Pablo Ibieta

Physics Ph.D. - Data Scientist

[alphaXiv @askalphaxiv]

"Mathematics in the Age of AI"

In an age of abundant AI-generated proofs, human understanding may become more valuable than proof generation itself.

This new paper from Terence Tao argues that AI could make solving hard math problems much cheaper and faster, but at the same time, creates a new problem.

If AI can generate thousands of correct proofs, mathematicians can’t possibly read, understand, verify, and teach all of them.

So the scarce resource in mathematics shifts from finding proofs to making sense of them.

The important work becomes deciding which results matter, explaining the key ideas, connecting them to existing theory, and turning them into knowledge other mathematicians can actually use.

https://t.co/otIGL0SQrq

[Sabine Hossenfelder @skdh]

Physicists may have found the origin of gravity. And that would finally explain why gravity is different from the other forces of nature.

[SIAM Activity Group on Dynamical Systems @DynamicsSIAM]

"Mathematics in the age of AI" (by Terence Tao): https://t.co/KDBuVUhbaN

[note: Submitted to the Proceedings of the ICM 2026]

[Yann LeCun @ylecun]

Major difference in my mind:
- an engineer, given a problem, invents and tries multiple solutions and stops when the solution is good enough. The goal is product innovation and shipping.
- a scientist asks new questions, proposes various new solutions, compares them (sometimes with old ones), and writes about it. The methodology must be sound or else peers will sneer. The goal is scientific breakthroughs and technological progress.

Both can be called "researchers". Many people can do both: these are activities, not identities.

Importantly, most product innovations are built on scientific breakthroughs and technological innovations that happened 2, 5, 10, or 20 years earlier.

[Pablo Ibieta @pabloibieta]

5/ Foundation post for the series: https://t.co/3pTPpB0woh
Prerequisite (mutual information vs. correlation): https://t.co/LW7kZRTy11

[Pablo Ibieta @pabloibieta]

1/ New post: conditional mutual information is just expected log-loss gain. The identity that turns CMI from an information-theoretic abstraction into a one-line addition to any training pipeline.

[Pablo Ibieta @pabloibieta]

4/ Once the identity is in hand, several decisions reduce to the same calculation with different conditioning sets:
– Switch the target? – Ensemble two scores? – At what interaction depth does my representation carry signal?
Each is its own post, coming up.

[Pablo Ibieta @pabloibieta]

3/ One catch — the log-losses have to be evaluated out-of-sample, otherwise the more flexible model overfits more, and the difference inherits the asymmetry. K-fold cross-fitting handles it. Same idea as Chernozhukov et al.'s debiased ML.

[Pablo Ibieta @pabloibieta]

2/ Train a model on Z. Train another on (S, Z). For each held-out example, take the difference of their log-losses. Average. That number is I(S; Y | Z).Why it works: for binary Y, expected log-loss at the truth equals the binary entropy of the truth.

[Sabine Hossenfelder @skdh]

String theory didn't just fail, it's far worse.

[Dmytro Krasun @DmytroKrasun]

I made https://t.co/Ihv3IPlwao to explain gravity to my kids. They can just play with mass and see how it bends space.

You don’t have to explain concepts anymore. You can build them!

[Pablo Ibieta @pabloibieta]

@sunilchandrase3 @predict_addict Of course there is, attention is just a cumbersome kernel trick.

[Mathematica @mathemetica]

max⁡(∣x∣,∣y∣,∣z∣,∣w∣)=1

You’re witnessing a 2D projection of a 3D shadow cast by a 4D tesseract.

As it rotates through the W-axis, the "inner" and "outer" cubes swap roles: a spatial inversion that feels like a glitch only because our biology is trapped in 3-space.
It’s a hauntingly beautiful reminder that our "reality" is often just a lower-dimensional cross-section of a much more complex structure. Perspective is everything.

This is used in high-dimensional data visualization, hypercube topologies in parallel computing networks, and exploring the geometry of extra dimensions in theoretical physics.

[Mathelirium @mathelirium]

A physical system, its phase-space state, and the energy function that governs the flow, all linked together.

The point of the animation is that these are not three different objects. They are three views of the same dynamics. The pendulum on the left is the physical motion. The curve on the floor is the evolution of its state. The translucent surface shows the energy landscape that organizes that motion.

[Pablo Ibieta @pabloibieta]

@predict_addict This paper overstates its results. The idea is good but they do no fully estimate the geometric product, which I think is the major flaw, the antisymmetric components are computationally expensive, further more if you go to higher dimensions.

[Michał Podlewski @trajektoriePL]

Terence Tao proposes what he calls a "Copernican view of intelligence".

Instead of buying into the common, one-dimensional narrative that artificial intelligence will simply evolve from "subhuman" to "superhuman" and ultimately make humanity entirely redundant, Tao urges us to look at the bigger picture.

Much like the Copernican revolution proved the Earth is not the center of the universe, Tao suggests we need to realize that human intelligence isn't the only, or necessarily the highest, form of intellect. Historically, we have treated other forms of storing or creating knowledge—like animals, books, and computers—as secondary. However, we actually exist within a much richer universe of intelligence.

Both human intelligence and computer intelligence possess their own distinct strengths and weaknesses. The true potential lies not in viewing them as direct competitors, but rather in focusing on collaboration. By working together, humans and computers can achieve additional things that neither could accomplish on their own, requiring us to think in much wider terms than just what humans or computers can do alone.

[Simone Conradi @S_Conradi]

New in the @marimo_io gallery!

How do you invert a matrix? Let a system reach thermal equilibrium and read the answer from its fluctuations.

Interactive notebook on thermodynamic linear algebra, based on Aifer et al. (2024).👇👇👇👇

[Valentin Ignatev @valigo]

Knuth dropped a paper that highlights a very cool AI usage that midwits, unfortunately, call vibemathing. It's the opposite of "vibe-" anything:

What happened is Filip Stappers gave the problem to Claude with a very tight self-reflecting loop, verbatim:

** After EVERY exploreXX py run, IMMEDIATELY update this file [plan md]
before doing anything else. ** No exceptions. Do not start the next exploration
until the previous one is documented here.

And then they meticulously verified each of 31 loops. And after the final loop, which seemed successful, Knuth wrote an actual formal proof theorem to fully verify the results. This is real math with using power tools that is super cool!

I think it's pretty safe to say that formal proof research will prove (haha got it) to be very important in the age of hallucinating machines.