I shall probably have to apologise to my potential readers who are not very fond of mathematics. The book I am currently going through – Christian Wolff's Anfangs-Gründe – is a textbook on mathematics and hence the investigation of it must almost inevitably deal with mathematical questions. I promise that I shall try to be less mathematically intimidating in my next posting, which also ends my investigation of this book.
This posting is then still about mathematics and more precisely about the so-called infinitesimal calculus. One might wonder why I have chosen to deal with such a non-philosophical question. Yet, the choice is not without precedents. For instance, George Berkeley criticised the inconsistencies implicit in the Newtonian understanding of the differential calculus. Even more importantly, Hegel dedicated a long passage in his Wissenschaft der Logik to the study and criticism of earlier opinions on infinitesimal calculus.
What then was so problematic in infinitesimal calculus? While regular algebra dealt with quantities in the usual sense, or as Wolff says, with finite quantities, infinitesimal calculus in its Leibnizian form suggested the idea of infinitesimal or infinitely small quantities. Although it may not be evident at first sight, the notion of infinitesimal quantity, as used by mathematicians of the time, was context-dependent. Thus, compared to lines, points were infinitesimals, but compared to planes, lines were infinitesimals: that is, a point could be regarded as an infinitely small line and line as an intinfinitely small plane.
Even the notion of inifnitesimals is somewhat suspicious, but even more suspicious is that infinitesimal calculus appeared to calculate with them as with ordinary quantities. For instance, if one takes two points in a curve described by some equation, the relation of the differences of the respective coordinates of the points describes the direction of the line connecting the two points. But then one assumed that the points were infinitesimally near one another and proceeded to make calculation as with ordinary quantities. To add an insult to an injury, the infinitesimals were finally just discarded – something that cannot surely be done with ordinary quantities. The result of this peculiar operation happened to be the direction of a tangent of the curve in that point, that is, a line touching, but not cutting the curve at that point, and one assumed that it was also the instantaneous direction of the curve at that point.
It was this dual role of infinitesimals that worried philosophers like Berkeley and Hegel. Nowadays mathematicians have constructed strict formal systems in which infinitesimals can be used, but such formal rigour was far from the undisciplined use of infinitesimals in 18th century. Furthermore, the original uses of the infinitesimal calculus were far removed from such abstract systems, and indeed, it is hard to see what physical sense one could make of such infinitesimals (e.g. points do not really have directions).
The first viable solution for the problems of infinitesimal calculus actually discarded the whole notion of infinitesimals. Instead, one spoke of limits. For instance, the forementioned ”direction of a point” could be taken as a limit for the direction of lines connecting the point to some point near it. Even Newton had used the notion of limit, but in the 19th century this notion was represented in a truly mathematical fashion. In effect, a limit for some operation in a given point is such that we can always choose an environment near the point where the results of the operation all fall within some arbitraty parameters. Thus, for instance, ”the direction of a point in a curve” is just a quantity such that we can find a part of the curve around this point where all the points connect to the reference point through straight line with direction that is arbitrarily close to this ”direction of a point”.
Well, how did Wolff then fare in dealing with the seeming incompatibility of the two ways to handle infinitesimals? We have seen that Wolff's dissertation handled differential calculus, but only on a very superficial level. Furthermore, the pragmatic tone of Anfangs-Gründe would suggest that Wolff would not really bother himself with difficult questions concerning the philosophy of mathematics. And indeed, Wolff merely presents the rather awkward analogy between infinitesimals and grains of sand: measuring a mountain has not failed, even if we have left one grain of sand unmeasured, because the quantity of the grain is so insignificant in comparison. Here Wolff confuses the insignificancy of small quantities with the complete immeasurability of infinitesimals. Thus, a grain of sand has some definite, albeit small quantity, while e.g. a point has no quantity. No wonder then that Hegel finds Wolff an example of the worst sort of muddle, when it comes to differential calculus.
The study of infinitesimal calculus ends Wolff's Anfangs-Gründe, but I still want to dedicate one blog text to this book. Still, one might rejoice that no mathematical questions are considered anymore. Next time we shall see what beauty means for Wolff.