How do you prove that 2+2=4? Is it enough to consider 2 objects, then 2 more then put them together and count them and get 4, or do you have to resort to fancy-schmancy methods?
Nov 18, 2012 7:45 PM
Re: How do you prove that 2+2=4? Is it enough to consider 2 objects, then 2 more then put them together and count them and get 4, or do you have to resort to fancy-schmancy methods?
Nov 16, 2012 1:05 AM
In article <firstname.lastname@example.org>, Charlie-Boo <email@example.com> wrote:
> On Nov 16, 12:33 am, William Hale <bill...@yahoo.com> wrote: > > In article > > <17eeb626-2a35-4177-ad68-92cf18be5...@c16g2000yqe.googlegroups.com>, > > > > Charlie-Boo <shymath...@gmail.com> wrote: > > > On Nov 15, 11:22 pm, donstockba...@hotmail.com wrote: > > > > Just askin. > > > > > Principia Mathematica BS says it takes 100 pages. I don't know > > > anybody who has tried to explain what is going on there (everyone just > > > sits in awe at the number of pages), but Peano Arithmetic proves it in > > > a few steps where 2 is 0'' and 4 is 0'''', based on the axioms x+0=x > > > and x+y' = (x+y)' where x' is x+1 ("successor of x"). > > > > > C-B > > > > You can view "Prinicpia Mathematica" at the link: > > > > http://archive.org/stream/PrincipiaMathematicaVolumeI/WhiteheadRussel... > > incipiaMathematicaVolumeI#page/n95/mode/2up > > > > More is being done than just proving that "2+2=4". For example, logical > > propositions like "q implies (p implies q)" are being proved. > > People always say "It takes 100 pages to prove 1+1=2!" as if it's > cool, contrary to Occam's Razor. So how many pages are needed for the > ultimate derivation of 1+1=2 (or 2+2=4 whatever)? > > C-B
Principia Mathematica is not trying to just prove "2+2=4". I believe that its original purpose was to show that all of mathematics could be derived from just logical principals (which I think even Whithehead himself eventually rejected). Principia Mathematica is developing theorems about logic and sets. From there, it shows how standard mathematics like arithmetic might be derived. Principa Mathematica is starting from axioms that are much more primitive than Peano's axioms for the natural numbers, so it takes longer to reach the point where you can prove "2+2=4".