Most people think that reality consists of objects existing apart from our minds, which we manipulate, in part, by the application of mathematics: I quite agree. But not all philosophers would be so quiescent: at least their logic, if it could speak for itself, would certainly beg to differ. The ideas of the great philosopher and mathematician Rene Descartes present a perfect case in point.
The widely accepted philosophy up to Descartes’ time was primarily Aristotelian--polished up and expounded upon by Thomas Aquinas—which was diffused into the cultural atmosphere by the Catholic Church, and eventually acquired the name Realism. This philosophy considered physical realities to be composed of substance and accidents, or, “secret energies” and appearances (Gilson 163).
The substances, or “secret energies”, were considered the very hearts of things. For example, an olive tree in the Garden of Gethsemane has accidental appearances which make us aware of it through our senses—it’s shape, color, texture, smell, and relation to it’s environment. But, according to this philosophy the appearances must “inhere” in something (Rizzi 365). That in which they “inhere”, the primary nature of the olive tree, is quite obviously not open to sense perception but known abstractly, by the intellect. Thus the substance, or nature, of the olive tree is a cause of the appearances, itself not visible.
The appearances, or in technical terms, accidents, were grouped into two main categories: quantity, or extension, and qualities. Quantity was considered fundamental, for every quality is an extended quality: redness, smoothness, square-ness -- all of these qualities exist quantified, as extended – even sound can be measured by periods of time. Thinking about extension in the abstract is, according to Realism, the basis of mathematics.
Enter Descartes. Descartes was a philosopher deeply impressed by the clarity of mathematics. In mathematics he could find “the certainty of its demonstrations and the evidence of it’s reasoning”, as Etienne Gilson quotes him (106). Gilson also notes Descartes’ increasing dissatisfaction with other types of knowledge; his Jesuit teacher Clavius had once written, in Gilson’s words, “There are innumerable sects in philosophy, there are no sects in mathematics” (104). This sentiment was likely to have helped Descartes combat the “complete skepticism” he found in the heavy influence (as Gilson contends) of the philosopher Montaigne (Gilson 110). “[Montaigne] had not found the key to universal knowledge” (Gilson 110). Such a clash between total philosophical skepticism and mathematical certainty can be easily seen to have birthed the brain child of Descartes’ “Universal Mathematics”, which, says Gilson, he would endeavor to apply to all fields of knowledge (Gilson 113). This philosophy will use as it’s “first principle” the method of clear (definitive) and distinct ideas (Gilson 122):
“all that can be clearly and distinctly known as belonging to the idea of a thing can be said of the thing itself… But what is it, to know something distinctly? When a mathematician knows a circle, he knows not only what it is [it’s definition]`, but, at the same time, what it is not. Because a circle is a circle, it has all the properties of the circle, and none of those that make a triangle a triangle, or a square a square. Philosophers should therefore proceed on the same assumption” (Gilson 122).
Gilson says that this principle would be the basis of all subsequent idealistic philosophies, for the idea of a thing was to be taken for the thing itself (122).
Up to this point quantity dealt with an aspect of a substance, the abstract consideration of which was called mathematics. Now the abstract definition was, in effect, to be the substance, and it’s “extension in three dimensions” it’s only attribute—it’s only physical property (Gilson 159). This was a move, says Gilson in so many words, that went no longer from physical substances to ideas, but from substantial ideas to physical attributes. (121)
The immediate result of this method in the physical world is, says Gilson, the deflation of all qualities, “such as weight, hardness, colour, and so on” thought to exist in extended things; this in addition to emptying the various words we use for “things” (tree, dog, flower) of any real meaning (160). Realists affirm that, at the very least, the primary qualities (known by touch and, incidentally, in some cases by sight) are necessary to experience the extension of a substance, such qualities being mathematically measurable: shape, weight, size, feel, relation to other objects. One would be inclined to think Decartes, as a mathematician, would be content with this affirmation. However, reversing the order of knowledge and starting from ideas, he could not be. Using the method of clear and distinct ideas you must find these qualities in the idea of extension, not extension by the sensible existence of these qualities (as the Realists say). Further, you must find a reason to think that the idea of extension corresponds to actual extension outside of your mind—including that of your own body. Such an application led to the well known dichotomy still prevalent today in many discussions of mind and matter: the idea of the “ghost in the machine.”
Decartes found justification for positing the idea that extension exists by appealing to the idea of God. These three ideas: “thought, extension, and God” -- no longer inferred from a given, substantial physical reality known first through the senses but, instead, existing as “distinct ideas”-- can be seen (if I may take some liberty with Gilson’s conclusions, 139, 148) winding off as three distinct philosophical paths (Gilson 115). Why? Here’s one major reason Gilson gives: Hume would say “if we have no adequate (clear and distinct) idea of ‘causality’ that can apply to matter, where could we get one to apply to God?” thereby detaching the idea of God from the idea of a reality outside the mind and from the mind itself (Gilson 174). Depending on which way, which idea you’re predisposed to assume, you could find yourself either an Idealist (“thought”—ideas are all that exist), an Empiricist (“extension”-- bodily sensations are all that exist), or an Ontologist (“God”—relying on the idea of God to secure the belief in a physical world). Either way you’re left trapped within your mind, and mathematics becomes confined--not to things, not to external reality, but to the relationships between ideas in our minds.
Works Cited
Gilson, Etienne. The Unity of Philosophical Experience. San Francisco: Ignatius Press, 1999.
Rizzi, Anthony. The Science Before Science: A Guide to Thinking in the 21st Century. Baton Rouge, LA: IAP Press, 2004.
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