Computation, AI and the future of mathematics

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Optimal stacking of oranges.

Computer tools in mathematics: From foot dragging to full embrace

The present author recalls, while in graduate school at Stanford nearly 50 years ago, hearing two senior mathematicians (including the present author’s advisor) briefly discuss some other researcher’s usage of computation in a mathematical investigation. Their comments and body language clearly indicated a profound disdain for such efforts. Indeed, the prevailing mindset at the time was “real mathematicians don’t compute.”

Some researchers at the time attempted to draw more attention to the potential of computer tools in mathematics. In the

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Pi Day 2024 crossword puzzle

Yes, Pi Day (March 14), that day of mathematical trivia and pie-eating, is almost here again.

As is the custom on this site, we present below a crossword puzzle constructed on a theme of mathematics, computing and the digits of pi. Enjoy!

This puzzle has been constructed in conformance with New York Times crossword puzzle standards. In terms of overall difficulty, my mathematician daughter and my genealogist spouse agreed that it would likely rate as a Tuesday (the New York Times grades its puzzles, with Monday puzzles being the easiest and Saturday puzzles being the most challenging).

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Using network theory to analyze Bach’s music

Johann Sebastian Bach; credit Wikimedia

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Bach as “mathematician”

Johan Sebastian Bach (1685-1750) regularly garners the top spot in listings of the greatest Western classical composers, typically followed by Mozart and Beethoven. Certainly in terms of sheer volume of compositions, Bach reigns supreme. The Bach-Werke-Verzeichnis (BWV) catalogue lists 1128 compositions, from short solo pieces to the magnificent Mass in B Minor (BWV 232) and Christmas Oratorio (BWV 248), far more than any other classical composer. Further, Bach’s clever, syncopated style led the way to twentieth century musical innovations, notably jazz.

There does seem to be a

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Overcoming experimenter bias in scientific research

History of measurements of the speed of light; from Henrion and Fischloff, “Assessing uncertainty in physical constants,” American Journal of Physics, September 1986

Reproducibility in modern science

Reproducibility has emerged as a major issue in numerous fields of scientific research, ranging from psychology, sociology, economics and finance to biomedicine, scientific computing and physics. Many of these difficulties arise from experimenter bias (also known as “selection bias”): consciously or unconsciously excluding, ignoring or adjusting certain data that do not seem to be in agreement with one’s preconceived hypothesis; or devising statistical tests post-hoc, namely AFTER data has already been

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Progress in troubled times: Good news in science, medicine and society

The CRISPR-Cas9 editing system (artist’s impression) for experimental treatments. Credit: Biolution GmbH/Science Photo Library; see

In today’s world, there is certainly no shortage of troubling news: full-scale war in Ukraine and Gaza; simmering conflicts in Sudan, Myanmar, Niger, Afghanistan, Colombia and Somalia; wildfires, droughts and floods connected to human-induced climate change, which world governments have been deplorably slow to address; outbreaks of Covid-19, malaria, RSV and other diseases; hunger and poverty afflicting millions worldwide; a troubling receptiveness for autocracy; continuing sexism, racism and xenophobia; and large numbers of people embracing utterly discredited pseudoscientific conspiracy theories (“your Zodiac sign holds

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Aliens made this rock: The post-hoc probability fallacy in biology, finance and cosmology

Updated 1 February 2024

Space aliens made this rock

Extraterrestrial aliens made this rock

While out hiking, I found this rock. The following table gives measurements made on the rock. The first two rows give the overall length and width of the rock. Each of the next six rows, after the first two, gives thickness measurements, made on a 3cm x 6cm grid of points from the top surface. All measurements are in millimeters:

Measurement or row Column 1 Column 2 Column 3 Length 105.0 Width 48.21 Row 1 35.44 35.38 36.54 Row 2 38.06 38.27

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DeepMind program discovers new sorting algorithms

Quicksort algorithm; credit: Abhilash Kakumanu

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The operation of sorting a dataset is one of the most fundamental of all operations studied in computer science. Literally trillions of sort operations are performed each day worldwide, more if one counts operations where a relatively small set of elements are merged into a larger set.

Many sorting algorithms are in use, including special routines for datasets of a certain size, and other routines optimized for specific hardware platforms and types of data.

One relatively simple algorithm, which is actually quite efficient, is the “quicksort” algorithm. It

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Is modern science socially constructed and forever tentative?

Is modern science socially constructed and forever tentative? Updated 7 April 2024 (c) 2024


Writers from the discipline known variously as “postmodern science studies” or “sociology of scientific knowledge” are often cited in discussions of science, philosophy and religion. Some of these writers, notably Karl Popper and Thomas Kuhn, have had significant impact on the field of scientific research.

Issues such as ensuring proper credit for the scientific contributions of non-Western societies, such as the ancient mathematics of China, India and the Middle East, as well as dealing with the chronic under-representation of women, racial minorities and indigenous

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Peter Borwein: A visionary mathematician

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Peter Borwein

Peter Borwein, former professor of mathematics at Simon Fraser University and director of the university’s Centre for Interdisciplinary Research in the Mathematical and Computational Sciences (IRMACS), died on August 23, 2020, at the age of 67, of pneumonia, after courageously battling multiple sclerosis for over 20 years.

The Notices of the American Mathematical Society has just published a memorial tribute, written by the present author, that summarizes Peter’s life and career. Here are a few highlights:

Peter Borwein is perhaps best known for discovering (often but not always with his brother Jonathan) new

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PiDay 2023 crossword puzzle

So soon? Yes, it is that time of year again — PiDay (March 14) is just a few days away. In honor of this mathematical holy day, we present once again a crossword puzzle with an appropriate pi-related theme.

This year’s puzzle implements a new design for a mathematical crossword, which to the present author’s knowledge has never before been employed. See, for example, clue 40 Across below. In all respects, though, the puzzle conforms to the standards of New York Times crosswords. In terms of overall difficulty (Monday = easiest; Saturday = most difficult), this puzzle most likely would

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