# What is the best proof that 1+1=2 by a person who was not Bertrand Russell?

The question says it all. Besides Russell and Whitehead, who are the philosophers and mathematicians who demonstrated successfully that 1+1=2? Extra bonus (i.e., a warm feeling inside) awarded for identifying the book or essay where the proof appears.

• This question is slightly misguided. When you talk about a proof of something like that, you are really talking about proving it using the axiomatic method, right? But the axiomatic method in a completely formal setting wasn't initiated until Frege. Before then, people didn't care about providing a rigorous foundation of mathematics in the same way that they did afterwards. So you aren't going to find a book where people rigorously work out arithmetical identities based off of first principles in the same way they did. – Not_Here Nov 26 '18 at 6:49
• Mauro's answer really points out another issue with the framing of this question. PM proves 1+1=2 in much the same way you do so in PA, how Dedekind and Peano did it, but it needs to prove much more first. PM isn't significant because it proved 1+1=2, it's significant because it was a full system of mathematical logic that was able to use a formal axiomatic method to prove all tautologies and many mathematical statements. Maybe you are really just concerned with who else has proven a very basic arithmetical identity, but I think this is a classic example of missing the forest for the trees. – Not_Here Nov 26 '18 at 10:03
• It's too bad Kant went straight for 7+5=12. But Peano did one better, he spent four pages proving that 1 is a number and put Pythagoreans to shame. For they held that a number is a multitude of units, and hence the unit is not a number. – Conifold Nov 27 '18 at 5:55
• @Conifold - "Peano spent four pages proving that 1 is a number". Where ? – Mauro ALLEGRANZA Nov 27 '18 at 9:30
• @Conifold - Thanks. Maybe... in Peano's Arithmetices principia (linked below) 1 ∈ N is Axiomata 1 (page 1). – Mauro ALLEGRANZA Nov 27 '18 at 9:50

In a modern axiomatic approach, it follows from the definition of 2 as the successor of 1 :

2=s(1) (also written : 2=1').

See Giuseppe Peano, Arithmetices principia: nova methodo exposita (1889), page 1.

The basic machinery was already in place in Richard Dedekind, Was sind und was sollen die Zahlen ? (1893), para.XI : Addition of numbers :

sum is completely determined by the conditions

m + 1 =m',

m + n' = (m + n)'.

Applying the first condition to m=1 we get :

1+1=1'.

Peano's approach is presumably derived from Leibniz's well-known proof of 2+2=4.

See New Essays on Human Understanding (1704), Book IV, vii,10 [English transl, page 414] :

Definitions. (1) Two is one and one.

(2) Three is two and one.

(3) Four is three and one.

Leibniz introduces only one axiom for equality :

Axiom. If equals be substituted for equals, the equality remains,

and thus the proof is incomplete, by modern standard, because it relies on the implicitly assumed associativity of sum.

Another source of Peano's work has been Hermann Grassmann.

Grassmann's Lehrbuch der Mathematik (1861) contains a reasonably complete axiom system for arithmetic.

See Hans-Joachim Petsche, Hermann Grassmann : Biography, Birkhauser (2009), page 198-on.

• That ''sum'' quoted section implies m+n'=(m+n)'=m'+n' where you subtract n' from both sides and you end up with m=m' LOL – EternalPropagation Nov 26 '18 at 15:17
• @EternalPropagation - (m+n)'=m'+n' is wrong. The successor of (2+3) is 6 while the succesor of 2 is 3 and the successor of 3 is 4 and 3+4=7. – Mauro ALLEGRANZA Nov 26 '18 at 15:23