An inscrutable 1000-page Lean proof may have low transmissibility amongst humans, yet extremely high transmissibility amongst AI mathematicians.
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.
I don't think Terence Tao sold out. However, just looking at it from OpenAI's perspective, this kind of advertising is almost certainly worth at least one order of magnitude more than $3M to the company.
I bought it on launch day, and I still use it at least a couple times a week. I also pretty much always take it with me when I travel (along with my MBP). Frankly, I'd use it even more, but because it's a fairly anti-social device I prefer to use it only when I have meaningful alone time. If I were living alone by myself, I imagine it could be a daily device for me.
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 original point of the stock market was to fund gigantic society-level projects (like railroads). Modern VC has replaced some of that at smaller scales but not all of it at the largest scales. So this could just be the stock market performing the function it was designed to perform -- helping fund something transformative on a societal level.
"Einstein cast gravity not as a force but as the geometric bending of space and time. In a popular analogy, the fabric of space-time is like the flat expanse of a mattress, and a massive object like a star is like a bowling ball sitting on top. The weight of the bowling ball compresses the mattress, forming a dimple — matter tells space-time how to curve.
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.
Imagine you are driving in a car coming up parallel to the sun on your left. Time moves a bit faster for you on the left side than the right side. This slight speedup makes your left side traverse space faster than the right side, which causes a slight drift to the left (and also makes you spin).
How does this cause a point particle to accelerate towards the sun? Must be something about the gradient, but how does the gradient of time cause you to curve towards the sun?
That's a great question. The answer is, the stuff you are reading in this thread is not right (you figured it out). The real version of the story is, there is this thing called the "Christoffel symbol," which tells you where, at every point in space, you would end up if you went in a certain direction, including which way you would be facing if you went that way. It relates three vectors: your direction of motion, the direction you are currently facing, and the delta to your direction of facing that would result from taking that direction of motion.
If you let your current momentum be your direction of facing, and let the same momentum also specify your direction of motion, the Christoffel symbol tells you what your momentum vector would be after an infinitesimal amount of motion. This can be integrated to find the version of a straight line appropriate for a curved surface (imagine an ant walking straight forwards on the surface of a cone or something), a geodesic. A changing momentum is like a force is acting, so that's gravity.
There is more to learn than that, of course. Many many many books have been written about general relativity and you can read them.
For a nice introduction to relativity, look at The Einstein Theory of Relativity: A Trip to the Fourth Dimension by Lillian R. Lieber. $15 on Amazon. Written in 1945 and still quite good.
No, but you can talk about changes in perturbations of fields over time in QFT (which has its own representational issues). A particle is a useful metaphor.
Their point in this case is that a wavefunction is spread out over space, which would cause it to be subject to a local clock gradient in curved spacetime. If you wanted to use particles, you'd need to use a Feynman-style "integrate over all possibilities" approach, which would again be subject to a clock rate gradient over space.
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.
I think it takes a very narrow understanding of physics discourse to call the particle a terrible metaphor. What's next, entirely discarding the teaching of newtonian physics in high school?
I’m going to ask the obvious next question… so if the sun and me in the car are next to each other but stationary, where is the attraction coming from now? As in, time may make the closer side slower, because we’re stationary, there’s no drift etc
You always have to define stationary when it comes to relativity.
There is no way to have a “zero speed orbit”. You’d be on a trajectory straight in to the middle of the sun or away from it (under your own power). The only way to stop is to push away with equal constant acceleration (which looks like “force”). This is what rockets do.
If a particle was dropped into the sun’s gravity (not with “horizontal” motion that might cause it to orbit), is it time dilation that causes it to accelerate toward the sun somehow?
My guess: In the reference frame where the particle is not moving, the sun would be either a) moving (with a perpendicular component) and be ever so slightly moving toward the particle or b) not moving but a third body would be, moving both the sun and the particle at different “strengths” (different mass and distance, different time dilation) thus the particle and the sun would appear to move closer to each other. That means in either case (sun in the middle or particle in the middle) the third body moving closer must make it look like the particle and sun are gravitationally pulling each other. If we then shift reference frame back to the third body being stationary and the particle and sun moving, we should see that. It would be really cool if we could simulate this to test it but I believe that would require solving the 3 body problem.
