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The world through the language of principal bundles is much simpler, but quite radical.

For instance, our usual intuition is that velocity is a derived notion. You start with distance/length → speed = distance/time → velocity.

But actually, you find out that the more unifying way of viewing physics starts with world lines as fundamental and then speed is less fundamental than velocity. It goes as follows: curve/worldline → tangent vector = velocity → metric measures velocity → speed → integrate speed → length → geodesics…

You’ve probably been told that the fundamental forces of nature are “fields” filling space(time?) and that gravity is just curvature.

Yes, gravity is interpreted as curvature in Einstein’s gravity but there are other equally valid gravity theories that have gravity not as curvature but as something called torsion and even nonmetricity, thus the curvature folklore is underdetermined by the data, much like most models.

Regarding the fundamental forces, this usually refers to “particle physics.”

Behind both of these pillars is something that has a beautiful name: principal bundles.

Once you allow yourself to view physics from the language of these bundles (and their connections, associated bundles), you’ll start to see the unification and beauty.

Actually, the past 200 years is a history of physicists rediscovering this one particular structure under different names. Vector potential. Gauge field. Christoffel symbols. The spin connection. Affine connection. Even photons…

Mathematicians, in another building, were studying these exact guys but as the “geometry” of bundles. Geometry is a funny word here, and it may confuse you. It certainly confused me for years because I thought of geometry as circles and lines, but you can generalize this to principal bundles, and that’s called differential geometry.

It turns out that in the mid 1970s, Tai-Tsun Wu and Chen-Ning Yang wrote the “dictionary” between the two vocabularies, and it became clear they had been describing one structure all along. This dictionary should technically be called a thesaurus… Math and physics have a poor history of naming conventions…

(superposition has nothing to do with supersymmetry which has nothing to do with a superspace nor superrenormalizability nor superconductivity etc.)

Anyhow, let’s get to it.

What a principal bundle actually is

The standard way this is told is that you have something called a “base” and then you “attach copies” of something, yada yada… You’ve probably heard that before.

This is confusing to the newcomer.

I think the better way to go about explaining it is to realize that any intricate object is built from smaller parts. For instance your kettle is built from some cylindrical-like object and then you glue a handle and then you glue a spout. Even my abnormally hairy legs can be thought of as a regular person’s legs, plus going to each little millimeter and placing a ghastly thick hair. Over and over.

My hairy leg is a fiber bundle (not yet a “principle” bundle… a disgusting bundle, yes, but not principle).

You can see this everywhere. The screen you’re reading this on now is at least RGB (red / green / blue), and on a little 2D surface. You can think of it like you create the 2D surface first and then you go to each point and you add the ability for it to be Red there. Green there and blue there and then you go to the next little millimeter or sub millimeter and then do the same and you just keep doing that across the entire surface.

If you’re thinking that this looks like a Cartesian product, then you’re exactly correct. There’s nothing here that I’ve said that’s different than a regular Cartesian product so far. A fiber bundle is a generalization of a Cartesian cross product.

Heck, even a dang Excel spreadsheet is a sort of bundle, where your rows are fibered with your columns! Hopefully you’re getting the hang of it. Part of the hindrance with apprehending some concepts is that we think they’re more complicated than they are.

When a physicist says you “attach copies” what they just mean is you form a Cartesian product, but close up (that’s why I said “millimeter” above… the technical way of phrasing this is to say “locally.” And you need the language of topology / open sets to specify what that means mathematically).

Similarly you can think of 2-dimensional space and then gluing time to it. This is what you do in these standard world line diagrams like the below that I generated.

Yes, spacetime is a fiber bundle! It’s a trivial one so far, but it’s a fiber bundle. I mean, Galilean so far, and not Minkowski but still.

Much of the time you’ll hear a lecturer talk about some extremely fancy object and then say “Ah, but you’ve known these before!”

For instance, they may talk about rings and then they may say, “ah, but you in kindergarten already knew what rings were because you can plus five and minus seven,” et cetera.

I personally have never found these helpful other than to just make me look momentarily foolish for not understanding the fancy object because apparently it was supposed to be obvious given that it was understood many years ago. So at this point, if you are confused and looking at all of the above thinking “Curt, like bro… I understand what spacetime diagrams are, maybe. Yes, I understand what a kettle is. And yes, I understand what your hairy leg is, (unfortunately), but…” well, then verily I say to you, you understand the general structure of a trivial fiber bundle.

In order to make this non‑trivial, all that’s required is something called “twisting.” By the way, I’ve never once heard twisting defined; it was just said aloud (at least to me). So I’m going to say it to you right now. When someone says that a bundle is twisted, all they mean is that the bundle is not isomorphic to a product bundle. Or using the phrasing above, that the bundle isn’t a global Cartesian product.

The most common object to display this feature is a Möbius band. It’s locally a rectangle but carries a half-turn no local picture can remove.

That is, if you are the proverbial ant on the Möbius band, at any given time you wouldn’t be able to tell if you were on a Möbius band versus say a regular sheet of paper untwisted. It’s only by traversing the entire band that you can detect this sort of twisting.

A principal bundle is the case where what you’re adding is a group of symmetries: rotations, phase shifts, Lorentz transformations.

Recall again in the above example

• for the kettle you added a handle plus a spout to a cylinder.

• for the legs you added hair to regular legs

• for spacetime you added time to space

The bolded parts above are what mathematicians would refer to as the fiber (in this analogy).

When you have a principal bundle, all you’re saying is that that bolded part is now something that is a group, a special kind of group called a Lie group but with extra conditions like it has no marked identity / origin, like a clock face with the numbers rubbed off. So, you can measure the angle between two hands, but there is no absolute twelve.

