Philosophical Investigations is one of Ludwig Wittgenstein's two greatest works, the other being the Tractatus Logico-Philosophicus. Wittgenstein had worked on the book for many years and it was published posthumously in 1953, originally in German (as Philosophiche Untersuchungen).
Considered by many to be one of the most important philosophical works of the 20th century, if not of all time, the Philosophical Investigations is unique in its approach to philosophy. Most philosophical texts read as histories of philosophy, summaries of philosophizing which has already occurred, a completed report on thought. Wittgenstein's book treats philosophy as a lab science, instructing the reader to undergo various thought experiments and do the actual work of philosophy. Rather than relying on the thinking of others, it insists that the reader do his own thinking. Wittgenstein asks the reader to imagine various worlds, and then to attempt to test the boundaries of that world, the advantages, the problems, etc. It is through these thought-experiments that the reader comes to philosophical conclusions by himself, rather than simply being told what philosophy has already discovered.
Wittgenstein seem to be convinced that many philosophical problems are the result of philosophers failing to properly understand the rules of language. For example a philosopher may ask "What is beauty?" and be convinced that there must be some essential thing which makes something beautiful. But for Wittgenstein, this is just a mistake in grammar occasioned by the form of the question "What is beauty?" As a matter of everyday experience, Wittgenstein would point out that we don't need to understand the essence of beauty to use the word properly, and in fact the search for the essence of beauty creates grammatical confusion about how the word should be used. In the end this grammatical confusion leads philosophers to say strange things which nobody else understands. Instead of searching for a mythical substrate which defines beauty, Wittgenstein suggests that we take our cues from the actual use of the word. In particular Wittgenstein asks us to look at the way in which we teach children to use a word.
This leads to the common gloss of Wittgenstein's argument in the Investigations -- "Meaning just is use." In other words, we don't define words by reference to things, but by the way they are used. This means there is no need to postulate that there is something called beauty which exists independent of any particular "beautiful object." This may be an accurate description of one line of thought in the book, but it is clearly a somewhat simplistic reading of the book as a whole.
Certainly the above gloss is correct insofar as it is true that the Investigations deal largely with difficulties of language and meaning. But the fact that Wittgenstein relies so strongly on indirect arguments makes understanding his project difficult.
A closer examination of one of the most influential sections of the book that deals directly with meaning and use reveals a far more complex view of language than is represented in the simple slogan above. As is common in the Wittgenstein's later works, he begins by asking the reader to perform a thought experiment. First he asks the reader to come up with a definition of the word "game". While this may at first seem a simple task, he then goes on to lead us through the problems with each of the possible definitions of the word "game". Any definition which focuses on amusement leaves us unsatisfied since the feelings experienced by a world class chess player are very different than those of a circle of children playing duck duck goose. Any definition which focus on competition will fail to explain the game of catch, or the game of solitaire. And a definition of the word game which focus on rules will fall on similar difficulties. The essential point of this exersize is often missed. Wittgenstein's point is not that it is impossible to define game, but that we don't have a definition, and we don't need one.
Everybody understands what we mean when we talk about playing a game, and we can even clearly identify and correct inaccurate uses of the word. All without reference to any "definition".
How exactly does this work? Why is it that we are sure a particular activity -- Olympic target shooting -- is a game while a similar activity -- military sharp shooting -- is not. Wittgenstein's explanation is tied up with a important analogy. How do we recognize that two people we know are related to one another? We may see similar height, weight, eye color, hair, nose, mouth, patterns of speech, social or political views, mannerisms, body structure, last names, etc. If we see enough matches we say we've noticed a family resemblance. It is perhaps important to note that this is not always a conscious process -- generally we don't catalog various similarities until we reach a certain threshold, we just intuitively see the resemblances. Wittgenstein suggests that the same may be true of language. Perhaps we are all familiar with enough things which are games, and enough things which are not games that we can instantly categorize new activities intuitively.
This brings us back to Wittgenstein's reliance on indirect communication, and his reliance on thought-experiments. If many philosophers are confused, it is because they aren't able to see the family resemblances. They've made mistakes in understanding the vague intuitive rules language uses (which Wittgenstein calls the rules of the language game), and have thereby tied themselves up in philosophical knots. He suggests that an attempt to untangle these knots requires more than simple deductive arguments which point out the problems with their particular position. Instead Wittgenstein's larger goal seems to be to try to divert them from their philosophical problems long enough to indirectly re-train their intuitive ability to see the family resemblances.
Perhaps the most celebrated argument put forward in the Philosophical Investigations is what is called the Private Language Argument, in which Wittgenstein asks if it is possible for us to have a language that nobody else can understand. Would such a language make any sense to me? How could I be sure that I had used the correct term to describe a sensation or object? Supposing I have a sensation S and mark it down every time S occurs, how can I subjectively be sure that I am relating the present S to the previous S? Wittgenstein says that is is like buying a hundred copies of the same newspaper to check that the first copy was correct.
The argument relies on the subjective nature of a possible private language, and essentially calls into question its objectivity. Once again Wittgenstein here calls on us to take language as it is used, but more substantially he raises many epistemological questions regarding the possibility of self knowledge and the nature of knowledge as a social phenomenon, whilst logically it is related to Gödel's incompleteness theorem.