International Mathematical Forum, Vol. 7, 2012, no. 10, 477 514 On the History of Differentiable Manifolds Giuseppe Iurato Department of Mathematics and Computer Science University of Catania, Catania, Italy iurato@dmi.unict.it Abstract We discuss central aspects of history of the concept of an affine differentiable manifold, as a proposal confirming the need for using some quantitative methods (drawn from elementary Model Theory) in Mathematical Historiography. In particular, we prove that this geometric structure is a syntactic rigid designator in the sense of Kripke-Putnam. Mathematics Subject Classification: 01A55, 01A60, 01A85, 03C07, 03C98, 03G27 Keywords: differentiable manifold, implicit function, Whitney's theorem, Lindenbaum-Tarski algebra, semantics, syntax, principle of virtual works, rigid designator 1 Introduction It is well-known (see, for instance, [39], [59]) that the origins of the modern concept of an affine differentiable manifold should be searched in the Weyl's work [65], where he gave an axiomatic description, in terms of neighborhoods (following Hilbert's work on the Foundations of Geometry), of a Riemann surface (that is to say, a real two-dimensional analytic differentiable manifold). Moreover, the well-known geometrical works of Gauss and Riemann1 are considered as prolegomena respectively of the topological and metric aspects of the structure of a differentiable manifold respectively in R3 and Rn, n ≥ 3 (see [6]). 1Nevertheless, following what has been said in the Introduction of [42], we may say that for a modern reader, it is very tempting to regard his [that is, of Riemann] efforts as an endeavor to define a "manifold", and it is precisely the clarification of Riemann's ideas, as understood by his successors, which led gradually to the notions of manifold and Riemannian space as we know them today. 478 G. Iurato All these common claims are well-established in History of Mathematics, as witnessed by the crucial work of E. Scholz (see [59]). Nevertheless, in this paper, we want to propose other possible viewpoints, about the same historical question, that are corroborated by some elementary methods of Model Theory applied to Mathematical Historiography. To be precise, we want to show that the Dini's works on implicit function theorems provide an essential syntactic tool, which were at the foundations of the modern theory of differentiable manifolds (see Examples 5 and 6, Section 1.1 of [26]). We may think the Dini's theory on implicit functions as a theory, in a certain sense, deductively equivalent (from the syntactical point of view) to the modern abstract theory of differentiable manifolds, via the fundamental works of H. Whitney. For a modern treatment of the theory of differentiable manifolds strictly related with Dini's and Whitney's theorems (and for other interesting imbedding results), see [43]. Furthermore, in this perspective, we want (logically) to relate these works of U. Dini with some arguments and statements of Lagrangian Analytical Mechanics, in such a way that the latter may be seen as necessary physical (hence, semantical) and formal motivations to the birth of the structure of differentiable manifold (as we know it today). At last (but not least), we prove that the geometric structure "differentiable manifold" is a mathematical entity that must be understood as a syntactic rigid designator in the sense of Kripke-Putnam. 2 The papers of Hassler Whitney With the papers [67], [68] and [69], Hassler Whitney began a detailed study of the structure of a differentiable manifold, mainly starting2 from the works of O. Veblen and J.H.C. Whitehead (see 7). Subsequently, he have improved and extended part of these results: for instance, his celebrated imbedding theorem was first stated in [68] for compact manifold, and extended to every paracompact manifold in [70]. In the Introduction to [68], he says that ¡¡A differentiable manifold may be defined in two ways: as a point set with neighborhoods homeomorphic with Euclidean spaces Rn (hence, according to Weyl), or as a subset of Rn defined near each point by expressing some of the coordinates in terms of others by differentiable functions (hence, according to Dini, as we will see). 