Mathematicians call this torsor. When your fibers are torsors, you have a principal bundle.

There’s a word that I mentioned before called a “connection”. This is used as an object. So, you and I have a “connection” (i.e. I love you, you love me) but we don’t have a principal fiber’s connection. At least not in this branch.

When your fibers are Lie groups, you can also think of them as their own space. So that means it’s like you have spaces, and then you “attach” spaces to spaces. But you don’t exactly have a sense as to if something is above or beneath or to the left of or what have you of another one in a completely different space.

I understand this is not clear. And that’s because to actually study this, you do need differential geometry. So rather than trying to derive some rigor from this, just understand this impressionistically.

Look, I’m standing here in Toronto in one particular place. You are probably in some other place. You’re not beside me, as far as I can tell (you stalker). So let’s just call the place where you are Hawaii. You are standing in some city on some beach, and I’m here in Toronto. When I say that something is above me, it could be technically below you, or to the side of you, or what have you. So “above” is not actually defined, it’s a relative term.

This is one of the reasons why some of the new atheists say that it’s a bit foolish for us to pray above, or say God is above, because what does that mean precisely? Is it just in the clouds? If so, then technically if you’re in China, you should be praying by looking at the ground. blah blah.

What we require in order to make the “As above, so below” quantitative (so to speak) is a “connection.”

This is a mathematical tool and again defined rigorously in differential geometry that tells you if something at two different (neighboring) points is actually above or below. You can compare. Without a “connection” you don’t have comparison.

That’s one way of thinking about what a connection is. It lets you compare!

The point is that if you go back to that spacetime diagram above. Without a connection you can’t actually say where any of those arrows point. Are they pointing above or beneath a different point in an adjacent particle? Who knows.

With no preferred origin in each fiber, there is no automatic way to call a state here “the same as” a state there. You need a rule to carry states from one fiberto the next without distortion. That rule is the connection, and it defines parallel transport.

So it’s the opposite of what you’re usually taught in general relativity.

This is another way where the intuition is reversed. So we’re taught this order:

• parallel transport / “same vector”

• → compare vectors

• → covariant derivative / connection

but actually, it’s

• principal connection

• → horizontal lift

• → parallel transport

• → holonomy; covariant derivative only for associated vector bundles!

There’s so much structure that needs to be in place before you even get to a parallel transport.

If you ever visualized a connection, it’s actually quite trippy. I like to think about them in terms of “horizontal subspaces.” (It turns out they are equivalent to a choice of horizontal subspace at every point, and you don’t need to concern yourself with what this means other than thinking that at every single point you can choose what is considered to be your neighbor.) Here is an example below.

Here geometry becomes physics. Carry a state around a small loop and it generically comes home rotated.

The rate at which tiny loops fail to close is the curvature. Why that and not something like bumpiness? We all know and love bumpiness (minus those on Accutane), but it turns out that the bumpiness is fully characterized by seeing what happens to abstract arrows as you move around a space.

I mean I can show you some sumptuous equation to show how learned I am, but … ah what the heck:

That Omega is the same as curvature and the Hol is “holonomy” (which means, how much do you change when going around a tiny loop).

So, in essence, curvature is just the answer to the question “how much does something change when you carry it around a super small loop?”

One last piece: If you pick a way for the group to act on a vector space, then the principal bundle spawns an associated bundle whose values are matter fields.

This means, an electron, a quark, is a “section” (so, a choosing of element per spacetime point, or what you abstractly think of as a “field”), except moved by the aforementioned connection.

The principal bundle holds the symmetry (group); the associated bundle holds what it acts on (a vector space).

That is the whole apparatus. Now watch how much of physics is exactly this.

Internal freedom

Another way of thinking about what these fibers are is as “internal dimensions.” Think about it like this. You can move about in space, right? You’re in some “external dimension”, but at any given point, wherever you land, you do have more freedom. For instance, you twirl in place. You can sing. You can dance the jig (as my uncle used to say). Etc. The place you are in space doesn’t fully characterize you. Why? There’s actually more that you can do, even once your spacetime point has been specified!

Something like that is going on with these internal dimensions. Namely, once you pick a spacetime point, you haven’t fully characterized everything, as there’s more room to do something. This “more room” part is the “fibers”.

One extremely clever idea is to think of these internal dimensions as actually spawned by something external, and that is something we’ll talk about another day. It’s something physicists have thought about for almost a hundred years…

From forces to curvature

Take electromagnetism. An electron’s wavefunction has a phase, a point on a circle: the group U(1). That phase has no absolute zero (only differences are measurable), a torsor at every point of spacetime.

To compare the phase here with the phase a moment later you need that connection we talked about. This can be given a symbol and physicists call it the vector potential A.

If you change which phase counts as zero (the origin) then A shifts to absorb it. For any group:

For the abelian circle this is just A ↦ A + dλ, with F = dA. That inhomogeneous term, added rather than merely rotated, is what physicists mean by gauge redundancy (which is different than gauge symmetry, which is also different than gauge invariance which is also different than gauge transformation… confusing, I know… I’ll write something about this in an upcoming Substack post on gauge theory to delineate these later).

The curvature F is the electric and magnetic fields, and half of Maxwell’s equations are then pure geometry, the identity every connection satisfies automatically.

The coupling of light to charge is already here. The electron is a section of the associated bundle, and the derivative that respects the connection,

is what differentiates it.

Minimal coupling is the covariant derivative doing its job, nothing added by hand.

I know this is tricky. I understand I just introduced a new word without defining it, namely covariant. But there’s only so much I can do in a Substack post. This is one of the reasons why I’m writing my book Edge of Everything. There are various ways physics is taught that I think are needlessly confusing. For instance, the wave function is neither a wave nor a function…

If you swap the circle for a larger non-commuting group then you get the strong force via an SU(3) connection.

