Repercussions from the Richard Paradox: Math Rules

By

Home / Repercussions from the Richard Paradox: Math Rules

Jules Richard, the French Mathematician

Jules Richard was born in France in 1862; he died in 1956. His statement of the paradox used the French language. The only picture of Jules Richard that was found online seems to be under copyright.

The Repercussions of Richard’s Paradox

One danger inherent in defining sets in natural language is that it is easy to assume that the definition is possible and does not contradict itself. This drove mathematicians toward formal logic. They would define symbols carefully and manipulate them according to certain rules. Giuseppe Peano attempted to restate Richard’s paradox in a more rigourous and formal manner, although this might not have been a satisfactory approach.

For All x, the Function of x, image by Mike DeHaan

Eventually a more useful solution came by setting limits on how sets or their elements are defined. The key was to forbid defining an element of a set by reference to other elements, so the digits of ‘f’ could not be defined by Cantor’s diagonal argument.

Other mathematicians have proposed yet further useful axiom collections. Natural language is simply not precise enough, and might not follow the rules created using formal axioms. In general, manipulating axioms in a formal manner avoids the Richard Paradox by disregarding any natural language definition for a set.

References:

Cantini, A. Paradoxes and Contemporary Logic. The Stanford Encyclopedia of Philosophy (Spring 2009 Edition, Edward N. Zalta, editor). Accessed January 26, 2012.

Feferman, S. The development of programs for the foundations of mathematics in the first third of the 20th century. Accessed January 26, 2012.

Lahr, D. and Rudel, D. The World According to Mathematics. (2002). Dartmouth University. Accessed January 26, 2012.

Click to Return to Page One: Repercussions From the Richard Paradox

Leave a Comment