2See footnote 2 of page 645 of [68]. On the history of differentiable manifolds 479 The first fundamental theorem is that the first definition is no more general than the second one; any differentiable manifold may be imbedded into Euclidean space. In fact, it may be made into an analytic manifold in some Rn¿¿. In [68], Whitney uses many results of [67] and, especially, he uses some approximation theorems of the Weierstrass type (see I.6. of [68]). In II.8. of [68], he proves (a first version of) the following, celebrated imbedding theorem (of Whitney) ¡¡Any Crmanifold of dimension m (with r ≥ 1 finite or infinite) is Cr-homeomorphic with an analytic manifold in Euclidean space Rm+1¿¿. There is another fundamental theorem proved by Whitney in [68], namely the Theorem 2 (exposed in II.8., after the above mentioned Theorem 1), that plays a crucial role in the proof of the various Lemmas to Theorem 1. In the proof of Theorem 2 of [68], many results of the theory of real analytic functions and their approximations, are used. Finally, we recall what he says in I.1. of [69] ¡¡Let f1, ..., fn−m be differentiable functions defined in an open subset of Rn. At each point p at which all fi vanish, let the gradients ∇f1, ...,∇fn−m be independent. Then the vanishing of the fi determines a differentiable manifold M of dimension m. Any such manifold we shall say is in "regular position" in Rn. Only certain manifolds are in regular position [...]. The purpose of the paper is to show that any m-manifold M in regular position in Rn may be imbedded in a (n−m)-parameter family of homeomorphic analytic manifold; these fill out a neighborhood of M in Rn. We may extend the above definition as follows: M is in regular position if, roughly, there exist n−m continuous vector functions in M which, at each point p of M , are independent and independent of vectors determined by pairs of points ofM near p. IfM is differentiable, the two definitions agree; the ∇fi are the required vectors. The theorem holds also for this more general class of manifolds¿¿. Clearly, the recalls to the Dini's work are evident. Moreover, as has been made after the works of Whitney (see, for instance, 1.1 and Theorem 3.2. of [27]; see also [43]), the Theorem 2 of [68], nowadays called regular value theorem, may be re-expressed and simplified through the implicit function theorem, starting from the original Whitney's proof, with a few modifications. 480 G. Iurato Further, the implicit function theorems are at the basis of the important notion of transversality, a modern differential topology tool (see [27]) that specifies the intuitive concept of "generic position" (drawn from algebraic geometry) of a manifold. However, we are mainly interested in the above fundamental Theorem 1, for the following reasons: we'll use this imbedding theorem to prove a certain logical (syntactical) equivalence between the theory of differentiable manifolds according to Weyl (that is to say, the modern one) and that deducible by the work of U. Dini. 3 The Implicit Function Theorem: a brief history The most complete work on the history of implicit function theorem, is [31] (for some aspects of this history, see also [41]). The germs of the idea for the implicit function theorem, can be traced both in the works of I. Newton, G.W. Leibniz, J. Bernoulli and L. Euler on Infinitesimal Analysis, and in the works of R. Descartes on algebraic geometry. Later on, in the context of analytic functions, J.L. Lagrange found a theorem that may be seen as a first version of the present-day inverse function theorem (see also, [32], 2, for the limitations of this theorem). Later, we'll return on this question in regard to the influences of the Theory of Analytic Functions and Algebraic Geometry in the birth of the modern notion of differentiable manifold. Subsequently, A.L. Cauchy gave a rigorous formulation of the previous semi-theories of implicit functions assuming that they were expressible as power series, a restriction then removed by U. Dini (see [8], p. 431). Indeed, from here on, the implicit function theorem has evolved until the definitive Dini's generalized real-variable version (see [16], [17]), related to functions of any number of real variables. Only with these Dini's works, we have a first complete, general and organic theory of implicit functions (at least, from the syntactic viewpoint). 