Actually, you can cross groups together! And it turns out the Standard Model is a connection on a principal bundle with group SU(3) × SU(2) × U(1), its quarks, leptons, and Higgs the sections of associated bundles.

That last line has so much packed into it…

Think about it like this. Every particle we’ve found is blind to a certain ℤ₆ subgroup of that product, so the group acting faithfully is the quotient

an isomorphism which you can learn more about in this John Baez and John Huerta article.

Which global version is the true one is not fixed by the particle list; several share the same spectrum, and telling them apart needs subtler objects, the allowed monopoles and line defects, still an open question.

The content underdetermines the bundle.

And when the Higgs takes a vacuum value it reduces the structure group, collapsing SU(2) × U(1) to the single U(1) of electromagnetism. This is sometimes called spontaneous gauge symmetry breaking but that’s also false because it’s the breaking of a “global” symmetry and gauge is a local symmetry (and a redundancy, so how can you break it?).

You can see this video by Richard Behiel (who is fantastic) for more on this.

The second place it hides: gravity

Gravity, too, is a connection on a principal bundle.

When I first learned it, I thought it must be that the next move is to call general relativity “just another gauge theory.” It turns out, there are ways of doing so but I haven’t convinced myself of them being correct (and that’s likely because I don’t understand it enough) so I won’t go into them here.

The reason gravity is different is that its bundle is the frame bundle: at each point, every basis (frame) of the tangent space.

The frames are then related by linear maps, so the fiber is GL(n, ℝ), which shrinks with a metric to the Lorentz group SO(1,3). A choice of frame, the tetrad or vierbein, is general relativity’s version of section.

So in particle physics, in the previous part talked about above, its sections are of associated bundles, and they are matter fields, whereas here the sections are perspectives (observer or light trajectories).

The tangent bundle is the associated bundle here. It’s automatically a vector bundle, which allows for calculus (so, that covariant derivative I didn’t define before)…

Interestingly, the analog of the electric potential for gravity is actually the metric. What physicists call the electric field is actually the electric field strength, and the metric field “strength” is actually the Riemann curvature tensor, involved in the Einstein equations.

Frederic Schuller goes further in his Theories of Everything conversation: requiring matter and spacetime to co-evolve and the gravitational action follows from the matter alone, with Maxwell in, Einstein-Hilbert out.

So why is gravity not merely Yang-Mills (particle physics bundle language) in a different hat?

The frame bundle carries something no internal bundle has. It has a canonical way to identify the abstract fiber with the tangent space you stand in, because the frames are frames of that space. You can “solder” the tangent space to the base…

Yes, solder is the technical term. Here the term is good! No complaints.

An internal gauge theory has no solder form θ, hence no torsion, so no length, and no metric. The electromagnetic bundle floats above spacetime disconnected whereas the frame bundle is basically welded to spacetime’s own tangent spaces.

It is tempting to call the solder form the thing that makes gravity special, and some (Jeeva Anandan, Richard Healey) have said nearly that.

James Owen Weatherall doesn’t think so (I’m going to speak with him on the podcast in a few months). He says a solder form lives on the frame bundle of any vector bundle, so its existence is not the dividing line. The sharper statement is what the bundle is soldered to. In gravity’s case, the frame bundle of spacetime is its own tangent bundle; its symmetry is the symmetry of the base and not an internal space lingering above it.

That is why gravity alone comes with geodesics, a metric, and the equivalence principle, and why its simplest action is linear in the curvature (although I explore quadratic gravity here with Neil Turok, and soon coming up Weyl gravity with Philip Mannheim).

Yang-Mills is quadratic (the embellished way of saying “squared” like x² is quadratic).

Gravity is a connection, but a soldered one, a Cartan connection. (It is also why energy stops being straightforward in general relativity, which I take up in What Is Energy, Actually?.)

One word of caution, if you’ve read this far and (to be punny) have the energy. Gravity has two different kinds of freedom. Both are actually called “gauge”.

One is rotating the frame at a point, which is called a local Lorentz transformation inside the fiber (special relativity) with the spin connection ω as its gauge field, gauge in the sense electromagnetism is.

The other is relabelling the points of spacetime. This is a diffeomorphism acting on the base itself (so, spacetime as a manifold itself). This, some would say deeper, kind of invariance is what makes general relativity strange.

Do not conflate these two kinds of “gauges.”

Why none of this is embellishments

So far the bundle is elegant repackaging. Over plain Minkowski spacetime, topologically trivial, every bundle is untwisted and all the physics sits in the connection.

The bundle language is necessary because most spaces, literally most, are not trivial.

This is what I talk about in my lecture for the Mind at Large conference about how the global can’t be inferred from the local.

It’s at this point that the standard physicists will start writing about Aharonov and Bohm. Instead of writing, you can listen:

Anyhow, bundles are also necessary to speak / make quantitative how the gluon field knots into integer classes, (the instantons) counted by

the second Chern number of an SU(2) bundle over the four-sphere.

Topology even decides whether a particle can exist on a spacetime! Like… Think about that. That’s nuts. To put a fermion on curved spacetime you have to “lift” the frame bundle from the Lorentz group to its double cover, Spin, because a fermion needs a 720-degree turn to come home.

Not every spacetime allows the lift; the obstruction is the second Stiefel-Whitney class (which is another fancy name, but basically asks whether the spacetime can even carry spinors).

Where it fails, a spinor field can’t be defined (a charged one can fall back on the weaker spin-c structure, which orientable spacetimes always admit), but the question stays topological through and through.

