On Mon, 11 Apr 2011 11:17:09 -0700, geremy condra wrote: > On Mon, Apr 11, 2011 at 2:10 AM, Steven D'Aprano > <steve+comp.lang.pyt...@pearwood.info> wrote: [...] >> Of course, some mathematics is obvious, or at least intuitive (although >> proving it rigorously can be remarkably difficult -- after 4000 years >> of maths, we still don't have an absolutely bullet-proof proof that >> 1+1=2). > > Erm. This is getting a bit far afield, but yes, we do. The statement you > provide above part of Presbuger arithmetic, which is both complete and > decidable.
Ah, I didn't know that! How wonderful! But in any case, Presburger arithmetic is much weaker than even Peano arithmetic. http://en.wikipedia.org/wiki/Presburger_arithmetic So, let me re-phrase my statement... in any realistically complex arithmetic that is consistent with operations performed for real-world applications (e.g. multiplication, division, exponentiation, ...), one cannot demonstrate a bullet-proof proof of 1+1=2. Better? :) Presburger arithmetic, Peano arithmetic, the Axiom of Choice... we're getting further and further away from "natural" mathematics, e.g. counting sheep in a field. [...] >> Part of the patent problem is that the distinction between discovery of >> a fact (which should not be patentable) and invention (which, at least >> sometimes, should be patentable) is not clear. The iPod existed as a >> Platonic ideal in some mathematical bazillion-dimensional abstract >> design space long before it was invented by Apple; does that make it a >> discovery rather than an invention? On the other hand, it is doing >> Apple a great disservice to ignore their creativity in finding that >> design point, out of the infinite number of almost-iPods that suck[3] >> or don't work. > > I agree. Of course, your post existed as a billion-point platonic ideal > beforehand, so you can't really claim credit (man, Plato figured > *everything* out!), but still. Damn Library of Babel, it has *everything* in it. Anyway, this is now getting off-topic even for the original off-topic post. Time to move on, methinks. -- Steven -- http://mail.python.org/mailman/listinfo/python-list