### Hard sciences

Here in Como the hard sciences are said to be physics, chemistry and mathematics, but

I have just been given a program by a student and I would like to know that it is correct. This is more difficult than writing a new program myself.

In mathematics instead it is usually much easier to be convinced of a proof than to construct a proof. Part of the reason is that mathematicians usually don't check a proof completely but try to find clues from the description and construct a proof themselves.

Reading a program and a proof are both difficult exercises. Proofs are usually deeper than programs, but verifying correctness, which is necessary with a program, is hard.

"I have spent a fair amount of effort during periods of my career exploring mathematical questions by computer. In view of that experience, I was astonished to see the statement of Jaffe and Quinn that mathematics is extremely slow and arduous, and that it is arguably the most disciplined of all human activities. The standard of correctness and completeness necessary to get a computer program to work at all is a couple of orders of magnitude higher than the mathematical community’s standard of valid proofs. Nonetheless, large computer programs, even when they have been very carefully written and very carefully tested, always seem to have bugs."

*not*computer science.I have just been given a program by a student and I would like to know that it is correct. This is more difficult than writing a new program myself.

In mathematics instead it is usually much easier to be convinced of a proof than to construct a proof. Part of the reason is that mathematicians usually don't check a proof completely but try to find clues from the description and construct a proof themselves.

Reading a program and a proof are both difficult exercises. Proofs are usually deeper than programs, but verifying correctness, which is necessary with a program, is hard.

**Added 2 February 2011**I have just seen this comment by William Thurston in an article (ON PROOF AND PROGRESS IN MATHEMATICS, BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 30, Number 2, April 1994, Pages 161-177)"I have spent a fair amount of effort during periods of my career exploring mathematical questions by computer. In view of that experience, I was astonished to see the statement of Jaffe and Quinn that mathematics is extremely slow and arduous, and that it is arguably the most disciplined of all human activities. The standard of correctness and completeness necessary to get a computer program to work at all is a couple of orders of magnitude higher than the mathematical community’s standard of valid proofs. Nonetheless, large computer programs, even when they have been very carefully written and very carefully tested, always seem to have bugs."

Labels: computing, mathematics

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