Something I’m trying to figure out how to explain simpler is how quantization can destroy a classical symmetry. This is called an anomaly. Just like your relationship with your ex, there are various fixes but they are not simple. What an anomaly is, is that some symmetry (called the global axial one) becomes non-zero when you “quantize.”

It leads to a chiral anomaly. This quantity counts instantons:

When you integrate it over spacetime, it is fixed by something called the index of the Dirac operator (its net zero modes, left-handed minus right-handed) through the Atiyah-Singer theorem. This latter theorem is something I’d like to write about more. I’m just trying to figure out how, but feel free to subscribe to get notified as to when. I do cover it a tad in the below.

Turn the same triangle onto the gauged currents and an anomaly there is fatal, so the Standard Model can exist only because its gauge anomalies “cancel”: across one generation the charges and representations of quarks and leptons conspire, suspiciously so, to sum to zero.

The principal fiber bundle geometry reaches down and disciplines which particles may coexist peacefully.

So what is actually real?

If the connection carries the physics, what is real? Here physicists hand off to philosophers. The slogan is that gauge symmetry is “not a symmetry, just a redundancy of description”. I’m still deciding whether this is profound, trivial, or false.

The reasoning is as follows. The potential A says more than nature does (many potentials give the same physics) and the field strength says less (Aharonov-Bohm).

Richard Healey (in Gauging What’s Real) says that what exists are the gauge-invariant holonomies of loops, not the local potential. This is the price of a world that is nonseparable. One that’s no longer built from independent local facts.

But “pure redundancy” is too quick. Carlo Rovelli will say (of course) that gauge variables are the real relational structure.

The edge-mode work of Donnelly and Freidel makes it precise, with the would-be redundant directions carrying physical charge at a boundary.

So… The resolution: transformations dying off at the boundary are redundancy, while those surviving at infinity are physical, the home of electric charge, large gauge transformations, the vacuum angle, the soft theorems.

If local gauge symmetry has any direct empirical significance, I don’t think we’ve found it. At least my perusal of the Stanford Encyclopedia’s accounting is that this is still an open question.

Is gravity, then, “really” a gauge theory? Well, the ordinary Einstein picture (interpreted via principal bundles) has it such that gravity shows none of the holonomy nonseparability that Yang-Mills does. So, not in this picture but maybe in another, as far as I can tell, though I’m still trying to understand this.

A force is not a thing that sits in space. It is the price the universe charges for carrying a description from one point to the next.

I want to hear from you in the Substack comment section below. I read each and every response.

—Curt Jaimungal

PS: Consider becoming a paying member on this Substack. This is how I earn a living, as I’m directly reader-supported. Moreover, you’ll get a slew of exclusive content such as early access to full podcasts. If you like the free content, you’ll love the members-only content.

PPS: I read The Economist and find their coverage of science, AI, and global affairs genuinely excellent. If you’d like to try it, my link gets you a huge discount. They don’t partner with just any podcast. In fact, I believe TOE is the only podcast they consistently partner with, thus you get a special 35% discount not offered anywhere else.

PPPS: You should also note that if you’re bewildered, you’re in great company; there’s no harm in rereading a section, nor in continuing to read despite not grasping everything. Recall, the point isn’t to drink from the firehose but simply to “get wet.” Each pass over will have you glean more. And more. As you progress through your mathematical, physical, philosophical (or whatever) education, through life in general, you’ll revisit the material and see it in a deeper, more integrated light each time. Don’t worry if something doesn’t make sense; understand it at a gestural level and feel free to move forward or backward as you please.

The gist is fine for now. In fact, this is effective field theory at its finest. In EFT, you learn there are different descriptions at different levels of reality. You may assume, like me, that to understand level 100 you must master level 1, then level 2, and so on (because everything is built from bricks). This is the conventional teaching in math and physics, where schools repeatedly stress mastering the foundations; that approach is indeed necessary if you aim to conduct research in a specific area since you need to know your subject inside and out.

However, wider leaps can be made much faster, and more enjoyable progress is possible without exhausting yourself if you adopt the effective‑field‑theoretical view that upper levels sometimes need not depend entirely on the full description of lower levels. Take solace in grasping the gist.

Simon Saunders is an Emeritus Professor of Philosophy of Physics at Oxford and an Emeritus Fellow of Merton College, and one of the people who took Everett’s “many worlds” from a fringe idea in a 1957 thesis into a position serious physicists and philosophers now argue about. His answer to a very old question (what is time?) turns out to be the key to his answer to a very strange one (are there many of you?).

We cover:

• Why time may be the one idea in physics we still don’t truly understand, even though we use it flawlessly every day of our lives.

• Stephen Hawking asked what “breathes fire into the equations.” The block universe, Saunders grants, has no answer, because nothing inside that block ever actually moves.

• Why Saunders finds it “almost insane“ to believe that this exact moment, with all of “its craziness and insanity,” was somehow necessary, built into the universe from its ve…

Read more

It’s not often that you reach a level of prestige and walk away from what gave you that status. I’ve been interested in some of these figures to see if there’s something that unifies them.

There’s Perelman, Grothendieck, Simons, Pascal... even Kaczynski to some extent. Is there anything that unifies these people or is it just the myth of the scorned unalloyed genius?

Grigori Perelman solved one of the most difficult problems in mathematics. He turned down a million dollars for solving it.

Actually, he had already turned down the Fields Medal, the closest thing his field has to a Nobel, refusing to be put on display “like an animal in a zoo.” He is still the only person ever to refuse it. I mean, Feynman talked about refusing the Nobel, but Feynman talked about plenty. And people talk plenty about Feynman. Sometimes wrongly.

