## Can computers do mathematical research?

AI comes of age

The modern field of artificial intelligence (AI) began in 1950 with Alan Turing’s landmark paper Computing machinery and intelligence, which outlined the principles of AI and proposed the Turing test. Although early researchers were confident that AI systems would soon be a reality, inflated promises and expectations led to disappointment in the 1970s and again in the 1980s.

A breakthrough of sorts came in the late 1990s and early 2000s with the emergence of Bayes-theorem-based methods, which quickly displaced the older methods based mostly on formal reasoning. When combined with steadily advancing

Continue reading Can computers do mathematical research?

## Peter Borwein dies at 67

Peter Borwein (sitting), with his brother Jonathan Borwein (courtesy Canada Foundation for Innovation)

Peter Borwein, retired Professor of Mathematics at Simon Fraser University (British Columbia, Canada) and former Director of SFU’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.

Peter was a prolific mathematician, with over 200 publications, including several books. His research included works in classical analysis, computational number theory, Diophantine number theory and symbolic computing. Many of these papers were co-authored with his brother

Continue reading Peter Borwein dies at 67

## Two mathematicians’ triple play

MathJax.Hub.Config({tex2jax: {inlineMath: [[‘$’,’$’], [‘\$$‘,’\$$’]]}});

The Erdős-Turán conjecture

Paul Erdős, one of the twentieth century’s most unique mathematicians, was known to travel from colleague to colleague, often arriving unannounced, and to immediately delve into some particularly intriguing research problem. See this article and this book for some additional background on this influential mathematician.

One of his more interesting conjectures is his “conjecture on arithmetic progressions,” sometimes referred to as the “Erdős-Turán conjecture.” It can be simply stated as follows: If $A$ is a set of positive integers such that $$\sum_{k \in A} \frac{1}{k} = \infty,$$ then $A$ contains arithmetic progressions of

Continue reading Two mathematicians’ triple play

## How old is the universe? New results clash

MathJax.Hub.Config({tex2jax: {inlineMath: [[‘$’,’$’], [‘\$$‘,’\$$’]]}});

The standard model

The standard model of physics, namely the framework of laws at the foundation of modern physics, has reigned supreme since the 1970s, confirmed to great precision in a vast array of experimental tests. Among other things, the standard model predicted the existence of the Higgs boson, which was experimentally discovered in 2012, nearly 50 years after it was first predicted.

Yet physicists have recognized for many years that the standard model cannot be the final answer. For example, quantum theory and general relativity are known to be mathematically incompatible. String theory and

Continue reading How old is the universe? New results clash

## Covid-19 and the worth of a human life

The scales of justice (courtesy Wikimedia)

Covid-19’s grim toll

The statistics are staggering: As of 1 June 2020, according to the Johns Hopkins University database, the U.S. had logged over 1.811 million confirmed cases of Covid-19 and over 105,000 deaths. The U.K. was next, with over 277,000 confirmed cases and over 38,000 deaths. Worldwide, over 6.3 million cases had been confirmed, with more than 376,000 deaths. If current trends continue, the U.S. death toll alone will soon exceed that of all wars in its history except for the Civil War and World War II.

The economic costs have been

Continue reading Covid-19 and the worth of a human life

## The origin of life in an inflationary universe

RNA (on left) compared with DNA (on right); courtesy Wikimedia

The abiogenesis problem

Exactly how life first emerged on Earth (the “abiogenesis” problem) remains a critical unsolved question in biology. Was it inevitable, given a favorable environment, or was it a fantastically improbable event? All we know for sure is that it occurred at least 3.8 billion years ago and possibly more than 4 billion years ago. The fact that life arose relatively soon after the surface of the Earth solidified indicates to some that abiogenesis was inevitable, but there is no way to know for sure. For further

Continue reading The origin of life in an inflationary universe

## Pseudoscience in the age of Coronavirus

The Covid-19 virus (courtesy U.S. Center for Disease Control and Prevention)

A pandemic is upon us

As this is being written (April 2020), the entire world is gripped in the throes of the rapidly spreading and deadly Covid-19 pandemic. International travel has been greatly curtailed worldwide; many businesses, large and small, have shut their doors; many K-12 schools and universities have closed; and entire regions and nations, encompassing well over one billion people, have been ordered to remain in their homes.

As of the current date (28 April 2020), the Johns Hopkins University Coronavirus Resource Center has tallied 3,062,000

Continue reading Pseudoscience in the age of Coronavirus

## “From Analysis to Visualization: A Celebration of the Life and Legacy of Jonathan M. Borwein”

We are happy to announce the publication of “From Analysis to Visualization: A Celebration of the Life and Legacy of Jonathan M. Borwein”, a compilation of research papers devoted to the memory of Jonathan Borwein. The book is the proceedings of a conference held in Borwein’s honor in September 2017 at Newcastle, Australia, near where Prof. Borwein taught for several years before passing away in August 2016.

The volume has been published by Springer, and is available for purchase from the Springer website, or from Amazon.com.

The individual papers are authored by many of Jonathan Borwein’s colleagues and collaborators. Here

## Why are people embracing astrology in an age of science?

An age of unparalleled progress

Though many do not recognize the fact, behind the disturbing headlines that dominate the news today, scientific progress marches forward, unabated and undiminished. Just within the past 100 years, researchers have discovered the theory of relativity, quantum mechanics and the standard model; unraveled the structure of DNA; sequenced the human genome; discovered the accelerating universe; observed extrasolar planets orbiting thousands of distant stars; and detected the collisions of black holes. See this Math Scholar article for additional details.

Continue reading Why are people embracing astrology in an age of science?

## PiDay 2020: A catalogue of formulas involving pi, with analysis

I have prepared a new paper containing a catalogue of 72 summation formulas, integral formulas and iterative algorithms for Pi. The catalogue contains both classical and modern formulas, ranging from Archimedes’ 2200-year-old algorithm to intriguing formulas found by Ramanujan and the quadratic, cubic, quartic and nonic algorithms of Jonathan Borwein and Peter Borwein, the latter of which double, triple, quadruple and nine-times, respectively, the number of correct digits with each iteration.

The catalogue of formulas and iterative algorithm is followed by results of carefully designed computer implementations, which enable one to compare the relative speed of these formulas.

Continue reading PiDay 2020: A catalogue of formulas involving pi, with analysis