Descartes, who was both a mathematician and a philosopher, presented his ideas regarding how to combine mathematical concepts to the human thought process in his book, Discourse on Method. The ideas he presents in this book revolutionized scientific and philosophical thought in his time. He directly rejected the dominant schools of thought of his time and proposed instead a more systematic approach that worked to remove the element of subjective experience.
His goal in doing this was to find a means of attaining perfect certainty of philosophical concepts. He bases knowledge not on what is thought to be known but instead insists that one must call into question everything one thinks one knows that has been learned through the senses of the body. These ideas are expressed in his first and second meditations.
As it is presented, Descartes gives four main rules of logic that must be addressed. The first of these is that one can only accept as true those things that are clearly and distinctly known to be true. Things can be clearly and distinctly known to be true by breaking down the problem under investigation into as many parts as are considered necessary to fully solve the issue. The third rule is that the logical process must proceed step by step from the simplest and easiest portion of the problem to clearly and distinctly know and progress in order of difficulty to the more complex. Finally, to be sure nothing has been omitted, Descartes instructs that one should always take an open view of the problem so other possibilities or seemingly unrelated issues might be considered.
Reading through these steps, the linkage of thought to mathematical methods of analysis can be traced.
Although his most famous statement, I think therefore I am, presented in his second meditation, rests to a large degree upon the sense that he is still thinking, the very fact that this sense still exists is proof for Descartes that there must be something in existence to realize the sense and therefore he, as a thinking entity, must exist. His simple summative statement is the result of a long discourse in which Descartes applies his principles of thought to carefully and analytically reject all knowledge he’s gained as a result of his sense or that he hasn’t direct knowledge of himself.
“It is some time ago since I perceived that, from my earliest years, I had accepted many false opinions as being true, and that what I had since based on such insecure principles could only be most doubtful and uncertain; so that I had to undertake seriously once in my life to rid myself of all the opinions I had adopted up to then, and to begin afresh from the foundations” (Descartes 42). Through this questioning process, he demonstrates how thought, not observation is the right foundation for knowledge.
Descartes bases his idea of clear and distinct knowledge on the sense that at such a deep level of knowledge, in which the only thing he is sure of is that he is a thinking being, impressions such as this must be real. In his experiment with the wax, Descartes acknowledges that the information we gain must necessarily come to us through the senses; however, our perception of these things depends entirely upon the intellectual processes of analysis and analytical connection. “We perceive bodies only by the understanding which is in us, and not by the imagination, or the senses, and that we do not perceive them through seeing them or touching them, but only because we conceive them in thought” (Descartes 54).
Given this new understanding of the separation of mind and body, the only logical solution is that Descartes’ clear sense that he must exist must be correct. As a result, Descartes manages to present his ideas, apply them to his thought and prove their superior quality.
Descartes, René. “Meditations.” Readings on the Ultimate Questions: An Introduction to Philosophy. Nils CH Rauhut & Renee Smith (Eds.). New York: Pearson, 2007.