Maybe I didn't do a good job explaining it, but the rest of my prior comment was about connecting AI-generated results back into human-style thinking. Inevitably, in the far future, it's not unreasonable to assume the world will be dominated by synthetic robots controlled by artificial intelligence, and there will indeed be a point where AI builds not just bridges but vast planetary, interplanetary, and space-based infrastructure projects beyond the ability of our current civilization. At that point, mathematics may permanently move beyond the grasp of the human species. You can't teach a dog general relativity. Surely, there are truths in mathematics (and possibly physics) you cannot teach a human. Not to digress, but for me, this kind of threshold is what a term like "superintelligence" means -- the point where an intelligence is discovering truths that cannot be taught back to humans because we're not smart enough. So far, our contact with this kind of intelligence has been limited to one-off, highly specialized cases (like chess) that have little grand implication for civilization, but that won't always be the case.

But, for today and probably at least our lifetime, to make them useful major AI advances in math will need to be "compressed" back into the specific network and "towers" of concepts and abstractions that human minds specifically can understand and intuit about. So I think both directions of formalization are equally important: translating natural language statements (theorems, lemmas, etc.) faithfully into Lean and letting a theorem prover run and decoding a dense Lean proof back into natural language (which, in some ways, is the more creative and open-ended problem -- there is no one right answer).

Probably AI mathematics needs a specially constructed or trained translation or compression system (likely also an AI system) that helps transmit dense Lean proofs back into human-style thinking. We may even see an entire field develop around creating human-comprehensible compressions of vast formal breakthroughs in mathematics. Such an activity would almost certainly be both art and science -- there's some objectivity in that certain abstractions or definitions inherently cover more ground more efficiently, yet there's also a deep creativity and artistry in finding compressions that are adapted to the specific 3+1D spatiotemporal intuition of the human mind. Perhaps with time this will keep a lot of the originality and creativity of research mathematics alive -- maybe with that work having even more centrality than it does today.

Instead of seeing this all as a loss of beauty in mathematics, I choose to see it as the beginning of a new age, which will bring entirely new problems to solve, yet also accelerate discovery at an exponential rate.

My main use cases are Mac Virtual Display, movies/entertainment, PS5 gaming [0], casual browsing, and -- most surprisingly -- reading. The first few are pretty self-explanatory, but reading is one of my favorite unexpected niche use cases. It's really nice having a floating book (via Apple Books) perfectly positioned at eye height in front of you in your favorite virtual environment, listening to music of your choice. This use case didn't really take off for me until the recent dual knit band fixed the comfort issue. I dabbled with reading in the Vision Pro before but the comfort level just wasn't quite there yet. The new band is good enough to make this one of my favorite ways to read today.

[0] I use the Portal app for this. It lets you stream PS5 games into a gigantic screen inside the Vision Pro. I combine it with a Dolby Atmos surround sound speaker setup in our upstairs game room. It's truly a stunning experience. The only reason I wouldn't declare this the gold standard way to play games is because it currently relies on WiFi streaming, which introduces some input lag. The lag tends not to be an issue with the games that I play, but it's enough that you wouldn't play competitive twitch shooters with it. If Apple had just allowed you to plug in an external device via HDMI, this would hands down be the most impressive gaming experience out there. I'm personally very sensitive to input lag thanks to years of low-latency PC gaming, but I know not everybody is. If you're not, you may be even more impressed by it than me.

The mathematics of this is a bit too complex to reproduce in a comment here, but in, say, the Earth's gravitational field, taking this effect into account (approximating GR as a field of locally varying clocks, then allowing, e.g., an electron's wavefunction to evolve on that spacetime) would reproduce gravitational acceleration / free fall towards the Earth.

Said differently: this is precisely the kind of nuanced scenario where getting sloppy with metaphors gets you into trouble very quickly. Quantum mechanics in curved spacetime is not to be dabbled with lightly.

If you view gravity as a mattress, you're stuck. There's nothing to do with it that you couldn't already do, because it's fundamentally wrong. Another way to say this is that it's actually an analogy for Newtonian gravity, not for GR, despite apparently including something curved.

If you view gravity as a field of local clocks that tick at different rates, you can make many correct predictions:

- Clocks at different heights will tick at different rates.

- GPS needs gravity corrections.

- Light climbing out of a gravitational field is redshifted.

- Radar signals passing near the Sun should be delayed (the Shapiro time delay).

- You can have no gravitational pull but still have time dilation (inside a perfectly spherical shell, Newtonian gravity seems to cancel out).

- From the outside, it appears to take an infinite amount of time for something to fall into a black hole's horizon.

- Aging can be path-dependent.

- Gravity affects every physical process: chemical reactions, radioactive decays, biological aging, atomic transitions, molecular vibrations, computer processors, pendulums, pulsars.

- A sufficiently precise clock can measure height.

- Objects in eccentric orbits should have periodic clock-rate changes.

- Quantum matter waves should accumulate gravitational phase shifts.

- Spectral lines from compact stars should reveal compactness.

- Thermal equilibrium in gravity should involve a temperature gradient.

The only plausible example I can think of that isn't purely theoretical / speculative would be gravitational waves.

