How Can Any Part of Mathematics Be Proven? The answer lies in demonstrating that a mathematical statement must be true if the underlying simpler math is also true. It is a great difficulty to show the increment from 1 to 2. Between 1910 and 1913, a three-volume work was published on this subject. Titled Principia Mathematica (The Principles of Mathematics), it imitated Isaac Newton’s 17th-century research. Its aim was to establish the fundamental basis of mathematics through logic. Authored by celebrated British philosophers Bertrand Russell and Alfred North Whitehead, this work is a cornerstone of the philosophy of mathematics. The first volume outlines the approach for the subsequent volumes, focusing on logical type theory. In type theory, every mathematical object is categorized within a hierarchy of types, each a subset of those above it. This categorization aims to prevent paradoxes, which often arise in logical systems. The second volume examines numbers. The third volume covers series and measurement. Despite its excellence, Gödel’s theorem would soon reveal that any attempt to prove the entire system of mathematics logically, including Russell and Whitehead’s effort, is itself a logical impossibility.

References

- Russell, B., & Whitehead, A. N. (1910-1913). Principia Mathematica. Cambridge University Press.
- Newton, I. (1687). Philosophiæ Naturalis Principia Mathematica.
- Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik und Physik, 38(1), 173-198.