4 The work of Ulisse Dini Ulisse Dini (1845-1918) was a pupil of Ottavio Fabrizio Mossotti (1791-1863) and Enrico Betti (1823-1892). The first was a physicist and a mathematiOn the history of differentiable manifolds 481 cian, deeply influenced by the works of J.L. Lagrange3, who taught Geodesy at the University of Pisa when Dini was a student. Betti was professor of Mathematical Physics at the University of Pisa and supervisor of the Dini's thesis4. In 1864, Dini published a paper on an argument of his thesis suggested by Betti; this first paper was followed by many other works on Differential Geometry and Geodesy. In that period, Dini was into a scientific friendship with E. Beltrami who took the Geodesy chair of the late Mossotti. At the same time, Dini was into touch with B. Riemann, then visiting professor at Pisa under Betti's interests. In 1865, Dini spent one year of specialization at Paris under the supervision of J. Bertrand, where he continued his thesis arguments, with further research in Differential Geometry, Geodesy, Algebra and Analysis. In 1866, Dini came back to Pisa, where he started his teaching career at the University, as professor of Geodesy and Advanced Analysis. Nearly seventies, Dini settled an important work on a rigorous revision of the mathematical foundations of Analysis, with his celebrated Lezioni di Analisi Infinitesimale (see [16], [17]) and the Fondamenti per la teorica delle funzioni di variabili reali (see [18]); in these works, it has been inserted many his original results and contributions: among these, the (so-called Dini's) theory of implicit functions, namely in [16], [17]. We are interested in Lezioni di Analisi Infinitesimale. These are the lessons given by the author in the Academic Year 1876-1877 at the University of Pisa, and there exist two contemporaneous autographed (or lithographed) editions: the edition published by the printing works Bertini, and the edition published by the printing works Gozani. Both editions are in a unique volume, divided into two parts: the first devoted to the Differential Calculus (with Chapters I-XXXII), the second devoted to the Integral Calculus (with Chapters I-XXIII). The Dini's theory on implicit functions is exposed in the following Chapter (of [16]) XIII. Derivate e differenziali dei vari ordini di funzioni implicite di una o pi variabili indipendenti, whereas, in the Chapter (of [16]) 3On the other hand, O.F. Mossotti was a close colleague and collaborator of G.A.A. Plana at Torino, and the latter, in turn, was a pupil of J.L. Lagrange at the Paris École Polytechnique. 4The possible influence of the work and teaching of Mossotti and Betti on the Dini's scientific training, might be traced, for instance, through a suitable adaptation of the socalled psycho-historical studies of E.H. Erickson. 482 G. Iurato XV. Cangiamento delle variabili indipendenti, Dini deals with certain forms of the so-called inverse function theorem. Finally, in the following Chapters (of [16]), Dini exposes some geometrical and analytical5 applications of some theorems of the previous Chapters XIII and XV. At the beginnings of the 20th century, Dini published a new revised and enlarged edition of the previous lessons [16], in two volumes (and each volume, into two parts). Nevertheless, as specified in the Preface to each volume, the new edition is different from the first only in notations and terminology, but not in the content: indeed, he notices that the Editorial publication of these lessons is motivated by the will of giving a historical evidence of his teaching in 1876-1877. For our purposes, we are interested in parte 1a and parte 2a of the vol. I of [17]; the parte 1a, of pages 372, contains the Chapters I-XVII, where the last Chapter has the following title XVII. Massimi e minimi delle funzioni di una o pi variabili indipendenti. The parte 2a, of pages 345, starts with the following headline − APPLICAZIONI GEOMETRICHE DEL CALCOLO DIFFERENZIALE − and contains the Chapters XVIII-XXXVI. It is completely devoted to the geometrical applications of the tools and methods developed in parte 1a: indeed, it is a very, organic treatise on Differential Geometry, fully based on the previous lessons [16]. Above all, in the Chapters XIX-XXXVI he uses extensively the theory of implicit functions (of the previous Chapters XIII and XV of parte 1a): for a modern (only) terminological reformulation of these Dini's (geometric) applications, see Cap. 2 of [2]. 