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Please do... In my freshman year of college, I asked my real analysis professor why 1+1=2 and he failed to provide an edifying explanation. He did, however, co

Please do... In my freshman year of college, I asked my real analysis professor why 1+1=2 and he failed to provide an edifying explanation. He did, however, commend me for asking -- I think it earned me some brownie points, which I redeemed by asking for extra clarification on more course-related topics later in the quarter. Anyway, it's always bothered me so if I can learn something about it then I would love to!

xcthulhu 16y ago
Check out the axioms of Peano Arithmetic: http://en.wikipedia.org/wiki/Peano_axioms#Arithmetic http://en.wikipedia.org/wiki/Peano_axioms#Arithmetic If we use Dedekind's recursive definition of addition, and the definitions 1=S(0) and 2=S(S(0)), then: S(0) + S(0) = S(S(0)+0) = S(S(0)) I know the computer proof assistant Isabelle/HOL, inspired by the Principia Mathematica, can prove simple theorems about arithmetic using a "rewrite" system similar to what I've done above.
jodrellblank 16y ago
Can you rephrase that question without using the word "why"? I have no idea what "why does 1+1=2?" is trying to find out.
RiderOfGiraffes 16y ago
As a taster ... What do you mean by "2"? What do you mean by "1"? What do you mean by "+"? What do you mean by "="? There's more than one way to get to the number 7. You can start at 0 and count upwards, or you can "add" the numbers "3" and "4". Why should it be that you end up in the same place? Slightly more complex/general ... Consider the number line, and divide the stretch between 0 and 1 into 9 equal pieces. Start from 0 and move along two of these pieces. Call the place you get to "T". Now consider the stretch from 0 to 2, and divide that into 9 equal sized pieces. Take just the first one, and call where that gets to "S". Why are they the same point?
- zackattack 16y ago
  >Now consider the stretch from 0 to 2, and divide that into 9 equal sized pieces. Take just the first one, and call where that gets to "S". Why are they the same point? This seems to me like it's simply rephrasing the question why does 1+1=2. Or, at least, I can't answer it without invoking the field axioms, or perhaps only the ring axioms. Please forgive me if my terminology is awkward.

xcthulhu 16y ago

Check out the axioms of Peano Arithmetic: http://en.wikipedia.org/wiki/Peano_axioms#Arithmetic http://en.wikipedia.org/wiki/Peano_axioms#Arithmetic If we use Dedekind's recursive definition of addition, and the definitions 1=S(0) and 2=S(S(0)), then: S(0) + S(0) = S(S(0)+0) = S(S(0)) I know the computer proof assistant Isabelle/HOL, inspired by the Principia Mathematica, can prove simple theorems about arithmetic using a "rewrite" system similar to what I've done above.

jodrellblank 16y ago

Can you rephrase that question without using the word "why"? I have no idea what "why does 1+1=2?" is trying to find out.

RiderOfGiraffes 16y ago

As a taster ... What do you mean by "2"? What do you mean by "1"? What do you mean by "+"? What do you mean by "="? There's more than one way to get to the number 7. You can start at 0 and count upwards, or you can "add" the numbers "3" and "4". Why should it be that you end up in the same place? Slightly more complex/general ... Consider the number line, and divide the stretch between 0 and 1 into 9 equal pieces. Start from 0 and move along two of these pieces. Call the place you get to "T". Now consider the stretch from 0 to 2, and divide that into 9 equal sized pieces. Take just the first one, and call where that gets to "S". Why are they the same point?

zackattack 16y ago
>Now consider the stretch from 0 to 2, and divide that into 9 equal sized pieces. Take just the first one, and call where that gets to "S". Why are they the same point? This seems to me like it's simply rephrasing the question why does 1+1=2. Or, at least, I can't answer it without invoking the field axioms, or perhaps only the ring axioms. Please forgive me if my terminology is awkward.