Note: this is still a draft.
Descartes sets out the fundamentals of his theory of symbols in Discourse II while explaining his fourth rule of science. There he fixes on mathematics as the only science which has so far produced demonstrations.1 This leads him to consider objects of these sciences.2 He notices of the objects of the science (multitude, magnitude, etc.), that the sciences do not consider the objects except in view of the relations or proportions obtaining between them.3 Thus, he discards all considerations except those of proportions in general from his new method.4 So, for example, their consideration of the distinction between the ratio of two planets would not be different from that of two numbers or two lines: insofar as they both have the same ratio and are proportional, they fall equally under his science.
Doing this presupposes that all relations between can be unified in a meaningful way. This is relatively easy for quantitative ratios. Numerical ratios and ratios of magnitudes have this correspondence: if a line is to a line as a number is to a number, when each line is divided into its corresponding number of parts, the divided parts are equal (e.g. if one line is to another as three is to two, then a third of the first line is equal to half of the second). If we ignore incommensurability, all quantities have the same kinds of ratios.5 Ratios of temperature and weight can be similarly led back to quantitative ratios: a thermometer makes a temperature known by the length of a column of liquid; a scale similarly makes weight known either by the number of weights necessary to balance the object or by the length at which the object balances. The use of instruments allows a large number of relations to be made quantitative, but how will fatherhood or sonship be quantified? It seems impossible to quantify such a relation.
Now, given that the method considers proportion in general, how it considers them must be thought out. Descartes takes them to be proportions of lines, there being “nothing simpler, nor anything that I could represent more distinctly to my imagination and senses.”[@Disc, CSM 121] One might object that the proportionality of lines is not clear and distinct. Given two lines, the proportion between them is not clear (in the case of curved lines, even Descartes holds that the proportion is unknowable). In the two lines which have a ratio, while it is clear that one is greater than another, it is not clear by how much it is greater. This lack of clarity, however, is a benefit, not a problem. Because a line can be divided into an arbitrary number of parts, any two commensurable lines can stand for an indefinite number of ratios and guarantees their sameness. A line three units long is to a line two units long not only as 3: 2 but also as 6: 4, 9: 6 and an indefinite number more. Since all these ratios can be represented by the same lines, one concludes immediately that 3: 2: : 6: 4: : 9: 6: : . . . Thus, its very lack of determinate divisions makes a continuous quantity most apt for representing proportions.
Such a use of lines, however, prescinds from the per se unity of a line. When two lines are said to represent the proportion, the two indicates that the lines are each one: one whole line, an undivided continuum. Cartesian representation takes this continuum and considers it as if divided into an indefinite series of chunks, ignoring its real unity. The advantage a line has for representation over a Euclidean number is that the number is actually divided in a determinate way: Two can never be Four. The line thus considered is already a symbol; it does not represent any particular quantity, but rather every quantity.(Klein 1968) Replacing the line with a letter (or reapplying the numeral)6 is only a slightly further step; such symbols are more conveniences than innovations.
Klein, Jacob. 1968. Greek Mathematical Thought and the Origin of Algebra. New York: Dover Publications.
- “Reflecting, to, that of all those who have hitherto sought after truthin the sciences, mathematicians alone have been able to find any demonstrations — that is to say, certain and evident reasonings — I had no doubt that I should begin with the very things that they studied.” (@Disc, CSM 120).↩
- The footnote in CSM includes astronomy and optics and the like under the title “mathematics.” Whether or not this is so, the conclusion I draw applies a fortiori to the objects of these sciences.↩
- “For I saw that, despite the diversity of their objects, they agree in considering nothing but the various relations or proportions that hold between these objects.” (@Disc, CSM120).↩
- I think I’m justified in using the word method here, as he uses further down on p. 121.↩
- This is true about the quantities mathematics studies. How such quantities relate to the quantities of material substances is a perplexing question.↩
- Having taken the numeral to represent the symbol-line, it is but a short step to symbolizing zero and the negative integers.↩