Is Leibnizian calculus embeddable in first order logic?
Piotr Blaszczyk, Vladimir Kanovei, Karin U. Katz, Mikhail G. Katz, Taras Kudryk, Thomas Mormann, David Sherry
aa r X i v : . [ m a t h . L O ] M a y IS LEIBNIZIAN CALCULUS EMBEDDABLE IN FIRSTORDER LOGIC?
PIOTR B LASZCZYK, VLADIMIR KANOVEI, KARIN U. KATZ, MIKHAIL G.KATZ, TARAS KUDRYK, THOMAS MORMANN, AND DAVID SHERRY
Abstract.
To explore the extent of embeddability of Leibnizianinfinitesimal calculus in first-order logic (FOL) and modern frame-works, we propose to set aside ontological issues and focus on pro-cedural questions. This would enable an account of Leibnizianprocedures in a framework limited to FOL with a small number ofadditional ingredients such as the relation of infinite proximity. If,as we argue here, first order logic is indeed suitable for developingmodern proxies for the inferential moves found in Leibnizian infin-itesimal calculus, then modern infinitesimal frameworks are moreappropriate to interpreting Leibnizian infinitesimal calculus thanmodern Weierstrassian ones.Keywords: First order logic; infinitesimal calculus; ontology;procedures; Leibniz; Weierstrass; Abraham Robinson
Contents
1. Introduction 22. Examples from Leibniz 22.1. Series presentation of π/ References 171.
Introduction
Leibniz famously denied that infinite aggregates can be viewed as wholes , on the grounds that they would lead to a violation of the prin-ciple that the whole is greater than the part. Yet the infinitary ideais latent in Leibniz in the form of a distinction between assignable and inassignable quantities [Child 1920, p. 153], and explicit in hiscomments as to the violation of Definition V.4 of Euclid’s
Elements [Leibniz 1695b, p. 322]. This definition is closely related to what isknown since [Stolz 1883] as the
Archimedean property , and was trans-lated by Barrow in 1660 as follows:
Those numbers are said to have a ratio betwixt them,which being multiplied may exceed one the other [Euclid 1660].Furthermore, Leibniz produced a number of results in infinitesimalcalculus which, nowadays, are expressed most naturally by means ofquantifiers that range over infinite aggregates. This tension leads us toexamine a possible relationship between Leibnizian infinitesimal calcu-lus and a modern logical system known as first order logic (FOL). Theprecise meaning of the term is clarified in Section 3. We first analyzeseveral Leibnizian examples in Section 2.This text continues a program of re-evaluation of the history of infini-tesimal mathematics initiated in [Katz & Katz 2012], [Bair et al. 2013]and elsewhere. 2.
Examples from Leibniz
Let us examine some typical examples from Leibniz’s infinitesimalcalculus so as to gauge their relationship to FOL.2.1.
Series presentation of π/ . In his
De vera proportione (1682),Leibniz represented π in terms of the infinite series1 −
13 + 15 −
17 + . . .
This is a remarkable result, but we wish to view it as a result concern-ing a specific real number, i.e., a single case , and in this sense involvingno quantification, once we add a new function symbol for a black boxprocedure (cid:3) called “evaluation of convergent series” (as well as a defi-nition of π ) (we will say a few words about the various implementationsof (cid:3) in modern frameworks in Section 4). EIBNIZIAN CALCULUS AND FIRST ORDER LOGIC 3
Leibniz convergence criterion for alternating series.
Thisrefers to an arbitrary alternating series defined by an alternating se-quence with terms of decreasing absolute value tending to zero, suchas the series of Subsection 2.1, or the series P n ( − n n determined bythe alternating sequence ( − n n . We will refer to such a sequence as a‘Leibniz sequence’ for the purposes of this subsection. This criterionseems to be quantifying over sequences (and therefore sets), thus tran-scending the FOL framework, but in fact this can be handled easily byintroducing a free variable that can be interpreted later according tothe chosen domain of discourse.Thus, the criterion fits squarely within the parameters of FOL + (cid:3) at level (3) (see Section 3). In more detail, we are not interestedhere in arbitrary ‘Leibniz sequences’ with possibly inassignable terms.Leibniz only dealt with sequences with ordinary (assignable) terms, asin the two examples given above. Each real sequence is handled in theframework R ⊂ ∗ R by the transfer principle, which asserts the validityof each true relation when interpreted over ∗ R .2.3. Product rule.
