Why is 1+1 =2 ?
This might seem arbitrary to most people, even intuitive. But, This is one of the deepest questions to have been answered in the entirety of mathematics.
In the thresholds of common day experience one might also find ‘things’ that do not seem to obey this proposition:
- 1 drop of water + 1 drop of water = 1 drop of water. 1 + 1 = 1.
- 1 cup of water + 1 cup of alcohol = about 1 ¾ cups of 100 proof alcohol. So 1 + 1 = 1 ¾
- 01 + 01 = 10 ( In binary )
- 1+1 =0 ( Modular addition by 2 )
- 1+1=11 ( concatenation of two strings )
But hey, 1 + 1 is 2. Conflicts arise due to improper usage of ‘1′, ‘+’ and ‘2′.
So, What does ‘1′, ‘+’ and ‘2′ mean?
Peanos axioms :
1) 1 is a natural number. *
2) Every natural number has a successor. *
3) No natural number has successor 1 (or 1 has no predecessor) *
4) Every natural number has a predecessor except for 1. *
5) You don’t need to know this one right now. *
We define addition a+b, for a,b natural numbers, as follows:
If a is 1, then a + b is b’s successor. *
Otherwise, a has a predecessor, denoted a’. Then, a+b equals the successor of a’ + b. * (Axiom 4)
For example :
3+2 = > successor ( 2+2 ) = > successor ( 1+2 ) = > successor( 2 ).
2 = > predecessor ( 3 ) = > predecessor ( 4 ) = > predecessor ( 5 ) .
If a is 1, and b is 1, then a = b is true.
If either a or b is 1, but the other is not, then a = b is false.
Otherwise, there exist a’ and b’, predecessors of a and b, respectively (Axiom 4). Then, if a’ = b’ is true, a = b is true, otherwise a = b is false.
The decimal number system and Arabic numerals are not part of this theory - they simply represent natural numbers. In any case, the successor of “1” is denoted “2”, the successor of “2” is denoted “3”, and so on. *
So given these definitions, it is clear that 1 + 1 = 2.
The Russel and Whitehead proof.
Believe it or not, it took Mathematicians Russel and Whitehead hundreds of pages to get to this result in the Principia Mathematica. This is considered to be a more rigorous proof for the proposition.
Going down that road is something only a truly dedicated mathematician would dare to endure.
Thanks for reading! Hope you enjoyed reading it as much as I enjoyed writing it.