The Philosophy of Descartes (1596-1650)
Descartes is considered to be the father of modern philosophy for a few reasons.
Firstly, as he participated in the Scientific Revolution of the 17th century, he was one of the first who grappled with the implications of the ongoing scientific breakthroughs in the context of the Christian scholastic beliefs predominant at the time.
Secondly, despite having a scholastic upbringing--where written pieces are usually addressed to other scholars--Descartes' own writing is unusually approachable. As Russell later put it, the pieces produced by Descartes read more like an explorer eager to share what he had discovered, than a teacher addressing pupils, or a pedantic scholar. This form of writing would mirror many later modern philosophers.
Lastly, Descartes' philosophy would bring a methodological break, even if not an ontological breakthrough. That is to say, while most of the particular answers brought in would not stand the test of time, his questions, problems and, above all, method of inquiry would.
Descartes' proposed approach was to not take for granted any assumption and to not accept any previous belief as true. He wanted a solid foundation, upon which he could build a consistent theory of understanding. This method is called the method of “extreme doubt”, or “Cartesian Doubt”.
A popularly recited piece of Descartes' philosophy is the maxim “I think, therefore I am”--“Cogito, ergo sum”--, but this is a conclusion that can be examined in a bit more detail.
While examining the “Cogito”, as the problem came to be known, he considered two possibilities:
(i) I “am” (exist), and I'm thinking these thoughts and experiencing these sensations, which are accurate reflections of the “real” world.
(ii) Let's say there is an “Evil Demon”--“Genius Malignus”--, who is deceiving me by supplying me with every sensation and thought, so as to make me believe those thoughts and sensations stem from a “real” world.
In either case, he concluded, the sole fact he is experiencing these thoughts or sensory data is the evidence for his existence: no matter whether he is being deceived or not; there is an “I” which would be the target of deception (in the latter case).
NOTE: Of course, Descartes' philosophy doesn't advocate for the existence of this “Evil Demon”; it merely uses the possibility of its existence as a rhetorical device (i.e., for postulating a thought experiment). The overall conclusion of the argument is indeed independent of whether it exists or not.
It's important to say, though, that later philosophers question whether this “I”--meaning “the self”--was warranted in the first place. Rigorously, the only assured truth is that “there is thinking going on” or, even, “there is the experience of thinking”. The “I” just “snuck in” due to some grammatical necessity, as it is said, and further justification would be warranted for whether thinking would presuppose “something that thinks”, let alone an independent continuous “self”.
Regardless, Descartes' work marks an important diversion from the previous body of thinking. By putting the source of knowable truth in one's own mind, he would place subjectivism in a central place of discourse. Moreover, by advocating for a dualistic division of nature, where there's mind (res cogitans) and matter (res extensa), Descartes would delineate central themes that later schools of thought, such as rationalists and empiricists, would elaborate upon.
Finally, his efforts showed that a rigorous skeptical approach to knowledge would necessarily require some kind of prior assumption. In Descartes' case, the assumption was a good, almighty God that would not deceive him--and thus allow him out of his “Cogito”. Therefore, skepticism has to stop somewhere--axioms, presuppositions, first-hand experience--, so as to build up any useful and practical body of knowledge.