We have d ( uv ) dx = dudx v + dvdx u and it looks like weneed quantification over pairs of functions ( u, v ). Here again we areonly interested in natural extensions of real functions u, v , which arehandled at level (3) as in the previous section.In [Leibniz 1684], the product rule is expressed in terms of differen-tials as d ( uv ) = udv + vdu . In Cum Prodiisset [Leibniz 1701c, p. 46-47], Leibniz presents an alternative justification of the product rule (see[Bos 1974, p. 58]). Here he divides by dx and argues with differentialquotients rather than differentials. Adjusting Leibniz’s notation, weobtain an equivalent calculation d ( uv ) dx = ( u + du )( v + dv ) − uvdx = udv + vdu + du dvdx = udv + vdudx + du dvdx = udv + vdudx . Under suitable conditions the term du dvdx is infinitesimal, and thereforethe last step udv + vdudx + du dvdx = u dvdx + v dudx , (2.1) P.B., V. K., K. K., M. K., T. K., T. M., AND D. S. relying on a generalized notion of equality, is legitimized as an instanceof Leibniz’s transcendental law of homogeneity , which authorizes oneto discard the higher-order terms in an expression containing infinites-imals of different orders.2.4.
Law of continuity.
Leibniz proposed a heuristic principle knownas the law of continuity to the effect that. . . et il se trouve que les r`egles du fini r´eussissentdans l’infini . . . ; et que vice versa les r`egles de l’infinir´eussissent dans le fini, . . . [Leibniz 1702a, p. 93-94],cited by [Knobloch 2002, p. 67], [Robinson 1966, p. 262], [Laugwitz 1992,p. 145], and other scholars.On the face of it, one can find numerous counterexamples to sucha principle. Thus, finite ordinal number addition is commutative,whereas for infinite ordinal numbers, the addition is no longer com-mutative: 1 + ω = ω = ω + 1. Thus, the infinite realm of Cantor’sordinals differs significantly from the finite: in the finite realm, com-mutativity rules, whereas in the infinite, it does not not. Thus thetransfer of properties between these two realms fails.Similarly, there are many infinitary frameworks where the law of con-tinuity fails to hold. For example, consider the Conway–Alling surrealframework; see e.g., [Alling 1985]. Here one can’t extend even such anelementary function as sin( x ) from R to the surreals. Even more strik-ingly, √ R ⊂ ∗ R framework. While it is not much of a novelty thatmany infinitary systems don’t obey a law of continuity/transfer, thenovelty is that there is one that does, as shown by Hewitt, Lo´s, andRobinson, in the context of first-order logic.Throughout the 18th century, Euler and other mathematicians re-lied on a broad interpretation of the law of continuity or, as Cauchywill call it, the generality of algebra . This involved manipulation ofinfinite series as if they were finite sums, and in some cases it alsoinvolved ignoring the fact that the series diverges. The first seriouschallenge to this principle emerged from the study of Fourier serieswhen new types of functions arose through the summation thereof.Specifically, Cauchy rejected the principle of the generality of algebra , EIBNIZIAN CALCULUS AND FIRST ORDER LOGIC 5 and held that a series is only meaningful in its radius of convergence.Cauchy’s approach was revolutionary at the time and immediately at-tracted followers like Abel. Cauchy in 1821 was perhaps the first tochallenge such a broad interpretation of the law of continuity, witha possible exception of Bolzano, whose work dates from only a fewyears earlier and did not become widely known until nearly half a cen-tury later. For additional details on Cauchy see [Katz & Katz 2011],[Katz & Katz 2012], [Borovik & Katz 2012], [Bascelli et al. 2014]. ForEuler see [Kanovei, Katz & Sherry 2015] and [Bascelli et al. 2016].2.5.
Non-examples: EVT and IVT.
It may be useful to illustratethe scope of the relevant results by including a negative example. Con-cerning results such as the extreme value theorem (EVT) and the in-termediate value theorem (IVT), one notices that the proofs involveprocedures that are not easily encoded in first order logic. These 19thcentury results (due to suitable combinations of Bolzano, Cauchy, andWeierstrass) arguably fall outside the scope of Leibnizian calculus, asdo infinitesimal foundations for differential geometry as developed in[Nowik & Katz 2015], [Kanovei, Katz & Nowik 2016].There are axioms in FOL for a real closed field F (e.g., real alge-braic numbers, real numbers, hyperreal numbers, Conway numbers).One of these axioms formalizes the fact that IVT holds for odd de-gree polynomials F [ x ]. In fact, one needs infinitely many axiomslike ( ∀ a, b, c )( ∃ x )[ x + ax + bx + c = 0]. Meanwhile, IVT in itsfull form is equivalent to the continuity axiom for the real numbers[B laszczyk 2015].3. What does “first-order” mean exactly?
