Mathematics

A phenomenal essay on a systematic approach to interactive visualization.

This just in: there are absolute, logical limits to the ability of any method (including the scientific one) for acquiring knowledge to produce a comprehensive theory of the world

Well, call off the search for a theory of everything. Physicist David Wolpert, in an article published in the prestigious Physica D (vol. 237, pp. 1257–1281, 2008), has shown that — at best — we can achieve a theory of almost everything. Wolpert’s work is very technical, but its implications are spectacular. Unlike the above mentioned limits to knowledge, which come out of empirical disciplines, Wolpert used logic to prove his point, following in the steps of the famousincompleteness theorem demonstrated by Kurt Godel in 1931. (An accessible summary of Wolpert’s discovery can be found in an article by P.-M. Binder in Nature, 16 October 2008.)

Basically, Wolpert — building on previous work by Alan Turing — formalized a description of “inference machines,” i.e. machines capable of arriving at inferences about the world (human beings are one example of such machines). Wolpert focused on what he calls strong inference, the ability of one machine to predict the totality of conclusions arrived at by another similar machine. Wolpert then logically proved the following two conclusions: a) For every machine capable of conducting strong inferences on the totality of the laws of physics there will be a second machine that cannot be strongly inferred from the first one; b) Given any pair of such machines, they cannot be strongly inferred from each other.

Of course, the article ends with the obligatory sideswipe at creationists.

Reference to original paper:

Physica D: Nonlinear Phenomena, Volume 237, Issue 9, 1 July 2008, Pages 1257-1281
David H. Wolpert

First Things has a great article on the intersection of science and faith. Definitely a great read.

If you’re interested in the topic, I highly recommend the book: “The Anthropic Cosmological Principle“.

If you haven’t watched the documentary on Fermat’s Last Theorem you should set aside 45 minutes as soon as you can. The documentary is a splendid testament to what happens when passion and perseverance are coupled with a stunningly sharp mind. Here are some notes from the producer:

There is a brilliant genius from the past who solves an apparently impossible problem. He dies without revealing the solution. This becomes buried treasure, and every subsequent mathematician goes in search of it. There are heroes, villains, rivals, rich prizes, a duel at dawn, a suicide and an attempted suicide, but after 300 years the problem remains intact. The greatest minds on the planet failed to solve it. Undaunted, however, a young boy promises to devote the rest of his life to solving this notorious problem. After thirty years he suddenly identifies a strategy that might work. For seven years he works in secret. He reveals his proof, only to learn that he has made a mistake. He hides away again, humiliated and ashamed, but he returns a year later, this time triumphant. The problem has been solved. His journey is over.

The documentary was about mathematics and mathematicians, but it was also about childhood dreams, ambition, obsession, passion, failure and triumph. Not surprisingly, there was a time when one of the Hollywood studios put in a serious bid to make a feature film, but somewhere along the line the project faded away.

The emotion of the documentary is clear from the first minute. The opening sequence shows Professor Andrew Wiles recalling the moment when he realised that he had solved Fermat’s Last Theorem and achieved his childhood dream. The memory is so moving that he begins to stumble over his words. He then pauses, takes a breath, tries to continue, but eventually he is overcome with emotion and turns away from the camera. There are other moments in the programme that are equally emotional.

I particularly like the final comments from Iwasawa, the Japanese mathematician who came up with the conjecture which was necessary to prove the theorem.

I have long held the theory that every conversation, if pursued long enough, naturally and necessarily ends with a discussion about the existence of God and our purpose on earth. A few years ago a long lunch conversation reached this point, and an engineer concluded with the statement: “My God is a set of equations.” To which I replied, “What do those equations describe?”

These days it is fashionable among the intelligentsia to declare with newly-discovered transcendence that religion is a good enough thing (if done in moderation) and science is self-evidently worthwhile but we should never, ever confuse the two. The intersection of science and faith, rationalism and mystery, is best left to the final pages of an epilogue in a serious book on science, or to footnotes in a book on faith.

And yet equations are merely an abstraction of the physical world and the Christian faith claims to worship a Jesus who walked in history and a God who created the physical. Why then, have the last few decades produced an intellectual movement so devoted to a separation of faith and science?

In 1931 Godel published a paper which challenged the basic assumptions underlying mathematics and became a milestone in the history of logic and mathematics. I believe that the world is still coming to terms with the philosophical implications of Godel’s Theorem of Incompleteness. Others have better summarized his theorem but it proscribes the limits of axiomatic logic and shows that provability is a weaker notion than truth; in short, there is Truth that we can’t logically prove.

Nagel and Newman, in their classic book on the subject, Godel’s Proof, conclude with the comment:

Godel’s proof should not be construed as an invitation to despair or as an excuse for mystery-mongering.

That might be. But it does, I believe, suggest that if God is a set of equations, those equations lie in a realm of mathematics about which we haven’t even begun to dream.