5 The paper of Henry Poincaré Following E. Scholz ([59]; see also [40]), in the paper [54] may be found another possible source of the concept of a manifold. In fact, H. Poincaré, in 1 and 3 of [54], has given a constructive definition of (unilateral/bilateral6) manifold as follows. 5Where, among other things, the author introduces the famous Dini's numbers of the Mathematical Analysis. 6The distinction between unilateral and bilateral manifolds is given in 8 of [54]. We refer to the bilateral case. On the history of differentiable manifolds 483 If x1, ..., xn are generic variables of Rn (n ≥ 2), then he considers the following system of p equalities and q inequalities (1)  F1(x1, ..., xn) = 0 .... Fp(x1, ..., xn) = 0 φ1(x1, ..., xn) > 0 .... φq(x1, ..., xn) > 0, with Fi, φj continuous and uniform functions, with continuous derivatives in such a way that J = ∥∥∥∂Fi ∂xk ∥∥∥ = 0 in each point of the common definition domain of the Fi. If p = 0, we have a domain. The system (1) defines a manifold of dimension m = n − p, that, when7 q = 0, it is possible to prove (see 3 of [54]) to be equivalent to a manifold defined by a system of equations of the following type (2)  x1 = θ1(y1, ..., ym) .... xn = θn(y1, ..., ym). Again, the (syntactic) recalls to the implicit function theory, are evident. However, the main historical interest of the paper [54] is known to be related to the origins of Algebraic Topology, and not to the (possible) concept of differentiable manifold (see [58]). 6 The work of Hermann Weyl The first definition of a complex two-dimensional topological manifold, as we know it nowadays, is exposed in 4 of [65], while in 6 of [65], the author gives the notion of a differentiable structure on such a manifold type. The Weyl's analysis starts from a geometrical representation of an analytic form (according to Weierstrass and Riemann), and attaining to a particular structure of (Riemann) surface8, through the new topological developments achieved by D. Hilbert and others. In particular, the local Hausdorff's concept of "neighborhood" of a point, has played a crucial role in the Weyl's construction of a topological manifold. 7Henceforth, if not otherwise stated, when we'll consider the equivalence between (1) and (2), it is understood that q = 0. 8This is not a surface, in the sense of Analysis Situs. 484 G. Iurato Moreover, some geometrical aspects of Complex Analysis at that time, have also played a fundamental (syntactic) role in the Weyl's work (as we'll see later). The central Weyl's idea is that of local homeomorphism of a manifold with Rn. Subsequently, Weyl introduced a differentiable structure on a topological manifold by means of such a local homeomorphism of this manifold with Rn, taking into account some previous works of F. Klein. For our purposes, it is necessary to examine such little known works of F. Klein on Riemann surfaces. F. Klein wrote a fundamental monograph9 on the concept of a Riemann surface, more general than the formulation used by Riemann in his studies on the theory of analytic functions. Klein based his work on the previous Riemann's studies on Abelian functions, on the fundamental 1870 paper of H.A. Schwarz10 on the integration of the bi-dimensional Laplace equation ∆u = 0, and on a 1877 paper of R. Dedekind. In all these works, there are some first results related to a particular class of Rn-imbedded surfaces, generated by analytic functions. Klein also knew other works on Rn-imbedded surfaces as, for instance, those of Alberto Tonelli (Atti della R. Accademia Reale dei Lincei, ser. II, v. 2, 1875), W.K. Clifford (1876), F. Prym (1874) and P. Koebe. As Weyl himslef said, these works of F. Klein seem to take an important role in the (Weyl's) definition of a differentiable structure on a manifold. Furthermore, Klein's Erlanger Program viewpoint seems to have been at the basis of the Weyl's definition of compatibility relations among local coordinate systems of a generic point of the manifold, since he introduces a group of local coordinate transformations Γ, that leaves fixed the origin of R2; such a group characterizes the manifold, and Weyl talks about surface of type Γ. 9Entitled Über Riemann's Theorie der algebraischen Funktionen und ihrer Integrale, Leipzig, 1882. See also F. Klein, Neue Beiträge zur Riemannschen Funktionentheorie, Math. Ann., 21 (1883). 