The adjective ‘first-order’ as we use it entails limitations on quan-tification over sets (as opposed to elements). Now Leibniz really didnot have much to say about properties of sets in general in the con-text of his infinitesimal calculus, and even declared on occasion thatinfinite totalities don’t exist, as mentioned above. Note that Leibnizarguably did exploit second-order logic in areas outside infinitesimalcalculus (see [Lenzen 1987], [Lenzen 2004]) but this will not be ourconcern here. Leibniz famously takes for granted second order logic informulating his principle governing the identity of indiscernibles . Whilesecond order logic is possibly part of Leibniz’s metaphysics it is not inany obvious way part of his infinitesimal calculus.Once we reach topics like Baire category, measure theory, Lebesgueintegration, and modern functional analysis, quantification over sets
P.B., V. K., K. K., M. K., T. K., T. M., AND D. S. becomes important, but these were not Leibnizian concerns in the kindof analysis he explored.In fact, the term “first order logic” has several meanings. We candistinguish three levels at which a number system could have first-order properties compatible with those of the real numbers. Note thatthe real numbers satisfy the axioms of an ordered field as well as acompleteness axiom.(1) An ordered field obeys those among the usual axioms of the realnumber system that can be stated in first-order logic (complete-ness is excluded). For example, the following commutativityaxiom holds: ( ∀ x, y ) [ x + y = y + x ].(2) A real closed ordered field has all the first-order properties ofthe real number system, regardless of whether such propertiesare usually taken as axiomatic, for statements which involvethe basic ordered-field relations + , × , and ≤ . This is a strongercondition than obeying the ordered-field axioms. More specifi-cally, one includes additional first-order properties, such as ex-traction of roots (e.g., existence of a root for every odd-degreepolynomial). For example, every number must have a cuberoot: ( ∀ x )( ∃ y ) [ y = x ], or every positive number have a squareroot: ( ∀ x > ∃ y ) [ y = x ].(3) The system could have all the first-order properties of the realnumber system for statements involving arbitrary relations (re-gardless of whether those relations can be expressed using + , × ,and ≤ ). For example, there would have to be a sine functionthat is well defined for infinitesimal and infinite inputs; the sameis true for every real function. To do series, one needs a symbolfor N , so as to define transcendental entities such as π or sine.We also introduce function symbols for whatever functions weare interested in working with; say all elementary functions oc-curring in Leibniz as well as their combinations via composition,differentiation, and integration.It follows from these examples that the first order qualification isconnected with the intended domain of discourse, so that any quantifierrelated to objects outside the domain of discourse is qualified as not afirst order one. It could be added however that all mathematical objectsare, generally speaking, (represented by suitable) sets from the settheoretic standpoint, and hence all mathematical quantifiers are first-order with respect to the background set universe (superstructure).The point with level (3) is that instead of quantifying over sequencesor functions, we relate to each individual sequence or function, and EIBNIZIAN CALCULUS AND FIRST ORDER LOGIC 7 make sure that it has an analogue in the extended domain. Such ananalogue of f is sometimes referred to as the natural extension of f .Then we can say something about the extension of every standardobject in our system, e.g., function, without ever being able to assertanything about all functions. Thus, the product rule for differentiationis proved for the assortment of functions chosen in item (3) above. Remark 3.1.
An alternative to the multitude of functional symbolswould be to add a countable list of variables u, v, w, . . . meant to de-note unspecified functions. The idea is to avoid quantifying over suchvariables, and use them as merely free variables. Then, for example,the product rule is the following statement: “if u, v are differentiablefunctions then the Leibniz rule holds for u and v ”, with u, v being freevariables.Note that we use FOL in a different sense from that used in formal-izing Zermelo–Fraenkel set theory (ZFC).When we seek hyperreal proxies, following the pioneering work of[Hewitt 1948] and [Robinson 1966], for Leibniz’s procedural moves, thetheory of real closed fields at level (2) is insufficient and we must relyupon level (3).Thus, Leibniz’s series of Subsection 2.1 is expressible in FOL + (cid:3) at level (3) but FOL level (2) does not suffice since π is not algebraic.Similarly, examples in Subsection 2.2 and Subsection 2.3 need symbolsfor unspecified functions which are not available at level (2).4. Modern frameworks
Based on the examples of Section 2, we would like to consider thefollowing question:
Which modern mathematical framework is the most ap-propriate for interpreting Leibnizian infinitesimal calcu-lus?
