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To the Netizen "Mathematics" Who Doesn't Even Understand Proof by Contradiction -- Please Change Your Name!

Chán Zhōng Shuō Chán

In mathematics, proof by contradiction is one of the most basic methods. Without proof by contradiction, there would essentially be no mathematics, and without mathematics, there would be no computers either. Yet this netizen calling themselves "Mathematics" doesn't even understand proof by contradiction -- please change your name!

After this ID posted "Netizen Mathematics, If You Have Any Academic Integrity, Stand Up and Admit Your Error!", this "Mathematics" person produced a post titled "Miss Chan's Academic Criticism of Me Is Quite Baffling," containing this statement: "To refute my view, one must specifically identify a law of motion that cannot be stored in a computer." Following the same logic, to prove that prime numbers are infinite, one must find a method, an actual program, to demonstrate that primes are infinite.

However, in mathematics, there is no need to find an actual program to prove that primes are infinite, because mathematics has a standard method called proof by contradiction: if primes were finite, multiplying all primes together and adding one would certainly not be divisible by any prime, which contradicts the divisibility properties of natural numbers -- therefore primes are infinite. Here, one can prove primes are infinite without even knowing a single prime number. Furthermore, even without knowing any transcendental number, we can use proof by contradiction to show that transcendental numbers are far more numerous than all algebraic numbers combined. This is the power of mathematical methods!

Since this "Mathematics" netizen doesn't even understand the most basic mathematical methods, I ask this "Mathematics" netizen to please change their name. The proof from this ID is appended below:

As for that "Mathematics" ID who regularly makes logical blunders, he recently proclaimed as if discovering a new continent: "From a mathematical theory standpoint, dichotomy, that is, using binary numbers, can express the entirety of human thought and laws of motion, because these thoughts can all be stored in binary computers." I have no idea what kind of "mathematical theory" this supposed Mathematics ID is referring to -- does merely claiming a Mathematics ID entitle one to spout mathematical nonsense? Very well, let us examine this so-called discovery or proclamation from this Mathematics ID in a thoroughly mathematical manner:

Let us assume this proclamation holds -- that all human thoughts and laws of motion can be represented using binary numbers. Then we line up these binary numbers representing all human thoughts and laws of motion. Now consider a binary number whose Nth digit differs from the Nth digit of the Nth binary number in our lineup (for example, if the latter is 1, the former is 0, and vice versa). Clearly, this binary number is not among the binary numbers representing all human thoughts and laws of motion, yet this binary number represents the following thought: "its Nth digit differs from the Nth digit of the Nth binary number in the lineup." In other words, this thought is not contained within the totality of human thoughts and laws of motion -- and this is a contradiction. By the simplest proof by contradiction, the assumption that "all human thoughts and laws of motion can be represented using binary numbers" does not hold. Not all human thoughts and laws of motion can be represented using binary numbers.

The proof strategy above is actually not an invention of this ID. It is a classical mathematical proof method with a popular name: the "diagonal method," which first appeared in Cantor's series of proofs on set theory in the 19th century. Leaving aside the mathematical developments of the 20th century, which exceeded all previous centuries combined, anyone with even a basic understanding of this 19th-century mathematical knowledge could not possibly proclaim such an absurd proposition.