The Peano axioms are a set of axioms for the natural numbers proposed by Italian mathematician Giuseppe Peano. From these axioms, one can define addition recursively and prove basic facts of arithmetic. A standard construction defines 2 as the successor of 1 and 4 as the successor of 3, and proves within this system that 2 + 2 = 4 as a theorem of arithmetic.

Logicism is the thesis that mathematics is reducible to logic, and hence that all mathematical truths are logical truths. This project includes giving rigorous derivations of elementary arithmetic statements such as that 2 + 2 = 4 from logical axioms plus definitions. Frege and later Russell and Whitehead famously attempted to derive arithmetic in this way in *Grundgesetze* and *Principia Mathematica*.

The identity 2 + 2 = 4 is frequently used for that discussion precisely because it is noncontroversial in all human cultures that have counting numbers and arithmetic up to 4. What makes the identity noncontroversial is that, if you count things in the world, four is what you get when you combine two with two. Indeed, from within the culture of contemporary professional mathematics, the claim that the identity is cultural is nonsense. Within that culture, it’s a universal fact, both theoretical and empirical.

This is a survey chapter about issues in the epistemology of elementary arithmetic. Given the title of this volume, it is worth noting right at the outset that I will assume that we do know that 2+2=4, and that other similar truths of elementary arithmetic are known as well. The question that guides this chapter is: how do we know such things? In particular, do we know them a priori, a posteriori, or some combination of the two?

Sabine Hossenfelder writes: "If you take two integers and use the standard addition law, then, yes, two plus two equals four." She then notes that the expression 2+2 can be interpreted differently in other mathematical structures: "The point is that two plus two is a symbolic representation for the properties of elements of a group. And the result depends on what the 2s refer to and how the mathematical operation '+' is defined." In the usual arithmetic of integers, however, the law is fixed and yields 2+2=4.

Khan Academy’s introductory article on basic addition explains that addition is about combining quantities. For example, it presents problems like "If we have 2 apples and we get 2 more apples, how many apples do we have now?" and notes that such problems are represented as 2 + 2 = 4. These examples illustrate the standard arithmetic fact that combining 2 objects with 2 more gives 4 objects.

The document introduces Peano's axioms and then defines addition on the natural numbers by two recursive rules: "(1) For all n ∈ N, n + 1 = σ(n). (2) For any n, m ∈ N, n + σ(m) = σ(n + m)."[4] Using these definitions and induction, it states: "Now proofs of all the familiar properties of addition and multiplication of integers can be carried out, by using the definitions and corresponding properties."[4] This includes basic identities such as 2 + 2 = 4, which are theorems in the system once numerals like 2 and 4 are identified as iterated successors of 1.

The video explains how to prove that 2+2=4 using the Peano axioms. After defining the natural numbers and addition recursively, the presenter concludes: "we've managed to show that 2 plus 2 is indeed equal to 4." The proof explicitly uses the definition of 2 as the successor of 1 and of 4 as the successor of 3 in the Peano framework.

In an informal discussion of formal systems, the article remarks: "The Peano axioms might have been convenient for deducing a set of theorems like 2 + 2 = 4, but really all of those theorems were true about the natural numbers long before anyone named them."[3] It continues: "And you could claim, accordingly, that '2 + 2 = 4' is 'valid' because it is an inevitable implication of the axioms of Peano Arithmetic."[3] This explicitly identifies 2 + 2 = 4 as a theorem logically implied by the Peano axioms.

In mainstream mathematics, the phrase "2 + 2 = 4" is understood inside the standard model of the natural numbers ℕ with its usual addition operation. Within this model, addition is a well-defined function satisfying the Peano axioms, and the equation 2 + 2 = 4 is a provable theorem. When mathematicians say that 2 + 2 equals 4, they are implicitly referring to this standard structure of arithmetic.

In this video on the Peano axioms, the presenter defines a successor function on the natural numbers and then defines addition recursively: "we'll define it recursively as follows a plus zero equals a and a plus the successor of b is equal to the successor of a plus b."[1] Using these definitions, he then explicitly applies them to show: "and with that we've proven two plus two is equal to four"[1], demonstrating a concrete derivation of 2 + 2 = 4 within the Peano framework.

In the discussion, one of the participants states: '2 + 2 does equal 4, and that is objectively real.' The conversation explores whether mathematical objects and truths such as '2 + 2 = 4' are discovered or invented, but the participants treat the equation itself as an uncontroversial mathematical truth. They argue that mathematical entities have stable properties that are true for all people who think about them, independent of our minds.