Define Math: Arising from the Highest Righteousness Standard: Rigorous Logical Proofs.

Language work: ear2eye.org · Physics work: continuous2discrete.org

Today's artificial intelligence can instantly generate Euclid's "Elements" from the axioms and postulates of Euclidean geometry. The same is true for other mathematical fields, except that it takes longer than producing "Elements". After all, all mathematics is a combination of mathematical concepts connected by rigorous logic. Computers are machines that make rigorous logical connections. All the concepts in mathematics, even if you take them all, are far fewer than the number of Chinese characters.

• Definition 1： Mathematics is Pure Form without Contents: Mathematics originated from counting the contents of nature, but has long lost these contents. As the highest or purest form of language, mathematics is the only pure form or content-free language that can accommodate any content. Thus, mathematics is to mathematical literature what poetry is to theorems and proofs of theorems are to music.

• Definition 2: Mathematics is the Only Righteous Discipline: Mathematics is proved correct by definitions and rigorous logic, which stands as the highest standard of righteousness. In contrast to all the other fields that rely on evidence, mathematics demands proof. Infinite examples mount to nothing. So, mathematics is the only discipline where being right is not a matter of evidence, consensus, or probability.

• Definition 3: Mathematics Started from Nothing: Mathematical genesis through the formal generation of the Daoist creation sequence: from nothing arises one, from one comes two, from two comes three, and from there, all things. In mathematical terms, nothing is the empty set. From the empty set comes the set of its subsets, and the rest. A mathematical statement—whether a claim, conjecture, or hypothesis—is considered true if it can be logically traced back to the foundational words and axioms from which mathematics begins. This idea reinforces the notion that mathematics is built upon itself, with each new word and theorem grounded in the established basics. For human mathematicians, the task is often difficult because they must navigate through complex layers of words and theorems. However, AI, which can process and recall foundational axioms effortlessly, can work directly from the beginning, reducing the need for complicated intermediate steps.

• Definition 4: Mathematics Is Everything We Already Knew: If mathematics has any information, knowledge, or anything of value, you already know. Your only problem is that you don’t know what you know. The mathematical proofs tell you what you learned or where to find what you already have. Mathematics started from the empty set. Specifically, everything in mathematics is already in the initial words and axioms. The fiber of rigorous logic weaves all the mathematics. Rigorous logic writes only two types of sentences: equalities and inequalities. The former maintains information and produces no new information, and the latter loses information. Some of us have real-life experiences in looking for things we already know, i.e., the Miranda warning is a legal requirement. So long as a suspect provides no new information. Follow rigorous logic; no guessing, no speculations. The police always only know what they already know. So, after mathematics, law should logically be the second subject that AI could potentially do better than humans." Please give me the shortest version, and if you have more, please give it to me too.

• Definition 5: Mathematics History: First discretize, then continue, then discretize again—this is the current continuous mathematics. In the future, we will move towards discretization. To calculate massive sums or products, infinity replaced massive quantities, and limits replaced finite ones. Continuous models have overshot discrete reality itself. In the future, starting now, using computers, we will transition from continuous ideal back to discrete reality.

• Definition 6: Mathematics History: Finite/discrete mathematics and infinite/continuous mathematics, the latter needs the proof by contradiction.

Papers & Preprints

Define Mathematics

• Mathematics, From Nothing to Everything

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A tiny lab by Dong Zhang. We start from nothing (∅) the empty set and only keep what can be proved by rigorous logical proofs. In physics we side with the discrete reality that made of atoms: atoms aren’t identical; field sums aren’t zero; photons live in an electromagnetic background.

Language work: ear2eye.org · Physics work: continuous2discrete.org

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