*Chiliogon in Descartes’ and Leibniz’s Theories of Knowledge: What is a Chiliogon?
In the fourth Meditation, Descartes uses the example of a chiliogon (a polygon with a thousand equal sides) as a thought-experiment to prove that we have at least two approaches to knowledge: imagination and conception. The chiliogon example illustrates this important distinction by showing that while we are able to conceive (or to think of) a chiliogon, we are not able to truly imagine (or to visualise) one. Our mental representation of a chiliogon would either be closer to a 20 to 30-sided polygon, or a circle. However this does not mean we do not possess the concept of a chiliogon.
“... I remark, in the first place, the difference that subsists between imagination and pure intellection [or conception]. For example, when I imagine a triangle I not only conceive (intelligo) that it is a figure comprehended by three lines, but at the same time also I look upon (intueor) these three lines as present by the power and internal application of my mind (acie mentis), and this is what I call imagining. But if I desire to think of a chiliogon, I indeed rightly conceive that it is a figure composed of a thousand sides, as easily as I conceive that a triangle is a figure composed of only three sides; but I cannot imagine the thousand sides of a chiliogon as I do the three sides of a triangle, nor, so to speak, view them as present [with the eyes of my mind].” (Descartes: Meditation VI)
In arguing that all things which “we clearly and distinctly perceive are true” (Meditations 83), Descartes attempts to understand how one can be led to make a false assertion. According to Descartes, one does not perceive everything around him distinctly, and yet one continues to make judgements based on his perceptions, regardless of whether they are clear or not. Although any assertion based upon a clear and distinct perception must be true, falsity can occur when one makes a judgement based on confused perceptions.
“And although, in accordance with the habit I have of always imagining something when I think of corporeal things, it may happen that, in conceiving a chiliogon, I confusedly represent some figure to myself, yet it is quite evident that this is not a chiliogon, since it in no way differs from that which I would represent to myself, if I were to think of a myriogon, or any other figure of many sides; nor would this representation be of any use in discovering and unfolding the properties that constitute the difference between a chiliogon and other polygons.” (Ibid.)
Leibniz also uses the example of a chiliogon in his metaphysics and epistemology, to illustrate the fourth division of knowledge, which shall be discussed later on. In his 1684 essay Meditations on Knowledge, Truth and Ideas, he sets out four divisions of knowledge. The first division is that all knowledge is either obscure or clear. Knowledge is obscure if it fails to provide its holder with enough information to identify the object of that knowledge, while clear knowledge is the opposite. [“Knowledge is clear, therefore, when it makes it possible for me to recognise the thing represented.” (p. 449)] His second division further sets clear knowledge into confused and distinct forms. Clear and distinct knowledge is that of which one is able to detail the features sufficiently to separate it from all others. According to Leibniz, we have such knowledge for “all concepts of which we have a nominal definition [nominalism asserts that abstract concepts, general terms, or universals have no independent existence but exist only as names], which is nothing but the enumeration of sufficient marks” (ibid). These sufficient marks refer to every detailed feature required to identify the substance or concept. The third division he claims is a sub-division of clear and distinct knowledge: it can either be adequate or inadequate. Clear and distinct knowledge can only be called adequate “when every ingredient that enters into a distinct concept is itself known distinctly, or when analysis is carried through to the end” (p. 250). Here Leibniz uses the example of gold to illustrate his argument: one may know the properties of gold well enough to separate it from other bodies and therefore possesses clear and distinct knowledge of gold. However, without carrying out an analysis to such an extent that every predicate of gold is understood distinctly, that clear and distinct knowledge of gold is still inadequate.
The fourth division is another sub-division within clear and distinct knowledge (independent of whether the knowledge is adequate or inadequate), which is the distinction between intuitive and symbolic knowledge. This division is employed when it comes to a complex concept. Here Leibniz uses the Cartesian example of a chiliogon to illustrate his fourth division of knowledge. While Descartes uses a chiliogon to explain the distinction between our two approaches to knowledge – imagination and conception (i.e. what we can imagine and what we can understand/conceive), Leibniz is not concerned with the ability to actually form a mental image (visualisation as imagination). For Leibniz, knowledge is intuitive when it is possible to perceive, clearly and distinctly, all of the parts within this complex concept. While knowledge is symbolic when one possesses clear and distinct knowledge of the entire concept, but fails to hold the same for all the individual parts of the complex whole. Leibniz's chiliogon aims to show how one can have knowledge which is clear and distinct in respect to the whole; yet also have knowledge of this object (a chiliogon) which is said to be symbolic, for it is impossible to think simultaneously of all the concepts involved in this extremely complex geometrical shape (p.450).
This is where Leibniz identifies the truth of an idea with the logical possibility of its existence, and falsity with an idea that contains a contradiction (p. 452). He attacks Descartes’ Cartesian position of establishing truth or falsity of a predication upon the distinctness and clarity of a perception. Leibniz claims that his precise definition and usage of clarity and distinctness are necessary in making useful the Cartesian axiom of “whatever I perceive clearly and distinctly in some thing is true, or may be predicated of it.” Leibniz also insists that an idea is not to be confused with an item of consciousness (a concept). An idea is the foundation of a concept (or an item of consciousness); in other words, concepts are produced by or founded on ideas.
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