Simple proofs of great theorems

Euler’s identity Credit:

Mathematics and beauty

Modern mathematics is one of the most enduring edifices created by humankind, a magnificent form of art and science that all too few have the opportunity of appreciating. The great British mathematician G.H. Hardy wrote, “Beauty is the first test; there is no permanent place in the world for ugly mathematics.” Mathematician-philosopher Bertrand Russell added: “Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music,

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New books and articles on the “great silence”

Credit: NASA

The great silence

As we have explained in previous Math Scholar blogs (see, for example, MS1 and MS2), the perplexing question why the heavens are silent even though, from all evidence, the universe is teeming with potentially habitable exoplanets, continues to perplex and fascinate scientists. It is one of the most significant questions of modern science, with connections to mathematics, physics, astronomy, cosmology, biology and philosophy.

In spite of the glib dismissals that are often presented in public venues and (quite sadly) in writings by some professional scientists (see MS1 and MS2 for examples and rejoinders), there

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Does the string theory multiverse really exist?

Credit: Berkeley Center for Cosmological Physics

String theory, fine tuning and the multiverse

String theory is the name for the theory of mathematical physics which proposes that physical reality is based on exceedingly small “strings” and “branes,” embedded in 10- or 11-dimensional space. String theory has been proposed as the long-sought “theory of everything,” because it appears to unite relativity and quantum theory, and also because it is so “beautiful.”

Yet in spite of decades of effort, by thousands of brilliant mathematical physicists, the field has yet to produce specific experimentally testable predictions. What’s more, hopes that string theory

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The 2018 Fields Medalists honored

Every four years the Fields Institute of Toronto, Canada announces its Fields Medal recipients. This year’s recipients are Caucher Birkar, Alessio Figali, Akshay Venkatesh and Peter Scholze.

The Fields Institute announces its awardees at the every-four-years International Congress of Mathematicians (ICM) meeting, which this year is being held in Rio de Janeiro. The awards, which are made to a maximum of four exceptional mathematicians under the age of 40, are often considered the “Nobel Prize” of Mathematics.

Here is some information on this year’s awardees and their work:

Caucher Birkar. Birkar was born and raised in a very poor farming

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The rise of pay-to-publish journals and the decline of peer review

Pi nonsense in peer-reviewed journals

In a previous Math Scholar blog, we lamented the decline of peer review, as evidenced by the surprising number of papers, published in supposedly professional, peer-reviewed journals, claiming that Pi is not the traditional value 3.1415926535…, but instead is some other value. In the 12 months since that blog was published, other papers of this most regrettable genre have appeared.

As a single example of this genre, the author of a 2015 paper, which appeared in the International Journal of Engineering Sciences and Research Technology, states, “The area and circumference of circle has been

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Does beautiful mathematics lead physics astray?


In a new book, Lost in Math: How Beauty Leads Physics Astray, Sabine Hossenfelder reflects on her career as a theoretical physicist. She acknowledges that her colleagues have produced “mind-boggling” new theories. But she is deeply troubled by the fact that so much work in theoretical physics today is disconnected from empirical reality, yet is excused because the theories themselves, and the mathematics behind them, are “too beautiful not to be true.”

Hossenfelder notes that there have been numerous instances in the past when scientists’ over-reliance on “beauty” has led it astray. Newton’s clockwork universe, with seemingly self-evident

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Fermi’s paradox and the Copernican principle

Distant galaxies magnified by a gravitational lens

Fermi’s paradox

As we have discussed on this forum before (see, for example, previous Math Scholar blog), Fermi’s paradox looms as one of the most profound and puzzling conundrums of science: Given that the universe is presumed to be teeming with intelligent life and technological civilizations, why do we see no evidence of their existence? Although the search for signals and other evidence from extraterrestrial (ET) societies continues (and is accelerating with new facilities and funding), nothing has been found in over 50 years.

Ever since Fermi first declared the paradox in

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Pseudoscience from the political left and right

Pseudoscience through the ages

Projected global mean sea level rise

Through the years, decades and centuries, the world has science has slowly turned back the tide of pseudoscience, with victory after victory against nonsense and ignorance. In the 16th and 17th century, the writings of Copernicus, Galileo, Kepler and Newton overturned the ancient cosmology. The revolting practice of bloodletting was overturned in the late 19th century. Astrology was scientifically defeated in the 18th and 19th century, although, incredibly, it continues to attract faithful adherents even to this day. Young-earth creationism was scientifically overturned by the early 20th century, and now,

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Chromosomes, DNA and human evolution


The Yunis-Prakash diagram comparing the chromosomes of humans, chimpanzees, gorillas and orangutans

Evolution in general and human evolution in particular continue to be bones of contention, so to speak, as evidenced by the ongoing efforts by some groups to prohibit or downplay evolution, or to mandate “equal time” for “intelligent design,” in state and local high school curricula. At of the present date (May 2018), just in the U.S., campaigns are active in Indiana, Louisiana, Oklahoma, South Dakota, Tennessee, Texas and Utah.

Although the role of chromosomes in heredity and evolution was recognized in the 19th century, it was

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Will experimental anomalies lead to new physics?

A proton: two up quarks and one down quark

Historical anomalies in physics

It is often said that experimental anomalies lead to new physics. This is actually a bit overstated. Actually, the vast majority of experimental anomalies turn out to have more prosaic explanations — errors in the experimental setup or analysis, or errors stemming from invalid applications of the theory.

Nonetheless, a few experimental anomalies in years past have led to important new advances in the field. A few examples are:

In 1887, Michelson and Morley compared the speed light in two perpendicular directions, hoping to measure the

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