Ilya Lipovsky (lipovsky@cs.bu.edu) on 2/20/08 wrote:
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>Personally, I do not completely disagree with your very global view on what "science"
>is. If I recall properly from several years ago, Kurt Gödel's monumental Second
>Incompleteness Theorem showed that you can have a true statement that is not provable
>from within the set of axioms in which it is true. That basically nullified Hilbert's
>program of showing completeness of mathematics (along with other things). That,
>basically relegated mathematics to the realm of science - e.g. true non-first-order
>arithmetic expressible statements (such as the 4-coloring theorem) were now not
>assured to be provable. Thus, mathematicians can still "discover" (like physicists)
>and "invent" (like engineers) new axioms. (By the way, from what I remember, it
>eventually turned out that 4-coloring was true, but it was proved with the help of a computer.)

That's an interesting (but reasonable) way of looking at it.

There are actually quite a few undecidable statements, with the continuum hypothesis and the axiom of choice being the two most famous ones. Wikipedia (I don't usually quote it this much, but these articles have actually been quite reasonable) lists a few more: http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems#Examples_of_undecidable_statements

This is actually quite interesting, because the self-referential nature of the particular statements that are used to prove incompleteness (effectively, "This statement is false") don't, to my mind, rule out the possibility that all "normal" statements are decidable. The fact that there are entirely reasonable statements that turn out to be undecidable as well is non-obvious to me, although someone far smarter might realise that there is a connection between them all.

>Nevertheless, I think even with all this philosophical stuff in mind, it's still
>necessary to distinguish between science and math as being in very different realms.

We normally use the term "empirical science" to describe what you have been calling "science"; maths and theoretical computer science are "formal sciences". Empiricists don't believe in anything except empirically-derived results, and you see them popping up from time to time complaining about string theory et al because an awful lot of work is put into something that they don't even consider "proper" science since we can't even predict what we would need to do to distinguish between string theory and the standard model. (Since we don't know which parameterisation of string theory matches our universe, we don't know what energies are required to produce the kinds of particles that would confirm its accuracy; the LHC might produce something that supports string theory (or supersymmetry at least), but then again it might produce nothing and the conclusion would be "Well, we just need more energy".)

>Therefore, "computer science" is informatics ;-).

QED? :-)