The origin of our Cartesian plane, and some Mathematical Philosophy
“Cogito Ergo Sum // Je pense, donc je suis // I think therefore I am”
Dare I make connections between my current liberal studies and my future with the much more mathematically based subject of engineering?
René Descartes has accrued accolades and criticisms alike since his initial prominence during the European Enlightenment. He is considered the single most important thinker of this period by some, a founder of modern philosophy who sought new ways to justify the science of his time. He is also a celebrated mathematician and physicist. It was Descartes who combined the disciplines of algebra and geometry to create the pair of intersecting axes that we use for our most fundamental mapping of functions – the Cartesian plane. As with most great thinkers, there is crucial context to Descartes’ success. Early on he studied under Jesuits; this is where he became so intrigued with the methods of acquiring knowledge over the actual content, or knowledge itself. This quickly developed into somewhat of an obsession, and coupled with his disappointment in traditional philosophy and the limitations of theology that he expresses in one of his works Discourse on Methods, it led to his primary philosophical inquiry: how do we know things to be true? Descartes valued anything that was, or could be utterly certain in itself or its truth. After his investigations and various works, European culture would change in such a way that subjective truth began to take epistemological prominence over objective truth. He started a trend towards skepticism within society that would gradually be built into every type of inquiry, raised the importance of method, and inspired the premise that the thinking mind was separate and more real than the body which houses it. One simple line from Descartes, known as his Cogito, encapsulates the essence of his influence and the origins of his philosophy. The famous saying “I think therefore I am” is the subject of the aforementioned accolades and criticisms, and integral to understanding how Descartes combines awareness of self, thinking, and existence in a way as to result in an intuitive grasp of truth.
Descartes’s methods are laid out in detail in the Discourse on the Method. As the title suggests, this work is an exploration of the methods of reasoning, and the contents are basically Descartes illustrating his approach to difficult, fundamental, and ontological questions. This is where he develops his beliefs in dualism; better known as the Cartesian, mind-body split, this was his theory that the mind is non-physical, but still exerts control over the separate, physical matter of the body. His Cogito can also be found in this text, and specifically came to be as a product of his quite radical skepticism. He set out to discover the fundamental and “absolutely primary subject of certitude for modern man”, and began with his consideration of the possibility that everything humans sense is illusory, meaning that all sensible objects – including the body – do not actually exist. He was skeptical that some universal evil genius was behind the scenes, manipulating everything we think we know, and this omnipresent deception makes it technically impossible to take anything as absolutely certain. Next he established some provisions: if you can't be sure of anything, then you should temporarily accept what the majority around you believes. Only once you arrive at certainty can you reevaluate. While Descartes was busy searching for this fundamental certitude, he realized that the very act of thinking was proof that he existed. Or rather, because he was consciously aware of his thought processes, he must exist! Personally, this was very unconvincing at first. It seems this sort of conclusion can be drawn from any action, why not “I type therefore I am”? However, Descartes would refute that “it is not the act which assures existence, but the thought or consciousness of the act.”
This connects to an acute awareness of self, thinking, and existence. The ability to channel such a fundamental skepticism and work backwards to assure yourself that you do indeed exist is a powerful exercise. However, this doesn’t necessarily make it an intuitive grasp of undoubtable truth. Descartes did help to establish that the conception of thought and being is profoundly influenced by the conception of truth,* but is “cogito ergo sum” – the Latin translation of his Cogito – undoubtable? It establishes that certitude is grounded in a man himself, such that you cannot think or propose anything new unless you are certain that you are actually thinking. In this way it is the basis of modern thought, and crucial for humans to be able to progress or accumulate knowledge. Descartes also emphasized that the capacity to think clearly distinguishes the characteristics that humans share with “lower animals” such as growth and nourishment. Only a single act of understanding and judgement is needed to establish that another is a real man. In my opinion, it is not the absolute proof of thinking, but instead a comforting reminder that our only fundamental knowledge is that of self-existence, so our immediate conscious experience can be satisfactory to ground ourselves and grow from there.
Because “I think therefore I am” exists with the pretext that our act of thinking is fundamental, it assumes there is no possibility that some larger force we are unaware of is behind our thoughts. If we assume this, then yes, Descartes presents an undoubtedly foundational truth. However, it depends on how radically skeptical you are. If you think, as Descartes often did, about the beauty of mathematics as “more simple and universal” seeming to “contain something that is indubitable”, or applied mathematics in physics, as well as geometry, these all require similar concessions. To build mathematical theorems, complete different proofs, and be able to manipulate these disciplines, there are inherent rules to be followed. “QED” (an abbreviation of the Latin words "Quod Erat Demonstrandum" which loosely translates to mean "that which was to be demonstrated") is usually placed at the end of a mathematical proof to indicate that it is complete. These declarations are satisfying, and mark a victorious end to some investigation, but they themselves rely on several assumptions along the way. Few have the time to question every minutia in a complex logical proof to determine how likely it is to be true, absolutely fundamentally true such that there is no underlying assumption to be checked against. Even the most basic proofs are built around other fundamental theories, axioms, and prior mathematics that have been instituted into a reservoir of universal acceptance. So, yes, if you start at the same basis that Descartes did, and take the experience of doubt to mean that you indeed exist, then his skepticism and the premises which follow are quite logical and convincing. After all, he had an incredible mind, and was arguably one of the most influential individuals of the enlightenment. His thoughts didn’t revolutionize the European way of life, and continue to influence prominent Western discourses today for no reason.
Whether or not Descartes’s skepticism and approach really works to insure us that what we know is real and true, it is extremely convincing, and coupled with his greater accomplishments as a philosopher and scientist, makes him a very respectable contributor to Western thought. The Cogito may not be one hundred percent sound, but it still sparked a fundamentally important method of thought and a skeptical approach, while highlighting that there will always be qualities to our knowledge that are beyond our grasp. If he taught us anything, it is the necessity of once in a while setting all your beliefs in doubt. This encourages a critical mindset that is inherently more progressive to complacency and misinformation. As one contemporary said in reference to Descartes work, “a decisive philosopher is someone who says not only more than one would have understood before him but more than one believes to have understood with him.” Descartes was a philosopher that raised questions no one ever thought of raising, insoluble questions that incite the future. In this way, he contributed to the efforts of keeping philosophy a continually open and endless game.
*for anyone interested in further reading, here is the link to my original paper with all of the footnotes and sourcing
