The significance of the paradox follows since, in classical logic, all sentences are entailed by a contradiction. In the eyes of many mathematicians it appeared that no proof could be trusted once it was discovered that the logic apparently underlying all of mathematics was contradictory. A large amount of work throughout the early part of this century in logic, set theory, and the philosophy and foundations of mathematics was prompted by this. Written as a defense of logicism, the book promoted wide acceptance of modern mathematical logic. Next to Aristotle's Organon , it is the most influential book on logic ever written.
Logicism is the idea of reducing all of mathematics to logic.
This theory lost much support over the next few years. However,by 1903, both Whitehead and Russell had come to similarconclusions regarding the theory of logicism. Whitehead's 1898 A Treatise on Universal Algebra and Russell's 1903 The Principles of Mathematics . Since their work was so similar, they began to work together on what would become Principia Mathematica . Unfortunately, after almost a decade of difficult work, Cambridge University Press informed the men that publishing Principia would costapproximately 600 pounds.