👉 Verified mathematics is grounded in rigorous logical reasoning and established axioms, ensuring that mathematical statements are true by definition and not by empirical observation. It relies on fundamental principles such as the Peano axioms for natural numbers, which define operations like addition and multiplication, and the Zermelo-Fraenkel axioms for set theory. These axioms provide a consistent framework for proving theorems and theorems about mathematical structures, ensuring that conclusions drawn are logically sound and universally applicable. For example, the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse equals the sum of squares of the other two sides (a² + b² = c²), is derived from geometric principles and can be rigorously proven using logical deductions from axioms, without needing physical measurement. This verification process ensures that mathematical truths are reliable and can be communicated and applied across various fields, from physics to computer science.