The frameworks we would like to consider are(A) a Weierstrassian (or “epsilontic”) framework in the context ofwhat has been called since [Stolz 1883] an
Archimedean con-tinuum, satisfying Euclid V.4 (see Section 2), namely the realnumbers exclusively; and(B) a modern framework exploiting infinitesimals such as the hy-perreals, which could be termed a
Bernoullian continuum sinceJohann Bernoulli was the first to exploit infinitesimals (rather
P.B., V. K., K. K., M. K., T. K., T. M., AND D. S. than “exhaustion” methods) systematically in developing thecalculus. The series summation blackbox (cid:3) (see Subsection 2.1) is handled dif-ferently in A and B. Framework A exploits a first-order “epsilontic”formulation that works in a complete Archimedean field. Thus, theconvergence of a series P i u i to L would be expressed as follows:( ∀ ǫ > ∃ n ∈ N )( ∀ m ∈ N ) " m ≥ n → (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m X i =1 u i − L (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ǫ . [Ishiguro 1990, Chapter 5] sought to interpret Leibnizian infinitesimalcalculus by means of such quantified paraphrases, having apparentlyoverlooked Leibniz’s remarks to the effect that his infinitesimals violateEuclid V.4 [Leibniz 1695b, p. 322].Meanwhile, framework B allows for an alternative interpretation interms of the shadow (i.e., the standard part, closely related to Leibniz’sgeneralized notion of equality) and hyperfinite partial sums as follows:for each infinite hypernatural H the partial sum P Hi =1 u i is infinitelyclose to L , i.e., ( ∀ H ) " H infinite → H X i =1 u i ≈ L . This is closer to the historical occurrences of the package (cid:3) as foundin Gregory, Leibniz, and Euler, as we argue below.In pursuing modern interpretations of Leibniz’s work, a helpful dis-tinction is that between ontological and procedural issues. More specif-ically, we seek to sidestep traditional questions concerning the ontologyof mathematical entities such as numbers, and concentrate instead onthe procedures, in line with Quine’s comment to the effect thatArithmetic is, in this sense, all there is to number: thereis no saying absolutely what the numbers are; there isonly arithmetic. [Quine 1968, p. 198] The adjective non-Archimedean is used in modern mathematics to refer to cer-tain modern theories of ordered number systems properly extending the real num-bers, namely various successors of [Hahn 1907]. In modern mathematics, this ad-jective tends to evoke associations unrelated to 17th century mathematics. Further-more, defining infinitesimal mathematics by a negation, i.e., as non-Archimedean ,is a surrender to the Cantor–Dedekind–Weierstrass (CDW) view. Meanwhile, trueinfinitesimal calculus as practiced by Leibniz, Bernoulli, and others is the baseof reference as far as 17th century mathematics is concerned. The CDW systemcould be referred to as non-Bernoullian, though the latter term has not yet gainedcurrency.
EIBNIZIAN CALCULUS AND FIRST ORDER LOGIC 9
Related comments can be found in [Benacerraf 1965]. If one couldseparate the “ontological questions” from the rest, then framework Awould be more appropriate than framework B for interpreting the clas-sical texts if and only if framework A provides better proxies for theprocedural moves found in Leibnizian infinitesimal calculus than frame-work B does, and vice versa.The tempting evidence in favor of the appropriateness of a mod-ern framework B for interpreting Leibnizian infinitesimal calculus isthe presence of infinitesimals and infinite numbers in both, as wellas the availability of hyperreal proxies for guiding principles in Leib-niz’s work such as the law of continuity as expressed in [Leibniz 1701c]and [Leibniz 1702a] as well as the transcendental law of homogene-ity [Leibniz 1710b]; see [Katz & Sherry 2012], [Katz & Sherry 2013],[Sherry & Katz 2014], [Guillaume 2014]. To what extent Leibnizian in-finitesimals can be implemented in differential geometry can be gaugedfrom [Nowik & Katz 2015].The question we seek to explore is whether the limitation of work-ing with first order logic as discussed in Section 3 could potentiallyundermine a full implementation of a hyperreal scheme for Leibnizianinfinitesimal calculus.With this in mind, let us consider Skolem’s construction of nonstan-dard natural numbers [Skolem 1933], [Skolem 1934], [Skolem 1955]; see[Kanovei et al. 2013, section 3.2] for additional references. It turns outthat one needs many, many nonstandard numbers in order to movefrom N to ∗ N , e.g., in Henkin’s countable model one has ∗ N = N + ( Z × Q ) . (4.1)Here we use Q to indicate that the galaxies are dense, so that betweenany pair of galaxies there is another galaxy (a galaxy is the set ofnumbers at finite distance from each other). Meanwhile Z indicatesthat each galaxy other than the original N itself is order-isomorphicto Z rather than to N , because for each infinite H the number H − ∗ N as in (4.1). Thus, the difference between the Leibnizian framework and a modern infinitesimal B-framework is large .On the other hand, a Weierstrassian A-framework may not coverall the moves Leibniz may make in his framework LEI, but one mightargue that the difference between (A) and LEI is small. Thus, onemay not necessarily have LEI ⊆ (A), but one might argue that thedifference (A) − LEI is small . This may be taken as evidence that (A)and LEI are more similar to each other than (B) and LEI are. Thiscould affect the assessment of appropriateness. Finally, could it bethat neither the Weierstrassian nor the modern infinitesimal accountis appropriate to cope with Leibnizian infinitesimal calculus?5.
