Advances in artificial intelligence raise major questions

Artificial intelligence in technology; credit: iStock-metamorworks

A brief history

The modern field of artificial intelligence (AI) arguably dates to 1950, when Alan Turing outlined the basics of AI in his paper “Computing machinery and intelligence” [Paper]. He even proposed a test, now known as the Turing test, for establishing whether true AI had been achieved. Early computer scientists were confident that true AI system would soon be a reality. In 1965 Herbert Simon wrote that “machines will be capable, within twenty years, of doing any work a man can do.” In 1970 Marvin Minsky declared, “In from three to

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Where are the extraterrestrials? Fermi’s paradox, diversity and the origin of life

Delicate Arch at night; credit:


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. This article presents background on Fermi’s paradox, explains why many of the proposed solutions are not viable, and mentions a few promising new results and directions.

Behind Fermi’s question was this line of reasoning: (a) Given the vast number of stars in the Milky Way (not to mention the larger

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PiDay 2022 crossword puzzle

So soon? Yes, it is that time of year again — PiDay, namely Mar 14 (from 3/14 in North American date notation) is less than two weeks away. Continuing a long tradition on the Math Scholar blog, we present a custom-constructed crossword puzzle to commemorate the occasion.

This year’s puzzle commemorates a well-known pi-related theorem, one of the most beautiful facts in mathematics, which was originally discovered in the 18th century. The theorem is stated, in full, in the completed puzzle (see clues 20A, 30A and 44A).

My spouse and one daughter, who solved the puzzle, agreed that in terms

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Latest experimental data compounds the Hubble constant discrepancy

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The standard model and the Lambda-CDM model

The standard model of physics, namely the framework of mathematical laws at the foundation of modern physics, has reigned supreme since the 1970, having been confirmed in countless exacting experimental tests. Perhaps its greatest success was the prediction of the Higgs boson, which was experimentally discovered in 2012, nearly 50 years after it was first predicted.

One application of the standard model, together with general relativity, is the Lambda Cold Dark Matter model (often abbreviated Lambda-CDM or Λ-CDM), which governs the evolution of the entire universe from the

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The brave new world of probability and statistics

Block function approximation to normal distribution


Today, arguably more than ever before, the world is governed by the science of probability and statistics. “Big data” is now the norm in scientific research, with terabytes of data streaming into research centers from satellites and experimental facilities, analyzed by supercomputers. “Data mining” is now an essential part of mathematical finance and business management. Numerous public opinion polls, expertly analyzed, guide the political arena. Covid-19 infection rates, immunization levels and r0 factors are a staple of nightly newscasts.

Yet the public at large remains mostly ignorant of the basic principles

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Computer theorem prover verifies sophisticated new result

Courtesy Maria Nguyen, Quanta Magazine

Computer discovery of mathematical theorems

In 1983 the present author recalls discussing the future of mathematics with Paul Cohen, who in 1963 proved that the continuum hypothesis is independent from the axioms of Zermelo-Fraenkel set theory. Cohen was convinced that the future of mathematics, and much more, lies in artificial intelligence. Reuben Hersch recalls Cohen saying specifically that at some point in the future mathematicians would be replaced by computers. So how close are we to Cohen’s vision?

In fact, computer programs that discover new mathematical identities and theorems are already a staple

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Bach as mathematician

Johann Sebastian Bach; credit Wikimedia

OK. Johann Sebastian Bach (1685-1750) was not a mathematician in a strict sense of the word. There is no “Bach convergence theorem” in real analysis, nor is there a “Bach isomorphism theorem” in algebra. Bach had no formal training in mathematics beyond elementary arithmetic.

But, as we will see, Bach was definitely a mathematician in a more general sense, as a composer whose works are replete with patterns, structures, recursions and other precisely crafted features. There are even hints of Fibonacci numbers and the golden ratio in Bach’s music (see below). Indeed, in this larger

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AI system finds counterexamples to graph theory conjectures


AI comes of age

After several decades of disappointment, effective artificial intelligence (AI) systems emerged in the late 1990s and early 2000s, with the emergence of Bayes-theorem-based methods, combined with steadily advancing computer technology.

One notable milestone came in March 2016, when a computer program named “AlphaGo,” developed by researchers at DeepMind, a subsidiary of Alphabet (Google’s parent company), defeated champion Go master Lee Se-dol, an achievement that many observers had not expected to occur for decades. Then in October 2017, Deep Mind researchers announced a new program, AlphaGo Zero, which was programmed only with the rules

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Muon result may rewrite standard model of physics

Muon g-2 experiment (courtesy Fermilab)

A new measurement of the magnetic moment of the muon may draw into question the standard model of physics, the reigning theoretical construct describing all known fundamental forces and particles of physics. The new result was released in a paper dated today (7 April 2021) with 240 authors, led by researchers at Fermilab in the U.S., but also including researchers from Italy, Germany, United Kingdom, Russia, South Korea, China and Croatia.

The standard model of physics is arguably is the most successful physical theory ever devised, explaining all known fundamental particles and all known forces

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Aho and Ullman receive the ACM Turing Award

Alfred V. Aho (courtesy ACM)

Jeffrey D. Ullman (courtesy ACM)

The 2020 Alan M. Turing Award, bestowed by the Association for Computing Machinery, has been granted to Alfred V. Aho, Professor Emeritus at Columbia University, and Jeffrey D. Ullman, Professor Emeritus at Stanford University. The ACM Turing Award, which is named after computing pioneer Alan Turing, is widely considered to be the most prestigious award in the field of computer science. Past recipients include many of the most accomplished figures in the field, including Richard Hamming, Donald Knuth, William Kahan, Edward Feigenbaum, Jim Gray, Tim Berners-Lee, John Hennessy and David

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