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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 fancyschmancy methods?
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8
Last Post:
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 fancyschmancy methods?
Posted:
Nov 18, 2012 7:45 PM


In article <b6d1ded83bed4f9f87da3b354f5d5a62@c17g2000yqe.googlegroups.com>, CharlieBoo <shymathguy@gmail.com> wrote:
> On Nov 16, 1:05 am, William Hale <bill...@yahoo.com> wrote: > > In article > > <cb47a428b69f46669c12d82b83a12...@4g2000yql.googlegroups.com>, > > > > > > > > > > > > CharlieBoo <shymath...@gmail.com> wrote: > > > On Nov 16, 12:33 am, William Hale <bill...@yahoo.com> wrote: > > > > In article > > > > <17eeb6262a354177ad6892cf18be5...@c16g2000yqe.googlegroups.com>, > > > > > > CharlieBoo <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"). > > > > > > > CB > > > > > > 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)? > > > > > CB > > > > 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.
The two axioms are your definition of addition. They are not the definition of addition for "Prinicpia Mathematica". Nor does "Prinicpia Mathematica" assume those two statements as axioms.
> 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?
From my brief search on the web for information on how "Principia Mathematica" did cardinal arithmetic, I found the following.
Let r be a set. The cardinality of r, denoted #(r), is the class {s  s is in 11 onto correspondence with r}.
Define 0 to be #({}).
Let r and s be sets. Define the "set sum" of r and s to be: r '+' s = {<{}, y>  y in r} union {<x, {}>  x in s}.
Define the sum of #(r) and #(s) to be: #(r) + #(s) = #(r '+' s).
Of course, this sum needs to be shown to be welldefined (as usually done in the standard way).
I want to show that x + 0 = x for all x. Of course, x must be a cardinality class (i.e., x = #(r) for some set r).
That is, I want to show that #(r) + #({}) = #(r) where r is any set.
Since #(r) + #({}) = #(r '+' {}) = #({<{},y>y in r}), I want to show that #({<{},y>y in r}) = #(r).
That is, I want to show that {<{},y>y in r} is in 11 onto correspondence with r.
Using the obvious 11 onto function, we see that the above is true.
> > 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.)
I don't think people are bragging about taking 100 pages to prove a trivial fact like 2 + 2 = 4. "Principia Mathematica" tries to show that most of mathematics can be derived from a few axioms.
Do you complain about Euclid "Elements of Geometry" taking pages to prove trivial facts?
> > CB > > > 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 > > > >



