Just like two two four

Probably every habrovchan at least once in his life heard this expression. Indeed, what could be easier? However, I knew a teacher of mathematical analysis, who, having heard this, smiled mischievously at his mustache and offered to prove this fact. After that, the speaker usually had cognitive dissonance.

And really, how to prove that 2 × 2 = 4? The answer is under habrakat.

Disclaimer

This article does not contain anything new for readers with a serious mathematical education. Also, it is likely that it will be uninteresting to people with a purely engineering mindset. This text was written in the calculation of those who are interested in the foundations of mathematics, but who still have not found the time and strength to figure them out.

Start over

What are natural numbers? Four out of five people met on the street will answer: "This is one, two, three, and so on." A stricter formulation of this answer, which I met in a school textbook, reads: natural numbers are members of an arithmetic progression starting with 1 and having a difference of 1. Another definition from the textbook: these are numbers that are used to denote the number of objects.
')
Until the end of the XIX century, natural numbers were determined approximately like this, or were not determined at all, relying on something taken for granted. And then perestroika began: the building of mathematics began to be transferred to the foundation of the theory of sets, and things that had previously seemed elementary suddenly demanded rigorous substantiation.

Peano Axiomatics

Comrade Giuseppe Peano, a great prankster and entertainer (which is at least a Latin-blue-flexion ), created a very simple and compact axiomatic of natural numbers, which is still used today. The natural numbers in his interpretation are similar to the " simply linked list " data structure - true, infinite.

So, the natural numbers are the set ℕ with the following function given on it: a → a ', which satisfy the following three axioms:

1. For each natural number a, there is a unique number following it a '.

This axiom means that our simply linked list is infinite. There is no such element that has null in the “next” field. It is also a list, not a binary tree: each element has only one next one.

2. There is one and only one number, not the next for any other. This number is called a unit. Each of the remaining numbers follows exactly one number (thanks to Kozy , I missed this phrase in the original version).

The list should have a head, and only one. The list should not go in cycles (the third element cannot be followed by the second).

3. The set of positive integers does not have its own subset satisfying axioms 1-2.

Without this axiom, we could, for example, add to the set of natural numbers one more uroboros number following itself. Or two more numbers that follow each other. In other words, Axiom 3 does not allow memory leaks that could arise from the isolated pieces of the list, which cannot be reached from the links if you go from the head. If you can throw something out of natural numbers, these are not natural numbers.

Addition and multiplication

Arithmetic operations in Peano's axiomatics are no less interesting. Addition is described by the following two properties:

1. (a + b) '= a + b'
2. a '= a + 1

- and multiplication - these two:

1. a × b '= a × b + a
2. a × 1 = a

Surprisingly, there is not a word about commutativity, associativity, distributivity and other properties of addition and multiplication, which are told in school. All of them are derived from these four basic ones.

2 × 2 = 4

Armed with knowledge, we can now proceed to the proof. However, you first need to understand two things: what is 2 and what is 4. Two follow the unit, therefore 2 = 1 '. The four follows the three, which, in turn, follows the two, which, as I said, follows the unit - therefore 4 = 1 '' '.

So, we need to prove the following: 1 '× 1' = 1 '' '.

First, we prove that two and two are two plus two. Really,

1 '× 1' = (1 '× 1) + 1' (the first multiplication property)
1 '× 1 = 1' (second multiplication property)
Therefore, 1 '× 1' = 1 '+ 1'.

Now we prove that 2 + 2 = 4.

1 '+ 1' = (1 '+ 1)' (first addition property)
1 '+ 1 = (1') '= 1' '(second addition property)
Therefore, 1 '+ 1' = (1 '') '= 1' ''

Conclusion

Any simple thing, if you look into it intently, after a while, it no longer seems simple. Natural numbers and operations on them are not an exception, but rather a vivid example. Even more complicated and interesting way in modern mathematics is the construction of sets of integers, rational and real numbers. But this is a completely different topic.

Post Scriptum