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 18, 2012 9:13 AM
> On Nov 16, 1:05 am, William Hale <bill...@yahoo.com> > wrote: > > In article > > > <cb47a428-b69f-4666-9c12-d82b83a12...@4g2000yql.google > groups.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.goog > legroups.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.) > > C-B > > > 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 > > > > > >
I think this goes back to Russel and Whitehead taking over one hundred pages in their set theory book to prove 1+1 = 2