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Introduction
Archimedes is widely regarded as the greatest mathematician of antiquity. He was a pioneer of applied mathematics, for instance with his discovery of the principle of buoyancy, and a master of engineering designs, for instance with his “screw” to raise water from one level to another. But his most farreaching discovery was the “method of exhaustion,” which he used to deduce the area of a circle, the surface area and volume of a sphere and the area under a parabola. Indeed, with this method Archimedes anticipated, by nearly
Continue reading Simple proofs: Archimedes’ calculation of pi
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Isaac Newton, Credit: sjisblog.com
Introduction: The fundamental theorem of calculus, namely the fact that integration is the inverse of differentiation, is indisputably one of the most important results of all mathematics, with applications across the whole of modern science and engineering. It is not an exaggeration to say that our entire modern world hinges on the fundamental theorem of calculus. It has applications in astronomy, astrophysics, quantum theory, relativity, geology, biology, economics, just to name a few fields of science, as well as countless applications in all types of engineering — civil,
Continue reading Simple proofs: The fundamental theorem of calculus
Top 500 supercomputer performance (orange = #1, blue = #500, green = sum)
Postmodern denials of scientific progress
The fact that scientific research has made immense progress over the past years, decades and centuries is taken for granted among professional scientists and most of the lay public as well. But there are others, from both the left wing and the right wing of society, who question, dismiss or even reject the notion that science progresses. One group, which is mostly rooted in the right wing of society, rejects the scientific consensus on evolution, as with the creationism and intelligent
Continue reading Is scientific progress real?
AI’s tortuous history
The field of artificial intelligence (AI) is actually rather old. Ancient Greek, Chinese and Indian philosophers developed principles of formal reasoning several centuries before Christ. In 1651, British philosopher Thomas Hobbes wrote in Leviathan that “reason … is nothing but reckoning (that is, adding and subtracting).” In 1843 century Ada Lovelace, widely considered to be the first computer programmer, ventured that machines such as Babbage’s analytical engine “might compose elaborate and scientific pieces of music of any degree of complexity or extent.”
In 1950 Alan Turing’s landmark paper Computing machinery and intelligence outlined the principles of AI
Continue reading 2018: The year that artificial intelligence came of age
This annual Supercomputing conference is a showcase for the field of highperformance scientific and mathematical computing, featuring a firstrate peerreviewed technical program, tutorials, workshops, and a massive exhibit hall where universities, national laboratories and computer vendors from around the world exhibit their research, hardware and software. This year’s conference, SC18, which is being held in Dallas, Texas, has attracted well over 10,000 attendees. The conference is cosponsored by the Association for Computing Machinery (ACM) and the Computer Society of the Institute of Electrical and Electronic Engineers (IEEE).
Awards
Four prestigious professional society awards are presented at the SC18 conference. This
Continue reading US leads but China rises in latest Top500 supercomputer list
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Introduction: Ancient Greek mathematicians developed the methodology of “rulerandcompass” constructions: if one is given only a ruler (without marks) and a compass, what objects can be constructed as a result of a finite set of operations? While they achieved many successes, three problems confounded their efforts: (1) squaring the circle; (2) trisecting an angle; and (3) duplicating a cube (i.e., constructing a cube whose volume is twice that of a given cube). Indeed, countless mathematicians through the ages have attempted to solve these problems, and countless incorrect “proofs” have
Continue reading Simple proofs: The impossibility of trisection
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The fundamental theorem of algebra is the assertion that every polynomial with real or complex coefficients has at least one complex root. An immediate extension of this result is that every polynomial of degree $n$ with real or complex coefficients has exactly $n$ complex roots, when counting individually any repeated roots.
This theorem has a long, tortuous history. In 1608, Peter Roth wrote that a polynomial equation of degree $n$ with real coefficients may have $n$ solutions, but offered no proof. Leibniz and Nikolaus Bernoulli both asserted that quartic polynomials of
Continue reading Simple proofs: The fundamental theorem of algebra
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Mankind has been fascinated with $\pi$, the ratio between the circumference of a circle and its diameter, for at least 2500 years. Ancient Hebrews used the approximation 3 (see 1 Kings 7:23 and 2 Chron. 4:2). Babylonians used the approximation 3 1/8. Archimedes, in the first rigorous analysis of $\pi$, proved that 3 10/71 < $\pi$ < 3 1/7, by means of a sequence of inscribed and circumscribed triangles. Later scholars in India (where decimal arithmetic was first developed, at least by 300 CE), China and the Middle East computed
Continue reading Simple proofs: The irrationality of pi
Euler’s identity Credit: Redbubble.com
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.” Mathematicianphilosopher 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,
Continue reading Simple proofs of great theorems
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
Continue reading New books and articles on the “great silence”

