Comment by heyodai
2 hours ago
Medieval astronomers were able to predict the orbits of planets surprisingly accurately, within 1/10th of a degree, even though they were assuming the Earth was the center of the solar system. They invented a very complicated system of deferents and epicycles (circles within circles). It was completely wrong but the outputs were surprisingly close to reality.
If we were using LLMs to analyze astronomy, we might just get increasingly complicated epicycles and never realize that the Sun is the real center of the solar system. We then never learn about the anomalies in Mercury's orbit that led to the theory of relativity.
So I agree. I seriously question how valuable LLMs can be in science/math beyond working as advanced search functions.
The epicycles model represents one arbitrary periodic function as the sum of multiple well understood periodic functions (uniform circular motion, i.e. complex exponentials). This is the same idea we use in a more systematic way today by way of the Fourier series, etc.
It's quite a good method for approximating any arbitrary smooth periodic function.
My best guess is that the originators of this model (perhaps as early as Hipparchus, but at any rate someone by the time of Ptolemy) didn't think there were literally circles being combined, but had a pretty clear idea that they were trying to make an approximate model to fit the data. A previous model, from Eudoxus, was also probably pretty accurate but more cumbersome to compute with, involving approximation of the visible planetary motion by the combination of uniform rotations of some imagined sphere(s) centered on the earth. (We don't know exactly because the books about it don't survive, so all we have are vague descriptions by other people.)
The introduction of LLM in mathematics is no different than the introduction of automatic calculators. Probably the calculators (computers) where even more groundbreaking if you think so (something that was impossible to calculate or simulate it was now possible).
Just don't believe at all the shitty marketing that AI companies are creating to make you believe that they have found the Holy Grail
If Epicycles made good predictions, then they were not "completely wrong". Nor is putting the sun at the center "right" so long as the predictions are still good. It is a matter of taste, not correctness (it is, however, very good taste).
I think you might be underestimating how transformative an "advance search function" could be. If the search engine knows how to design and carry out experiments in order to synthesize new evidence that allows it to ensure that the answers it provides are well supported, then humans are essentially out of the "truth" part of science, leaving them only to advice on the "beauty" parts, which might be ok, but it's a pretty big shift.
It might have been a matter of taste in the 17th century. The Church was willing to accept a model where the sun goes around the earth, and the planets around the sun. This model agreed with observations, and was in good taste as well, as it avoided the problem of why it doesn't feel like we're in motion, and why we don't get flung off the earth.
Then Newton came along and blew all of the models out of the water. Nobody's at the center of the universe [0]. A single theory of gravity models both planetary motion and why we don't get flung off the earth. And we can "feel" the rotation of the earth by experiments which came along later such as Foucault's pendulum and the Coriolis force.
[0] A puzzle I like to taunt my friends with: The earth is in fact equidistant from the edges of the observable universe in all directions, so we must be in the center after all. ;-)
sorry, saying that "it's a matter of taste" whether the earth revolves around the sun is completely ridiculous. making useful predictions is one thing, but actual truth is not a "matter of taste".
They both revolve around their barycenter. So neither one revolving around the other is ground truth. In practice either one can be the right choice (earth at center for video game, sun at center for astronomy predictions).
My point is that the laws of physics work just fine in a wide variety of mathematical framings. It's sensible to pick a framing that keeps the complexity of the equations as low as possible, but you're not incorrect if you pick a frame that puts the moon in the center or any other point. The accuracy of the theory's predictions are completely separate from where it applies labels like "center".
I mean to be completely accurate the Earth doesn't strictly orbit the Sun. Both the Sun and Earth orbit each other around a point at their "combined" center of mass, the barycenter - https://en.wikipedia.org/wiki/Barycenter_(astronomy). This is something you need to take into account for all the planets in the solar system and as a result, the "center" of the whole system is constantly moving. The solar system's barycenter is often not even inside the Sun.
I think that sort of proves the point that the OP was making, which is that you can create a functional model around any arbitrary location and it can be valid (as long as valid means: it can make useful predictions). When you put the center of the Sun at the center of the model, you still need your models to account for the "orbital wobble" happening as a result of the Sun doing a little dance with Jupiter (and all the other masses that orbit it) in the equations for the determining the relative positions of the Sun and Earth.
Using the Sun as the center versus the solar systems barycenter are both somewhat "arbitrary" in that way, and depend on what you're trying to do with the information - how accurate you need it to be, and how complicated you're willing to make the calculations to hit higher accuracies.
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Epicycles weren't untrue, they're just an unnecessarily complex model that can be used to describe what picking a different point of reference can yield as a much simpler model. Not all heliocentric models got everything right just because they picked a simpler reference point for producing a model. Judge a model based on its accuracy, not its point of reference.
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> but actual truth
there is no "actual truth" - there is only what you can measure reliably
https://en.wikipedia.org/wiki/Epistemology
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Precession of perihelion of Mercury was not the motivation of the discovery of general relativity theory. It's only an evidence that convinced why the theory ought be true.
Of course. But we might have found out the sun is the center of the solar system much sooner. Who knows!
I feel like there's a lot of "humans are special" in this thread. There's a good chance we're not.
What you have to believe is that somehow the entire field of math was just leaving open problems on the ground that were actually easily solvable and were essentially free Fields Prizes, tickets to a lifetime of Fame and Academic superstardom. Orr LLMs are actually doing something novel.
> It was completely wrong but the outputs were surprisingly close to reality.
that's not how physics works. epicycles weren't wrong. there is no wrong/right - modern scientists don't believe in platonism. physics is a collection of models which are empirically tested. if epicycles makes predictions to more decimal places than relativity then epicycles are "right" and relativity is "wrong".
now think about how this translates to LLMs...
> physics is a collection of models which are empirically tested. if epicycles makes predictions to more decimal places than relativity then epicycles are "right" and relativity is "wrong".
Most models have relation to each other so it does not suffice to take one in isolation. It's one continuous world (in our knowledge) so almost everything relates to one other. The truthiness of something is not whether it fits some particular phenomena, but how does it generalize.
> Most models have relation to each other so it does not suffice to take one in isolation
You don't know what you're talking about (most people on here do not).
https://en.wikipedia.org/wiki/Ansatz
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