The Reality Oriented Mathematician

By

Dennis P. Allen, Jr., Ph.D.

Chapter 1: Where We Are Now

Dr. Thomas E. Phipps, Jr. in his excellent book “Heretical Verities” in discussing mathematics says: “People who value cactus cannot be prevented from seeking it out and wallowing in it. … People like G.H. Hardy (‘A Mathematician’s Apology’, Cambridge, 1969), who form the chief role models for modern pure mathematicians, have charted just this regrettable course – with a cost to mathematics that can never be reckoned. Hardy incidentally uses the word ‘significance’ where I use ‘fruitfulness’. His ‘mathematician’s apology’ consists in dividing mathematics into two disjoint halves, one ‘trivial’ or ‘useful’ which he consigns to perdition, the other ‘real’, useless, and incomparably superior on both aesthetic and moral grounds. Writing in 1940, he says that ‘No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity’, and by such reasoning place these superior subjects on the moral plane of the angels along with all ‘real’ mathematicians.” But, of course, relativity’s consequential E = m c2 played a singularly horrific role in the mighty, mighty U.S. Air Force’s campaign in the Pacific during WWII, commanded by a true warrior (if there ever was one) by the name of General Curtis Le May. And these forces, under Le May’s leadership, dealt a fiery death to countless fanatical, yet hapless Japanese, both in and out of the military, as such weapons as napalm and magnesium incendiaries do not distinguish between the two, nor, of course, do deadly, deadly nuclear weapons. Further, Hardy’s philosophy as set forth in his above mentioned book is fanciful in other ways too, as for example in his (with E.M. Wright) “An Introduction to the Theory of Numbers” (fourth edition) on page 39, he ascribes the proof that the square root of two is irrational – this being the first irrational number to be discovered – to Pythagoras. However, history disputes that claim, and assigns this discovery that not all numbers are rational to the Greek mathematician Hippasus of Metapontum, who was a member of the Pythagorean sect or religion. This religion’s creed was heavily dependent on their numerology involving rational numbers, unlike the Hebrew numerology which only concerned integers. And, of course, the “priests” of this sect had thoroughly drummed into their worshipers that all numbers were rational as a cornerstone of this rational number numerology. Thus this surprising and momentous discovery was immediately viewed as a deadly heretical threat to their religion by its leaders. They conferred and decided on a two step program. Step one was to execute poor Hippasus, and step two was to swear themselves to secrecy concerning the matter. And, indeed, history records that they actually succeeded in keeping the existence of irrational numbers secret for centuries! Now, of course, this execution should not surprise anyone for, as Stephen F. Hayward points out in an article in ‘National Review’ (May 16, 2011) “Tyrants understand themselves quite well – this is one of the clear teachings of Xenophon whose Heiro tells Simonides that the tyrant “lives night and day as one condemned by all human beings to die for his injustice.” And, by way of example, he mentions that “Poland’s foreign minister and former ‘National

Review’ correspondent Radek Sikorski reported in 2005 about a conversation at a diplomatic dinner in Havana involving Fidel and Raul Castro, during which the former rebuffed a speculative suggestion from his brother that Cuba consider liberalizing its economy, arguing that they’d both end up swinging from a lamppost in a matter of months.” But getting back to Hardy, the whole very sordid story of poor Hippasus obviously did not “fit” into this vision of a morally and aesthetically superior utopian society of pure mathematicians engaged in “useless” pursuits such as number theory … as the school of the Pythagoreans proved to be a ruthless pack of murderers intent on preserving their meal-ticket numerological religion at all costs … even the execution of one of their sect who had greatly advanced their number theory, but in a way that threatened their very comfortable and extremely convenient life-styles!