(originally written up at TLT - 2005.04.25)
I've been thinking about Science and Mathematics lately, and how they interact. There was a very interesting paper mentioned on http://lambda-the-ultimate.org/, The Unreasonable Effectiveness of Mathematics in the Natural Sciences, which is a great read. This and a few other random encounters have led me to contemplate the relationship between mathematics and science.
One thing, which I cannot repeat often enough, is a simple yet profound difference between the process of the two disciplines. Mathematics is about proof, while science is about disproof. This has surely been written elsewhere, but it is nice to contemplate the idea on its own. In mathematics you come up with a theorem, and then attempt to prove that the theorem is correct or incorrect though logical argument. In experimental science, on the other hand, you come up with a theory which explains all known data points. You then try to collect further data points or make a logical argument in an attempt to disprove the theory.
That framework in mind, lets turn our attention to the concept of completeness in mathematics. From Goedel we have the incompleteness theorem, which states that (in our current system) there are truths of mathematics which are not reachable by logical argument from existing truths. That is -- there are some statements which we cannot prove nor disprove, but for which there is no situation in which the statement is false. In other words, there are true statements of mathematics which cannot be proved.
Now lets turn our attention to science. Are there, then, statements of science which cannot be disproved, in a corresponding way? The most direct instance of such a concept is that of God. God is here defined as something completely outside our physical system, yet which exists non-the-less. Existence without physical manifestation seems like something we surely cannot disprove. This seems to put the question outside the reach of science.
Yet we have one more bit of thought from the mathematicians which can be borrowed. How did they react to the incompleteness theorem? They kept right on trying to prove everything under the sun. Even with problems which took years and years to solve, they continued. Even with the threat that these problems may in fact be un-provable. Many of these problems have yielded great results and even proofs... but others are left unsolved. When do we give up?
Now apply this question to science.