Wednesday, January 13, 2010

Everything and More A Compact History of Infinity by David Foster Wallace

David Foster Wallace (DFW) writes in “Everything and More A Compact History of Infinity” that the mathematician Cantor peered at the problem of infinity until it drove him mad. Well, sort of. He explains that would be the romanticized view of what drove Cantor insane while explaining that in actual fact Cantor suffered from bipolar disorder. As the cliché reads there is a fine line between genius and madness. Certainly Nietzsche, to give another example of a deep thinker, went insane although modern doctors suggest that he had syphilis. And DFW himself in 2009 after having written this quite complex book on mathematics and his well-received novels and short stories wrapped a noose around his neck and hung himself from the patio of his house. This makes one wonder whether it is good for the mental health to spend too much time thinking about math, metaphysics, and philosophy. Be careful or you could upend the emotions.

I write that this book on mathematics is “complex” because I found it complicated to read even though I have a bachelors of science degree in math having earned a solid C for my efforts. What I learned in college is that there is a limit to one’s intelligence and that certain people will be better able hold aloft multiple notions at once which together comprise one idea and then be able to distill those into one elegant proof. But this is no reason to avoid the complex. One’s ability with math can increase with practice and one’s ability to think deeply will improve as well. Thus enlightened by DFW’s book and a little practice one can appreciate the subtle and complex and the beauty of the ideas found by the geniuses of the past.

Wallace was not a professional mathematician but an autodidact who read and the distilled the famous works of modern and ancient mathematics and compiled his findings of one aspect of math, the number ∞, into this book. We should be grateful that he has produced this anthology of sorts because it would probably take months if not years to do the same yourself if you can understand, for example, Aristotle’s “Physics”, Euclid’s “Elements”, and more complex works of the past few hundred years. Wallace says you don’t need much mathematical training to understand his book but no doubt it would help even while he provides you a glossary to help you along.

He starts by explaining how the Greek Pythagoreans (which he calls the DBP or “Divine Brotherhood of Pythagoras”) were stymied by the discovery that not all numbers could be expressed as a ratio of two numbers. The Greeks following the principles of Euclid translated all mathematical ideas to geometry and were thus were able to map out on paper the area of a triangle and more. But what were they to do when the Pythagorean Theorem showed that the hypotenuse of a right triangle whose opposite and adjacent sides had length 1 was equal to √2 which cannot be expressed as a rational number? (Recall from highschool the a^2 + b^2 = c^2 .) Aristotle looked at the decimal expansion of √2 and other irrational numbers dismissing as only “potentially” possible that this series of digits 1.421…. could run on forever, i.e. doubting that there was such a notion as an infinity. Wallace walks you through all these ideas.

Aristotle and others had contemplated such problems as Zeno’s paradox in the years BC without finding any solution. To wit, Zeno had said if you are standing n feet from a wall and then proceed half the distance to the wall, stop, then walk half of the remaining distance, stop, then do it again, you would never reach the wall as there would always be another ½ of the remaining distance to cover. That is, if the distance to the wall is n then the series 1/2 n, 1/(2 ) (1/2)n,… and so forth that expression would never sum to 1. The same type of thinking suggested that an arrow in flight never had any forward motion at any single instant of time t. Even Leibneitz and Newton had not solved these problems and written down solid proofs of ideas of real and irrational numbers and the limit of a function when they invented the calculus showing that the function lim(h→0)(f(x+h)- f(x))/h ) was the derivative of the function or the tangent to the curve thus making possible modern engineering and physics. These two 16th century mathematicians simultaneously suggested that when x grows large the ratio 1/x goes to zero but then x is close but not equal to zero they could dismiss it as nothing without encountering the nonsensical 1/0. Wallace found this contradictory and says that notion is not properly explained in freshman college math.

Mathematicians would have to wait until the 18th century for Cantor and Dedekind to explain away some of the paradoxes. Namely they learned how to deal with infinite sets and concluded that the general principles of equality and so forth do not apply when dealing with sets that are not infinite. Dedekind cleverly showed that he could cut the number line in half such that there is a maximum rational number in one section and a minimum in the other. Thus in between the two there are the numbers which are not rational, i.e. the irrational numbers. What Cantor did was to close the gaps in Dedkinds thinking with solid proofs and to explain that two infinite sets could be different. Wallace quotes Betrand Russell as saying that the ordinary mortal would have problems with these ideas of infinity when they can cannot be counted or contemplated in a manner which is easy to see. You think?

After reading this book I dove back into my college calculus book to explore some of the fascinating ideas that Wallace explained that I had either not studied in college or not understood the first time around. (Plus I needed to catch up with my son so I could help him with his AP Calculus that he studies in high school.) For example Dedkind showed that any continuous function can be expressed as a infinite series. This leads to such elegant looking constructs as, for example, sin x = x - x^3/3! + x^5/5!+ x^7/7! + … and π = 4/1 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11 …That all these common expressions can be expressed as an infinite series of numbers and that, moreover, thanks to Cantor and Dedkind these series can be solved is fascinating to behold. Way cool.

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