There Is No “Why” in the World#9

Proof in Deduction

Prove that √2 is irrational.

Then we should begin like this.

Assume that √2 is not irrational (that is, assume it is rational).
→ a contradiction occurs
→ therefore √2 is not rational
→ therefore √2 is irrational.

This method is called reductio ad absurdum.

Instead of proving a proposition P directly, we assume its negation ¬P, derive a contradiction, and return to the conclusion that ¬P cannot hold. In other words, reductio is less a technique for explaining truth, and more a technique for making falsehood unable to survive.

So this proof does not explain the reason why √2 is irrational.
It only shows that the claim “√2 is rational” collapses.

There is an important hidden premise here.
We must already possess an intelligence that can reconstruct the whole.

Only if we already know the classificatory scheme “if it is not rational, then it is irrational” can we move straight from the conclusion “it is not rational” to “it is irrational.”

The standard proof

Assume √2 = p/q (where p and q are coprime).

Square both sides:
2 = p²/q² → p² = 2q²

If p² is even, then p is even:
p = 2k

Substitute:
4k² = 2q² → q² = 2k²

So q is even as well.

But if both p and q are even, that contradicts the assumption that they are coprime.

Here we did not look into the true nature of √2.

What we actually used was not √2 itself, but rules about evenness and oddness, and the stipulation “coprime.” Using those rules and that stipulation, we showed that the assumption “√2 is rational” cannot withstand the structure it creates.

Proof without reductio

Of course, one can also prove things without reductio.

Some mathematical propositions look like constructive proofs. They reach the conclusion “it is so” while also presenting a procedure that constructs what is being claimed.

For instance, to prove that the sum of two even numbers is always even:

Let an even number a = 2m and an even number b = 2n. Then

a + b = 2m + 2n = 2(m + n)

This directly produces an expression showing that the result is even.

But even so, it is hard to say that we have arrived in a fundamentally different world.

Constructive proofs, too, ultimately run on definitions such as “an even number is a multiple of 2,” rules such as “treat identical things as identical,” and inference rules such as “this kind of calculation is permitted.” The method may differ, but the floor beneath it does not.

The floor beneath deduction

Deductive proof always runs on some foundation:

• Axioms

• Definitions

• Rules of inference

What are axioms?
They are what we decide to begin with without proving.

What are definitions?
They are the act of deciding to handle a concept according to certain rules.

What are rules of inference?
They are procedural rules that say, “It is logically acceptable to say this.”

These are not objects of proof. They are the floor that makes proof possible.

So the question “Why do we believe this axiom?” cannot be pushed all the way through from inside the system.

Because we decided so.
Because it is useful to decide so.
Because deciding so reduces contradictions.

But these are not proofs. They are reasons for a choice.

Where proof stops

If a mathematical system is sufficiently strong, consistent, and effectively axiomatized, then it is known that the system has the following limitations:

• There will be statements that are true but cannot be proven within the system.

• The system cannot, in general, prove its own consistency.

Deduction is powerful, but that power is taken for granted, and it is exercised only where we have chosen to stand.

We set out to build proofs in order to answer “why?”
But at the very bottom of proof, we are forced to face sentences like these:

From here on, it cannot be proven.
We decided to begin here.