10Perhaps, it may be suitable to observe that H.A. Schwarz and U. Dini had a scientific epistolary exchange just in the seventy years of the 18th century. In the same period, Dini published Sull'integrazione dell'equazione ∆2u = 0, in Annali di Matematica Pura ed Applicata (2 (5) (1871) pp. 305-345), a paper on the same argument of the Schwarz's one. Moreover, at the end of chapter XV of [16], the author exposes the so-called (geometric) Legendre transformation, asserting that it may be useful to some questions related to the integration of ∆2u = 0. We will return on these last arguments when we shall talk about the role played by the principle of virtual work in our historical issue. On the other hand, these last references on some 19th century works related to the Rnimbedded problems of surfaces, are little known in the relative scientific literature (see, for instance, the historical notes of [24], for the metric case). On the history of differentiable manifolds 485 Later on, in [66] the author makes some applications of what is said in [65], in the context of General Relativity. 7 The works of O. Veblen and J.H.C. Whitehead O. Veblen and J.H.C. Whitehead, in the paper [63] (and, more extensively, in [64]), introduce two definitions of a n-dimensional (regular) affine manifold through three groups of axioms. In the Introduction, the authors define ¡¡a manifold as a class of elements, called points, having a structure which is characterized by means of coordinate systems¿¿, where the notion of (local) coordinate system is the same of the Weyl's one. Next, they introduce the notion of regular transformation by means of Dini's implicit function theorem (see page 552 of [63]). This notion is put at the foundation of a definition of regular manifold, through the further notion of pseudo-group of transformations (see [44], [30] or [11]), via three groups of axioms that, on the whole, characterize the concept of manifold (see 5). The next sections of [63], are devoted to the consistency and independence of the previous groups of axioms, to some topological considerations and to few analytic applications. In this case too, Dini's implicit function theorems play a crucial role in the definition of manifold, since this is characterizable as an abstract entity locally diffeomorphic to Rn, via allowable − through regular transformations − local coordinate systems11 (see Examples 5. and 6., Section 1.1 of [26], and also the next paragraph). 8 The role of Dini's theory of implicit functions in differential geometry In this paragraph, we want claim attention on the existence of important logical (and historical) links between the theory of implicit functions, settled by Ulisse Dini, and the construction of the abstract theory of a (topological) affine manifold. 11Furthermore, the authors devote 2 of Chapter III of [63], to explain the Implicit Function Theorem as a fundamental tool that will be used in the remaining text. 486 G. Iurato It is possible to build up a theory of affine manifolds in Rn, by means of Dini's implicit function theorems and the inverse function theorem: see, for instance, the excellent and organic treatment given by one of the most thorough and complete Italian treatise on Mathematical Analysis, that of Bruno Pini12 (see [52], parte I, Capitolo 2, 2 and parte II, Capitolo 7, 3), or the exposition of [15], secondo volume, Cap. V. The implicit function theorem and the inverse function theorem, characterize the local structure of any manifold (see the "parametrization" technique of [60], Cap. 5; see also Chapter 5 of [25]): to this end see, for instance, Theorems 3 and 4, Chapter 5, of [62]. Moreover, a manifold (in Rn) may be thought in a certain sense, as given by the set of zero values of a given system of functions of the type (1) (equivalent to (2)), discussed in the previous 5. Here, we do not develop the detailed calculations connected with these claims, since we have other aims. Nevertheless, it is necessary to recall the main definitions and theorems related to such a question, following, respectively, the exposition given by [22] in Chapters VII and VIII, and by [55] in Chapter 4. According to [22], the local character of Dini's implicit function theorems led to important applications still