Three Math Books

Erdos never owned a car or a room of his own...

"THE MAN WHO LOVED ONLY NUMBERS" - The Story of Paul Erdos, by Paul Hoffman, Hyperion, 77 West 66th St, 11th floor, New York, NY 10023 1998, HB $22.95, ISBN 07 868 6362-5 PB $12.95 ISBN 07 868 8406-1

THIS IS A WARM and delightful biographical treatment of Paul Erdos, a mathematician whose knowledge of prime numbers was the stuff of legends. It is a small book and a great read. Author Paul Hoffman sprinkles his book with mathematical lore, as well as the sayings and doings of the eccentric, peripatetic Erdos.

Born in Hungary, Erdos never owned a car or a room of his own, but lived in the worldwide subculture of mathematics, from meeting to meeting, staying with friends. Erdos was fragile and very limited in many ways, yet his outrageous humor, intensity and mathematical brilliance made him beloved of peers.

Erdos published almost 1,500 papers with 485 collaborators, who can now claim an Erdos number of 1, based on the proximity of their co-authorship. (A co-author with one of the original collaborators can claim an Erdos number of 2, and so on.) Erdos quips are legion: Rather than good morning he might say that his mind was open. If he misplaced something, he would blame it on the SF, or Supreme Fascist, his term for God. Children were epsilons, wives bosses. Though he slept only a few hours daily, thanks to coffee and amphetamines, he died at 83 (in 1966); his epitaph, which he chose himself, reads to the effect that he is at last no longer getting dumber. This is a book for everyone, not just math hounds.

"MATHEMATICS FOR THE MILLION" - How to Master the Magic of Numbers, by Lancelot Hogben, W. W. Norton & Co. Inc., 500 Fifth Ave, New York, NY 10110 1993 (multiple reissues from 1937), PB $16.95 ISBN 0-393-31071-x

THE FIRST HOGBEN BOOK I read was called The Mother Tongue (1967), now long out of print. It is a coherent exposition of the inter-relationships among Indo-European lan-guages, which provides an insight into the diverse grammars of English itself. The British Isles were conquered and ruled for centuries, by Romans, Nordics and French, each layering their grammar and syntax onto native Celtic.

Each invading language remains embedded in English, contributing to the richness of the "Mother Tongue" as well as its systematic vagaries of spelling. Lastly, in the 16th Century the English made things simpler by discarding Latin grammar. Because of these advantages, Hogben makes the case for English to become the lingua franca of the modern world. Voila! …(Sorry, Francophiles!)

However, Hogben first became widely known for Math for the Million. Since 1937, it has gone through more than 40 editions. It is a like a textbook, but has a broader message. The author notes that every advance or new development in mathematics is the result of people doing things. It is people doing things who change thought, rather than people thinking things who change what we do. People counted, measured the size of their fields, sailed the globe. Counting led to the number system, and to the concept of zero. Debt to negative numbers. Measuring fields, led to Euclidian (once more aptly called plane and solid) geometry. Sailor/navigators needed to measure the earth by the stars or sun, and expanded to trigonometry, and so on.

No Platonic purist therefore, Hogben introduces each development in mathematics within its historical or cultural context, rather than through didactic or theoretical concepts. This makes his book good reading as well as good math. It is filled with historical detail, related with typical British humor.

In the end though, it gets right down to serious math. Each chapter is followed by exercises, which can stretch the reader's mind and patience. I enjoyed the book, though I felt like I was back in college and needed to do the homework, even though I am unlikely to use the skill. Come to think of it, that's how it was with most of what I learned in college. Maybe reading math books is a form of entertaining regression.

"GOD'S EQUATION" - Einstein, Relativity, and the Expanding Universe, By Amir D. Aczel; Four Walls Eight Windows, 39 West 14th St, # 503, New York, NY 10011 1999, HB $22.00 ISBN 1-56858-137-8

THOUGH WRITTEN IN A CLEAR, clean style, Aczel's book is peppered with jargon, sometimes explained using more of the same. Reading explanations was like using soapy water to rinse dishes: they were as slippery and opaque as the original. The scope of the book is remarkable. We go from Berkeley to Babylon and back again. From Euclidian to Non-Euclidian, to topologic mathematics; from Pythagoras to Newton, Planck, Einstein and Hawking, with frequent stops in between. He does not neglect astronomy, cosmology or quantum physics, or the cosmological constant.

Finally the reader is deposited on some far shore, presumably in sight of "God's Equation," a formula that unites the physics of the universe within the holy sphere of mathematics. The equation is, of course, not merely E=mc2. Its terms are serially explained in the text, but not until the end (p 219) brought together. Even then, I confess to being only vaguely aware of what the terms stand for, let alone their significance. I quote:

"This is Einstein's field equation with the cosmological constant, which is our best estimate of God's Equation:

where  is the Ricci tensor, R is its trace, is the cosmological constant,  is the measure of distance-the metric tensor of the geometry of space, G is Newton's gravitational constant,  the tensor capturing the properties of energy, momentum and matter, and 1/2 and 8 are numbers." Well, at least I understood the numbers 1/2 and 8. Maybe.

I was sometimes seasick during the voyage, and would not suggest it for casual reading. There were two aspects of the book that I enjoyed thoroughly. First, this odyssey of math contained interesting biographical details of Einstein's life, and his interaction with his peers.

These include three ecliptic expeditions to confirm that light is curved around massive objects. Einstein's effort to demonstrate this during an eclipse in the Crimea was ruined by the outbreak of WWI (does Sarajevo sound familiar?); its failure led to estrangement between Einstein and his colleague Freundlich, who actually was arrested by the Russians as a spy suspect, along with all his telescopic and photographic gear.

The other expeditions by the British astropyhsicist, Arthur Eddington, were organized after Einstein managed to smuggle a copy of his work to England during WWI. Along the path of the eclipse, in Brazil and Africa, Einstein's predictions were confirmed, though Einstein was never informed of the result by Eddington or his colleagues.

Also interesting is the finding that the universe is apparently expanding at an ever increasing rate, so that the farthest objects we can see are moving away from us at 0.95 times the speed of light, faster than the near galaxies.

A rubber band analogy (p 178), however, implies that there is no center to our universe, or the center is everywhere, so that we too are moving near the speed of light. There is not enough mass in the universe to make it fall back in upon itself through the forces of gravity, yet there may be a force that accelerates dispersion. This leads some to conclude that the universe will never be pulled back together, but simply age into a series of lost old black holes.

Yet in my innocence, and ignorance, I wonder. If time moves more and more slowly as velocity approaches that of light, could it not be that our expanding universe, now exploding at nearly light speed, could reach and pass that limit, so that time would stop and then run backwards? If so the universe would collapse into itself inevitably back to the Big Bang, if that is where it began. If infinity means what we suppose, this cycle has occurred an infinite number of times.

I'll call it Loofbourow's hypothesis, comforted by the thought that it won't soon be tested, even though some may say it is mathematically impossible. Will I be, have I been, a better man in other universes, or do I make the same mistakes over and over? Very probably.

Very probably. Very probably.

lufboro@jps.net