That's a very cool analogy but I might not be understanding something here. Why then do objects that have no light have gravity? If 99% comes from time dilation, why am I stuck to the earth rather than drifting toward light sources?
The point is that mass bends space-time. The amount of bending is dependent on the size of the mass and on the distance from the mass. Even though the Sun is incomparably heavier than the Earth, it is also MUCH farther away from you. So, space-time around the Earth is curved much more towards the center of the Earth than it is towards the center of the Sun. In the mattress analogy, consider a large mattress, with a bowling ball and a car sitting on it. The car will obviously bend the mattress much more, but if you're close to the bowling ball, you'll still fall towards the bowling ball first before both you and it fall towards the car.
So, say you're in an airplane moving directly forward, with the Sun just overhead (and the Earth obviously just below you). The Earth curves spacetime towards it a lot in this area, while the Sun curves it towards itself just a little bit. The overall curvature is such that time still moves more for the bottom of the plane (closer to the Earth) than the top of the plane (closer to the Sun). So, the bottom side moves a little slower than the top side, but the structural integrity of the plane pulls the top side towards the bottom, causing a slight motion towards the Earth - gravity [note that the GP's explanation got the signs a little wrong - time flows slower, not faster, closer to a big mass]. Conversely, if the Earth disappears from the picture and only the Sun remains, now the top part of the plane will move slightly slower, pulling the bottom part towards it, and thus towards the Sun.
If the sun is on my left, doesn’t that mean time moves a bit slower on my left and the slowdown on the left means I’ve traveled less on my left side? Thus I turn left toward the sun.
All objects move through spacetime at the speed of light, but a stationary object is moving in the time direction. (And the time dimension has opposite sign to spatial dimensions, so (Lorentzian) rotation's effect on length works opposite to what you'd expect from Euclidean rotations: https://commons.wikimedia.org/wiki/File:Spacetime_diagram_of....) Suppose we drop a test mass from the top of the leaning tower of Pisa. The "forwards through time" direction takes the object deeper into the local gravity well: as far as the test mass is concerned, it's just moving forwards through time according to Newton's First Law, and everything else is accelerating towards it for no apparent reason.
It may help your intuition to consider the extreme case of a black hole. The event horizon is where time is so warped that no possible future trajectories lead outside of the black hole, and you need a magical time machine to escape. (Of course, the best way to gain intuition is to work through the mathematics, either symbolically or with diagrams, rather than reading English-language descriptions.)
There is a sense in which an orbit is a straight line. Obviously, an orbit is not a straight line through space (unless you count the perfect and unobtainable orbit of a beam of light around a black hole, some distance from the event horizon), but we often think of them as spirals through spacetime: there's an argument that really we should think of them as straight lines through spacetime, much like how a great circle is a straight line along the earth's surface.
fortunately that video is more gentle but the math in that youtube channel absolutely melts my brain some days, I can keep up for the first minute but then all bets are off as he dives in and I realize there are some insanely brilliant people out there
It means clocks tick at different rates depending on where they are.
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.
That's right -- the atmosphere stays attached to the Earth mostly thanks to gravity, and the Earth's gravity in GR is almost entirely the gradient in clock rate near the Earth.
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.
I’ve seen this described before in terms of how GPS is able to do what it does, so not surprising. But still haven’t explained why time would so dominate the space dimension.
Edit: The response below is dead for some reason, please vouch.
Everything we experience is far larger in time than space, so of course time effects dominate on scales we perceive.
But this just raises the question of what it means to be larger in time than space. You can look at it in terms of multiples of Planck distance or time, but I think there's a more enlightening way to look at it. If you express the speed of light in those Planck units, it's 1. But the speed of light is also the maximum speed of causality. Any causally-bound system must run long enough for chains of causation to propagate, usually far below the speed of light in practice. This means that basically anything that exists within the bounds of our manipulation must be happening at scales where there is far more time involved than space.
We all exist below the diagonal because the diagonal is the bound at which the ways chemistry and biology work no longer even are theoretically possible.
You contribute to the stress-energy tensor by having mass or energy. That directly influences spacetime curvature. The Einstein field equations tell you how precisely.
I'm having a bit of an issue teasing apart space and time when it comes to reasoning what bending time even means without space, let alone comparing the magnitude of the two. They're inextricably linked—as I understand it, both space and time can be seen as a magnitude of causality.