When the Clay Mathematics Institute offered him its million‑dollar prize in 2010, Perelman objected that the award slighted Richard Hamilton. The reasoning is that, in math, you’re constantly building on others’ work. Suppose someone came after you spent decades painstakingly building a mountain, and that person takes that little hat from the Abu, crowning the mountain, then they get the majority of the accolades for solving the problem; not you.

Perelman was the “they” here, and while most would be glad for the hype, he found that unfair.

It is difficult not to romanticize this.

A mind powerful enough to settle a 100 year long conjecture, walking away from money and esteem on principle? It’s like a New Testament parable.

Speaking of Christianity, then there’s Alexander Grothendieck (the subject of one of my favorite conversations on the podcast, with Colin McLarty). This man rebuilt the foundations of geometry, won the Fields Medal in 1966, and then, across two decades, left nearly all of it.

He resigned his research institute in 1970 (the trigger he named was its partial funding from the French defense ministry). Then moved to a life of private mysticism.

He is said to have burned 25,000 pages of his own writing, and lived as a near-total hermit for the last 23 years of his life. People would write to him and he would send those letters back unopened. Most of us at least skim what we ignore. He was the original boss.

The Door Most People Use

This sort of “principled renouncer” isn’t the usual case of a great mind leaving academia. Why? Because for every Perelman, there are literal thousands who leave without making a fuss. They mostly leave through the same exit: the one marked money.

I know (both personally and from colleagues) several PhD’s in theoretic physics who went into finance, for instance. Same with math. For instance, James Simons is the patron saint of this exit.

A first-rank geometer (his name is on the Chern-Simons invariants which I covered in my String Theory Iceberg). Simons left his chair at Stony Brook to start what became Renaissance Technologies. Its Medallion fund reportedly returned around 39 percent a year after fees from 1988 to 2018 (the firm discloses nothing, so that figure comes from Gregory Zuckerman’s reporting rather than an audit).

Simons is the cinematic version. The larger flow is a tad duller: math and physics PhDs pouring into software, data science, and the quant desks at firms like D.E. Shaw and Jane Street. Only about 40 percent of new physics PhDs land in even potentially permanent positions, and most never get the research post they trained for. So they leave. Nothing mystical about that. Having a job is better than not having one.

When you picture a great mind walking away, you picture Perelman. At least I do / did. But the numbers basically describe the brilliant engineer at a mundane hedge fund instead of the supramundane parable.

Where the Drama Comes From

Why does the dramatic kind of departure read as a mathematics-and-physics story?

The cynical answer is because many think of themselves as geniuses. When one sees someone walking away, one identifies with the exclusion. They think to themselves, “so am I.”

The less cynical answer is just a sampling bias. These fields make some renunciations blatant (a Field’s medalist walking away), while the majority of the cases (like just going into industry, or having other life plans / priorities) unheard.

Actually, the first material is the myth of the early peak. G.H. Hardy, in A Mathematician’s Apology, wrote that mathematics, “more than any other art or science, is a young man’s game.”

Think about it. Galois dead at 20, Abel at 26, Ramanujan at 32. But interestingly, if you look at the data, the age at which major scientific work gets done has risen across the 1900s. Here’s a study about that. (though, this study is about Nobel winners doing their best work in late 40s, and not about mathematicians)

What may matter more than the fact is the belief: a mathematician convinced that their best work is behind them at forty faces a real crisis even when the conviction is false. Intellectual life is replete with feeling horrible about one’s lack of accomplishments.

The second material is a collision between the ideals of purity and the reality of credit. When one works in math, you get the feeling that you’re pursuing of kind of ideal impersonal truth. But then you look at the sociology of it and some of it runs, like every human institution, on priority and status. Notice I said “some” and not “most”! It doesn’t matter what the percentage is, as long as it’s non-trivially above zero, because zero is the ideal. A non-trivial deviation from that ideal is experienced as corrupt.

To those who have been excluded, pointing to that corruption is an easy scapegoat. It doesn’t mean there isn’t corruption; I’m not making a positive case in either direction. I’m talking primarily about why this myth / meme of the lone excluded genius is one that resonates and propagates.

Grothendieck’s late bitterness was about exactly this corruption. He despised that his school took funding from a military connection (as he was an ardent pacifist). He also had the conviction that his work had been “buried” by former students (his old colleague Jean‑Pierre Serre wrote back to tell him, plainly, that he had the particulars wrong).

Perelman’s objection was sharper and stranger. He wasn’t protesting outright fraud but a community full of people who are “more or less honest, but they tolerate those who are not honest.” To a temperament drawn to mathematics for its cleanliness, that ordinary compromise this is a betrayal that stings more than the accolades are sweet.

The largest study of the question (Kyaga, more than a million people) found that people in creative professions as a whole, mathematicians and scientists included, were no more prone to mental illness than the general population, with the single exception of bipolar disorder, and were actually less likely to suffer depression, anxiety, or schizophrenia.

The across-the-board elevation appeared for one profession only: authors. Writing about the legend that set theory drove Georg Cantor mad, David Foster Wallace called the cliché both “wrong” and “insulting”... I think he was right.

What Stays Sealed

The urge to explain what happens leads us to think that corruption forced them out or genius sits too close to madness. But as Judge Judy cautions says “that calls for the operation of another person’s mind.” I love Judge Judy. I watch her each day.

We don’t actually know why Perelman walked away. He hasn’t explained himself to anyone in years, and the most cited book about him was written by someone he never agreed to meet. We don’t know what Grothendieck found, or fled, in the Pyrenees. What we have is the documented part: a resignation letter, a refused prize, an archive that’s burnt away.

