It is possible to formally prove that 2+2=4 using Peano's five axioms for the natural numbers and elementary set theory (actually a long and tedious process). Is it possible to prove it based on the definitions and axioms for numbers given in Euclid's Elements?

  • "Numbers are so familiar that it hardly occurs to us that the theory of numbers needs axioms, too. In fact, that field was one of the last to receive a careful scrutiny, and axioms for numbers weren’t developed until the late 19th century by Dedekind and others. By that time foundations for the rest of mathematics were laid upon either geometry or number theory or both, and only geometry had axioms." From aleph0.clarku.edu/~djoyce/java/elements/bookVII/bookVII.html.
    – user3164
    Jun 21 '14 at 6:51
  • The question fits Math.SE best.
    – user132181
    Jun 21 '14 at 8:31
  • 2
    The proof of 2+2=4 in PA is only long and tedious in the same way that many people consider 2 miles a long distance to run: the only people who care about proving 2+2=4 (and a good number of people who do not) find it an easy jog down to the corner. Perhaps you had in mind Principia Mathematica? Jun 21 '14 at 10:21
  • @NieldeBeaudrap I sketched an outline just now at math.stackexchange.com/questions/842314/prove-that-22-4/… Jun 22 '14 at 5:50
  • My comment still stands. This may not be the one-liner one might optimistically hope for elementary arithmetic, but anyone who has done propositional logic in a formal system can tell you that this is essentially short and sweet. Jun 22 '14 at 12:09

On the issue of Euclidean Arithmetic, see by Ian Mueller, Philosophy of Mathematics and Deductive Structure in Euclid's Elements (1981 - Dover reprint).

All Ch.2 is devoted to this topic; see page 58 :

In books VII-IX Euclid develops the subject of arithmetic in almost complete isolation from the remainder of the Elements. [...] [In contrast to previous books, we] find no specifically arithmetic postulates in the Elements.

We have definitions regarding numbers in Book VII [see Euclid's Elements] :

Book VII : Definitions

Definition 1 : A unit is that by virtue of which each of the things that exist is called one.

Definition 2 : A number is a multitude composed of units.

Definition 3 : A number is a part of a number, the less of the greater, when it measures the greater;

Definition 4 : But parts when it does not measure it.

Definition 5 : The greater number is a multiple of the less when it is measured by the less. [...]

But there ara no Axioms; see Mueller, page 59 :

Euclid does not prove that 2+2=4 or that a 2 and a 2 combined yield a 4, nor does he even have the apparatus for doing so. Such facts, insofar as they are used in the Elements, are used without proof.

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