having local character: among these, there are the inverse function theorems (or local invertibility theorems). Roughly speaking, a differentiable manifold is a subset Γ ⊆ Rn that may be locally represented as a set of zeros of functions of more variables whose Jacobian matrix has maximum rank. For example, we may consider a surface Γ ⊆ R3 given by g(x1, x2, x3) = 0 with ∇xg = 0 for each x = (x1, x2, x3) ∈ Γ, or the geometric entity Γ given by the nondegenerate intersection of p (≥ 2) hyperplanes Γ1, ...,Γp of Rn. In the latter case, if Γi is represented by the linear function gi(x) = ∑n j=1 aijxj , x = (x1, ..., xn), i = 1, ..., p, then Γ is represented by the zeros of the linear function g(x) = (g1(x), ..., gp(x)), so that, if Γ = Ker g, then dim Γ = dim Ker g = n − dim Im g = n − rank A where A = ∥aij∥; moreover, we suppose that det A = 0. Finally, if we want that such a Γ has dimension m ∈ N with m < n, then we must impose that both p = n−m and rank A = n−m, or that p = rank A = n−m. Therefore, if we extend these last examples to the case in which g is non-linear, then we should impose that its Jacobian matrix has maximum rank, and since this is variable with the points of Γ, it follows that the representation of Γ as set of function zeros can only have local nature. Generalizing, we have that Dini's theorem implies that a manifold locally may be thought both as a non-degenerate intersection of diagrams of regular functions (definition 1) and 12See [50], [51], [52], [53], and other his monographs on Advanced Mathematical Analysis. In passing, Bruno Pini (1918-2007) was one of the major Italian mathematicians of the 20th century; inter alia, he was the discoverer (contemporaneously, but independently, with J. Hadamard) of the so-called Pini-Hadamard inequality and of the Pini-Hadamard parabolic analog of Harnack's second theorem (see [29] and [57], Chapter 18, 1). For more on scientific biography of Bruno Pini, see [34]. On the history of differentiable manifolds 487 as images of regular functions (definition 2), in both cases the Jacobian matrices having maximum rank; further, from the pointwise variability of the Jacobian matrix, it follows the possibility of introducing local coordinate systems. Hence, a first definition of manifold arises when we consider it as the result of gluing together many pieces each of which is a curved (due to the non-linearity of the various functions g) subset of Rn obtained intersecting a subspace (of Rn) with an open set (of Rn). Precisely, we have the following Definition 1. Let Γ ⊆ Rn, m ∈ N with m < n, and k ∈ N or k = ∞. Then we say that Γ is a Ck-manifold of Rn with dimension m, when, for each x0 ∈ Γ, there exist an open neighborhood I of x0 and a function g ∈ Ck(I,Rn−m), such that Γ∩I = {x;x ∈ I, g(x) = 0} and rank J(g)(x) = n−m for each x ∈ Γ ∩ I. Here, J(g)(x) is the Jacobian matrix of g computed in x. The following definition of a manifold arises when we consider it locally identified as the image of regular functions. To be precise, we have the following Definition 2. Let Γ ⊆ Rn, m ∈ N with m < n, and k ∈ N or k = ∞. Then we say that Γ is locally the diagram of a m variables Ck-function when, for each x0 = (x10, ..., xn0) ∈ Γ, there exist two open neighborhood I ′ and I ′′ respectively of the points (x10, ..., xm0) and (x(m+1)0, ..., xn0), and there is a C k-function h : I ′ → I ′′, such that, setting I = I ′ × I ′′, we have Γ ∩ I = {(x1, ..., xn); (x1, ..., xn) ∈ I, (xm+1, ..., xn) = h(x1, ..., xm)}, unless unessential permutations of x1, ..., xn. In such a case, we say that Γ has a structure of a Ck-manifold with dimension m. The latter is the definition of a manifold via parametrization, which is the result of a formalization of the geographical mapping that put into bijective correspondence a geographical chart C with a certain zone C ′ of the Earth surface; in such a way, it is evident what basic role the tools and methods of the Geodesy have played in developing the intuitive idea of what a manifold can be13. Indeed, from this last point of view, we reach the following Definition 3. Let Γ ⊆ Rn, m ∈ N with m < n, and k ∈ N or k = ∞. If x0 ∈ Γ and Ω is an open set of Rn, then a local m-chart of class Ck of Γ at x0 is an injective Ck-function r : Ω → Rn such that there exists an open neighborhood I of x0 in such a way that Γ ∩ I = r(Ω) and rank J(r)(t) = m for any t ∈ Ω. A m-atlas of class Ck of Γ is a family {ri}i∈Ξ (Ξ is a set of indices) of local m-charts of class Ck such that the union of the related image sets is Γ. Finally, Γ is a Ck-manifold of dimension m if it has a m-atlas of class Ck. The parametric map r−1 : Γ ∩ I → Ω provides a local coordinate system in such a way that, if x ∈ Γ ∩ I, then the Cartesian coordinates t1, ..., tm of r−1(x) are said to be the local coordinates of x with respect to the given local coordinate system. Now, we consider a particular, simple situation that allows us to put into evidence that certain conditions, imposed on the rank of the various Jacobian matrices, are needed in order to prove the equivalence among the above mentioned definitions of a manifold. Let n = 3,m = 2 and Γ be a plane of R3 containing the origin of R3. Such a plane may be considered as the set of zeros of a suitable linear operator with rank n − m = 1; let a1x1 + a2x2 + a3x3 = 0 such an operator with (a1, a2, a3 =) = 0, and let a3 = 0, for instance. Hence we have x3 = px1+ qx2, so that such a plane is also the diagram of a linear operator from R2 to R (that is to say, from Rm to Rn−m); finally, the same plane has the following parametric equations x1 = t1, x2 = t2 and x3 = pt1+ qt2, so that it is the image of 13Till up the first middle of the 20th century, the Geodesy was a usual subject-matter taught in the mathematical faculties. 488 G. Iurato the linear operator (t1, t2) → (t1, t2, pt1+ qt2), operating from R2 to R3 (that is to say, from Rm to Rn), with rank 2 (that is to say, m) whatever p, q. From here, in the more general case in which Γ ⊆ Rn is a subspace of dimension m(< n), it is possible to prove that Γ may be represented as set of zeros of the linear map associated to a certain (n −m,m)-matrix with maximum rank, as the diagram of a certain linear operator from Rm to Rn−m, and as image of a certain linear operator from Rm to Rn with rank m. Finally, if Γ is a manifold, then instead have to do with a linear operator (like in the previous examples, in which such a linear operator globally represents Γ), we'll have to do with regular non-linear operators providing (in general, only) a local representation of such a manifold. The fundamental tools that allow us to get such a local representation are just the Dini's theorem and the inverse function theorem. Indeed, as is seen in the above mentioned example concerning a plane of R3, it was necessary to solve an implicit equation with respect to one of its three variables, so that, in the general case, it will be necessary to solve a system of the type g(x) = 0 with respect to n − m of its n variables, and the implicit function theorem is the most natural tool for solving such a problem. This theorem, nevertheless, may only provide local representations, also in the case in which the manifold is globally given as set of zeros of a unique function. In Theorem 1.11 of Chapter VIII of [22], it is proved the following fundamental result: Theorem 1. Let Γ ⊆ Rn, m ∈ N with m < n, and k ∈ N or k = ∞. Then, the following conditions are equivalent: 1. Γ is a Ck-manifold of dimension m, according to the Definition 1; 2. Γ is locally the diagram of a m variables function of class Ck, according to the Definition 2; 3. Γ is a manifold having a m-atlas of class Ck, according to the Definition 3. In the proof of implication 1 . ⇒ 2 ., it is used Dini's implicit function theorem, whereas in the proof of the implication 3 . ⇒ 1 ., it is used inverse function theorem. Among other things, the above theorem 1. provides a useful criterion to verify whether a certain subset of Rn is a manifold. Lastly, we observe that Dini's implicit function theorem and inverse function theorem are strictly correlated between them. The above exposition drew from [22], starts with Dini's theorems toward inverse function theorems. Instead, according to [55], it is possible to start with inverse function theorems toward Dini's theorems. For instance, Chapter 4 of [55] begins with problems concerning possible inversions of differentiable functions between Rn-type spaces, hence with problems of local inversion of functions of many variables. The first historical methods related with this type of problems concern with the class of differentiable functions, since the differential of