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).
Prima calculi magnitudinum elementa
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 a work where I supplement in some way what I began to write at some point as an Introduction to Mathesis Universalis; and I have already redacted a few things pursuing the project; in particular, the proofs of the foundations of calculus, which seem to me of primary necessity in order to give to science its complete solidity” (To Schmidt, 3(13) March 1699; A I, 16, 633).
Leibniz sent some drafts on this topic to his secretary R. C. Wagner (who Schmidt had introduced to him) in April 1699 and told him that Schmidt was willing to have them transcribed by a famulus, as he already had for the first manuscript on Mathesis Universalis (A III, 8, 96). The very same day, Wagner went to Schmidt with these “pages on Mathesis universalis and in them the proofs of the calculus of magnitudes” (A I, 16, 730-731). Thanks to a subsequent letter, we can identify these documents as being the Prima calculi magnitudinum elementa (GM VII, 77-82). Indeed, in a letter dated 14/24 April 1699 (A III, 8, 103), Wagner mentions to Leibniz that he should delete a redundant expression in the second page of the draft (“In plagula 2da sub initium (ni) (24) delenda erunt verba si subtractio vel signum – evanescat”) – a sentence which is clearly in the manuscript of the Prima calculi under the said number and that Leibniz forgot to erase after correcting the passage.
This remark holds for the copy of the manuscript (LH 35, IV, 13, fol. 1-4, the original draft by Leibniz being in fol. 5 and 6), which was presumably written out by Schmidt’s famulus and on which Leibniz indicated several (sometimes substantial) corrections. These corrections induced some inconsistencies in the numbering of sections (and of subsequent internal references to these sections: we indicate the corrected numbers in red in brackets). We also have several other drafts, which clearly belong to the same set of preparatory versions (in particular the Calculus magnitudinum, LH 35, IV, 13, fol. 11-12 and the Caput primum. De aequalitate, fol. 9-10). They are also very close to what can be found in the manuscript Mathesis generalis (LH 35, I, 9, fol. 9-14) – itself close to “numerus integer…” (LH 35, I, 9, fol. 7). The version copied by Schmidt’s famulus with Leibniz’ corrections is what Gerhardt transcribed in his Mathematische Schriften. Awaiting a critical edition of this set of manuscripts, we propose here a translation of Gerhardt’s version.
Mathesis Generalis
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 Elémens du calcul would put to completion). Finally, this text is also more complete in that it presents proofs concerning multiplication, which Prima calculi did not deal with.
A first transcription of this text was made by Emily Grosholz and can be found in [Grosholz and Yakira, 1998], p. 89 under the title Mathesis generalis est scientia magnitudinum. However an entire folio (LH 35, I, 9 fol. 14v) dealing with the axiomatization of multiplication is missing from this edition. A complete transcription and commentary, as well as a list of variants, can be found in [Leibniz 2018, pp. 157-180].
Numerus integer est totum ex unitatibus collectum
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 sequens semper fit ex praecedente adjecto seu 1. Nempe praecedens sit p et sequens sit S, eritque generaliter p +1 idem quod S) – where Leibniz seems very close to defining natural numbers by the notion of successor. This group of texts is also important as they give an immediate context for the famous proof that “[latex]2+2=4[/latex]”, based on definitions and identical axioms, which was presented in Nouveaux Essais pour l’entendement humain (IV, 7 § 10; A VI, 6, 413-414).
Elemens du calcul
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 one characterizing the reflexivity of equality and the rule of substitution. This is Leibniz’s favorite characterization of equality. As he was well aware, one can demonstrate on this basis the other usual properties of equality: transitivity and symmetry (see A VI, 4, 831, prop. 1 and 3). This is what he does in fol. 7r, which presents the “propositions” deriving from our list of axioms. Then he characterizes addition by the following properties: commutativity (item (3) [latex]a+b=b+a[/latex]), invertibility (item (4) [latex]a-a=0[/latex]) and the fact that zero acts as neutral element (item (5) [latex]0+a=a[/latex]). He then does the same with multiplication describing commutativity (item (6) [latex]ab=ba[/latex]), invertibility (item (11) : [latex]\frac{a}{a}=1[/latex], considering, as expressed in item (12), that [latex]\frac{a}{a}[/latex] is a notation for than [latex]a\cdot\frac{1}{a}[/latex]) and the fact that 1 acts as a neutral element ([latex]1\cdot a=a[/latex]). Finally, he also specifies the property of distributivity of multiplication over addition (item (10): [latex]ab+c=ab+ac[/latex]).