That’s very interesting, but the analogy is not wrong, because the mattress is analogous to space-time, not just space. Essentially it is a 2D analogy for a 3+1D reality.
The beauty of a useful analogy is it allows you to make correct inferences, even without a full understanding.
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.
It's just an analogy, you're not supposed to think too deeply about them.
The main take away for a lay person is that _like_ the mattress space is being deformed. That's where the analogy stops. Taking it further, like with all analogies, breaks the analogy.
If the analogy was a perfect one, then it would just be the reason and not an analogy.
My main gripe is how hard for most people it is to extrapolate that deformed mattress into a 3d space.
I am not an expert at all but I believe the rough idea is that in General Relativity, everything follows a geodesic. The straightest possible path through curved spacetime.
They use that to explain why you don't feel acceleration when you jump off a building or out of a plane.
They say you DO feel it when you are standing, because the earth is impeding you and pushing you away from the geodesic you naturally want to follow.
So it is counterintuitive. the standing still person is being accelerated/pushed. The freefall person is free.
John wheeler said "Spacetime tells matter how to move; matter tells spacetime how to curve". I think it explains the circular nature.
Mass curves spacetime, that changes how mass moves , mass moving to a new position changes the curvature, that changes how the mass will move next.
The dimple in the mattress staying with the mass.
You've got it right. Our normal environment is an accelerating reference frame. The resting frame (i.e., free fall) is easier to analyze using physics, but is unfamiliar (indeed tend to feel alarming) to us.
I agree. The analogy is using our intuition of how objects move because of gravity to explain gravity. It has always seemed circular to me. I bet there is a better analogy.
That is a harder question than it sounds. The answer might actually be "there are no such examples", but I'm not confident enough to jump to that with any certainty by any means. Near and inside black holes, there can certainly be significant warping of space, but it's unlikely to be near exclusive warping of space.
The only plausible example I can think of that isn't purely theoretical / speculative would be gravitational waves.
time and gravity are the same thing, the history of understanding physics is basically of the same nature, understanding that two things are actually one thing, which is more like philosophy but with physical confirmation
It's been speculated for at least a decade now that geochemistry spawned biochemistry and life as we know it. This appears to be the latest instance of this pattern. One of the most notable examples is geothermal processes simply creating calm energy gradients that are stable for billions of years (e.g., underwater alkaline vents), which can then essentially "manufacture" organic compounds, which naturally assemble into more complex compounds like magnetic Lego blocks, which ...
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, ?).
Yet, there is only one form of life on earth exhibiting cellular metabolism and DNA/RNA replication. That original life form formed as soon as the earth became suitable for life. In the 3+ billion years since, there has been no new life form created that we know of despite the ongoing unfathomable computations.
That is not unexpected. There are three forms worth considering, actually: bacteria, archaea, and eukaryotes. The first two share DNA/RNA replication, but they operate in completely different ways. The third is dramatically more complex than either of the former two, yet emerged from them. Once the first two existed, they rapidly filled almost all possible niches available at that time, and there was no space for a third form to emerge (lest it be immediately consumed by the first two), unless that third form was exceptionally competitive (like eukaryotes were and are). The first two forms did emerge relatively early on. The third form, representing one of the most stunning advances in the history of life on Earth, took over two billion years of 10^35+ ops/second of continuous computation to emerge. In terms of total compute, that's about 10^25 greater than today's largest known frontier training run. After that point, evolutionary selection pressure began operating at higher levels of abstraction, selecting for complex multicellular morphological form and later on intelligence, culture, and beyond, over several additional billion years, while bactera and archaea continued to consume all available microscopic niches.
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.
On 1 February 1871 Charles Darwin wrote about these publications to Joseph Hooker, and set out his own speculation that the original spark of life may have been in a "warm little pond, with all sorts of ammonia and phosphoric salts,—light, heat, electricity &c present, that a protein compound was chemically formed". Darwin explained that "at the present day such matter would be instantly devoured or absorbed, which would not have been the case before living creatures were formed." [1]
"There's strikingly little agricultural innovation in this corner of France" they mused... as the ground shook from tanks and shrapnel bursts.
