# Reducing Mathematics to Logic

June 26, 2021Russell’s first work on the philosophy of mathematics was An Essay on the Foundations of Geometry in 1897. This work is generally written in a Kantian line. Russell later thought that this approach contradicted Albert Einstein’s understanding of space-time and completely abandoned the views he advocated here.

Russell’s interest shifted to the definition of number in the following period and he examined the views of mathematicians and philosophers such as George Boole, Georg Cantor and Augustus De Morgan. In the same period, it was determined by the researches of historians that he had an idea about the works of Charles Sanders Peirce and Ernst Shröder. In 1900, at a congress he attended in Paris, he met the Italian mathematician Giuseppe Peano (1858 – 1932). At that time, Peano continued to work on arithmetic in an axiomatic system. Accepting the terms “zero”, “number” and “consequence relation” and the English determinant “the” as simple (undefined), he aimed to deal with all axioms related to numbers and the theorems of arithmetic based on them in a system. Russell wondered whether these basic terms could also be defined logically, and he focused his work on this. Until 1903, he continued his studies on these subjects. From 1897 to 1903 he published three separate works: On the Notion of Order, Sur la logique des relations avecles applications à la théorie des séries and On Cardinal Numbers.

Russell thought that the foundations of mathematics could be dealt with in a logical system today called high-order (higher-than-first-order) logic. Russell encountered Frege’s work during this period and saw Frege attempting to define number with concerns similar to his own. As we have mentioned before, he discovered a paradox in Frege’s system and reported it to Frege in a letter. On the other hand, he presented this paradox and wrote a solution proposal in the appendix of his work, The Principles of Mathematics, which he was continuing at that time. At the same time, Russell was working on another proof of Georg Cantor. Cantor gave a proof that the greatest numerical number does not exist, and this theorem became known as the Cantor Paradox. At first, Russell thought this proof was flawed. It was later discovered that this paradox was a special case of the Russell Paradox. Thereupon, Russell concentrated on his work on sets and classes. Russell’s proposed solution in that appendix was based on Russell’s views on clumps. Russell later developed these ideas and developed what is now called the theory of types. The main purpose of this theory was to axiomatize set theory in such a way as to prevent the emergence of these paradoxes.

The famous Gödel Incompleteness Theorems (1931), on which mathematical logic and calculus are based today, assume the formal system presented in Principia Mathematica. Gödel showed that in a formal system strong enough to represent arithmetic propositions, there are true propositions that neither the proof nor the non-prove can be given, and in this respect, the formal systems have a deficiency in representing the true propositions of arithmetic. The second theorem states that the consistency of a formal system strong enough to represent arithmetic propositions cannot be given within the system.

Russell wrote the famous Principia Mathematica together with Alfred North Whitehead, using the results he obtained in the theory of types. This work, the first volume of which was published in 1910, was designed to consist of four volumes and to include the basics of geometry in the fourth volume, but the fourth volume was never published.

Russell’s last important work in the philosophy of mathematics is his book, Introduction to Mathematical Philosophy, which he wrote while serving a prison sentence for his anti-war attitude during the First World War. In this work, Russell presented his studies and philosophical results up to that point.

Prepared by: Sociologist Ömer YILDIRIM

Source: Omer YILDIRIM’s Personal Lecture Notes. Atatürk University Sociology Department 1st Year “Introduction to Philosophy” and 2nd, 3rd, 4th Grade “History of Philosophy” Lecture Notes (Ömer YILDIRIM); Open Education Philosophy Textbook