In 1896, a 24-year-old mathematician challenged Ludwig Boltzmann with a remarkable theorem. Ernst Zermelo argued that a mechanical system could eventually return arbitrarily close to its original state. But if entropy always increases, how could that happen? Boltzmann accepted the mathematics. His answer was probability: such returns may be possible, but for ordinary gases the waiting time would be unimaginably long. Their disagreement helped clarify why the second law of thermodynamics is statistical rather than absolute.

Full interview: youtu.be/anAmTGcewX8?si=D2mE…

Sam Altman Says Mathematicians Are Beginning to Accept AI's Role in Mathematical Discovery.

When functions begin to dance, even the world’s greatest dancers lose the rhythm of their steps.

In 1985, Carl Sagan warned the US Senate that increasing greenhouse gases could transform Earth's climate. He used Venus, with its surface temperature near 470°C, to explain the physics.

Postcard from Albert Einstein to David Hilbert, Berlin, 18 November 1915 Dear Colleague, The system of equations you have presented agrees, as far as I can see, exactly with what I have found in recent weeks and submitted to the Academy. The difficulty was not in finding generally covariant equations for the g₍μν₎, since this can easily be accomplished using the Riemann tensor. Rather, the difficulty was in recognizing that these equations represent a generalization, indeed a simple and natural generalization, of Newton's law. I succeeded in recognizing this only in the last few weeks (I sent you my first communication), although three years ago, together with my friend Grossmann, I had already considered the only possible generally covariant equations, which have now proved to be the correct ones. We abandoned them only with great reluctance, because my physical analysis had apparently indicated that they were incompatible with Newton's law. The important thing is that these difficulties have now been overcome. Today I am presenting to the Academy a paper in which, using general relativity and without any auxiliary hypothesis, I quantitatively derive the perihelion motion of Mercury discovered by Le Verrier. No theory of gravitation has succeeded in doing this until now. With best regards, Yours, Einstein

“Mathematics Is a Subject for the Young” In a new essay published on Terence Tao's blog, NYU mathematician Tristan Buckmaster offers a notably optimistic perspective on the future of mathematics after recent breakthroughs in AI-generated proofs. His central argument is that the next generation of mathematicians, particularly graduate students and postdoctoral researchers, will shape the future of AI-assisted mathematics. Buckmaster compares this moment with IBM's Deep Blue defeating Garry Kasparov in 1997. Chess did not disappear when computers became stronger than humans. Instead, powerful engines transformed how people studied and played it. He believes something similar could happen in mathematics, although mathematical understanding presents a different challenge. He also raises concerns about the concentration of advanced AI capabilities inside a handful of corporations and argues for wider access, better attribution, and stronger protection of researchers' work. An interesting observation from the essay: "Mathematics has never been a list of true statements. Certifying a result settles the question, not the search for understanding."

Joseph Bertrand exposed this paradox in 1889.

On July 17, 2014, a 67-year-old retired mathematician was brushing his teeth when an idea suddenly struck him. Thomas Royen had found a proof of the Gaussian correlation inequality, a problem that had resisted mathematicians for decades. He wrote the proof in Microsoft Word and published it. Almost nobody noticed. It took years for the mathematical community to fully recognize what he had accomplished.