Separating entities from procedures
What would it mean exactly to separate ontological problems fromprocedural problems? A possible approach is to attempt to accountfor Leibniz’s procedures in a framework limited to first order logic,with a small number of additional ingredients such as the relation of infinite proximity and the closely related shadow principle for passingfrom a finite inassignable quantity to an assignable one (or from a finitenonstandard number to a standard one), as in 2 x + dx pq x . As far as Skolem’s nonstandard extension N ⊂ ∗ N is concerned,anything involving the actual construction of the number system andthe entities called numbers would go under the heading of the ontology of mathematical entities. Note that the first order theories of N and ∗ N are identical, as shown by Skolem (for more details see Section 3). Inthis sense, not only is one not adding a lot , but in fact one is not addinganything at all at the level of the theory .What about the claim that Leibniz did not have this perspective ?It is true that he did not have our perspective on the ontological is-sues involved in a modern construction of a suitable number systemincorporating infinite numbers, but this needn’t affect the procedural match.What about the claim that one has to build up a considerable con-ceptual machinery to emulate Leibniz’s probably rather modest arsenalof procedural moves; that is to say, we may be able to emulate all ofLeibniz’s moves in the modern framework, but it also enables us to On occasion Leibniz used the notation “ pq ” for the relation of equality. Notethat Leibniz also used our “=” and other signs for equality, and did not distinguishbetween “=” and “ pq ” in this regard. To emphasize the special meaning equality had for Leibniz, it may be helpful to use the symbol pq so as to distinguish Leibniz’sequality in a generalized sense of “up to” from the modern notion of equality “onthe nose.” EIBNIZIAN CALCULUS AND FIRST ORDER LOGIC 11 carry out many moves that Leibniz would have never dreamt of ? Asmentioned above, this is not the case at the level of first order logic.What about the claim that the difference between Leibniz frameworkand the infinitesimal framework is large ? At the procedural level thisis arguably not the case.What about the claim that the Weierstrass framework may not coverall of Leibniz’s, but the difference , (A) − LEI, is small , indicating that(A) and LEI are more similar to each other than (B) and LEI are,affecting the assessment of appropriateness ? What needs to be pointedout here is that actually the considerable distance in ontology between(A) and LEI is about the same is the distance between (B) and LEI. TheWeierstrassian punctiform continuum where almost all real numbersare undefinable (so that no individual number of this sort can ever bespecified, unlike π , e , etc.) is a far cry from anything one might haveimagined in the 17th century.As far as the question Could it be that neither Weierstrass nor the in-finitesimal account is appropriate to cope with Leibniz? this is of coursepossible in principle. However, we are interested here in the practicalissue of modern commentators missing some compelling aspects of in-terpretation of Leibniz’s work because of a self-imposed limitation toa Weierstrassian interpretive framework.6.
A lid on ontology
It could be objected that one cannot escape so easily with the generalargument along the lines of “Let’s Ignore (ontological) Differences,” orLID for short (putting a lid on ontology, so to speak).The LID proceeds as follows. We start with the ‘real’ L, i.e., themathematician who lived, wrote, and argued in the 17th century. Itseems plausible to assume that L based his reasoning on a mixture offirst and second order arguments, without clearly differentiating be-tween the two.In a reconstruction of L’s arguments, one replaces the cognitiveagent L by a substitute L1 who argues only in a first order frame-work. This entails, in particular, that L1 cannot distinguish between N and ∗ N .However, it seems likely that L could distinguish the two structures,simply because he did not distinguish between the first and secondorder levels. In other words, the LID recommendation does not helpbecause the distinction between first and second order does not onlyaffect the ontology but also the epistemology of the historical agentsinvolved. In sum, a modern infinitesimal reconstruction of L deals with a first-order version of L, namely L1, and not with L. In line with his posi-tion on geometric algebra, [Unguru 1976] could point out that L1 isa modern artefact, different from the “real” L. Therefore additionalarguments are needed in favor of the hypothesis that L and L1 areepistemologically sufficiently similar, but this seems difficult. In anycase, a purely ontological assumption does not suffice.To respond to the L vs L1 distinction, note that the tools one needsare almost limited to first order logic, but not quite, since one needsthe shadow principle and the relation of infinite proximity. Rather thanarguing that L = L
1, we are arguing that L = L ǫ .Now the difference between calculus and analysis is that in calculusone deals mostly with first order phenomena (with the proviso as inSection 5), whereas in analysis one starts tackling phenomena thatare essentially second order, such as the completeness property i.e.,existence of the least upper bound for an arbitrary bounded set, etc.It seems reasonable to assume that what they were doing in the 17thcentury was calculus rather than analysis.As far as Unguru is concerned, he is unlikely to be impressed by in-terpretations of Leibnizian infinitesimals as quantified propositions orfor that matter by reading Leibniz as if he had already read not onlyWeierstrass but also Russell `a la Ishiguro, contrary to much textual ev-idence in Leibniz himself. We provide a rebuttal of the Ishiguro–Arthur logical fiction reading in [Bair et al. 2016] and [Bascelli et al. 2016].7. Robinson, Cassirer, Nelson
Robinson on second-order logic.
