Considering how essential math and science is for the prosperity of mankind (not even speaking about the cultural value) the question of how to reward people working and contributing in these fields effectively and appropriately is of extreme importance. (And I think the current decline in our societies is to no small degree caused also by our utter failure to address that issue.)

It is also fascinating, because I don't think there is any solution within our existing system, at least not any I know of. Theorem ownership is not a good solution (and neither are patents in general). Probably the most achievable (or rather the least unachievable) solution is a kind of communist utopia, where people can dedicate their time to a pursuit of any endeavor they see fit, as resources for a decent life are abundant and excessive power capture impossible. (The other option, somewhat dystopian, and which would not require humanity to change too much in its current mode of conduct, would be a totalitarian or caste-like capture of society by the scientific community.)

Incidentally, if AI proves as powerful as some expect it to become, it could bring about another solution of that issue by making all human science and mathematics obsolete, pushing its true market value to zero.

(With apologies for rambling.)

Almost every country in the world has a patent system for a reason...and they are pretty essential for big pharma to function. Granted there should be better calibration of duration - particularly in software, but rewarding first movers is wise in many market conditions. My point regarding "math as a hash list of theorems" is that Theorems will never lose their value, regardless of how powerful AI becomes; they represent compressed knowledge, and as such it will be more efficient for a more advanced AI to query known theorems than reconstruct each one from scratch - in that gap lies a marginal compute saving; an api service or tool call that someone might charge for. Even AI can benefit from specialists...

Good luck!

We are at the point where the way in which humans do math and science changes significantly, and I have no good idea at all in what state is it going to settle down. But you are one of (many, I suppose) people exploring the new wilderness, so I wish you best.

If we get AI singularity, then humans will stop doing scientific progress altogether. But if we don't, then it's pretty predictable what's going to happen - things will be much the same as now, except everyone will be using AI for proofs, data analysis, theoretical models and designing experiments, so important discoveries will happen more often. It's also possible that after the AI craze dies down, we'll have enough computational capacity to solve protein folding.

Yes, I think so. I remember Eric Cline mentioning this hypothesis in a lecture or a podcast about a decade ago.

Sea people as refugees displaced by drought, famine, war or just a general societal collapse.

In any case, the temporal coincidence of the bronze age collapse and the wide-spread adoption of iron working, which was inferior to bronze at first but more readily accessible, is fascinating*. I am not sure if we have a full picture of the role of that change yet. (Was it just a consequence of the collapse of trade and tin imports, or did it cause the destruction of the bronze age empires in the first place, or just prevent their return, or was it more complicated?)

(*) I do not mean the superficial tautology of "iron age" following the "bronze age" but the fact that the bronze age, in this region, ended in a societal collapse.

Exactly. And I am appalled by the level of injustice and stupidity (I will explain both) demonstrated by so many comments here.

Injustice, because they refuse to see the other side of the coin, or the full picture, and a decision ignoring the full picture can be just only by accident (at least in common moral systems).

Stupidity, because it betrays a lack of touch with reality. In this regard I am reminded of a recent thread here, where an article about artificial meat was being discussed. An author of the article expressed their uncomfortable feeling when eating the artificial meat, caused by their thought about the unnatural, "lab" origin of that substance. If that article was sincere, that person have probably never visited a slaughterhouse, or killed an animal by their own hands in order to eat it. This kind of cessation of contact with underlying reality behind our everyday experience is quite common in our modern societies.

In order to be succinct, let me just say that historically it was common for elderly people to live in multigenerational households, sharing their space with a younger family and helping (as long as they could) with chores and taking care of children. While that is not the only viable way, and I am in no way against the modern pension system in general (although its current implementations in most if not all western societies have their problems; and also make many people to lose "touch with reality" in the way I alluded to above), it was quite a natural way of handling the issue of old age that reflected constraints under which a human society existed (and still continues to exist today).

Since I feel that the dominant sentiment of the commenters here is based on emotions and "morality" in a way that I believe is inherently evil, I will close this comment with a refrain from emotions and morality, and state a cold-hearted truth: A society that discards its old people might be a bad one to live in but might be able to survive. A society that discards its young people is destined to die. [To avoid misunderstanding: I do not argue that we should discard old people.]

You are mistaken on several grounds, but the most glaring issue is that you do not properly understand the concept of property/ownership.

