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There is nothing esoteric about topos theory, it is a formally defined concept: https://en.wikipedia.org/wiki/Topos

Thought it was important to state this, as Grothendieck also wrote rather esoteric texts, Récoltes et Semailles as well as La Clé des Songes, which haven't been translated or even published.

ValentinA23on Rubi: Symbolic integrator based on an extensive system of integration rules

nothing https://www.livescience.com/49610-scientists-unboil-egg.html

ValentinA23on Diffusion models are real-time game engines

What you're pointing at mirrors the same kind of limitation in using LLMs for role-play/interactive fictions.

ValentinA23on Removing stuff is never obvious yet often better

I get what you mean. However other comments raised some valid points too. It's just a few hours of works . I think what matters a lot more is when these kind of anticipated features do not add up to constant gains (which in your case, setting aside the fact that this senior engineer rolled back your tactful additions, would be the time it takes to run the prod DB migrations, since this is what differs between your early and late implementation). More often than not an architectural choice will impact n features orthogonally with respect to other potential features. If it takes you 2 days of work to implement that particular transversal feature, the time it will take you to implement it across all "base" features will be 2n days of work. Stuff like class inheritance will allow you factor that into, say 2 days + 10n minutes. I witnessed disasters because of situations like this where stuff that would have taken me days to implement (or even hours) took 6 months. And the main reason for this is that another team was tasked with doing this, and they didn't know that part of the code base well, which, because everything had to be super "simple" ("easy" would fit better), no class hierarchy, everything super flat, each base feature taking up thousands of lines across several microservices in the hope that anybody could take over, was a slow, dreadly, soul crushing job. Members of the previous team said it would have taken them 2 weeks (because they had a long experience with the spaghetti dish). I re-implemented the program at home in a weekend (took 3 month to code): the required change would have taken me an hour to code, and a fews day of integration to change those n base features (but because I had a leetcode-complex, very expressive architecture, and knew the domain very well). It took the new team 6 months. 6 months ! And I think they only managed to implement one feature, not all of them.

Result: disgruntled senior employees quitted and were replaced with juniors. 3 month later, overtime was distributed across the tech department, killing the laid back 4-days week culture that was put in place to attract talents, the employee unionized, some more quitted, and about a year later, upon failing to hire back these productive elements, the COO was fired.

ValentinA23on A wonderful coincidence or an expected connection: why π² ≈ g

There is relationship between the metric system and the French royal system. The units used in this system have a fibonacci-like relationship where unit n = unit n-1 + unit n-2.

    palm : 7,64 cm
    span : 12,36 cm
    handspan : 20 cm
    foot : 32,36 cm
    cubit: 52,36 cm

cubit/foot =~ 1,618 =~ phi, el famoso Golden ratio. foot/handspan =~ phi too. And so on.

From this it turns out that 1 meter = 1/5 of one handspan = 1/5 x cubit/phi^2

Another way to get at it is to define the cubit as π/6 meters (= 0.52359877559). From this we can tell that

1cbt = π/6m

π meters = 6 cubits

Source: https://martouf.ch/crac/index.php?title=Quine_des_b%C3%A2tis...

ValentinA23on A wonderful coincidence or an expected connection: why π² ≈ g

If I come up with my own measuring unit, let's call it the sneezle (whatever the actual length I assign to it) I will be able to also define a duration unit (say, the snifflebeat) based on the time it takes for a pendulum one sneezle long to complete a full oscillation, and vice versa I can define the sneezle by adjusting the length of a pendulum so that it oscillates in two snifflebeats. Here are the maths:

T = 2π√(l/g)

T/2π = √(l/g)

(T/2π)^2=l/g

g = l/(T/2π)^2

g = l/(T^2/4π^2) = 4π^2xl/T^2

Now substitue T with 2 and l with 1 an you get

g = 4π^2x1/2^2 = π^2

It doesn't matter what the pair of units assigned to T and l are. However, they'll be interrelated.

There is nothing arbitrary, and no coincidences behind g =~ π^2. It just requires to do some history of metrology and some basic maths/physics.

If you want to discuss coincidences, may I suggest you to comment on this remark I made and which hasn't received any attention yet ?

https://news.ycombinator.com/item?id=41209612

ValentinA23on A wonderful coincidence or an expected connection: why π² ≈ g

Only because you're using metric seconds instead of "imperial seconds" (the time it takes for a 1 foot long pendulum to complete a full oscillation).

