Mathematics, computing and modern science
Illustration of the planar unit distance problem. Credit: Noga Alon, et al., 2026, OpenAI Introduction By several measures, the past year has been a banner year for advances in artificial intelligence in many fields, and perhaps most remarkably in mathematics. While these software systems still have many shortcomings, and some perform significantly better than others, nonetheless the overall…
The march of human progress There is much to celebrate in modern society: Crime rates are declining worldwide; worldwide extreme poverty continues to fall; worldwide life expectancy continues to rise; computer and information technologies continue their remarkable forward march; artificial intelligence is on the rise, which, while carrying some risks, also holds great promise; and advances in…
MathJax = { tex: {inlineMath: [['$', '$'], ['\\(', '\\)']] }, svg: {fontCache: 'global'}}; Credit: Vatican Museum 2023-2025: Can chatbots prove mathematical theorems? Three years ago (23 Feb 2023), the present author tested ChatGPT, which then was the state-of-the-art in AI-based large language models (LLMs), with requests to produce rigorous mathematical proofs for four well-known but nontrivial…
credit: Vera C. Rubin observatory Where are the extraterrestrial civilizations? (Fermi s paradox) Updated 23 May 2026 (c) 2026 Introduction In 1950, while having lunch with colleagues Edward Teller and Herbert York, who were chatting about a recent cartoon in the New Yorker depicting aliens, physicist Enrico Fermi suddenly blurted out, Where is everybody?, a question now known as Fermi s paradox.…
Happy PiDay! Just in time for PiDay, a team of researchers at the Israel Institute of Technology have announced the results of using AI-based computer tools to automatically catalog a large body of existing pi formulas. In particular, these researchers found underlying structure in these formulas, in particular a conservative matrix field (CMF) functions as a sort of fingerprint, enabling…
Yes, in just a few days we will once again celebrate Pi Day (3/14 in North American date notation). Our custom on this site has been to offer a crossword puzzle on a math or Pi theme. This year we offer two puzzles (take your pick or do both). Some notes: 1. Each puzzle is written to standards of The New York Times daily puzzles. 2. In each puzzle, a handful of the squares contain two letters. 3.…
Source: New York Times A discouraging year Today, as a year ago when we posted a similar report, there is no shortage of discouraging news headlines, among them (as of 1 Jan 2026): A war in Ukraine that has continued for nearly four years, killing or injuring hundreds of thousands on each side. A war in Gaza that, despite a shaky cease-fire, raises serious questions about the long-term stability…
Google data center in Ohio Artificial intelligence s march of progress Without doubt, artificial intelligence (AI) and machine learning (ML) systems have scored numerous remarkable successes over the past few years: Defeating the reigning world chess champion. Defeating two champion contestants on the American quiz show Jeopardy. Defeating the world s best human Go player. Defeating all other…
Credit: Yuichiro Chino/Getty Updated 10 June 2025 Artificial intelligence s march of progress One by one, large AI- and machine learning-based systems have scored numerous successes over the past few years; it is clear that we have witnessed (and are witnessing) a revolution in the field. Among these successes are the following: Defeating the reigning world chess champion. Defeating two champion…
Credit: Lawrence Berkeley National Lab MathJax.Hub.Config({tex2jax: {inlineMath: [['$','$'], ['\\(','\\)']]}}); Updated 2 April 2025 The Hubble tension One experimental anomaly that has bedeviled physicists and cosmologists for years is the discrepancy in values of the Hubble constant based on different experimental approaches. This discrepancy is now known as the Hubble tension. The Hubble…