For all practical purposes property/ownership is a service provided by society, not a static "attribute" of stuff (the only exception is when you live entirely without any sort of contact with other people, in which case, however, the concept of property becomes void, at least practically). And any service carries a cost with it.

For instance, in our kind of societies, the institution of property requires, at the very least: (1) legislature that defines what property is and all the related rules, (2) the administrative part that handles property transfers and other issues, and keeps the record of who own what, (3) the judiciary system that interprets the legislation and resolves disagreements and other issues related to property, (4) the police or some other body that enforces and protects the whole system within the society, (5) the army that protect the property of the society against the claims of external groups. In our societies many of these institutions carry also other functions, but enabling the system of ownership is an important part of their job. And this service is very expensive --- so not paying any taxes on property means that those costs must be covered by some other sources (e.g. selling natural resources, income taxes, ...) and, usually, lead to a transfer of wealth within the society.

So the idea that you just "own" something (your house for instance) and any kind costs of that imposed on you by the society/government are unjust is completely and patently misguided. If anything, the opposite it true.

(That does not mean that the tax system must be necessarily constructed to reflect that, as there is no single and obvious correct way to design a tax system, and other important considerations enter that issue.)

An observation from within the system:

A year or so ago I completed a PhD in theoretical high energy physics. I did that after working for about 5 years outside of academia [the reasons for that were financial issues in my family]. Because of this hiatus (I suppose) and the fact that I had troubles getting reference letters (my master thesis advisor passed away at quite a young age right at the time when I tried to start applying for positions; we actually agreed to meet in person to discuss next steps but the meeting never happened) I had issues getting accepted into PhD programs and I ended up with an offer from a rather weak institution in my home country (which is in EU). I had no other options to choose from (and I could not wait another year) so I accepted. However, I also looked at the publishing record of the team that I was about to join. Their works did not seem stellar (and I did not expect that) but they seemed to publish regularly, on several topics, and with several collaborating institutions.

When I started I quickly realized that the knowledge/expertise level was much lower than I expected (and I did not expect too much). The group consisted of the group leader and three senior researchers, and I am pretty sure that my knowledge of QFT when I *started* working with them was the best in the group. (Now, I had studied in my own time during some part of the years I was outside of academia, and I think my knowledge at that time exceeded that of an average 1st year hep-theory PhD student in Europe, but it was also nothing spectacular; definitely not what I would call "competent"; which is admittedly a high bar in quantum field theory / hep theory, but something I would expect senior researchers to more or less satisfy.)

I realized that the one topic on which the group was publishing on their own was just a continual rehash of the same thing (applied to various different problems, so perhaps not completely useless) and the other topics, those which seemed more advanced when I originally looked at the publication record, were all done essentially outside the group by other people. The group members contributed with some non-essential help (often resembling a work done by a student, such as finalizing a manuscript or double checking calculations) or with nothing; and were added as coauthors due to some other reasons (I suppose: past loyalties, friend groups, advantage of having foreign or external institution co-authors). [Note: I was not included into any of those collaborations so I am not speaking with a full knowledge of the inner workings there.]

During this time I saw that people being added as coauthors had a very noisy relation to what they did or didn't do on any given paper. I was added as a co-author on a paper that I made essentially no work on [not fair], I was added as a co-author on papers which drew upon some of my earlier results [fair, but I did not subscribe to the overall spirit of the paper or even to the paper being published at all], I was a co-author of papers where number of authors and my positions in the list roughly corresponded to my contribution [as it should be], I was a co-author of papers that were done basically by myself but there were several other co-authors included who made a little to no contribution [and the ordering of the author list was always alphabetical not reflecting the relative contribution].

There is also another aspect of this: If you are an early career researcher (PhD student, PostDoc, and even non-tenured professor) you might have a very limited control over who gets included on the publications you work on, and on what publications your name appears. In my case, I am a very disagreeable person when I think things are being done wrongly and yet I have not managed to refuse from being included on papers I did not want to be a co-author of, or to prevent people who contributed nothing to be included on my papers. [I mean the only way to achieve that meant escalating conflict into levels which (a) would likely disable any further cooperation with the group, and (b) might be even morally questionable concerning the level of distress it would make to other people who just seemed to be fine about "the normal way" things were being done.]

