Random reflections on mathematics.

In my youth, I would make jokes claiming that 2+2=3 or 2+2=5. My teachers disapproved, and I was, of course, fundamentally incorrect.

We begin with the numbers: 0, 1, 2, 3.

This is how we learned to define quantities, such as stating that there are five people in a group or that we have killed one mammoth

We then learn addition: 1+1 = 2, 1+1+1 = 3. Occasionally, we learn the inverse operation, such as 1 = 2 + (-1). This is how we learned to count

For instance, if one more person joins us, we now have 5+1 = 6 people attempting to hunt a mammoth.

Next, we learn multiplication: 1 * 2 = 2, 2 * 2 = 4; and sometimes division: 6/2 = 3, along with the inverse operation: 1/2 is the inverse of 2. For example, one mammoth is sufficient to feed 16 people, meaning one person can consume 1/16 of the mammoth.

Modern mathematics studies these concepts as follows:

• Formal Arithmetic: We have 0 and 1, and the operation of increment (+1).
• Group Theory: Studies the operation (+) and certain constraints (e.g., a+b = b+a).
• Field Theory: Studies the operation * and sometimes /.
• Field Extensions: Studies the addition of new elements, such as i, where i * i = -1. Sometimes, this can be expressed as ℂ = ℚ(√-1), yielding the well-known complex numbers, or more advanced structures like quaternions ℍ, where i * i = j * j = k * k = i * j * k = -1.

This progression leads to: ℤ⁺ ℤ ℚ ℂ ℍ …

Studying field theory provides a highly abstract concept: one should never think about numbers, but only about operations. For example, one can define a seemingly unusual multiplication operation as i * i = -1, which makes no sense in the context of hunting mammoths. Similarly, quaternions are very abstract, yet they remain essential for physics.

The next key point is as follows:

1. We have one of the great inventions of the 20th century: the calculator. It is excellent for performing +, -, /, and * operations on numbers.
2. Now, we have AI. AI is excellent for performing +, -, /, and * operations without specifying what +, -, /, and * mean.

For example, on elliptic curves, there is a sum of points, where you take a point and find the next point on the elliptic curve, creating the sequence: P, P + P, 3P, 4P … You then take another point Q and create the sequence: Q, 2Q, 3Q. This addition operation is defined by the tangent to the point on this elliptic curve.

To the surprise of many, this task requires computation - specifically, knowledge of how to construct the tangent and how to find the next point. It is noteworthy that our generation often cannot even multiply mentally, let alone handle elliptic curves used in the most advanced cryptography.

Therefore, despite the extreme complexity and abstractness of mathematics, particularly addition on elliptic curves, there is also multiplication on them, which is even more complex for humans.

However, today, in an era characterized by:

1. Computers that allow for the rapid verification of whether addition on an elliptic curve is correct
2. AI that can perform formal reasoning

We obtain mega-calculators for tasks such as computing minimal elliptic curves with maximal rank, as mentioned above.

Do you understand the implication? We have formalized arithmetic, addition, and multiplication, moved away from digits, and created a mega-calculator that is also capable of formally deriving proofs.

In other words, AI has not invented anything new; rather, we have begun solving a new class of problems—more abstract, but still using the same method: the calculator of the 21st century.

And depends on the task: count mammoths, or lock a super secure safe locker 🔒, you have a sum function that the calculator can use.

And what about 2 + 2 = 3, and 2 + 2 = 5.

I was wrong. In standart arifmetics it is 2+2 = 4, but you can also consider Z_3, where all arifmetics is by module 3, and write 2+2=1