You can follow a proof. But you cannot follow a person all the way to the edge of a choice. (I’ve written before about why you may never fully know even your own mind; another mind, at that altitude, is darker still.)

I want to hear from you in the Substack comment section below. I read each and every response.

—Curt Jaimungal

PS: Consider becoming a paying member on this Substack. This is how I earn a living, as I’m directly reader-supported. Moreover, you’ll get a slew of exclusive content such as early access to full podcasts. If you like the free content, you’ll love the members-only content.

PPS: I read The Economist and find their coverage of science, AI, and global affairs genuinely excellent. If you’d like to try it, my link gets you a huge discount. They don’t partner with just any podcast. In fact, I believe TOE is the only podcast they consistently partner with, thus you get a special 35% discount not offered anywhere else.

PPPS: I’m slowly recovering from my health problems of extreme insomnia. Thank you for your kind words and support. I also just put out a new banger interview with Neil Turok on something called “quadratic gravity.” It’s an alternative method to quantizing gravity in 4 dimensions, without strings.

Why is it that something we can be so confident about is one of the words we can define the least precisely?

This video is about existence. I cover:

• Russell’s escape route: stop treating existence like an ordinary property of individuals. Treat many existence claims as claims about whether a description is instantiated.

• “Batman” becomes “the caped crusader of Gotham.” If no such unique person is instantiated, no nonexistent vigilante needs to be referred to.

• Kripke’s challenge: names and descriptions behave differently in modal contexts. Peter Parker may have avoided the radioactive spider bite. “The guy bitten by a radioactive spider” does not move the same way.

• Meinong’s tempting move (at least to me): objecthood and existence can come apart.

• How existence and non-existence is everywhere: software and AI give analogies for nulls, fictional truths, and real-wo…

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This video is my investigation into defining a notoriously vague concept.

• Why fundamentality is not reductionism.

• The ungrounded. Not the tiniest constituent, but the thing nothing else gets to take credit for.

• Many idealists and materialists share one habit: assume reductionism, then declare it’s all consciousness, or all matter. (The “all information” crowd too.) Same reflex with a different hat.

• What do connected, boundaryless topological 1-manifolds have to do with this all?

I want to hear from you in the Substack comment section below. I read each and every response.

—Curt Jaimungal

PS: I read The Economist and find their coverage of science, AI, and global affairs genuinely excellent. If you’d like to try it, my link gets you a huge discount. They don’t partner with just any podcast. In fact, I believe TOE is the only podcast they consistently partner with, thus you get a special 35% discount not offered anywhere else.

…

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Secondly, we just finished editing this episode with Neil Turok and I think you’ll enjoy it.

Turok, the Higgs Chair of Theoretical Physics at the University of Edinburgh and former director of Perimeter Institute, returns to quadratic gravity in four dimensions...

He (and his PhD student Sam Bateman) found a way to circumvent what is ordinarily thought to be a death blow to a theory (namely, negative norms). He basically embraces them and reinterprets them.

We cover:

• The Hilbert-space assumption: not as a casual rejection of quantum mechanics, but as a question about what this construction actually requires.

• Krein spaces (or the mathematical setting where an indefinite inner product can be treated systematically)

• Why curvature-squared gravity was attractive in the first place: renormalizability, ultraviolet behavior, and a four-dimensional path that looked too sim…

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For the past three months I’ve been going through some health problems. It’s been debilitating, and I’ve been getting extremely upset, which further makes it a vicious cycle. It’s insomnia. This is something I’ve dealt with on and off in the past, but this is the longest flare‑up I’ve ever had. It’s frustrating because I feel so much pressure to work on the podcast, put out great content, write here on Substack, work on the CORE contest (for graduate explainer videos), and I have a book I’m writing with Penguin Press. I ordinarily can handle, but the past couple months in particular have taken their toll.

FYI, this is not a huge channel. It’s just myself, my video editor, and my wife, who is considerably lovely, of course, and she’s helping me manage all of this, alongside my doctor. That’s three people.

Anyone who’s battled insomnia knows how damn depleting virtually every aspect of it can be. In case you’re wondering, yes, nutrition, workouts, CBT, sleep hygiene, meditation, etc., everything is on point, fortunately, but unfortunately it’s not enough. It’s extremely excruciating. I have some tests coming up to help diagnose wtf is going on.

In the meantime, your prayers, patience, kindness, and support in any form are appreciated while I take a step back from content creation over the next month.

TOE’s small, support genuinely keeps it alive, so I’ll be direct: if the show has meant something to you, here are a few ways to help.

1. Watch something on the channel / recommend: (browse here as there’s so much new content)

2. PayPal one‑time donation

3. Crypto one‑time donation

4. Join on paid-Substack here (if you’re not already)

Watching content helps feed the algorithm and thus helps feed TOE. Whatever you do—even reading all the way until the end here—is fantastic. None of it is expected. And I thank you dearly.

—Curt

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So, almost every interview with Roman Yampolskiy opens with the apocalypse. I wanted to do something different today…

Roman popularized the term “AI safety,” and his career has shifted to proving the specific ways a superintelligence cannot be controlled. His latest book is titled, with no irony, AI: Unexplainable, Unpredictable, Uncontrollable.

The simulation he thinks we are probably living inside. Whether the models he studies are already, in some thin sense, conscious. What free will could possibly mean if a little quantum randomness is the only thing saving it from determinism. Somewhere in there I told him I agree with his spirit but disagree with his text, and we spent the rest of the conversation finding out which of us was the fool. NOTE: Probably me.

We cover:

• The strongest thing he believes, stated flatly: “you cannot indefinitely control something smarter than you.”

