ok. I suppose that makes Einstein the exception. He thrived during an uneventful spell in Bern. Didn't suffer any turmoil. Went from heaven to heaven. Take Schrödinger, or others in the 20's-30's. My argument stands. And please focus on the big picture of it. Does social/political turmoil (paradoxically and positively) influence the highest creativity?

Not GR, though.

I'm taking you up on this mentioning science by its big names... I personally can't think of any valid reason why nobody has appeared in the last 100 years to fill the shoes of the Newtons, Galileos and Einsteins of yore. Being full of admiration for such people myself, I must say I don't think there's anything supercalifragilisticexpialidocious in the appearance of certain types of individuals. Those able to solve especially challenging problems that take some highest-order, highly-conceptual sorting out. Statistically, it must just happen every so and so. More frequently so the more humans are born. So why hasn't it happened? Here's a frightening idea: Newton had to take refuge in Woolsthorpe after the Great Plague of 1665, Einstein had to take refuge in America after the Nazi persecution of Jews, and Galileo suffered persecution from the Roman Inquisition due to his non-orthodox views on astronomy. Could it be that we need more... --dare I say it?-- persecution? Of course, I'm just trying to be constructively provocative. My reflective point being that there could be something about times of great strife* that brings the best of human beings and make really top-notch ideas be born. Does any of that make sense? * In a milieu that has the proper cultural seeds planted, of course.

Exactly. The most significant critical experiments are far fewer and farther between. This gives more than enough time for people to fall into what I would call theoretical "bubbles" of (perhaps) unnecessary hidden assumptions. You could call this "theoretical daydreaming". After a while a dangerous area of misleading terrain forms. People involved embrace theoretically very promising, very interesting ideas, but with a lot of semi-digested ancillary junk that maybe shouldn't be there in the first place. Good topic. I did notice Hossenfelder's pessimism too.

That's the beauty part, right? 😬

When is an insect bite not best avoided? Thanks for the scale information. Do you happen to know the species?

Spotting obvious solutions first can sometimes be very helpful to obtain the non-obvious ones when the equation is polynomial and has degree > 4. Example: x5=x What would you do?

You mean two sevenths of a pie are different from the 2/7 of the other portion? Oh, OK, I get it. Children appear to be able to count the exact number of atoms in a portion of pie.

Oh, that's so Italian! No, com'on, those recollections I'll keep to myself. But you sent me down memory lane now.

Thanks for the very interesting historical note. I didn't know it went by the name of 'hexagonal aether'. I do remember Feynman's Lectures on Physics mentioning it along the lines of 'Maxwell conceived of an aether made up of gears' or something like that.

Friends fairly sharing pizza has always worked for me. I seem to remember some explanation with pie when I was a kid. It's possible my brain has edited my memories and it was all more formal than I care or dare to remember. Pies work because rational fractions make immediate intuitive sense, IMO. I lost my innocence when I had a teacher at university who said something like 'definitions are not to be understood; definitions are definitions!' which, let me say, I think is completely wrong. Definitions should be motivated. For this man definitions were like a thunderbolt from mathematical heaven. Marcus du Sautoy explains Egyptian, Babilonian, Indian and Chinese mathematics with beans, and peas, and eggs, and things like that, in a wonderful documentary about the history of mathematics.

Exactly. And it would share cosmological equation of state with ordinary matter. It would be another garden-variety type of matter to be detected in scattering experiments. Why bother with superconductivity if the idea doesn't even leave the ground?

Yeah, we got it. To boldly think what no one has ever thought.

One problem with this configuration is that it is anisotropic. And what keeps the spheres from collapsing due to electrostatic attraction? You need a constraint, like rigidities in mechanical problems. Or guess what... quantum mechanics. Dislocations, or kinks, or twists in this grid would not propagate equally in every direction, I think. You should also be aware that field theory does all of this much more simply and elegantly. Plus it gives the right predictions. And that people have tried for ages to build something in the way of a mechanical model, without success...

has a better chance? Yes, it seems to go in the direction of helping kids understand positive integer multiples of (-1) as sequential subtractions. It's very much in the spirit of what @John Cuthber said.