Christian Wolff: Elements of all mathematical sciences - Infinitesimals = grains of sand?

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.

Christian Wolff: Albegraical dissertation on infinitesimal differential algorithms (1704)

As promised, I shall begin my blog with Christian Wolff, the major figure in the arena of German philosophy before Kant. I am sure that many of you have heard the phrase Leibniz-Wolffian philosophy, which would indicate that Wolff merely copied Leibniz and had nothing original to say. Furthermore, many critics, like Hegel, have stated also that Wolff is a pedantic and boring writer. Both of these statements are far too critical. Even before this project, I had read German-language versions of Wolff's logic and metaphysics and have nothing to complain about his style: his writings are at least as exciting and engaging as writings of an average analytic philosopher.

In addition, we should not think Wolff as a mere faithful disciple or even imitator of Leibniz. Wolff and Leiniz corresponded mostly about mathematcial questions and Wolff could not even have known Leibniz's philosophy in a deep manner: the majority of the works of Leibniz were published only after his death and after Wolff had begun his own philosophical career. Indeed, we might as well speak of a Wolff-Leibnizian philosophy: Wolff was the true beginner of the philosophical tradition in Germany, while Leibniz's works influenced this tradition only later.

I have chosen to take Wolff's dissertation Dissertatio algebraica de algorithmo infinitesimali differentiali as a suitable beginning for his career, although the work is not philosophically important. In fact, the dissertation is not about philosophy at all, but as the title says, studies mathematics and particularly differential calculus. Wolff began as a mathematician and what we would call a physicist and in a later book he even congratulates himself on having made important innovations in the field of aerometry or the study of measuring air pressure.

The subject matter of the dissertation is from a modern viewpoint rather elementary. Wolff derives some easy formulas of differential calculus, such as how to differentiate a product of two functions – nowadays a clever high school kid could do this. Although differential calculus was still somewhat of a novelty, I can find nothing in Wolff's dissertation that either Leibniz or Newton wouldn't have achieved already – Wolff even himself makes copious references to his predecessors in the thirty-odd pages. If this was a standard dissertation at the time, I must say that the criteria have been tightened from those days.

What I find most interesting in the whole dissertation are the short corollaries at the end of the work. After rather mathematical considerations Wolff merely enumerates certain further consequences such as ”matter is infinitely divisible” and ”God is to creatures as our mind is to entities of reason”, without offering any argument or explanation as to how these rather philosophical sentences related to the question of differential calculus. This might remind one of certain analytic philosophers who do a whole article or presentation full of clever logical tricks and then end up by telling that all of this has rather interesting philosophical consequences, but that this is so evident one hardly needs to spell the argument.

This is undoubtedly all we need to say about Wolff's dissertation, but we are still not yet through with his mathematical writings: the next few texts shall describe Wolff's mathematical masterpiece that purports to go through the elements of all mathematical sciences.