Sunday, January 24, 2010

The Limits of Knowledge, Part II

(continued from Part I)

Bertrand Russell (1872-1970)

One way of ensuring your name will always be remembered is to have some important scale or number named after you. Slightly less surefire, but still valid, is to make a prediction so fundamentally wrong that people will delight in repeating it generations later. Lord Kelvin passes on both accounts with the Kelvin temperature scale and this (possibly apocryphal) quote: "There is nothing new to be discovered in physics now; All that remains is more and more precise measurement." Five years later, Einstein would publish his paper on Special Relativity.

I bring this up to point out the scientific and mathematical philosophy of the 19th century. Both were progressing at unprecedented rates, and it seemed possible that man would master all their secrets. With each scientific success, people believed that some fundamental truth was coming closer. Quantum Mechanics deals with a fundamental indeterminacy in the universe (see Schrödinger's Cat), but Newtonian physics is theoretically solvable. If you know the position and velocity of each particle you can predict the future as far forward as you care to. There was a novel philosophy growing that man wasn't a creature of free will, or a Calvinist puppet of the divine, but simply the inevitable progress of a well defined and understandable set of physical laws.

In contrast to the physical sciences, mathematics and logic were in no threat of being exhausted. There is no upper limit to the length of a conjecture, so you can always find a new one to prove. But there was a similar optimism in the field. Even if the statements in math are infinite, it was believed that they ultimately encompassed all possible truths. Any truth could be mathematically proven, any falsehood thoroughly contradicted. And the world of math was seen by its practitioners as one of very clearly defined truths and falsehoods. The everyday world is filled with shades of gray, but math is a Platonic, quasi-religious realm of absolutes. It was even believed that you could build a machine that would take in any mathematical statement and print out a proof of that conjecture's truth or falsehood.

Each field of mathematics derives from its own set of base axioms. You cannot do geometry without definitions about points and lines. Integer arithmetic is very different from arithmetic over real numbers. To many, this was an unsatisfactory state of affairs. They believed since there is only one set of all true statements, they should all derive from the same basic facts. Gottlob Frege, a German mathematician tried to unify math, using set theory as a basic set of axioms to derive the other fields of mathematics. Set theory concerns itself with groupings of objects together. The set of even numbers starts 2,4,6.... The set of nations of the world include Germany, Australia, etc. Frege was able to show that with some basic axioms about sets you can derive geometry, arithmetic, etc.

As his second edition of the book was in the final preparatory stages for publishing, Bertrand Russel sent Frege a letter alerting him to the fact that his axioms allowed in paradoxes. Frege had to add an appendix at the last moment, starting "A scientist can hardly meet with anything more undesirable than to have the foundations give way just as the work is finished..." Frege had begun to put math on a highly formalized setting, but in doing so had created a system with infinite valid statements that could not be proven one way or another. Such a mathematical system would not be a Platonic home of all truths, but one filled with uncertainties.

The paradox comes from the fact that sets can contain themselves. "The set of all non-Empty sets" is itself non-empty, and thus contained by itself. In contrast, the set of nations is not a nation, and so is not contained by itself. Does "the set of all sets not containing themselves contain itself?" If it does, then it must have the property of not containing itself, which is a contradiction. But if it doesn't contain itself then it meets the requirements for the set, and therefore does. Whether you put it in the set or leave it out you get a contradiction. There are a number of less mathematical ways to formulate this problem, such as the following: "There's a barber in a town who shaves all men that do not shave themselves. Who shaves the barber?" It seems like a silly game, building nonsense sentences. It seems odd that something as trivial as question about a barber should shock the foundations of the mathematical world. And yet it did...

Russell was purportedly crushed by his paradox. He had grown up with the quasi-religious view of mathematics as a bastion of truth, and hated to now be associated with an attack on that foundation. But as the good mathematician, he could not deny the mathematical truth that set theory did lead to paradoxes. For many years after he left the field of mathematics, unable to come to terms with the damage he had done to the field. Finally he returned, determined to re-establish the foundations of mathematics. With the help of Alfred Whitehead, Russell wrote a three volume book, "Principia Mathematica" designed to retrace Frege's steps while avoiding the paradoxes Russell had discovered. It was an extremely detailed effort. The book contains the famous remark "From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2."...379 pages in. The book is viewed as one of the most important ever written on mathematics, clearly defining the basic logic that validates it as never before. But did he succeed in clearing the paradoxes out of mathematics? In 1931 the extremely eccentric mathematician Kurt Gödel would prove once and for all that anyone setting out on the endeavor of formalizing mathematics in a way that avoids paradoxes was doomed to failure. And with that, what I would view as the modern age of mathematics would begin.


  1. Paul, first let me say that I think you might be the smartest person I know (and I fancy myself to know a fair few intelligent people).

    I also want to register my approval that you're writing about those crazy 19th century folk who tried to circumscribe anything and everything into a set of rules. God bless their misguided little hearts. And of course, I'm always glad to hear about my old friend Bertrand Russell.

    Well done overall, my good chap!

  2. Thanks Meredith! They do warm your heart a little, don't they? You pass from belief in an all powerful and incomprehensible God to a mathematically incomprehensible universe, but for that brief time everything was just bouncing billiard balls, and there was no reason you couldn't explain all of existence in a long enough book. I'm partial to Leibniz's belief that if you assign the correct words to numbers, you can replace imprecise thinking with math.