Descartes establishes the ideas of intuition, deduction, and enumeration as a set of tools used to acquire knowledge with a significant degree of certainty. However, the definitions of the three ideas are neither mutually exclusive nor exhaustive in their coverage of all knowledge acquisition tactics. These ideas may build upon, overlap, or even oppose each other, making their relationship more ambiguous than simply red, blue, and yellow.
For the sake of conceptual clarity, consider a continuous chain of thought to consist of nodes which represent self-contained facts and logical links that span between nodes. Intuition is the most fundamental of the three: a simple and indivisible certainty. Deduction is a means by which one’s thought can traverse across nodes and arrive at a distant conclusion whose connection with the starting node is not immediately apparent, thus giving rise to complex thought. Reconciling the two, Descartes explains that pursuit of knowledge should start “with the intuition of the simplest ones of all” and “try to ascend through the same steps to a knowledge of all the rest” (CSM I 20, AT X 380). Somewhat separate from the other two, enumeration is the act of laying out all relevant information pertaining to a particular subject and can be thought of as identifying the important nodes that logical links may be drawn between. Descartes emphasizes the prudence with which enumeration should be conducted, describing the process as “an investigation which is so careful and accurate that we may conclude with manifest certainty that we have not inadvertently overlooked anything” (CSM I 25, AT X 388). Descartes’ loose definitions and ambiguous examples of the three ideas have led to interpretive difficulties that are the most apparent in distinguishing deduction from the other two.
Descartes does not make clear whether the relationship between intuition and deduction is analogous to component and assembly or seed and tree. The argument for the seed and tree relationship is tempting because Descartes establishes the notion of the absolutes (referring to intuition) and the relatives (referring to deduction) “which we can relate to the absolute and deduce from the absolute in a definite series of steps” (CSM I 21, AT X 382). Though it is clear that a deductive chain must stem from an absolute, intuitive fact, what is neglected in the seed and tree interpretation is the fact that discrete steps along the chain of deductive reasoning are also composed of other intuitive facts. For example, take the problem 1453 + 672: two randomly chosen integers whose sum is far from intuitive. 1453 + 1, however, is simple enough to be considered intuitive. Subsequently, this complex problem that requires deduction can objectively be broken down into 672 iterations of an intuitive problem. In fact, all arithmetic problems involving summing can thus be considered relative to a basis of absolute problems of the type N + 1.
Furthermore, there is no clear boundary between what is considered simple enough to intuit and what is complex enough to necessitate deduction. Is there an absolute scale by which to measure problem complexity, and does Descartes allow interpretive room for overlap between the two? Consider expert mathematicians who “acquire through practice the ability to make perfect distinctions between things” (CSM 33, AT X 401). Is the Pythagorean Theorem, a nontrivial fact that the typical person must deduce, considered simple enough to be called an intuition to the expert mathematician? In short, the answer is no: the Pythagorean Theorem is still a deduced concept. Though problem complexity may seem to vary across different levels of mastery and intelligence, the only true variation being observed is the speed at which logical links can be drawn between nodes. Regardless of subject familiarity, the simplicity of fundamental nodes stays universally constant, consistent with Descartes’ idea of the absolutes. Thus, intuition and deduction do not overlap, for what is fundamental by definition cannot also be composed of connections between the “more fundamental”. As a result of a clear binary classification, there is an absolute scale to measure complexity that transcends the ‘ease’ by which one comes to understand the concept at hand, a metric often incorrectly confused with complexity.
Lastly, interpretive difficulties between deduction and enumeration are perhaps the most convoluted. Is enumeration an extreme form of lengthy deduction where “every single thing relating to our undertaking must be surveyed in a continuous and wholly uninterrupted sweep of thought” (CSM I 25, ATX 388) lest we lose sight of where we started? Branching off of this idea, is enumeration simply the act of laying out the facts without drawing lines between them, hence serving as an extension of deduction designed to encompass empirical, observation-based studies? The most accurate relationship between deduction and enumeration is that of the trailblazer pioneering and the hiker following a predetermined set of landmarks. Returning to the node-link visual, deduction is the process of starting at a specific node on an empty field and using logical reasoning to “find” the next node. When the desired end node is reached, one can backtrack and see the formation of a singular, continuous trail of knowledge. On the other hand, enumeration, also called induction, starts with an array of observed nodes and tries to weave the most rational set of links between desired nodes, utilizing intermediate nodes in between to explain what otherwise would be considered a gap in reasoning. As an example, Descartes uses induction to prove a circle as the area-maximizing geometry given constant perimeter. Consistent with the trailblazer/hiker relationship, this inductive proof disregards the classic analytical method of ‘finding’ the solution with discrete logic. Instead it establishes all geometries that must be disproved as counterexamples and weaves an elegant, domino-like explanation as to why none of them can cover more area than the circle.
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