# Bertrand Russell’s Understanding of Logic and Mathematics

June 26, 2021Russell’s views on mathematics follow a logician (“logicalizing”) line. Logicism is the view that mathematics can be reduced to logic, in other words, that mathematics can be derived from logic.

Russell showed that the most fundamental axiom of the logic system, introduced by Gotlob Frege, the founder of the logician school, as being able to derive arithmetic in his book The Fundamental Laws of Arithmetic, published in the second volume of 1903, is inconsistent (the Frege system, which allows the existence of a set in which all sets that are not members of themselves are members, such contained the contradiction that a set could exist if and only if it did not exist; Russell showed this.)

Although Russell’s contribution, also known as the “Russell Dilemma”, destroyed Frege’s logician system, it did not prevent him from defending a new logicism and publishing “The Principles Mathematics” (Principles of Mathematics, 1903), one of the most valuable works of this field.

Then, in “Principia Mathematica” (1910, 1913; 3 volumes), which they wrote together with the Cambridge mathematician and philosopher A. N. Whitehead, Russell and Whitehead presented a system that did not fall into contradiction with Frege’s system. Although the Russell-Whitehead system has its own problems, the important place of this work in fields of study such as the philosophy of mathematics and the foundations of mathematics is indisputable.

The typical feature of Russell’s philosophy is that modern formal logic, of which he is considered one of the founders, is celebrated as a guiding method in almost every philosophical problem. Working in the field of philosophical logic in the early 1910s, Russell believed that everyday language had many flaws that could be overcome by examining the logical forms of everyday language propositions in the light of modern quantification logic.

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