The success of epistemology, the philosophy of knowledge, as a branch of philosophy probably owes more to Bertrand Russell and Alfred North Whitehead's Principia Mathematica than to any other single work. What we are going to discuss now is the first international epistemological school of philosophy, the Logical Positivists. Without the "Principia" and the impact it had on the philosophy of the turn of the century there would probably have been no successful logical Positivist school, and modern analytic philosophy might never have developed. What Russell and Whitehead attempted was a complete reduction of mathematics to logic. And the attempt was nearly successful. Whatever the shortcomings of the work, it, and Russell's own manner of speaking, were undoubtedly part of the kindling that set off the Positivist movement. Russell and Whitehead's Principia failed to accomplish what they attempted, the reduction of all mathematics to logic simply because it was impossible task. Godel later showed that no consistent set of logical relationships could be proven within the system itself. To do so would add another proposition outside the field which would then create another set which could not be proven without stepping outside that. Bronowski applied that to scientific knowledge. He explained it using the analogy of a paradox presented by the ancient Greek Epimenides. What Epimenides said was "All Cretans are liars." But, since Epimenides himseelf was a Cretan, if the statement is true, then his making the statement makes it false. Therefore you can neither prove the statement to be true, nor show it to be false. The problem comes up whenever you try to prove the truth of any scientific statement. As Bronowski put it, whenever you use the phrase "the statement is" you are no longer using the language to describe things. You are in a different universe of discourse about statements about things and you can never get out of it. This fact, Bronowski said, was proven by Alfred Tarski in 1931.

In 1931 Tarski proved quite simply that all this amounts to saying that there is no complete language of science. If you look at Tarski's proof, it essentially consists in showing that, as soon as you not only exhibit the statements but add to them the statement "is true," you are in trouble because you are bound to be landed in contradictions like the one which arises in Epimenides paradox.

The logical positivists began their search prior to this development and thus had much more optimism than would have been if they had realized it. Keep in mind that Russell does not fit neatly into the Positivist tradition. However, he encountered the same problem. He came to philosophy from mathematics. What he did was to elaborate the role of logic in philosophy, to elevate it in fact into a science. In  "Logical Atomism" he stated that logic is what is fundamental to philosophy and that schools should be characterized by their logic rather than their metaphysics.

There are a few basic problems that plagued post-Humean empirical philosophy. The first dealt with the problem of experience itself. John Locke's description of experience began with experienced things which were reduced to ideas in the mind. Hume reduced experience to passions and impressions. Later empiricists saw Locke's ideas as mysterious occult entities. At the same time, Hume's passions and impressions lacked the necessary mechanical connection with the outside world that were the basis of Descartes' passions and impressions. They were devoid of knowledge on their own. His "fork" ensured that they could never be directly connected with things in a world outside the mind. In general positivists rejected the transcendental approach of Kant. Thus what the problem of knowledge came down to was the discovery of the meaning of sense perceptions. Russell never even assumed that physical entities existed in the world. By that I don't mean that he denied their existence, but that his approach neither assumes their existence nor requires it. What his approach finally came down to was to make philosophy a subject indistinguishable from science. He thought that the idea that it might be different was an unfortunate legacy from theology.

The premises of mathematics, he said, deduced as they are from a set of propositions developed as assigned premises, cannot be believed solely on the truth of these premises. The premises are less obvious than some of their consequences. They are thus believed on the basis of those consequences. But, he insisted, this is what science is. Scientific concepts, Maxwell's equations for example, are believed because of the observed truth of their consequences. As a result it is not the logically simplest propositions that are the most obvious. Why should we believe any given set of propositions? Our reasons for believing logic and pure mathematics are partly inductive, partly Probable in spite of the fact that in their logical order the propositions follow from the premises by pure deduction. Mathematics, then, may be true, but to prove it true would require other considerations.

Again, beginning with the empiricists view that all knowledge must be derived only from experience, the problem of knowledge becomes the problem of the reduction from sense experience. Russell found two steps necessary to accomplish this reduction. The first would be a technically pure logical language which could express "Atoms", of sense content. By this he meant sentences which express hypotheses concerning the possible meaning behind the content of sense impressions. The second requirement would be a formal logical language which could express the logical relations between these Atoms of sense content. From that point on philosophy would be simply another form of science searching for ways to verify the hypothesized relationship. This was what he called Logical Atomism, and became the basis for a more general approach called the "Hypothetical-Deductive" approach to knowledge.

Of course the idea that knowledge could only be derived from experience is Aristotelian as well as empiricist. But Aristotle maintained that we derive knowledge from experience through inductive reasoning. David Hume showed that inductive reasoning was illogical, that it could never result in knowledge. Just as important was the concept of necessary cause that was so fundamental to all pre-Cartesian philosophy. This, as we have seen already, was replaced first by the Galilean turn to experimentation, and second the Humean rejection of necessary cause. So, as Russell said, philosophy, should be simply another science beginning with atoms of sense content then deriving knowledge from the logical relationships behind this sense content.