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 17, 2012 9:23 PM
On Nov 16, 1:05 am, William Hale <bill...@yahoo.com> wrote: > In article > <cb47a428-b69f-4666-9c12-d82b83a12...@4g2000yql.googlegroups.com>, > > > > > > Charlie-Boo <shymath...@gmail.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,
You cannot prove the 2 axioms for addition because they are the definition of addition. You only prove something about a function or relation after having defined it and its properties, then you can derive further conclusions about it. Peano defines addition and multiplication, and as a result when a+b=c we can prove that a+b=c. This is because the relation a+b=c is recursive so truth coincides with provability.
How do you think they derived the fact that x+0=x for all x? How many pages does that take? Can you substantiate your claim?
It is just silly to brag about taking 100 pages to prove a trivial fact that is nothing more than by definition of the numbers 1 and 2 (or 2 and 4.)
> so it takes longer to reach the point where you > can prove "2+2=4". > > Here's a tinyURL for "Principia Mathematica": > > http://tinyurl.com/cgmfzfb > >