Just like two two four

Probably, every Khabrovsk citizen 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 maliciously in his mustache and offered to prove this fact. After that, the speaker usually experienced cognitive dissonance.

And really, how can one 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 not be interesting to people with a purely engineering mindset. This text was written for those who are interested in the foundations of mathematics, but who still have not found the time and energy to understand them.

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 more rigorous wording of this answer, which I met in the school textbook, says: 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 indicate the number of objects.

Until the end of the 19th century, natural numbers were determined approximately like this, or were not determined at all, relying on something for granted. And then perestroika began: the building of mathematics began to be transferred to the foundation of set theory, and things that previously seemed elementary suddenly required a rigorous justification.

Axiomatics Peano

Comrade Giuseppe Peano, a big mischief-maker and fanatic (which is worth at least Latin-blue-flexion ), created a very simple and compact axiomatics of natural numbers, which is still used today. The natural numbers in his interpretation are similar to the data structure of a “ simply connected list ” - though infinite.

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

1. For each positive integer a there is a unique number a 'following it.

This axiom means that our singly linked list is endless. There is no such element that has null in the "next" field. Also, this is a list, not some binary tree: each element has only one next.

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

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 natural numbers does not have its own subset satisfying axioms 1-2.

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

Arithmetic operations in Peano's axiomatics are determined no less interesting. Addition is described by the following two properties:
1. (a + b) '= a + b'
2. a '= a + 1

- and multiplication - with these two:
1. a × b '= a × b + a
2. a × 1 = a

Surprisingly, there is not a word here about the commutativity, associativity, distributivity and other properties of addition and multiplication, which are described 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, first you need to understand two things: what is 2 and what is 4. Two follows 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 twice two is two plus two. Really,

1 '× 1' = (1 '× 1) + 1' (first property of multiplication)
1 '× 1 = 1' (second property of multiplication)
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 closely at it, after a while it ceases to seem simple. Natural numbers and operations on them are not an exception, but rather a vivid example. In an even more complex and interesting way, in modern mathematics, sets of integers, rational and real numbers are constructed. But this is a topic of a completely different conversation.

Post script

Why is everything written above - nonsense and demagogy
As you know, the same theory can be based on completely different systems of axioms. The same axiomatics Peano has a bunch of options that differ in wording, but fundamentally similar. So how is axiomatics of natural numbers introduced at school?

This is not pronounced aloud (yes, schoolchildren at that time do not yet know scary words such as “set” and “function”), but in fact a lot of natural numbers in a school is defined as a set of lines of special characters called numbers. Lines must be finite, non-empty, and must not begin with a character called zero.

Relations of equality and inequality, addition, subtraction, multiplication and division - all this is determined through operations on strings of characters. For strings of one character (i.e., for individual digits) there are special tables - tables of addition and multiplication. For longer lines, special rules allow you to reduce actions on them to actions on individual digits. These rules and tables are the school axiomatics of natural numbers.

In this understanding of natural numbers, “2 × 2 = 4” is part of the axiomatics, since this identity is contained in the multiplication table. Then, really, nothing could be easier. But Peano's axiomatics are still not harmful to know.