In any case, while there are still some really good people in academia (actually the best people I met in my life were nearly all in academia), in overall (by no means fully indicated in the above paragraphs) I feel very bitter about it, and I think it is so dysfunctional and morally corrupted, that it is nearly inevitable that it collapses in future. The current and future shockwaves of the change in the public sentiment and funding, the evolution of demographics, and the consequences of AI, just hasten the process that would have probably unfolded, sooner or later, anyway.

I wholeheartedly sympathize with this experience

Nitpicking: It is not possible to derive the theory of relativity from medieval knowledge. At least not in any way resembling how the development actually happened, that is, heavily influenced by observations (or rather an iteration of speculation and observation). [Unless one counts both Galilean relativity and some parts of the theory of electromagnetism as being contained in medieval knowledge.]

Perhaps it would be possible if the classically observed reality turned out to be possible (consistent under some reasonable conditions) only as a consequences of some deeper, sufficiently determined theory; but that would go far beyond our current level of knowledge, at least as far as I can say.

The earliest you could reasonably derive special relativity is probably 1881 with the first version of the Michelson–Morley experiment, which showed that speed of light is the same no matter how you move through space. Which, on the scale of discoveries, is a pretty short timeline to the 1905 publication of special relativity.

Which doesn't really stop you from training an LLM on 1904 knowledge and having it derive special relativity. LLMs trained with knowledge cutoffs that far back is a fun exercise, and one I've also dabbled in in the past. But you don't have anywhere near enough training data to reach SotA levels of intelligence, even if you had the money for those training runs. And lobotomizing all modern information out of LLMs doesn't seem viable either. So I don't see how you could ever turn that into a viable benchmark

This is something of an aside on your first paragraph and lh712's. The theory of special relativity (and field theory in general) can be "derived" (as done in e.g., Landau, Lifshitz _Classical Theory Of Fields_) without empirical observation using concepts known to Galileo/Newton (if they count as "Medieval") - just with different assumptions/ideas about inertial frames. If you assume there is any phenomenon with a fixed observed speed in all "inertial frames" then you get special relativity with Einstein's gestalt-switch. After that it is, like so much in physics, a matter of thinking of an experiment to distinguish what matches capital-N Nature best.

Thinking otherwise mistakes a fundamental error that the only way for something to happen is how it historically happened. Reconciling observations forced relativity historically, but it could have been sussed out without those. That there were very fast but finite speed phenomena (which could motivate relativity reducing to Newtonian models) was seen in 1676 by Rømer with the speed of light and eclipses of Io, a Galilean moon of Jupiter. It probably could have been done with Galileo's telescopes in 1610. (That is just one example of a phenomenon with a very fast yet finite speed to suggest other experiments to test that "hypothetical relativity".)

TBH, this all relates to teaching "physics without calculus" and that sort of thing. Is the most clear presentation/derivation that which mirrors the history of our muddled yet ever demuddling ideas or that which starts from our most thoroughly demuddled ideas, best notations, etc.? The momentum is certainly the historical approach, yet there are notable exceptions.

Well, exactly, and I think that that is in line with what I originally wrote, and which is that you need the general relativity principle (by general I don't mean the general relativity but the concept of the equivalence of inertial frames) and the theory electromagnetism. (By the way, I am familiar with the Landau--Lifshitz textbook.)

(1) The relativity principle is exactly what you refer to yourself. You can't apply "Landau--Lifshitz"-like arguments without it. (And I don't think these principles count as medieval knowledge.)

(2) I mentioned electromagnetism, because you need some clue for the concept of an absolute speed that is same for all inertial observers. This is very counterintuitive from our everyday experience, and counterintuitive from the point of view of somebody living in the time of "Galilean" or Newtonian mechanics. Theory of electromagnetism is the only thing that I am aware of, that is nearly (by a stretch) accessible at the level of "medieval" knowledge, from which the concept of constant speed follows. (Historically: Maxwell's completion of previously inconsistent equations of electromagnetism yielded a wave solution propagating with the constant speed of light. This was interpreted in terms of the ether originally, but it is a strong hint in itself for Einsteinian principle of relativity; and a reasoning along the lines you alluded to can them be applied, at least in principle.)