The following quote is fromRobinson’s
Non-standard Analysis :The axiomatic systems for many algebraic concepts suchas groups or fields are formulated in a natural way withina first order language. . . , However, interesting parts ofthe theory of such a concept may well extend beyondthe resources of a first order language. Thus, in thetheory of groups statements regarding subgroups, or re-garding the existence of subgroups of certain types will,in general, involve quantification with respect to sets ofindividuals. . . [Robinson 1966, section 2.6, p. 19]This appears to amount to a claim that the local ontology may indeedbe often formulated in first-order terms, while the global ontology isdeeply infected by second-order concepts. The latter may typicallyinvolve objects and arguments qualified as second or higher order with
EIBNIZIAN CALCULUS AND FIRST ORDER LOGIC 13 respect to the former, which nevertheless are of the first-order typewhen considered as related to the background set universe.
Ernst Cassirer.
Does the equation L = L ǫ not amount to anunderestimation of the historical Leibniz? Is it reasonable to assumethat he only invented the calculus, and not analysis? According toCassirer, the basic concepts of analysis were deeply soaked with phi-losophy, i.e., for Leibniz mathematical and philosophical concepts wereintimately related: Leibniz himself asserted that the new analysis has sprungfrom the innermost source of philosophy, and he assignedto both regions [i.e., analysis and philosophy] the taskto confirm and to elucidate each other. [Cassirer 1902,p. xi]If L = L1 + epsilon, i.e., if the historical Leibniz was mainly deal-ing with calculus , this may appear hardly compatible with Cassirer’sperspective; see [Mormann & Katz 2013]. This impression would be,however, a misunderstanding. In order to forestall it, it merits be-ing pointed out that developing the calculus was a great mathematicalachievement of philosophical relevance. It is only today that the term calculus possesses a connotation of routine undergraduate mathemat-ics, but not in the 19th century.As far as Cassirer is concerned, Leibniz was indeed doing analysis asl’Hˆopital called it. It is not even sure Cassirer was aware of the moreadvanced analysis. Leibnizian calculus only seems “trivial” from thestandpoint of properly 20th century mathematics. It is an advance inunderstanding when we make a distinction between Leibnizian calculusand analysis. We don’t mean to diminish Leibniz’s greatness by thisdistinction, nor do we suggest that Cassirer was wrong . He was merelyusing the term analysis in its 17–19th century sense rather than thesense in which we use it today.7.3. Edward Nelson.
As far as the passage from [Robinson 1966] isconcerned, we find the following comment at the end of the paragraph:The following framework for higher order structures andhigher order languages copes with these and similar cases.It is rather straightforward and suitable for our pur-poses. [Robinson 1966, section 2.6, p. 19] In support of this claim, Cassirer refers here in particular to Gottfried WilhelmLeibniz, Die philosophischen Schriften, hrg. von Carl Immanuel Gerhardt, 7 Bde.,Berlin 1875–1890, Bd. VII, S. 542. (Cassirer 1902, p. xi) In the original: “Leibniz selbst hat es ausgesprochen, daß die neue Analysis ausdem innersten Quell der Philosophie geflossen ist, und beiden Gebieten die Aufgabezugewiesen, sich wechselseitig zu best¨atigen und zu erhellen.”
EIBNIZIAN CALCULUS AND FIRST ORDER LOGIC 15
Robinson then proceeds to develop a solution, which roughly corre-sponds to level (3) as outlined in Section 3. The claim that we aredealing with first order logic plus standard part is in a sense a mathe-matical theorem, undermining the contention that “this seems hardlycompatible, etc.”Edward Nelson demonstrated that infinitesimals can be found withinthe ordinary real line itself in the following sense. Nelson finds infinites-imals in the real line by means of enriching the language through theintroduction of a unary predicate standard and an axiom schemata (ofIdealization), one of most immediate instances of which implies the ex-istence of infinitely large integers and hence nonzero infinitesimals; see[Nelson 1977]. This is closely parallel to the dichotomy of assignable vs inassignable in Leibniz, whereas Carnot spoke of quantit´es d´esign´ees [Carnot 1797], [Barreau 1989, p. 46]. Thus, we obtain infinitesimals assoon as we assume that (1) there are assignable (or standard) reals,that obey the same rules as all the reals, and (2) there are reals thatnot assignable.In more detail, one considers the ordinary ZFC formulated in firstorder logic (here the term is used in a different sense from the rest ofthis article), adds to it the unary predicate and the axiom schemata,and obtains a framework where calculus and analysis can be done withinfinitesimals. For further discussion see [Katz & Kutateladze 2015].The passage from Robinson cited above does indicate that secondorder theory may often be interesting. However, in the case of thecalculus/analysis as it was practiced in the 17th century, we are notaware of a single significant result that cannot be formulated in a sys-tem of type FOL + (cid:3) (see Section 2 for examples of results that can).Arguably it was calculus (rather than analysis) that Leibniz invented,in the sense that there don’t appear to be any essentially second orderstatements there.It may seem surprising that there could be a kind of pre-establishedharmony between a modern logical category, namely, first-order results,and a historical category, namely, results of 17th century calculus. Thisidea suggests further questions: does this only hold for the calculus, oris it always or often the case that a historically earlier realization of atheory covers only the first-order part of its successor. How do arith-metic and geometry behave in this respect? Would it really make senseto systematically distinguish between Leibniz and Leibniz1, Euclid andEuclid1, etc.? Not standalone.
