## Monday, January 23, 2012

### "Proving something does not make it true"

This weekend I accidentally stumbled on a page for science teachers. http://digipac.ca/chemical/proof/index.htm

I was appalled by the twisted message, that reduces logic to a suspicious rhetorical device. Money quotes:

Proof does not equal truth

Here's a definition of what it means to prove something: "Proof is arriving at a logical conclusion, based on the available evidence." Notice that this has absolutely nothing to do with being right or wrong.

Proving something does not make it true. It just means that you have convinced other people that the evidence supports your conclusion.

I was particularly stung by the third quote, because I use similar terms to try to explain how to write a proof: I usually say that it should be just formal enough, but not too formal, and that "just formal enough" depends on the recipient - just enough to convince the reader that you indeed have a valid proof.

1. What a strange website. Of course, he is correct with respect to the use of the word "proof" outside of mathematics. Proof in a court of law is different only in degree, not in kind, to proof in the empirical sciences. Proof in mathematics is different.

2. `Beware of bugs in the above code; I have only proved it correct, not tried it.'' -- Knuth

3. Science is different from math in this respect. The classic book on scientific writing "A Manual for Writers of Research Papers, Theses, and Dissertations" by Booth, Colomb and Williams for example claims that empirical evidence makes a more convincing argument than logical deduction.

For example, if your thesis is that "Needle exchange programs promote drug use", and you deduce this by saying that "Needle exchange implies that it is safer to use drugs implies that more people use drugs", then your argument is less convincing than if you went out and surveyed a bunch of people, and they agreed that they use drugs more after the introduction of a needle exchange program.

I suppose real life phenomena is often imperfectly modeled by math, and there is always scope for loopholes. Which is why in the sciences we need both mathematical deduction and empirical evidence to solidify a claim.

4. Last anonymous pegged the issue.

Proofs in empirical sciences (even physics) are proofs in a mathematical model that is supposed to describe a physical system, but necessarily imperfectly. Since the statement that you really care about is the physical system, the "proof" is just another piece of empirical evidence, and does not really prove that the assertion is true of the physical universe.

In mathematics, the statements that we really care about actually are about the mathematical models that we are studying, and so correspond to "truth". (If truth is defined only w.r.t. the model, if the formal system is sound, ...)

5. This is an old topic; for a CS perspective, see the classic paper "Social processes and proofs of theorems and programs" by DeMillo, Lipton and Perlis, from CACM vol 22, no 5 (1979).

6. The other key phrase is "...based on the available evidence." In math, we define the universe of "available evidence". In the physical sciences, it is always possible that more measurements (or more precise measurements) will refute the current theory.

7. Two more similar sayings are as follows:

"The virtue of a logical proof is not that it compels belief but that it suggests doubts. The proof tells us where to concentrate our doubts."

(Henry George Forder, 1927, page viii, the Foundations of Euclidean Geometry)

Possibly a derivative of the above, and better known is:

"It is one of the chief merits of proofs that they instill a certain skepticism about the result proved." (Bertrand Russell)

8. That isnt really a question...the thing you have written is the formula for quadratic equations...I was appalled by the twisted message, that reduces logic to a suspicious rhetorical device.

9. I have half a suspicion (based on no evidence, just a gut feeling) that the author of the website has a hidden agenda, either to promote creationism or to deny global warming.