A glass of sea water seems so peaceful... with its turbulent combat hellscape of voracious protists and viral shrapnel, where you're lucky to make it through a day without being eaten or lysed.
Network effects + energetic fficiencies. On an energy landscape that includes integration over very short and very long lifetimes, the thalweg of utility/energy rests right about where the current codon optimizations are. And any schemes that deviate don't get to share in the others' bounty. Reusing your foods' effort saves a lot, metabolically.
As I understand, earlier forms of life used RNAs as building blocks (instead of proteins encoded in DNA->RNA), so protein-based life _was_ a completely different form of life.
Some of the oldest replication machinery in our cells still uses the good old rusty RNA building blocks at its core (however nowadays they're propped up with proteins), and the newer machinery is almost entirely "high tech" proteins.
So you could say that in the billions of years, entirely new life forms were created, and they just completely displaced the older, less effecient ones. Probably pure-RNA life forms were not even the first ones, and they completely displaced even more primitive prior biotechnology when they appeared.
Is that still the case with the discovery extremophiles that exist on chemical vents deep under the ocean and far away from the sun? Or rather, how sure are we that they're the same form?
Yes. All known life shares an (assumed) last common universal ancestor (LUCA, presumed extinct), and there is significant evidence pointing that way.
We can infer properties and function by looking at genes shared between archea and bacteria that most likely came from such an ancestor; this paints a picture of a DNA-based anaerobic thermophile (think hydrothermal vents) with a membrane and simple anti-virus defenses (CAS).
> 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.
Or "I read the Hitch Hiker's Guide to the Galaxy".
I almost agreed with your comment, but then I remembered there are countless planets with conditions unsuitable for life (as we know life). We have found a couple planets that are optimisticly closer to Earth conditions, but very few, and there is usually some characteristic that makes it a stretch still.
With that said, if Earth was compared to a super computer, the initial conditions and perturbations (weights and biases, or probabilistic inference) are very important, as most planets that are also performing ~10^38 ops/sec will never succesfully manufacture biochemistry/life.
I've actually thought about this. My personal interpretation (which I admit is absolutely a bit playful, but I also find it very fulfilling) is that gravity acts like a resource allocator. It clumps matter together into stars, planets, etc., forcing it to interact, while keeping those objects far enough apart that they generally can run independently of each other for very long periods 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.
> gravity is just launching countless trillions of parallel instances of the same computation, with nearly all possible initial starting conditions
Great take and conclusion, no issue with the playful language here. It makes sense that, entropy being a required output with time, and 'infinite' near isolated cases would find just about every way to create that entropy, to include the efficiency at creating it (life, as you said).
Lovely take. Earth is a supercomputer. And to play on sibling responses to your first comment, this supercomputer is solving the answer to life, the universe, and everything. The answer is entropy though, not 42
As Crime Pays But Botany Doesn't always says... botany is just applied geology, so it probably isn't a big stretch to extend that to the underlying chemistry.
On my long-term todo list, is making a didactic simplified tree-of-life, compressed reflecting highly conserved microRNA families (perhaps also Hox clusters and TF transcription factor families) as a regulatory complexity budget, expanding and refining. Traditional presentations obscure the stacking, for example making primates seem just another mammal, instead of a "WTF happened there".
> 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.
This approach is great for peacetime and for when the team is already reasonably functional and performing. The really hard leadership problems occur during wartime (the business is in crisis, or shrinking, or responding to a serious competitive threat, or must aggressively cut costs, or must integrate an acquisition, or...) or when the team you have is routinely underperforming at scale.
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).
Wartime is exactly when centralized control breaks down the hardest, because conditions start changing incredibly fast and communication breaks down. There's a reason the phrase is "fog of war" and not "fog of peace"!
In management, what it means is having to repeatedly make decisions that are in the best interest of the company, but not necessarily in the best interest of the people on your team. This could mean needing to fire people, conduct layoffs, merge teams together and remove redundancies, strip a manager of their direct reports or reduce their scope, replace a leader, drive a major re-org that changes people's jobs in ways they may not like, shut down an entire project or team that isn't succeeding even though it's very popular or well-liked in the organization, own a technical decision that hurts one or two teams but helps the overall organization enough to offset it, etc.
Counterpoint: Name a great victory where top leadership mattered very little.