Let us return to the comment
In the contextof Skolem’s construction of nonstandard natural numbers and some re-lated stuff, one is impressed by how many nonstandard numbers oneneeds to move from N to ∗ N , e.g., in Henkin’s model ∗ N = N + ( Z × Q ) and it goes without saying that Leibniz did not have this perspective ,that was addressed briefly above. One could elaborate on the “im-pression” concerning “how many nonstandard numbers” one needs todefine ∗ N consistently and conveniently.An infinitely large number say H is not a standalone object, butrather lives in a community of numbers obeying certain laws whichmathematicians anticipate as a goal of the construction of a nonstan-dard number system ∗ N . Such a commitment to anticipated laws forcesSkolem and others to add to N a suitable entourage of H along with H itself. What are the laws involved?Modern specialists in Nonstandard Analysis (NSA) stipulate that ∗ N should satisfy the axioms of Peano Arithmetic and moreover, satisfythe same sentences of the language (not necessarily consequences ofthe axioms) that are true in N itself. This is called (the principle of)Transfer today. Mathematicians of the 17th (or even 19th) century hadneither this perspective nor the tools consistently to define ∗ N or ∗ R .On the other hand, one can argue that there is no need for actu-ally rigorously defining ∗ N in order to make use of its benefits. Onecan argue that it is sufficient to have some idea of Transfer on topof an acceptance of infinitely large numbers per se (possibly as use-ful fictions , to borrow Leibniz’s expression). We have argued that theLeibnizian Law of continuity is closely related to the Transfer principle;see [Katz & Sherry 2013].Therefore the claim that
Leibniz had not the slightest idea of this stuff (the “stuff” being the modern technique of building nonstandard mod-els) is perhaps technically true, but it does not reflect all the aspectsof the interrelations within the Leibniz/Weierstrass/NSA triangle.8.
Conclusion
The vast oeuvre of Leibniz is still in the process of publication. Inprinciple a lucky scholar might one day unearth a manuscript whereLeibniz tackles a property equivalent to the completeness of the reals(after all the existence of the shadow is so equivalent), involving quan-tification over all sets of the number system and therefore second-order.However, this is unlikely in view of Leibniz’s reluctance to deal withinfinite collections, as mentioned above. If level (3) of first order logic isindeed suitable for developing modern proxies for the inferential moves
EIBNIZIAN CALCULUS AND FIRST ORDER LOGIC 17 found in Leibnizian infinitesimal calculus, as we have argued, thenmodern infinitesimal frameworks are more appropriate to interpretingLeibnizian infinitesimal calculus than modern Weierstrassian ones.
Acknowledgments
M. Katz was partially funded by the Israel Science Foundation grantnumber 1517/12.
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Selected papers of Abraham Robinson . Vol. II. Non-standard analysis and philosophy. Edited and with introductions by W. A.J. Luxemburg and S. K¨orner. Yale University Press, New Haven, Conn.[Sherry & Katz 2014] Sherry, D., Katz, M. “Infinitesimals, imaginaries, ideals, andfictions.”
Studia Leibnitiana (2012), no. 2, 166-192. (The article was pub-lished in 2014 even though the journal issue lists the year 2012.)See http://arxiv.org/abs/1304.2137 [Skolem 1933] Skolem, T. “ ¨Uber die Unm¨oglichkeit einer vollst¨andigen Charak-terisierung der Zahlenreihe mittels eines endlichen Axiomensystems.” NorskMat. Forenings Skr., II. Ser.
No. 1/12, 73-82.[Skolem 1934] Skolem, T. “ ¨Uber die Nicht-charakterisierbarkeit der Zahlenreihemittels endlich oder abz¨ahlbar unendlich vieler Aussagen mit ausschliesslichZahlenvariablen.”
