Fundamenta Calculi
In this section, we present a series of texts dedicated to the axiomatic foundations of algebra. At first sight, it may seem that it was a topic that had interested Leibniz for many years, due in particular to his early interest in the building of abstract forms of calculi. A closer look at the manuscripts however, tends to indicate that this work on “foundations” (fundamenta calculi is one of the terms used by him) was typical of a late period. It appears to have started, to be more precise, in the wake of a treatise on mathesis universalis which Leibniz undertook in the spring of 1699 (on the context of this project, see Leibniz (2018), texts [3] and [4]) and which he wanted to complement by some rigorous “proofs” (see, in particular, the letter to Johann Andreas Schmidt from 3(13) March 1699 (A I, 16, 633), where Leibniz announces a treatise on fundamenta calculi following what he has already sent on mathesis universalis).
This goes against a common belief according to which Leibniz – like English algebraists in the 19th century – would have passed smoothly from the axiomatization of basic algebra (on numbers and magnitudes) to more “abstract” calculi, such as those he devised for logic in the middle of the 1680s (calculus de continente et contento or calculus coincidentium). Interestingly enough, the process seems to have been the other way round: the axiomatization of logical calculi made him aware of the problems raised by the axiom of idempotence, which does not hold in the usual treatment of magnitudes and of the subsequent impossibility of building a genuine “universal” calculus – as he had first dreamt. This could have motivated the wish to render the axioms of the calculus magnitudinum explicit – maybe in the hope of extracting a more abstract structure holding for all mathematical calculi (see the characterization of ars combinatoria in A VI, 4, 510 or 922-923). Another motivation, as explained in several texts from the beginning of the 1690s, came from physics, and more precisely from the formalization of Dynamica. Indeed, the “estimation” of forces necessitated enlarging the mathematical treatment of magnitudes so that “intensive” quantities could be dealt with. By contrast this forced, once again, the necessity of characterizing explicitly the particularities of the usual (or “ordinary”) calculus on magnitudes (See Leibniz 2018, text [3a] where the philosopher explains how he was led to return to the foundations of the treatment of magnitudes because he had to clarify the use of ratios and proportions in the realm of physics).
The first attempt in the axiomatization of algebra is the much celebrated text: Prima magnitudinum calculi elementa, which was edited by Gerhardt at the end of the 19th century and for which we now possess a precise date (see the presentation). This text is very similar to another entitled Mathesis generalis. An interesting difference between the two is that the latter tries to provide a foundation for natural numbers too – an enterprise for which we also possess the draft Numerus integer est totum ex unitatibus collectum (LH 35 I 9 Bl. 7). Yet it is important to keep in view that in this group of texts the central concept remains that of “magnitude” (and not that of “number”). The famous “analytical” proof of “2 + 2 = 4” from definitions (and an axiom of identity), which Frege took as a prototype of a logical view on numbers, takes place, in fact, in this context, where “number” is something needed for the measurement of magnitudes (and not a logical entity given a priori). The texts on the “Elémens du calcul”, written in French, are in the same spirit, but were written later (between 1707 and 1710, considering the type of paper used by Leibniz).
Overviews
Overview of Prima calculi magnitudinum elementa
Thanks to the progressive edition of Leibniz’ papers, and in particular of his correspondence, we now have access to the circumstances surrounding the writing of the Prima calculi magnitudinum elementa. The occasion leading to this text was the arrival in Helmsdted of the theologian Johann Andreas Schmidt in 1695. Schmidt was asked to take on the teaching of mathematics and, since he was not a specialist of this topic, he turned to Leibniz for help (Schmidt to Leibniz, 31 August/10 September 1697; A I, 14, 467). Leibniz worked on this project in the following year and at the end of 1698 sent to Schmidt the beginning of a treatise on Mathesis Universalis (A I, 16, 295; 341 and 393; see [Leibniz 2018, p. 113-120] for a French translation and a presentation of this draft, which was edited by Gerhardt in GM VII, 53-76). Schmidt was very pleased with the result and urged Leibniz to complete his project (A I, 16, 393 and 607-608). It is in this precise context that Leibniz announces at the beginning of 1699 a continuation of his drafts containing the proofs for the foundation of calculus: “I will give... view
Overview of Mathesis Generalis (1699 – 1700)
The manuscript "Mathesis generalis" belongs to the group of texts that Leibniz produced in 1699-1700 to provide rigorous demonstrations for the "foundations of calculus" (see the presentation of Prima calculi magnitudinum elementa). One point of difference with Prima calculi is that here Leibniz gives a prominent role to the notion of natural number and to the famous proof, which he would take up again in New Essays on Human Understanding, of "2 + 2 = 4" (see the text Numerus integer, which seems to be a first version of the beginning of our text). Another point of divergence is the fact that Leibniz first gives interpretations of computations with negative numbers before formulating purely formal definitions by means of the properties of the inverse (an approach that the manuscript ... view
Overview of Numerus integer est totum ex unitatibus collectum
Numerus integer est totum ex unitatibus collectum seems to be a first draft for the opening part of mathesis generalis (besides the similarity in content, it was also written on a type of paper which is used for the second manuscript). The manuscript has been transcribed with variants by Emily Grosholz (Grosholz & Yakira 1998). It presents an attempt in which the notion of natural number is presented before that of magnitude. By contrast, the final version returns to the idea that magnitude is the central concept of algebraic calculus and that natural numbers should be defined, as measure of magnitudes, in terms of parts (Numerus integer est totum ex unitatibus tanquam partibus collectum). Another very interesting aspect of the text is a comment Leibniz wrote in front of the notations for the first nine digits and which he finally erased: “Where the following is produced from the preceding by adding 1. Let the preceding be p and the following S, we will have in general that p+1 is the same as S” (Ubi... view
Overview of Elemens du calcul
The Elemens du calcul is remarkable for the list of axioms it provides. Compared to the axioms isolated at the end of the 1680s in the calculus coincidentium or later in texts such as Prima calculi magnitudinum elementa, the first striking difference is the sheer number of listed items. Whereas Leibniz usually posits “[latex]a=a[/latex]” as the main axiom (see, for example, Prima calculi magnitudinum elementa, GM VII, 77), and sometimes adds idempotence ([latex]a+a=a[/latex]) to specify logical calculi (A VI, 4, 834 (with commutativity) and 848 (with “[latex]a-a=0[/latex]”)), here he proposes a list of no less than 15 axioms and one “demand”. Moreover, this list is very similar to the one we would nowadays provide for the characterization of a number field (to which Leibniz adds rules for exponentiation).
The first two axioms are the usual... view