tiistai 16. elokuuta 2011

Christian Wolff: Elements of all mathematical sciences - The axiomatic method in Wolff

We have seen how Wolff differentiates between nominal definitions, which merely explain what a word means, and real definitions which tell how to generate a thing corresponding to a definition. In the actual text of the Anfangs-Gründe, all the Erklärungen are mere nominal definitions, while no explicit real definition is mentioned. We shall see in a while that despite this the work does contain many real definitions.

From mere nominal definitions one cannot really deduce anything substantial and no philosopher has ever denied this. Well, Aristotle appears to say in his Posterior analytics that one could literally deduce all theorems from mere definitions, but Aristotelian definitions are not supposed to be nominal, but reveal the essence of something, so this might not actually be an exception. In any case, we do need some basic propositions (Grundsätze, as Wolff says in German) and from these basic propositions together with nominal definitions we can then derive other propositions, which Wolff calls Lehrsätze.

In addition to this division of basic and derived propositions, Wolff dstinguishes propositions merely describing a state of affairs from propositions stating a method or a need of a method for doing something. This division resembles the division of definitions to nominal definitions describing how to use a word and real definitions telling how to manufacture a thing of a certain kind. Similarly, basic propositions can be divided into axioms describing certain fundamental states of affairs from which proofs can begin and into postulates describing certain fundamental methods for doing things. Finally, derived propositions can be divided into theorems that need to be proved and into problems that need first to be solved and then these proposed solutions need to be proved to solve the problem. We might thus classify the constituents of the axiomatic method in a following form:

Unfortunately, this classification oversimplifies Wolff's thought. Firstly, while nominal definitions can be separated from axioms and theorems, real definitions are not a type of statements different from postulates and problems. Indeed, real definitions tell us how to find or manufacture certain things and thus are included either in postulates or in problems.

Secondly, Wolff notes that we can actually do without the problems altogether, because they could be turned into theorems. Theorems, Wolff says, are always of the form ”if p, then q”, whereas problems could always be translated into this form in the following manner: if the solution is followed, the problem is solved.

The interesting question undoubtedly is how Wolff suggests the axioms to be justified, that is, what is the ultimate basis of his demonstrations. Now, the Anfangs-Gründe investigates also handicrafts like architecture or artillery and Wolff admits that he must use information from experience as a justification of some theorems in these sciences. Yet, in case of pure mathematics, Wolff suggests that just by looking at the (nominal) definitions we can immediately see that certain basic propositions are valid. In Kantian terms, Wolff thnks that axioms are analytic.

Wolff's suggestion appears rather outdated for anyone who is acquainted with how axiomatics are dealt in modern mathematics: axioms are chosen not because we would be somehow immediately certain of their truth, but because we are interested in finding out what propositions follow from these axioms. Indeed, when Wolff suggests as an axiom that between two points we can draw only one straight line, it is difficult to see how this could be decided merely from Wolff's definition of straight line: straight line is a line where a part of the line is similar to the whole.

Actually one might criticise even Wolff's definition: how can he be sure that only the supposed straight lines are such that their parts are similar to the whole? Indeed, we might well imagine a sort of fractal line where the details of the whole would be repeated ad infinitum in smaller and smaller pieces of the whole. Yet, we should not be too harsh on Wolff, who surely could not have been aware of the exotic geometric shapes in modern mathematics. Furthermore, Wolff's definition is probably derived from tradition, because it appears to be an improved modification of Euclid's definition of straight line as a line which lies evenly with the points on itself.

Even if we accepted Wolff's definition, it would appear that his axiom presupposes characteristics of space that are not included in the definition of straight line. In Kantian terms we would say that Wolff's axiom is actually synthetic, and indeed, this is what Kant said of a related proposition that a straight line is the shortest line connecting two points. In other words, geometry requires according to Kant something more than mere conceptual analysis.

Although Kant's viewpoint on the axioms of geometry seems a clear improvement on Wolff's, it is interesting to note that on Hegel's opinion the very axiom on straight lines that Kant had considered is actually analytic. This does not have to mean that Hegel would have returned to Wolffian idea of axiomatics. To see this, we must first consider why Kant would have called the axiom synthetic. He probably had an independent notion of a straight line, perhaps something similar to Wolff's definition.Then Kant just needed to notice that this definition did not imply the axiom, and the syntheticity of the axiom became evident.

Hegel, on the other hand, apparently does not begin from an independent definition of a straight line. True, he begins his justification of Kant's axiom from the idea that a straight line is the simplest of all lines, but here this simplicity indicates not so much a characteristic, but a search for a characteristic. The most basic type of line is to be called a straight line and we need to choose what sort of line are we to take as the most basic. Kant's supposed axiom is then just the required definition of a straight line: the minimality of the length of a line is chosen as a criterion for its simplicity.

Understood as a definition, Kant's supposed axiom is undoubtedly analytic: being a shortest line between two points is just what being a straight line means. The analyticity of the axiom would still not undermine the syntheticity of geometry. We would still have to take as an axiom, firstly, that there is something corresponding to this definition – that there is no infinite series of shorter and shorter lines connecting two points – and secondly, that a line determined by two points is unique – that there are not more than one lines between two points equalling the lower limit of all lines connecting the two points (note that elliptical geometry does not satisfy the latter condition: for instance, there are infinite numbers of equally short routes from the North to South Pole on the surface of the Earth).

Interestingly, if we accept Kant's axiom as the definition of straight line, as Hegel suggests, Wolff's axiom becomes in a sense also analytic: indeed, there can be only one unique shortest line connecting two points. That is, if there just is any shortest line: as we have seen, this is not guaranteed by the definition.

We are still not completely through comparing Wolff's idea of mathematics with the ideas of later philosophers. Next time, we shall see whether Hegel's criticism of Wolff's notion of differential calculus was justified.

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