Fundamenta Mathematicae , 150-161.[Skolem 1955] Skolem, T. “Peano’s axioms and models of arithmetic.” In Mathe-matical interpretation of formal systems, pp. 1–14. North-Holland Publish-ing, Amsterdam.[Stolz 1883] Stolz, O. “Zur Geometrie der Alten, insbesondere ¨uber ein Axiom desArchimedes.” Mathematische Annalen (4), 504–519.[Unguru 1976] Unguru, S. “Fermat revivified, explained, and regained.” Francia ,774–789. Piotr B laszczyk is Professor at the Institute of Mathematics, Peda-gogical University (Cracow, Poland). He obtained degrees in mathe-matics (1986) and philosophy (1994) from Jagiellonian University (Cra-cow, Poland), and a Ph.D. in ontology (2002) from Jagiellonian Univer-sity. He authored
Philosophical Analysis of Richard Dedekind’s memoir
EIBNIZIAN CALCULUS AND FIRST ORDER LOGIC 21
Stetigkeit und irrationale Zahlen (Cracow, 2008, Habilitationsschrift).He co-authored
Euclid, Elements, Books V–VI. Translation and com-mentary (Cracow, 2013), and
Descartes, Geometry. Translation andcommentary (Cracow, 2015). His research interest is in the idea ofcontinuum and continuity from Euclid to modern times.
Vladimir Kanovei graduated in 1973 from Moscow State Univer-sity, and obtained a Ph.D. in physics and mathematics from MoscowState University in 1976. In 1986, he became Doctor of Science inphysics and mathematics at Moscow Steklov Mathematical Institute(MIAN). He is currently Principal Researcher at the Institute for In-formation Transmission Problems (IPPI), Moscow, Russia, and Pro-fessor at the Moscow State University of Railway Engineering (MIIT),Moscow, Russia. Among his publications is the book
Borel equivalencerelations. Structure and classification , University Lecture Series 44,American Mathematical Society, Providence, RI, 2008.
Karin Katz (B.A. Bryn Mawr College, ’82); Ph.D. Indiana University,’91) teaches mathematics at Bar Ilan University, Ramat Gan, Israel.Among her publications is the joint article “Proofs and retributions,or: why Sarah can’t take limits” published in
Foundations of Science . Mikhail G. Katz (B.A. Harvard University, ’80; Ph.D. ColumbiaUniversity, ’84) is Professor of Mathematics at Bar Ilan University.Among his publications is the book
Systolic geometry and topology ,with an appendix by Jake P. Solomon, published by the AmericanMathematical Society; and the article (with T. Nowik) “Differentialgeometry via infinitesimal displacements” published in
Journal of Logicand Analysis . Taras S. Kudryk (born 1961, Lviv, Ukraine) is a Ukrainian mathe-matician and associate professor of mathematics at Lviv National Uni-versity. He graduated in 1983 from Lviv University and obtained aPh.D. in physics and mathematics in 1989. His main interests arenonstandard analysis and its applications to functional analysis. Heis the author of books about nonstandard analysis (in Ukrainian andEnglish) and textbooks about functional analysis (in Ukrainian) co-authored with V. Lyantse. Kudryk has performed research in non-standard analysis in collaboration with V. Lyantse and Vitor Neves.His publications appeared in
Matematychni Studii , Siberian Journal ofMathematics , and
Logica Universalis . Thomas Mormann is Professor at the Departamento de L´ogica yFilosof´ıa de la Ciencia de la Universidad del Pa´ıs Vasco UPV/EHU (Donostia-San Sebasti´an, Spain). He obtained a PhD in Mathemat-ics from the University of Dortmund, and habilitated in Philosophy,Logic, and Philosophy of Science at the University of Munich. Hepublished numerous papers in philosophy of science, history of philos-ophy of science, epistemology, and related areas. He is the editor ofCarnap’s “Anti-Metaphysical Writings” (in German). Presently, hismain research interest is in the philosophy of Ernst Cassirer and, moregenerally, in the Marburg Neo-Kantianism.
David Sherry is Professor of Philosophy at Northern Arizona Uni-versity, Flagstaff, AZ. E-mail: [email protected]. He has researchinterests in philosophy of mathematics, especially applied mathematicsand nonstandard analysis.
P. B laszczyk, Institute of Mathematics, Pedagogical University ofCracow, Poland
E-mail address : [email protected] V. Kanovei, IPPI, Moscow, and MIIT, Moscow, Russia
E-mail address : [email protected] K. Katz, Department of Mathematics, Bar Ilan University, RamatGan 52900 Israel
E-mail address : [email protected] M. Katz, Department of Mathematics, Bar Ilan University, RamatGan 52900 Israel
E-mail address : [email protected] T. Kudryk, Department of Mathematics, Lviv National University,Lviv, Ukraine
E-mail address : [email protected] T. Mormann, Department of Logic and Philosophy of Science, Uni-versity of the Basque Country UPV/EHU, 20080 Donostia San Sebas-tian, Spain
E-mail address : [email protected] D. Sherry, Department of Philosophy, Northern Arizona Univer-sity, Flagstaff, AZ 86011, US
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