• Whether the models he studies are already conscious. He would not let himself off the hook: “ei…

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- Go to https://vanman.shop/curt to get 15% off your order

- Go to https://www.plaud.ai/curt to get a Plaud device today - I personally subscribe to The Economist. TOE listeners get 35% off the annual subscription. No other podcast has this! https://economist.com/TOE Janna Levin — Claire Tow Professor of Physics and Astronomy at Barnard College, founding Director of Sciences at Pioneer Works, and co-host of Quanta's The Joy of Why — works with Brian Greene to explore whether hidden Klein-bottle dimensions could explain why matter beat antimatter after the Big Bang. The geometry does the work — no fudged parameters. FOLLOW: - Spotify: https://open.spotify.com/show/4gL14b92xAErofYQA7bU4e - Substack: https://curtjaimungal.substack.com/subscribe - Twitter: https://twitter.com/TOEwithCurt - Discord Invite: https://discord.com/invite/kBcnfNVwqs - Crypto: https://nowpayments.io/donation/TOE - PayPal: https://tinyurl.com/paypalTOE TIMESTAMPS: - 00:00:00 - Universe as Gödel Sentence - 00:05:24 - Unknowable Initial Conditions - 00:10:28 - AI and Non-Computable Consciousness - 00:15:36 - The Hard Problem Paradox - 00:25:25 - Klein Bottle Topology - 00:32:04 - Pin Structures and Chirality - 00:39:56 - Breaking Matter-Antimatter Symmetry - 00:51:26 - Topology and Dark Energy - 00:57:00 - Black Hole Information Paradox - 01:07:04 - ER=EPR and Firewalls - 01:13:37 - Black Holes as Particles - 01:18:42 - Emergent Gravity and Holography - 01:23:43 - Rejecting Physical Infinities - 01:31:41 - Narrative Truth vs. Axioms - 01:41:22 - Insomnia and Mathematical Madness - 01:47:22 - Scientific Mysticism and Honesty - 01:53:34 - Biological Morality and Advice LINKS MENTIONED: - Janna's Substack: jannalevin.substack.com - Janna's Books: amazon.com/stores/author/B001IXTNZQ - Janna's Papers: inspirehep.net/authors/1000438 - Pioneer Works: pioneerworks.org - Whales Don't Want to Go to Mars [Article]: jannalevin.substack.com/p/billions-of-exoplanets-zero-aliens - Black Hole Blues [Book]: amzn.to/4cPOcfr - A Madman Dreams of Turing Machines [Book]: amazon.com/dp/1400032407?tag=toe08-20 - Gödel Incompleteness Theorems: plato.stanford.edu/entries/goedel-incompleteness - Gödel Numbering: en.wikipedia.org/wiki/G%C3%B6del_numbering - Wave Function of the Universe [Paper]: isidore.co/misc/Physics%20papers%20and%20books/Classic%20Papers/Wavefunction%20of%20the%20Universe%20(Hartle%20&%20Hawking).pdf - Janna & Penrose at Oxford: jannalevin.substack.com/p/penrose-and-mein-oxford - Hard Problem of Consciousness [Paper]: consc.net/papers/facing.pdf - Klein Bottle: mathworld.wolfram.com/KleinBottle.html - Klein Bottle Cosmology [Paper]: arxiv.org/abs/2511.23447 - Brane-World Motion in Compact Dimensions [Paper]: arxiv.org/abs/1103.2174 - Dark Energy & Extra Dimensions [Paper]: arxiv.org/abs/0707.1062 - Particle Creation by Black Holes [Paper]: link.springer.com/article/10.1007/BF02345020 - BH Complementarity [Paper]: arxiv.org/abs/hep-th/9306069 - Black Holes: Complementarity or Firewalls? [Paper]: arxiv.org/abs/1207.3123 - Thermodynamics of Spacetime [Paper]: arxiv.org/abs/gr-qc/9504004 - Most Abused Theorem in Math [TOE]: youtu.be/OH-ybecvuEo - Roger Penrose [TOE]: youtu.be/sGm505TFMbU - Roger Penrose [Part 2]: youtu.be/iO03t21xhdk - Neil Turok [TOE]: youtu.be/zNZCa1pVE20 - David Chalmers [TOE]: youtu.be/5r9V1ryksnw - Brian Greene [TOE]: youtu.be/O2EtTE9Czzo - Leonard Susskind [TOE]: youtu.be/2p_Hlm6aCok - Ted Jacobson [TOE]: youtu.be/3mhctWlXyV8 - Juan Maldacena [TOE]: youtu.be/6LbRHMvyrik More links at https://curtjaimungal.substack.com Guests do not pay to appear. #science

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Kurt Gödel personally sponsored his last paper for the Proceedings of the National Academy of Sciences. He won the Alan T. Waterman Award in 1984 for “the revitalization of the foundations of mathematics.” He founded Reverse Mathematics in 1974, delivered the Tarski Lectures at Berkeley in 2007, and received an honorary doctorate from Ghent in 2013. He’s been featured in TIME, New Scientist, and Nautilus.

Like, man… I can keep going!

Anyhow, he’s spent most of his life on one question: can ordinary finite mathematics be trusted? His theorems say no. This episode is an exploration into why.

We cover:

• T…

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Relevant to today, she’s also the co-author, with Brian Greene, Daniel Kabat, and Massimo Porrati, of two new papers about the shape of the universe.

The papers compactify the extra dimensions of the universe on a Klein bottle. The non-orientable topology, it turns out, breaks CP symmetry (for free). This is the first long-form podcast treatment of that work.

Soon, I’m going live on Substack with Prof. Janna Levin in the coming days… You can join and ask questions! Stay watching my notes here to find out when. Should be in a week or so.

In today’s episode, we cover:

• The cold open: is the universe a Gödel sentence about itself?