This terminology is, of course, anachronistic and one could doubt, in particular, that Leibniz had a clear view of the role of “inverse” elements. A few remarks are therefore in order to motivate a comparison. Firstly, it can be seen that the property [latex]a\cdot\frac{1}{a}=1[/latex] is proved in fol. 7r (Elemens du calcul ordinaire. Propositions) by the simple conjunction of axioms 11 and 12. This was supposed to be proposition III, then transformed into IV and then rewritten (without change) as prop. V : “[latex]a\cdot\frac{1}{a}=1[/latex] Dem: [latex]a\cdot\frac{1}{a}=[/latex] (per ax. 12) [latex]\frac{a}{a}=[/latex] (per ax. 11) 1”(LH 35 4 12 Bl. 7r). More importantly, maybe, it can be noted that division is not introduced in our text as an operation sua generis. In the first part of the manuscript, explaining the definitions (called “Significations”), division is defined as a number, noted [latex]\frac{a}{b}[/latex], which, when multiplied by b, gives a (and, in the same way, subtraction is defined as a number, noted [latex]a-b[/latex], which, when added to b gives a). This contrasts starkly with other texts, such as the Prima calculi magnitudinum elementa, in which Leibniz defines subtraction (more rarely division) as the operation of taking away something (logical calculus itself is sometimes called “plus and minus calculus“).
The only axiom that appears to be missing for us would be associativity. This was certainly the conclusion drawn by Frege when looking in Leibniz for an “analytical proof” of elementary number statements such as “[latex]2+2=4[/latex]” (Grundlagen der Arithmetik (1884) § 6). Yet the reason this axiom is missing is that it appears in the definition as a rule for the use of parentheses: “[latex]a+b+c[/latex] est autant qu’ [latex](a+b)+c[/latex], ou qu’ [latex]a+(b+c)[/latex] et abc est autant qu’ (ab)c ou qu’ a(bc)” (item (10) in “Significations”). Interestingly enough, Leibniz thought of positing this as an axiom. In the version sketched in fol. 11-12, he first writes two axioms listed 13 and 14 stating that: [latex]a+b+c=(a+b)+c[/latex] and [latex]abc=(ab)c[/latex], before changing his mind and putting it as a characterization of the use of parentheses in fol. 1.
The same could be said about the fact that Leibniz did not specify that [latex]\frac{1}{a}[/latex] exists only when a is supposed to be different from 0. When one looks at the other documents, one sees that this was supposed to be specified later, as a proposition (see, for examples, LH 35, 4, 12, fol. 7r : “[latex]\frac{a}{0}[/latex] est une absurdité”). Indeed, for reasons we cannot detail here, Leibniz considered that this was true not in full generality, but “only if a is not 0”. As explained in a scholium on fol. 7r, this was related to the fact that the expression [latex]\frac{0}{0}[/latex] may appear in Differential Calculus where, under some specific conditions, it can have a meaning (See. S. Bella, “De l’intraitable à l’indéterminé : entre calcul et géométrie, réflexions leibniziennes autour de [latex]\frac{0}{0}[/latex] (1700-1705)”, to appear in Philosophia Scientiae, June 2021).
The comparison with other attempts such as that of Prima calculi or of Mathesis generalis immediately shows that some statements, which are axioms here are theorems there. For example, as stated in Mathesis generalis, “[latex]a-a=0[/latex]” can be derived from the definition of subtraction. More interesting maybe, this is also the case in our text. But this should be no surprise: Leibniz always claimed that mathematical axioms are demonstrable (from the one and only genuine axiom, i.e. the principle of identity, and definitions). It should be noted, in passing, that this is another way of saying, in a view which is often ascribed to the turning point of the 19th century, that axioms (except identities) are definitions in disguise. The definition of subtraction in our text is that the sign “[latex]a-b[/latex]” signifies a number such that when it is added to b it gives a. Henceforth, since [latex](x-a)[/latex] is defined as the number giving x when added to a, one has [latex](x-a)+a=a[/latex], if and only if [latex]x=a[/latex]. By axiom 5, we have [latex]a=0+a[/latex] and henceforth [latex](a-a)+a=a=0+a[/latex], and finally [latex]a-a=0[/latex]. As can be seen in this example, axioms do not act here as “logical” foundations, but as a way to characterize the “calculus magnitudinum” as a structure. It is all the more impressive that Leibniz precisely chose the properties we would also consider as characteristic of a field (in this case, of magnitudes).
Another quite remarkable feature of this text is the introduction of logical signs as primitive. They appear in a corner of fol. 11v as a way of marking consequentia and the fact that it can be symmetrized in the case of what we would call logical equivalences.
In fol. 6v, Leibniz even proposes introducing two different signs: one for consequentia (and consequentia mutua) and one for inferences (If A then B).
The context is a series of algebraic identities one can derive from the axioms by using substitution and the fact that most of them are “if and only if” conditions:
In the fragment called “Propositions” (fol. 7r and v), he simply remarks in passing that all these propositions are “convertibles”, a fact he announces that he will develop in more detail later. But in this latter text we can witness another interesting development: these reflections are put under the heading of the treatment of “formulae” (in this case the fact that “deux grandeurs sont égales quand les formules qui les experiment ne different que par deux grandeurs ingredientes égales”). Leibniz then introduces symbols for formulae (capital letters) and for magnitudes (small letters), in order to indicate the compatibility of mathematical equality with logical equality through substitution (what we would call a “congruence”). This constitutes the climax of a “logical” approach to mathematics, which Leibniz announced in the first texts on mathesis universalis as a Logica mathematica, conceived as a logic of mathematics (see Leibniz (2018), text [3b] and Michel-Pajus & Rabouin (2017)).