Yeah. My point was mostly to expand upon your original post - sorry if it sounded like I disagreed. You say the same in your original "iteration of speculation and observation". Everything else is about what "derive" and "knowledge" might mean (I agree "medieval" usually means pre-Renaissance and Galileo is modern-era), how much empirical "proof" is in "derive", etc. However, all you need for "possible" is "the idea", some "consequences", and ways to test.

So, to push back a tad on your more recent "only thing that I am aware of" and to maybe explain my above point better, I do think there was enough information/ideas in the abstract in Galileo/Newton/Leibniz' times to suggest the idea. Leibniz himself pushed back hard on Newton's absolute space/time (long before Mach). For Leibniz, it would have been counterintuitive space/time vs. counterintuitive fixed speed. So, that pushback itself could have been enough of a "clue" in your terms -- in some alternate timeline -- to drive a speculation-observation cycle starting from different inertial concepts - with Galilean moon eclipses then enough a clue that fast speeds existed to fool our slow-speed intuitions. (And all this in the "modern era".)

So, I continue to think it "not impossible" that the relativity principle could have arisen before any EM theory at all -- it just didn't. That matters for these kinds of speculative questions about what information horizons support what developments. A single "fast enough" fixed speed is is not that wild & crazy. The modern world has a zillion obscure physics theories like that (mostly just because so many more people work on that stuff, but that's a probability thing, not a possibility thing, and I suppose also partly inspired by how physics turned out - so not truly independent). History is littered with things that could have happened, but didn't.

P.S.: and apologies for "mistakes a fundamental error". I of course meant "makes a", if you wanted any evidence that I was not an LLM. ;-)

Thank you for your reply! Yes, I fully agree that meaning of "derive" and other related concepts are non-trivial. (And I actually changed my own "working definition" between my two posts; while in the first one I was leaning more towards theories anchored by observation, even if indirectly, in the second one I slipped into more mathematical/philosophical perspective. The second part of my first reply makes sense only from the first kind of perspective, because in that part I imagine that the observed reality, even at the medieval level of knowledge, might perhaps necessitate a particular kind of underlying theory, from which the relativity principle would then "fall out". That was the only way I could think of at that moment how SR could have been "derived" without further observations.)

Your points about Leibniz are very interesting, and while I find that line of thought fascinating I must admit my own lack of knowledge of the historical context. In any case, thank you for pointing out that idea!

You're welcome.

Another lesser known wrinkle along these lines is that if more of Aristarchus of Samos' work on heliocentricity had been developed into a "calculational framework" for planetary motion by a contemporaneous Ptolemy competitor (a la Copernicus in 1543), various relativity ideas might well have started in 270BC instead of 1600 AD.

Aristarchus was already WAY ahead of a few games - making a guess that "absolute rest" was illusory and tiny stars were similar to giant Sol. If not for Archimedes' serious star power, we might not even know of Aristarchus' heliocentric ideas! Relativity ideas of various kinds are short intellectual hops from "the absolute rest of your intuition is illusory".

Aristarchus and his supporters just had to assert The Stars were very distant -- far enough for there to be NO PARALLAX! Various "coincidences" all conspired to suppress Aristarchus' plausibility, like: A) Just HOW DISTANT stars are/local galactic stellar density &| B) poor human VISUAL ACUITY relative to C) Earth ORBIT vs. Sun LUMINOSITY (parallax baseline) & maybe existence of Moon to vent atmo &| D) VERY SLOW (3 millennium) development from near prehistoric glass (1500BC) to grinding lenses for human eyes in the 1300s AD leading to Galileo's telescopes & etc. If ANY of (A)..(D) were about ~10x better (ALL of which are imaginably so), the day that universe changed could've been much earlier. Heck, even if Aristarchus just had a really good spin doctor like a major religion pushing the plausibility of "Sun=A close Star", that might've been enough.

Archimedes almost invented the underlying ideas of integration and limits and all that, too. Close but no cigar, but (had that been in hand) analytic geometry and differential equations are short steps away. I mean, Newton surely noted the equivalence of gravitational and inertial "mass" (charge vs. kinematics) which is the weak principle of equivalence of GR, after all. Like I said - "littered". ;-) There are probably whole books written on the topic (or adjacent topics) of "All the Things Humanity Nearly Figured Out Earlier" that have a more historical bent than the usual "sci-fi tilt".