The followers of Aristotle were called the "peripatetics", and often his general "school" of philosophy is called "peripatetic" or "peripateticism." This name comes from the one of two sources. On the one hand, his school in Athens had colonnades in which the instruction perhaps occurred. The Greek for "colonnade" is peripatoi. On the other hand, however, it is also said that Aristotle walked while lecturing. The word for "a walker" in Greek is peripatetikos. Hence, he would walk back and forth with his students, lecturing. From this one little habit (or, perhaps, the aforementioned colonnades), there arose a title that was affixed to philosophers for over two millennia!

In the Bible, the books of Exodus through Deuteronomy tell of the departure of the Hebrew people from Egypt. The classic moment in this sojourn is their time at Mount Sinai. This is where the so-called Ten Commandments were said to be presented by God to Moses. Whatever might be the historical accuracy of this overall tale, this is an important fact to know, as the experience of the Hebrew people in the desert was pivotal for their self-identity. This would remain a continuing motif throughout their scriptures as well as in the Christian scriptures as well, which would present Jesus as a kind of second Moses.

In the Bible, the books of Exodus through Deuteronomy tell of the departure of the Hebrew people from Egypt. The classic moment in this sojourn is their time at Mount Sinai. This is where the so-called Ten Commandments were said to be presented by God to Moses. Whatever might be the historical accuracy of this overall tale, this is an important fact to know, as the experience of the Hebrew people in the desert was pivotal for their self-identity. This would remain a continuing motif throughout their scriptures as well as in the Christian scriptures as well, which would present Jesus as a kind of second Moses.

The Pre-Socratics are so named merely because of their place in history. They came before Socrates. Socrates is like a "dividing line" for Greek philosophical history! In any case, the Pre-Socratics like Thales and Anaxagoras were interested in the basic elements from which all things were made and how those elements became what we know of in our experience. Other thinkers like Heraclitus and Parmenides were interested in more esoteric questions about being and non-being. In some ways, these thinkers were interested in "metaphysics"—especially Parmenides and Heraclitus. More broadly, the group was interested in cosmology or the way the world is to be understood philosophically. Hence, the best overall category for them is that of "natural philosophy", understanding that all of these terms were coined after the time of the pre-Socratics. With Socrates, philosophy took on a very human-centered and ethical set of concerns.

The philosophy of Plato is known for many things. One of his most famous doctrines is the so-called theory of "Forms" or "Ideas." This refers to the notion that everything in the world "participates" in some kind of universal and separate Idea. Thus, we can only call this or that tree a "tree" because those various individual trees participate in the universal notion of "treeness."

This position is often called "extreme realism" because it supposes that our ideas are so real that they actually exist. It is like there is a kind of "heaven" (loosely speaking) in which the Ideas exist. A radically contrary position is sometimes named "nominalism", which indicates that there are no universal Ideas that are really "out in reality." Instead, a strictly "nominalist" position would state that universal ideas like "tree" and "dog" only exist because we give name to individual things.

The notion of the "liberal arts" is notoriously slippery. In different ages, different things are said to be part of the liberal arts. As the curriculum of the medieval university solidified, however, the seven liberal arts took up a basic form that became normalized by the twelfth / thirteenth century. These were divided into the "trivium" and the "quadrivium"—or, the "three subjects" and the "four subjects." The trivium was devoted to topics needed for all other learning, so it focused on grammar, rhetoric, and logic. These introduced the quite young students to the tools needed for later studies. The quadrivium was made up of primarily mathematical courses, namely arithmetic, geometry, astronomy (like physics today), and music (a study of harmonies and proportions).

The philosophy of Plato is known for many things. One of his most famous doctrines is the so-called theory of "Forms" or "Ideas." This refers to the notion that everything in the world "participates" in some kind of universal and separate Idea. Thus, we can only call this or that tree a "tree" because those various individual trees participate in the universal notion of "treeness."

This position is often called "extreme realism" because it supposes that our ideas are so real that they actually exist. It is like there is a kind of "heaven" (loosely speaking) in which the Ideas exist. A radically contrary position is sometimes named "nominalism", which indicates that there are no universal Ideas that are really "out in reality." Instead, a strictly "nominalist" position would state that universal ideas like "tree" and "dog" only exist because we give name to individual things.

During the second half of the 13th century, there were several standard philosophical styles. One was the style of literal commentary, which was used for interpreting the texts of Aristotle and other accepted authorities. In addition, within theological circles, there was the development of the so-called "summae"—lengthier summaries of doctrine, sometimes quite systematic in nature.

These large texts, however, were made up of questions that were written in a style quite peculiar to the Middle Ages, namely that of "disputed questions." These texts would set forth a question, take options "pro" and "con" and then make a conclusion based upon both sides of the question. Sometimes, in shorter works like this, the author would only consider the options that were contrary to his own, helping the reader to understand some possible objections to the text.

This style of disputation was used in many of the treatises that began to be written late in the 13th century and into the 14th. It remained a style that was used especially in Catholic thought for some time. It was, however, most vigorously used as a writing style during this period of the Middle Ages.

(It should be added that there were also short treatises written during this time. They are not, however, the most unique style of writing during the time period.)

In his The Course of Positive Philosophy, Auguste Comte famously divided the history of societies up into three main periods. These periods are said to be characterized by their manners of explaining the causes of human experience. The first is the theological phase. In this stage, causes are explained primarily in terms of gods or a god. Then, in the "metaphysical" stage, abstract notions are used to explain the causes of things. Finally, in the "positive" stage, only "positive" (or posited) facts and scientific observations are the primary mode of explanation. Comte thought the final type of society was the highest and most developed. By doing so, he set up his own time's philosophical conception of knowledge as the pinnacle of human thought.

The theologian Thomas Aquinas wrote much in his brief forty-nine years of life. Many people are introduced to him through a brief passage in his Summa theologiae in which he proposes five possible "ways" for proving God's existence. In various manners, these five proofs are based on ways that someone can start with human experience and prove from that finite, changing experience how there must be an unchanging God.

Many of the other thinkers listed in this question as potential answers had interest in matters similar to this as well. Most directly pertinent for this matter is Anselm of Canterbury. Anselm is well known for his so-called "ontological argument" for God's existence (though the title is a bit of a later attribution). His general idea was that so long as you can have an idea of, "Something than which nothing greater can be thought," you can prove that such a great thing must indeed exist—precisely because it is so great that it is perfect and hence has the perfection of existence.