Near Earth’s surface, clocks lower down tick very slightly slower than clocks higher up. The change in tick rate is on the order of 10^(-16) per meter. While extremely small, that's enough to generate the familiar 9.8 m/s^2 spatial acceleration we experience. Such a small gradient in clock rates generates macrosopically noticeable spatial accelerations because the "translation" factor is c^2, a tremendously large number.

Now, if I wanted to cover all my bases here, I'd need to point out that gravity does also bend space -- that is just not a relevant factor for "ordinary" gravity acting on relatively slow moving matter (like the Earth itself, or the Earth's atmosphere). For instance, for light itself, spatial bending is just as important (in fact, the gravitational deflection of light by a weak static gravitational field is controlled by a near 50/50 split between spatial and temporal effects). Near a massive black hole, it's not that simple and can't meaningfully be understood in terms of "time" and "space" effects being independently separated.

Imagine spacetime as a field of local clocks. Far from the Sun, clocks tick faster. Near the Sun, clocks tick slower. A freely moving object tries to follow the straightest possible path through spacetime. But because the “time axis” changes from place to place, what counts as “straight ahead into the future” tilts slightly inward near the Sun. So the Earth’s path through spacetime curves toward the Sun.

Earth’s spatial speed around the Sun is about 30 km/s. But through spacetime, its “timeward” motion is basically c, 300,000 km/s. So even a tiny tilt in the time direction creates a significant spatial acceleration. That is why the time-warping term dominates for slow massive bodies.

In this analogy, a planet is like a smaller ball. If it rolls close enough to the bowling ball, its path will be altered by the dimple in the mattress — space-time tells matter how to move."

This analogy is wrong in a way that even people who've studied physics often don't realize.

On an everyday scale like the Earth orbiting the Sun, almost none of that gravitational interaction is from the bending of space. Far beyond 99% (actually, about 99.999999%) of it is from the bending of time.

If you allow me to exercise some creative liberty with language, it's almost as if gravity is just launching countless trillions of parallel instances of the same computation, with nearly all possible initial starting conditions. Some of those initial conditions allow the local compute capacity to "descend" into finding more and more optimal ways to increase entropy and heat dissipation by exploiting local energy gradients (i.e., life).

In terms of the frequency of life, I'd expect basic microscopic life to be somewhat common, as it's "just" a way of exploiting geochemical energy gradients for local entropy maintenance. That doesn't necessarily even mean fully functioning cells, genetic codes, etc. It really just means molecular compounds or assemblies that exploit or create energy gradients. However, generally once that's kicked off, it's reasonable to consider that this generally would lead to the kinds of selection pressures that favor the development of what we'd know as basic cellular machinery and replication.

However, complex life I would expect to be almost vanishingly rare. The Earth only managed to figure it out a single time so far as we know (generating eukaryotes from bacteria/archaea) in billions of years. How many other planets feature roughly the same chemical computation which just never explored the right niche of chemical possibility to give rise to that complexity? This suggests to me the universe must expend unbelievably vast amounts of computation to overcome the threshold to complex life. I don't think it'd be unreasonable to assume that it requires thousands of separate "planetary computers", each with basic life, for only a single one to generate something like complex life (eukaryotes or equivalent), and that's to say nothing of the millions or billions of planets that don't generate any life at all.

Beyond that, life itself modified the environment that produced the original process of abiogenesis. The early Earth featured a carbon-rich acidic ocean. After life emerged, metabolism began altering the planet’s redox chemistry, consuming available chemical free energy, transforming atmospheric and ocean composition, and eventually oxygenating the surface environment. In other words, the machinery that produced life was not left running in the same state. This is why I called it a self-modifying search engine -- search accelerants operate by changing the search landscape that the engine operates over.

I like to think of the Earth as a supercomputer running a vast self-interactive chemical computation of unfathomable scale for an unfathomably long amount of time. In this view, the Earth is roughly a ~10^38 ops/sec dissipative self-modifying search engine, of which life captures roughly ~10^35 ops/sec into metabolism, heredity, ecological competition, and evolutionary search. Once proper biological evolution kicked in, with some bumps along the road, it has had a general tendency to reallocate that immense compute capacity in a way that increases search adaptivity per joule by finding and stacking "search accelerators" (prebiotic geochemistry/biochemistry, replicators, cells, DNA/RNA/protein systems, mitochondria, sexual reproduction, multicellularity, nervous systems, intelligence / brains, language / culture, science / technology, ?).

These two situations require different techniques. Applying peacetime techniques during wartime does not work: you'll rapidly accumulate debt from unsolved organizational problems, politics you've lost control of, competitive pressure you failed to respond to decisively enough, or an underperforming team you've failed to correct enough. Or all of the above.

But, similarly, applying wartime techniques during peacetime also does not work. You'll alienate your high-performing team and suffocate critical innovation that will grow the business.

Confusing the two situations is a major category error that managers often make. It often happens because they've only experienced one of the two categories before, they were successful previously, they don't fully appreciate the extent of the existence of the other category, and when they encounter it for the first time they rely too much on their prior experience and have slowed down their own learning too much (because of said prior success).