Over a succession of Tuesday mornings in the spring of 1962 my brother Robert and I met infinity and took its measure. We wasted no time getting ready for school on Tuesdays, because the early morning math class taught by Mr. R.W. McCarley started at 7:30.
Mr. McCarley was younger than most of our teachers. Whether he came in early for us and a few other students from pure enthusiasm, or possibly gained some compensation from the school or our parents, I never thought to question.
Whoever had the idea for that extra class must have received an enthusiastic response from our father. When he was a senior in high school in Oxford, Mississippi, in the 1930s, the faculty decided that he had mastered everything they had to teach. So they let him study books on electronics and physics, aided by tutoring from professors at the University.
In Mr. McCarley’s class we were exposed to Boolean set theory and other subjects well beyond the usual high school geometry, algebra, and pre-calculus. Most of all I remember that he introduced us to different orders of infinity. Mr. McCarley told us that the infinite series of rational numbers, which includes all the numbers represented by fractions of two integers, is no larger than the infinite series of integers. This is true despite the fact that between any 2 rational numbers is an infinitely divisible gradation. The crucial insight is that all rational numbers can be mapped onto the infinite set of integers (see Box).
Although it might seem that there are more rational numbers than integers, the order of infinity is the same. Not so for the set of real numbers. Real numbers include all rational numbers and also numbers that cannot be expressed as a fraction of 2 integers (an example is “pi”). Real numbers cannot be mapped onto the infinite set of integers. Therefore, the set of real numbers has a higher order of infinity. Wow!
Mr. McCarley invited us to try to bisect, then to trisect an angle drawn on a sheet of paper, using a compass and ruler. The first operation is simple. The second, dividing a given angle into 3 equal smaller angles, is impossible. After letting us sweat it out for half an hour (before 8:15!), Mr. McCarley confessed as much, and he added that the impossibility of trisecting an angle had been proven only recently in the history of mathematics.
Counting, arithmetic, and geometry developed all over the world in concert with trading as well as religious observation of the heavens. Multiplication tables appear on clay tablets dated as early as 2500 BCE in Sumeria. An Egyptian papyrus from about 1650 BCE is an instructional text for students of arithmetic and geometry. We count from 1 to 10 and then start counting again, 11 to 20, and so on. The Babylonians counted from 1 to 60 before starting over. Today we divide an hour into 60 minutes and a minute into 60 seconds, thanks to the Babylonians. The decimal system of numbers with the critical inclusion of “0” for zero first appeared in India. This knowledge was transferred and enhanced by Persian and Arab scholars before reaching the West. “Algorithm” and “algebra” are words derived from Arabic.
In ancient Greece a remarkable enthusiasm flared up for elucidating angles, figures, and numbers. For some it became much more than a game or pastime. Pythagoras, son of a Mediterranean trader in the late 6th century BCE, may have caught the spark from Egyptian priests or from travelling Chaldeans in Tyre. Either Pythagoras or his followers devised a kind of geometrical algebra. Their contributions included what we still call the Pythagorean theorem, which proves that the square formed on the hypotenuse of a right triangle is equal in area to sum of squares of the 2 sides (see header image for this blog).
The Pythagoreans applied measurement and number to the lengths of vibrating strings, demonstrating a quantitative theory of music. They formed secret societies devoted to ritual practices, dietary restrictions, philosophical purification, and geometry. The inner circle, those who learned to prove the theorems, were called mathematikoi, and from them our word “mathematics” derives. Aristotle later wrote, “The Pythagorean … having been brought up in the study of mathematics, thought that things are numbers … and that the whole cosmos is a scale and a number.”
Mathematics has long been connected with notions of ultimate discovery and true reality. My encounter with infinity in the early morning math class gave me a small sense of what this means. The Pythagoreans took it as far as they could, and others extended the trail. It is no coincidence that René Descartes, who devised the Cartesian coordinate system giving algebraic expression to geometry, also reasoned his way to a set of truths which, he said, could not be doubted.
With the advent of computers, transistors, and integrated circuits during my lifetime, mathematics has escaped almost every imaginable expanding cage of computational limits. Smartphones in the hands of children do billions of calculations per second.
It’s important to remember that mathematics emerged first and prepared the way for digital application, as soon as the engineering physics of germanium and silicon, especially silicon, reached a stage to allow it. For more than 4 millennia until the first electronic computer and the first transistor appeared, mathematics performed manually, a product of pure human thought, earned its place as the central organizing core of science, and arguably its most advanced expression.
During this time the Pythagorean dream inspired leading thinkers in every age to approach the possibility, subliminal yet guiding, that mathematics might describe ultimate reality.
Today mathematics rises to new heights on a surging crest of technology. In the next blog we’ll examine how the mathematical foundation for our digital age – a transition from classical mathematics to statistics and simulation – was laid more than a century ago in the mid-1800s.
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Header image (c) William B. Faulk, CC by SA 4.0 Intl, Wikimedia Commons.
Sources various, especially www-history.mcs.st-andrews.ac.uk from School of Mathematics and Statistics, University of St Andrews, Scotland.
 O.Connor, J.J. and Robertson, E.F. Pythagoras of Samos, accessed 6/20/2016 at http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Pythagoras.html.