ValentinA23on A wonderful coincidence or an expected connection: why π² ≈ g

https://www.ukbiblestudents.co.uk/Great%20Pyramid/chapter%20...

>“It was contended," says Dr. Peacock, " by Paucton, in his Mẻtrologie, that the side of the Great Pyramid was the exact 1/500th part of a degree of the meridian, and that the founders of that mighty monument designed it as an imperishable standard of measures of length.

https://www.theguardian.com/science/2020/dec/06/revealed-isa...

>Newton was trying to uncover the unit of measurement used by those constructing the pyramids. He thought it was likely that the ancient Egyptians had been able to measure the Earth and that, by unlocking the cubit of the Great Pyramid, he too would be able to measure the circumference of the Earth.

ValentinA23on A wonderful coincidence or an expected connection: why π² ≈ g

A Martian meter defined by martians should relate their average size, the number of fingers they have on their hands and some basic measure of the planet.

I mean, one meter is defined as 1/10^7 of the distance between the equator and the poles which leads to a round number in base 10.

A unit system is not just something that matches objective reality but something that has some cognitive ergonomy.

ValentinA23on A wonderful coincidence or an expected connection: why π² ≈ g

A bit out of topic, however

https://www.youtube.com/watch?v=0xOGeZt71sg

Note: I'm more inclined to think this is a coincidence given that it establishes a link between the most commented text and the the most commented building in history. However I don't think these kind of relationships based on "magic thought" should be discarded right away just because they are coincidences, and I'd be very interested in an algorithm that automatically finds them.

ValentinA23on A wonderful coincidence or an expected connection: why π² ≈ g

Okay so this one has an explanation. But what about these ?

https://en.wikipedia.org/wiki/Mathematical_coincidence

One french royal cubit ≈ one egyptian cubit ≈ about π/6 meters. One royal span ≈ 1/5 meter = 20cm.

I'm wondering whether some of these coincidences could be explained by the anthropic principle, which deals with these quasi-equalities, for instance:

>An excited state of the 12C nucleus exists a little (0.3193 MeV) above the energy level of 8Be + 4He. This is necessary because the ground state of 12C is 7.3367 MeV below the energy of 8Be + 4He; a 8Be nucleus and a 4He nucleus cannot reasonably fuse directly into a ground-state 12C nucleus. However, 8Be and 4He use the kinetic energy of their collision to fuse into the excited 12C (kinetic energy supplies the additional 0.3193 MeV necessary to reach the excited state), which can then transition to its stable ground state. According to one calculation, the energy level of this excited state must be between about 7.3 MeV and 7.9 MeV to produce sufficient carbon for life to exist, and must be further "fine-tuned" to between 7.596 MeV and 7.716 MeV in order to produce the abundant level of 12C observed in nature.

Source: https://en.wikipedia.org/wiki/Triple-alpha_process#Improbabi...

The idea goes like this:

1. A more fundamental aspect under the anthropic principle which underpins the existence of complex life and intelligent observers is the quasi-alignment of values such as the fundamental constants in physics within a short margin.

2. If you consider the universe to be the product of a random sampling process over these constants (either real or virtual, it occurred many times or just once), and given the fact we exist, which implies an abundance of coincidences, the maths seem to tell us that we should expect to observe superfluous coincidences that are non-functional for the appearance of complex life, rather than the strictly minimal set of functional coincidences necessary for its emergence.

3. This implies that coincidences and pattern seeking are not just features (or bugs) of our complex minds but are present in the universe latently since it is not just fine-tuned for the emergence of complex life but for the presence of coincidences such as these https://medium.com/@sahil50/a-large-numbers-coincidence-299c....

4. It may be even testable by running computer experiments relying on genetic programming/symbolic regression to see whether there is something special about the value of physical constants in our universe when compared to the value they would have in other universes. I think such experiments should factor the fact that not all equations with the same mathematical complexity (number of operands and operators) have the same cognitive complexity. Indeed, if you look at the big equation in the link above, you'll remark that it can be further compressed into a/b = c/d (where a is the photon redshift radius for instance). So I guess you'd also have to throw into the mix Kolmogorov algorithmic complexity to assess this aspect (which is in fact used in some cognitive theories of relevance to tackle this kind of stuff to the tune of "simpler to describe than to generate")

Thoughts ?