Or, for that matter, a massive upset where top leadership did not truly contribute significantly.
The "fog of war", AFAIK, tends to refer to general breakdown of communication (as you noted), but even fully localized control (terrorist cells, I suppose) are not highly effectual without coordination and informed assessment of the overall picture. The horrific triple (almost quadruple) attacks of 9/11/2001 would have been greatly diminished, probably by 2/3, if the attacks weren't centrally coordinated.
Wartime is exactly when centralized control is most needed.
Leaders can matter—a lot—even if they do not exercise granular, top-down control. They can provide the right context and motivation, articulate high-level aims, resolve issues and create the kinds of systems and cultures that do not need explicit direction.
A great illustration of this idea in context is Andrew Gordon's book on the Battle of Jutland, The Rules of the Game[1]. He mostly contrasts the leadership philosophies in the British Navy shortly before and during the Battle of Jutland. The British Navy became very top-down and procedure-oriented during peacetime, which did not generalize well to operating in battle.
Synthesizing a ton of inputs to help clarify a decision or set of options is exactly one of the easiest and most powerful use cases for AI agents right now.
I don't think that part is true, either. The average human could be trained to use an agent to synthesize information in their job to help make product decisions. The average human could not be trained to evaluate whether a reasoning model produced a correct proof in research-level mathematics. To be sure: reviewing a candidate proof at this level written by AI is significantly easier and faster than writing and creating it from scratch. But it's still not something hardly any humans could credibly do.
Anthropic looks a lot more like early Google -- not the first mover, but "lightning in a bottle" culture, talent, focus, and product direction that causes them to become a dominant, enduring figure.
OpenAI looks a lot more like early Yahoo -- earlier, quite a spectacle at first, definitely a game-changer and disruptor, but overspent, less focused, and subject to slow collapse under its own fragmentation and lack of overwhelming clarity of mission and purpose.
All that said, history rhymes but does not repeat, and trying to map present-day companies onto previous generations is an exercise in futility. The future is fundamentally unique.
I'm not religious and haven't been since 2008. However, the world today is very different from then. It's fragmented, far more authoritarian, much more dangerous, with "us vs them" mentalities just gaining more and more traction in general in so many countries. There are almost no political leaders left in the world offering a vision that is distinct from mere survival instinct or domination or some mixture of the two. In the last decade we've seen the rise of multiple world-historical tyrants. Meanwhile, many major religions have lost all moral credibility due to continued decades of horrible violence. I can't believe I'm saying this, but it'd be nice to see some real, genuine world leadership from the Pope right now.
I have long come to the conclusion, backed by data, that presidential and semi-presidential systems are deeply flawed.
There's a reason why not a single country turning authoritarian in the last 50 years has been a representative parliamentary democracy. The last one has been Sri Lanka in the 70s. Not a single one since then.
Electing single individuals to power instead of parties and coalitions is a terrible idea.
They are all, and I want to emphasize all, presidential or semi presidential. From Belarus to the Philippines, from Russia to Nicaragua, from Turkey to Tunisia the list is entirely composed by presidential or semi presidential republics.
There are several reasons why this happens, and why it tends to kill pluralism and proper democracy with winner-takes-all mechanics (which also tends to aggregate people across very few/two parties).
In 1971 the government declared an unlawful state of emergency that stayed in force for 6 years suspending civil liberties, suppressing press freedom and giving the executive wide powers. The constitution was updated, by the parliament, the same constitution prolonged the current government mandate for two years without elections.
What you're talking about are the events of 1978.
And, as you probably know, during the 80s, that presidential and authoritarian shift only got worse.
But my point stands, 1970s Sri Lanka, is still to date, the last parliamentary republic to turn authoritarian. Didn't cite this randomly.
In 1971 there was an insurrection that nearly succeeded in overthrowing the government so the emergency was initially justified.
I think we have different definition of authoritarian: yours is broader. Sri Lanka did continue to have elections and changes of government even at its worst, but I would only call it authoritarian in the period when there was clearly a lack of freedom of the press (when journalists risked being disappeared).
My definition strengthens your point as by that time Sri Lanka had a presidential system.
We didn't have nearly the prosperity gospel and doomsday cult of christians we have today in 2008 (or at least they were kept much more at bay instead of running the country)
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.
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