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.

William of Ockham (1825-1347) is best known in the popular mind for the so-called "Ockham's razor," a common sense idea that we shouldn't just suppose that there are many kinds of beings in the world necessary for explaining our theories. He is, however, also known (for related reasons) as a "nominalist," that is, as someone who thinks that universal notions like "dog" and "tree" do not subsist in reality but only arise in the mind, which compares individual things that are maximally similar—hence, calling those individual things by a given universal name.

He also wrote the highly influential Summa logicae, which outlined a form of logic that was congenial to this outlook (as well as other aspects of his theory of mind and metaphysics). If you ever doubt that there was sophisticated philosophy in the middle ages, crack open Ockham's works, and you will see that he (along with many others) were quite agile minds indeed!

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.

Immanuel Kant was the inheritor of the great pedagogical program of German scholasticism, drawing on a number of thinkers such as Wilhelm Gottfried Leibniz (especially through the works of Christian Wolff) and many, many others. At a certain point in his career, however, Kant came to the conviction that the excesses of these so-called "rationalistic" philosophers could not provide an adequate grounding for the sciences and for the moral life.

Therefore, Kant undertook a change of perspective that led to the publication of his three best known works: The Critique of Pure Reason, The Critique of Practical Reason, and The Critique of Judgment. These three texts sought to explain just what could be known within the bounds of finite human reason—thus providing a critical perspective regarding what he took to be the excesses and emptiness of the philosophy that he had taught for many years.

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).

Immanuel Kant was the inheritor of the great pedagogical program of German scholasticism, drawing on a number of thinkers such as Wilhelm Gottfried Leibniz (especially through the works of Christian Wolff) and many, many others. At a certain point in his career, however, Kant came to the conviction that the excesses of these so-called "rationalistic" philosophers could not provide an adequate grounding for the sciences and for the moral life.

Therefore, Kant undertook a change of perspective that led to the publication of his three best known works: The Critique of Pure Reason, The Critique of Practical Reason, and The Critique of Judgment. These three texts sought to explain just what could be known within the bounds of finite human reason—thus providing a critical perspective regarding what he took to be the excesses and emptiness of the philosophy that he had taught for many years.

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.