Ludwig Wittgenstein's considerable fame as a philosopher comes on the strength of just two books. Tractatus Logico-Philosophicus was a slim book published in 1921, while Wittgenstein's Philosophical Investigations were compiled after his death from his notebooks and published in 1953. The Philosophical Investigations were largely a refutation and rebuttal of Wittgenstein's earlier work.

It was Edmund Husserl who inaugurated the movement known as Phenomenology. He was closely concerned with issues pertaining to the foundations of mathematics and logic as well as numerous questions in psychology being discussed in his time. Phenomenology became a wide and varied field, incorporating many thinkers throughout the 20th century. It remains a major school of thought, though its influence has become more diffuse. Husserl believed that he was providing a form of philosophy that overcame the modern problem of Idealism, allowing philosophers once again to discuss the "things in themselves." Phenomenology became a study of the way that things "appear"—how they come into awareness and just how they are constituted by the human knower. He is known for works such as Logical Investigations, Ideas (in numerous versions), Cartesian Meditations, and Formal and Transcendental Logic.

Perhaps you do not know all of these thinkers, though the names are likely to be somewhat known to you, at least from lists in texts. Sadly, America has not existed long enough to create a large group of philosophers as was the case in ancient Greece, the High Middle Ages (or even the so-called period of "Silver Scholasticism" in Spain and the Low Countries), or modern Europe. Still, there have been some, but among their number has never been Bertrand Russell. Lord Russell is an Englishman—a great logician and mathematician but not an American philosopher.

The 20th century political philosopher, Leo Strauss, believed that many political and philosophical writings contain esoteric meanings. He came to this conclusion based on his interpretations of the writings of the Muslim philosopher al-Farabi and the dialogues of Plato. Plato's dialogues are particularly well known for the use of many layers of irony, wit, and story to express deep philosophical truths. Strauss (and his followers, who are called "Straussians") came to use this method for interpreting many texts.

He was very influential, given his time as a professor at the University of Chicago, and is often associated with the American political movement known as neoconservatism. This association is not quite true, however. Really, if his followers share anything in common, it is a devotion to a close reading of texts, often those of Greek thinkers but also including others in the canon of philosophy and political philosophy. Their readings do tend to be so close as to look for many such hidden shades of esoteric meanings.

Analytic philosophy refers to a school of philosophy that grew in England during the early part of the twentieth century, thanks to thinkers such as Bertrand Russell and G.E. Moore. Analytical philosophy is so-called because of its focus on logical clarification, an analysis of logic itself, and the positing of logical problems. Analytical philosophy, in contrast to so-called Continental philosophy, also rejects grand sweeping philosophical systems or theories and focuses on the small details of philosophical problems.

Charles Sanders Peirce is largely credited with developing and defining pragmatism in the late nineteenth century, and fittingly was a chemist. Pragmatists described thought not as mirroring reality, but as instead being used to fully predict and plan in problem solving. Peirce's ideas about pragmatism were picked up and built on by early-twentieth-century American thinkers like John Dewey and William James.

Alfred North Whitehead famously said that all of Western philosophy was nothing more than a series of footnotes to Plato. This statement is a bit deceiving. There were many creative Western philosophers after Plato, bringing unique ideas into the history of philosophy. The general idea that he was trying to express was that Plato was the first lengthy writer, one who was very influential on another great philosopher who was his student, namely Aristotle. Every future philosopher in the West took up some position that was somewhat positive or negative regarding these figures.

Thus, even many centuries later, the early modern rejection of Medieval philosophy was actually also a rejection of Aristotle in many ways, for Aristotle was a key component of medieval learning. Interestingly, at this same time, there was a great uptick in studies of Plato among humanist scholars. Still, even in utter rejection of some previous thinker, one must have considered that thinker at some point and at some depth. This keeps some "root" in the past even while rejecting it. Thus, even thousands of years later, philosophers write "footnotes of footnotes of footnotes" concerning the thought of Plato—at least in a broad sense of speaking!

In developing his own theories on capitalism and society, Karl Marx was heavily influence by the phenomenology of Georg Wilhelm Friedrich Hegel. In particular, Marx used Hegel's dialectic, which stressed that any idea unfolds along a three part process. First, a thesis is presented, which leads to a reaction in the form an antithesis, which is combined with the thesis in a resolution called the synthesis.

Immanuel Kant was a profoundly influential philosopher who helped reshape philosophical trends in Europe around the turn of the eighteenth to the nineteenth century. Key to his thought was that moral imperatives needed to be absolute and grounded in reason. Kant referred to this imperative as the "categorical imperative," which was an argument that there is always an absolute moral right that should be followed.

The French nobleman, Alexis de Tocqueville, came to the United States in 1831 ostensibly to study prison reform in the new nation. During his time in the United States, he took copious notes and undertook some travels to help him understand the newly formed democratic republic. After his return to Europe, he penned the text Democracy in America, which gave a kind of "outsider's look" at the new nation and its institutions.

Although written by a foreigner, it was quite perspicuous on many trends in American governance and culture. To this day, it remains a text that various political factions (of differing allegiances) quote seemingly to their own personal advantage. Though certainly "high level" and dated, it remains an important text for understanding the founding of the United States of America.