Perhaps the most dominant theme in modern philosophy is the consequences of rigorous mathematical investigation of the world. Many of the early modern philosophers (e.g., Descartes, Leibniz, Hobbes, Newton) were at the forefront of the "scientific revolution" that followed the Renaissance. That the physical world is subject to mathematical treatment is a central doctrine of a sect of philosophers centered around Pythagoras in the 6th century BCE. Pythaogras himself had learned mathematics in Egypt. The core doctrine of the Pythagoreans was that number is the basis of all reality. We can apply mathematics to the world because the world is itself numerical. As will be seen, something like this thesis (though limited to the physical world) was adopted by Descartes.
The views of the Pythagoreans heavily influenced Plato in the 4th century BCE. In his cosmological work Timaeus, which was known during the Middle Ages, Plato described the molding of the the primal matter of physical universe by a god, who used eternally existing numbers (among other things) as a model. Again, we have an account of why mathematics is applicable to the physical world.
A key feature of Platonism is the separation of the physical world from an ideal realm of forms (which includes numbers). Physical objects are modeled on forms and only approximate their perfection. On Plato's view, these ideal objects are better-known than physical objects. The human intellect can grasp the forms in a more secure way than the human senses can grasp physical objects, due to the stable nature of the forms and the fluxuation of the physical world. This "rationalism" was to become a key feature in much of early modern philosophy, especially through such thinkers as Descartes, Spinoza, and Leibniz.
A number of ancient philosophers claimed that reality is unitary, not separated into separate realms. Some, such as Parmenides, held that physical reality is in a certain sense illusory. But others, most notably the atomists, took the physical world to be the fundamental reality. In the early modern period, the best-known atomist was Epicurus, a generation younger than Aristotle. The early atomist Democritus, a 4th century BCE contemporary of Socrates, claimed that gross sensible objects are composed of tiny indivisible parts, whose activity explains what is observed through the senses. This led him to a position similar in a way to that of Parmenides, that what is presented to the senses is not the way things really are. The characteristics of objects that can be observed by the senses exist only relative to the senses, while the characteristics of the atoms are real in a non-relative way. The explanation of physical things by the behavior of their insensible parts, and the division of the characteristics of things into the sense-relative and the absolute, is characteristic of nearly all the modern philosophers of the 17th and 18th centuries.
The philosophy of Aristotle was heavily influential in the late Middle Ages, due in large part to the work of Thomas Aquinas in the 14th century. Aristotle's basic way of understanding the world was through the way in which it functions. He recognized that living things are organized in ways that promote the accomplishment of certain ends, such as motion or nutrition. A key theme in his philosophy was the generalization of the "teleology" found in biological organisms to the entire physical world. The idea that everything function so as to bring about pre-determined ends was very amenable to Christian theology, according to which the physical world was created to fulfill God's purposes. Nonetheless, it was the application of teleological explanation to the physical world which came under heavy attack by most of the modern philosophers (who at least nominally professed Christianity). It is here that a decisive break from "the ancients" was made by "the moderns."
Having just been exposed briefly to three very different views about the nature of reality, the reader might be inclined to wonder how one might choose between them. What evidence could be brought forward that any of them is right? Things would appear to be just as they do no matter which of these views of reality were adopted. The ancient skeptics held that we should restrict our claims to the way things appear, and should give up claims to knowledge of any underlying or separate reality or purpose. (They extended this caution to claims about good and bad, right and wrong.) Any argument that could be made in favor of one or another will always be opposed to another argument favoring something contradictory to it, and how are we to choose? Moreover, things appear differently to different people. How are we to say which, if any, of these appearances reflects the way things really are? Skepticism flourished in the period after Aristotle, and it was revived in the Renaissance. Descartes considered it a serious threat, and much effort was made by modern philosophers to combat it. Some, such as Hume and Kant, embraced it to a more or less limited extent.