Bertrand Russell

The fundamental cause of the trouble is that in the modern world the stupid are cocksure while the intelligent are full of doubt. Bertrand "Bertie" Russell (1872—1970) was a British mathematician, liberal, pacifist, socialist, philosopher and logician. He also wrote books and essays for popular audiences; a good many of these were full of his signature snark about religion. Russell's Teapot is a good example of this. Along with Gottlob Frege and Ludwig Wittgenstein, he is considered one of the founders of modern and one of the greatest logicians of the 20th century.

Contributions to mathematics
He coauthored Principia Mathematica with A. N. Whitehead (not to be confused with Isaac Newton's Philosophiæ Naturalis Principia Mathematica), which attempted to use symbolic logic to derive mathematics from basic axioms (1+1=2 is proved in Volume II). It is highly regarded by many mathematicians, but is pretty unreadable unless you have a brain the size of a planet, or no life (and that's still no guarantee). The book was an attempt to make Hilbert's Program a reality by its use of a finite and small set of axioms with which one could (in principle) derive all of modern mathematics. PM is essentially an attempt to make all of mathematics constructive.

Austrian logician Kurt Gödel eventually proved that PM is incomplete, which means that it does not have the ability to produce theorems that prove everything within its particular domain (all of mathematics in this case). The reason for this was essentially that Russell and Whitehead overlooked meta-mathematics, and did not believe (or it did not ever occur to them) that their system could be used to reason about meta-mathematical concepts. Gödel observed that you could in fact use PM with meta-mathematics, and he promptly invented Gödel-numbering, which allows one to prove things about PM itself in PM. Contrary to popular belief, he did not prove that it was "wrong", "inconsistent", "useless", or any other adjective that implies incorrect.

Theory of definite descriptions
Russell's most notable contribution in the philosophy of language is his theory of definite descriptions, used to provide a structure of meaning to existential statements like:

“The present King of France is bald”.

The significance of such a sentence is that there is no present king of France and so this sentence does not refer to anything. If we were to take a naïve view of language that suggests meaning comes from what an expression denotes (as J.S Mills did) then such a statement would be meaningless. This is counter-intuitive as humans seemingly talk of non-existent things meaningfully all the time. Russell proposed that the meaning arises from the proposition expressed by a given statement which can be represented in symbolic notation used in first order logic. In this case such a sentence would take a form like:

'“∃x((Kx & ∀yKy → x=y) & Bx)”

What this says is there exists a king of France, at most there is only one such king, and that king is bald. Of course nothing actually satisfies the truth conditions of this statement so it is false, but it is still meaningful. To illustrate what the meaning is symbolically is the purpose of such notation for definite descriptions.

The use of such definite descriptions to assign a logical meaning to statements that refer to nothing real is not controversial, but to suggest that all proper names are merely a concealed set of definite descriptions is very controversial. Russell did indeed suggest that all proper names were merely references to sets of definite descriptions and this was a point of attack from the philosopher Saul Kripke. If you take something like the proper name Aristotle to be series of definite descriptions like “the student of Plato, teacher of Alexander the Great, etc.” We can still talk about Aristotle counterfactually — for example, talking about what would happen if Aristotle died when he was two years of age. To talk about such modal possibilities makes reference to an Aristotle not met by the definite descriptions we apply to Aristotle in our actual world. This may seem to be an esoteric point, but think of the possibilities of making discoveries about the life of Aristotle that overturned what is accepted about his life currently; so this isn’t a trivial manner. Kripke suggests an alternative theory, arguing that proper names function as rigid designators referring to the same subject/object across all possible worlds. This doesn’t make definite descriptions from Russell’s theory to represent statements like “the unicorn was majestic” irrelevant however

Logical atomism
Russell developed a theory called logical atomism which doubled both as a metaphysical view and a view on appropriate philosophical methods; it was the view that the world consists only of particular objects bearing certain qualities or standing in relation to other particular objects. With that the world would come to bare certain “atomic facts” that could be described by atomic sentences in formalized logic. This was developed in attempt to develop an idealized language that could in principle come to describe every truth about the world (at least in theory).

The view came to influence Russell’s student Ludwig Wittgenstein and is reflected in Wittgenstein’s early work the Tractatus Logico Philosophicus. Wittgenstein however came to reject this view in his later work the Philosophical Investigations.

Logical atomism can be contrasted by logical holism. The former view describing the world being made up of indivisible simple atomic facts that are knowable, each coming together to build complex fact that can then go on to describe the world. Logical holism rejects such knowable atomic facts and argues that every knowable statement can only come to be known when one is aquatinted with the system of facts as whole.

Logical holism was espoused by philosophers like W.V.O. Quine

His arguments for materialism
Russell has provided arguments in response to radical skepticism and metaphysical idealism in his 1912 work "Problems of Philosophy" that provide a notable example of Inference to the Best Explanation. This is in appealing to the explanatory power of materialism when providing explanations to his sense experience when seemingly observing his cat meow for food, and use the litter box. For Russell this becomes his argument against solipsism. To Russell there is no real distinction between metaphysical idealism and materialism as both ascribe the existence of objects that exist independently from his specific subjective experience. He also comes to reject Descartes' "I think, therefore I am" arguing that the only thing he can know for certain is that sense experience/thought is currently happening, and that there is no justification to believe there is an “I” having such experiences (echoing David Hume). This has lead some to characterize Russell as a kind of foundationalist whereas abductive reasoning is an acceptable form of inference to other forms of knowledge within Russell's epistemology.

It should be noted that "Inference to the Best Explanation" is a controversial form of inference, especially in Philosophy of science due to "dynamic Dutch book" challenges provided by Epistemic Bayesians.

Other stuff
Russell's crowning achievement was having the character of Professor Yaffle from the much loved UK children's television series  modelled on him.

Well-known publications

 * Why I Am Not a Christian (1929). Haldeman-Julius Publications ISBN 9780415325103
 * History of Western Philosophy (1946) Routledge Classics ISBN 0-415-32505-6
 * Essays in Skepticism (1963). New York: Philosophical Library ISBN 9780806530116