• “The shape of space might have a lot to do with the universe that we live in.” Topology as the all-stop shop: possibly dark matter, possibly dark energy…

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We cover:

• Whether ER equals EPR is a literal identity

• DESI’s hint that dark energy may be time-varying, and why Maldacena says w less than minus one would be “the biggest news in the last hundred years”

• When you put one observer with a clock inside the universe, the de Sitter partition function goes from imaginary to real-positive. Basically, the observer is what gives geometry something to count. We need observers for quantum gravity to work.

• The quantum extremal surface, the Page curve, and the island formula as one variational principle that resolves Hawking’s information paradox.

• Maldacena calling himself “a dinosaur” on AI, and what he is doing about it: a current follow-up project with Alfredo Guevara to a recent Strominger paper that used GPT to derive a new gluon amplitude formula.

• And what it means that the architect…

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I sat down with , who teaches at the University of Ljubljana, holds a senior research post at the Birkbeck Institute for the Humanities in London, and is a global distinguished professor at NYU.

For those who know him only through his political writing, this conversation will read like a different author. I asked him at the start to keep it apolitical. He agreed immediately, and the next two hours went to what Žižek actually cares about: ontology, quantum mechanics, consciousness, and the materialism he has spent a decade reading out of Hegel.

His claim is compact. Quantum indeterminacy is the substrate being, as he puts it, less than nothing. The universe is unfinished. Freedom lives in that gap.

Of course, I press him (politely and with respect) on these claims, and we have some fun.

We cover:

• “Freedom is a self-chosen form of necessity.“ Why falling in love is the cleanest example of a free act.

• The pushback: I press Žižek …

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Why can't local agreement scale to global truth? At the Mind at Large consciousness conference, hosted by the Center for Process Studies, I make the case using sheaf theory and physics — breaking down consciousness, free will, and the hard problem to show why sincerity beats any polished, totalizing philosophy. FOLLOW: - Spotify: https://open.spotify.com/show/4gL14b92xAErofYQA7bU4e - Substack: https://curtjaimungal.substack.com/subscribe - Twitter: https://twitter.com/TOEwithCurt - Discord Invite: https://discord.com/invite/kBcnfNVwqs - Crypto: https://nowpayments.io/donation/TOE - PayPal: https://www.paypal.com/donate?hosted_button_id=XUBHNMFXUX5S4 TIMESTAMPS: - 00:00:00 - Intellectual Slogan Fatigue - 00:06:11 - The Reverse Elephant Metaphor - 00:11:12 - Local-to-Global Extension Failures - 00:16:13 - Three Senses of Irreducibility - 00:21:16 - Sheaf Theory and Obstructions - 00:26:30 - Hard Problem Formalizations - 00:31:36 - Averted Vision and Consciousness - 00:36:36 - Taxonomy of Everything - 00:41:39 - Pre-articulate Scientific Intimations - 00:46:45 - Speculative Philosophy and Humility - 00:51:47 - Ontological Significance of Vagueness LINKS MENTIONED: - Matthew Segall [TOE]: https://youtu.be/DeTm4fSXpbM - Jenann Ismael [TOE]: https://youtu.be/7kvXihDAOi0 - Aephraim Steinberg [TOE]: https://youtu.be/cOZ3Kto6NIc - The Third Option [TOE]: https://youtu.be/tJsghrZQaYU - Scott Aaronson & David Chalmers [TOE]: https://youtu.be/7PlmOXQ18jk - Curt Debunks "All Possible Paths" [TOE]: https://youtu.be/XcY3ZtgYis0 - Consciousness Iceberg [TOE]: https://youtu.be/65yjqIDghEk - Stephen Wolfram [TOE]: https://youtu.be/FkYer0xP37E - Karl Friston [TOE]: https://youtu.be/uk4NZorRjCo - Michael Levin [TOE]: https://youtu.be/c8iFtaltX-s - Geoffrey Hinton [TOE]: https://youtu.be/b_DUft-BdIE - Noam Chomsky [TOE]: https://www.youtube.com/playlist?list=PLZ7ikzmc6zlORiRfcaQe8ZdxKxF-e2BCY - Anna Ciaunica & Michael Levin [TOE]: https://youtu.be/2aLhkm6QUgA - Mind-at-Large Project: https://ctr4process.org/mind-at-large/ - Process and Reality [Book]: https://amazon.com/dp/0029345707?tag=toe08-20 - The Structure of Science [Book]: https://amazon.com/dp/0915144719?tag=toe08-20 - Sorites Paradox: https://plato.stanford.edu/entries/sorites-paradox/ - The Mathematics of Self [Article]: https://curtjaimungal.substack.com/p/the-mathematics-of-self-why-you-can - Gödel, Escher, Bach [Book]: https://amazon.com/dp/0465026567?tag=toe08-20 - Sheaf (Mathematics): https://en.wikipedia.org/wiki/Sheaf_(mathematics) - Counterexamples in Topology [Book]: https://amazon.com/dp/048668735X?tag=toe08-20 - Process Philosophy: https://plato.stanford.edu/entries/process-philosophy/ - Consciousness Explained [Book]: https://amazon.com/dp/0316180661?tag=toe08-20 - The Conscious Mind [Book]: https://amazon.com/dp/0195117891?tag=toe08-20 - Matter and Memory [Book]: https://amazon.com/dp/1420937804?tag=toe08-20 - Boiling Frog: https://en.wikipedia.org/wiki/Boiling_frog - Peano's Axioms: https://mathworld.wolfram.com/PeanosAxioms.html - Aharonov-Bohm Effect: https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect More links at https://curtjaimungal.substack.com Guests do not pay to appear. #science

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