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Ludwig Josef Johann Wittgenstein (1889 – 1951) was an Austrian-British philosopher, professor in philosophy at the University of Cambridge (1939-1947). He worked in foundations of mathematics and on mathematical logic, the philosophy of mind, and the philosophy of language. He played a central, if controversial, role in 20th-century analytic philosophy.

Ludwig Josef Johann Wittgenstein (1889–1951) was an Austrian-British philosopher, professor in philosophy at the University of Cambridge (1939-1947). His work, technically separated into three periods but generally divided into the 'early' and the 'later' Wittgenstein, was primarily focused on the philosophy of mathematics and the philosophy of language.

Originally a student of engineering, Wittgenstein always had a keen interest in mathematics and was keen on the idea of designing and flying aeroplanes. After receiving a diploma in engineering he encountered the works of Bertrand Russell and Gottlob Frege on the foundations of mathematics and became deeply intrigued in formal logic, visiting Frege who in turn recommended he go to Cambridge to study under Russell. Wittgenstein would become Russell's fiercest pupil, seeking to challenge the logician at every opportunity.

Initially a collaborator in the logical atomist movement pioneered by Gottlob Frege and Bertrand Russell, Wittgenstein's concern for the limitations of language led him to became one of the movement's critics, before Gödel formally proved the futility of the logical atomist project. Wittgenstein's first work, Tractatus Logico-Philosophicus, explores the limitations of the logical atomist project and of philosophy itself, though rooting his thesis in formal logic and the propositional approach to philosophy and knowledge. One of the most significant ideas in the Tractatus is the distinction between what can be said and what can be shown, an idea that is surely rooted in his conception of mathematics as a language with only a syntax and no semantics or factual reference, and thus as a language which can be grasped through its syntax alone and can only generate rules. Ironically, the Tractatus was somewhat misunderstood by the logical positivists of the Vienna Circle, who basically sought to weaponise its concluding statement:

7. Whereof one cannot speak, thereof one must be silent.

against metaphysics, in a bid to silence the obscure, unscientific and often verbose continental philosophers with whom their movement took issue.

Frustrated with the lack of understanding he received from his contemporaries and the futility of even his own work, Wittgenstein focused on lecturing and his ideas about the foundations of mathematics would change dramatically. Technically this would be his 'middle' period but because he essentially instructed, and concerned himself with, the foundations of mathematics during this period, and because many of his philosophical views would later be compiled in his Philosophical Investigations after his death, this period tends to be overlooked by philosophical academics. His middle period is best captured in a collection of his lectures on the foundations of mathematics, which are demonstrative of his shift from logical atomism and his lofty ambitions to complete Russell's Principia Mathematica to a sceptical finitism.

War played an odd role in Wittgenstein's life. During the first world war, Wittgenstein composed his Tractatus Logico-Philosophicus and afterwards became the foremost philosopher of his time, while during the second he become very critical of philosophy and withdrew from academia, working as a nameless assistant in a hospital. Eventually he would return to philosophy, though he resigned from lecturing to focus on writing two texts that would come to influence a generation of analytic philosophers and would remain of significant interest up to the present day: the Philosophical Investigations and On Certainty.

These books represent the culmination of Wittgenstein's later period, and while the controversy surrounding the Tractatus was significant, it is this later work for which Wittgenstein is best remembered and in which he contributed some of his most significant insights. Though Wittgenstein was always a distinct character in philosophy, during this later period Wittgenstein is unique among his contemporaries in analytic philosophy for taking a somewhat more continental approach to philosophy.

In the Philosophical Investigations, Wittgenstein updates his original views about meaning from a static, pointing system of simple names and composites referring directly to facts, to a dynamic and mutating grammar that comprises life and situations as its metalanguage. He described these situations as forms of life or language-games that embody a rudimentary language that our actual language comes to represent through rule-following or a natural grasp of the abstract 'game' in a given form of life.

Wittgenstein not only resembled continental philosophy in his move away from formalisation, however. He was always cautious—or at least reticent—about the subject/object distinction, a central dogma of the Cartesian tradition that his contemporaries were all too ready to affirm. He avoided speaking about a distinct and foundational I or a cogito except for illustrative purposes. His emphasis fell rather on the facts themselves, the experience, the behaviour of people and the world in what he called 'forms of life' or 'language-games'. Even in his early work, which can be interpreted prima facie as a potential weapon against continental schools of thought, Wittgenstein rather described and visualised the experience, the nature of reality as pictures that are not absolutely distinct from the observer, the observer ever a part of the world and necessarily limited by its bounds.

For Wittgenstein, language was not simply descriptive, it embodied human experience and much of his later work can be said to be a phenomenological project. Seldom will you find a philosopher who is interested in the philosophy of mind/phenomenology who is not at least familiar with Wittgenstein's later ideas.