a function is the first, natural linear approximation tool for these functions, and we have a large class of results for linear applications (like the differential map) suitable to answer to the above mentioned inversion problems. Therefore, the principle of the method consists in a generalization of what is known about linear maps toward the more extended class of differentiable maps. Because of the local nature of the differential map, it is clear that the obtained results from this generalization have a local character as well. In 2 of [55], the author deals with some problems concerning local inversion of maps. If Ω ⊆ Rm and Λ ⊆ Rn are non-void open sets, then let f : Ω → Λ be a continuous map; if x ∈ Ω, then let ȳ = f(x). We say that f is locally injective on x if there exists a neighborhood U of x such that f|U is an injective map. We say that f is locally surjective on x if, for any neighborhood U of x, f(U) is a neighborhood of ȳ. On the history of differentiable manifolds 489 As regard the local inversion problems of maps, let us consider the following examples. Given two open sets Ω,Λ ⊆ Rn, for a14 C1-map f : Ω → Λ to be invertible in a point x ∈ Ω, it is necessary and sufficient that its Jacobian matrix in x, say J(f)(x), be not singular when n = m); so, we obtain a characterization of the local invertibility of a C1-function in the case n = m. The general case of arbitrary n,m ∈ N, is as follows. Let f : Ω → Rn be a C1-map defined on an open set Ω ⊆ Rm; if we want to locally study f in a neighborhood of a point x ∈ Ω, then we have need to consider the rank of the Jacobian matrix J(f)(x) of f at x (that represents the differential of f in x). If rx = rank J(f)(x), then rx ≤ min{n,m} for every n,m ∈ N and x ∈ Ω, the local invertibility of f in x being possible only when rx is highest. Therefore, we first consider the case rx = min{n,m}, in such a way that it remains maximum in a neighborhood of x, since f ∈ C1(Ω,Rn). Let rx = m < n. In such a case, we have the following inverse function theorem Theorem 2. Let f : Ω → Rn be a C1-map defined on an open set Ω ⊆ Rm, and let rx = rank J(f)(x) = m. Then, f is locally injective in x, and, moreover, the image, through f , of an open neighborhood of x, is a regular15 Cartesian graph having as a base an open subset of Rm. For a proof, see Theorem 4.3 of [55]. Instead, if rx = n < m, then f is locally surjective, so that we have the problem of studying the inverse image of every point y ∈ Rn that lies in a neighborhood of f(x). To this end, Dini's theorems are a fundamental tool for the resolution of such a problem. For instance, if we consider the case study m = 2 and n = rx = 1, then it holds the following Dini's implicit function theorem Theorem 3. Let R2 = R′ × R′′ with R′ ∼= R′′ ∼= R. Let f : Ω → R be a continuous function defined on an open set Ω ⊆ R2 with fy continuous on it; let P0 = (x0, y0) ∈ Ω be a point such that f(x0, y0) = 0 and fy(x0, y0) = 0. Then f is locally surjective in P0. Moreover, there exist a neighborhood U of x0 on R′ and a neighborhood V of y0 on R′′, such that the set of zeros of f on U × V is a regular Cartesian diagram with base U , that is to say, there exists a neighborhood W of 0 on R such that, for each z ∈ W , the set {(x, y); (x, y) ∈ Ω, f(x, y) = z} ∩ (U × V ) is a regular Cartesian diagram with base U . Such a theorem is applied to the study of the set of zeros of a real function f of two variables: for example, if f is a function verifying the same hypotheses of the previous theorem, Γf = {(x, y); (x, y) ∈ Ω, f(x, y) = 0} and if there is a point (x, ȳ) ∈ Γf such that (fx(x, ȳ), fy(x, ȳ)) = (0, 0), then Γf , in a neighborhood of (x, ȳ), is of the form y = φ(x) or x = ψ(y), for certain C1-functions φ or ψ. This result fails in the degenerate case fx(x, ȳ) = fy(x, ȳ) = 0, that is to say on the singular points of Γf . In the general case, we have the following Dini's theorem Theorem 4. Let f : Ω → Rn be a C1-function defined upon an open set Ω ⊆ Rm, with n < m. If rx = rank J(f)(x) = n in a point x ∈ Ω, then f is locally surjective on x. Moreover, there exist a neighborhood V of ȳ = f(x), a neighborhood U of x and a (m−n)-dimensional open set V ′′ of Rm such that, for every ȳ ∈ V , f−1({ȳ})∩U is a regular Cartesian diagram with base V ′′. For a proof (making use of the above mentioned theorem 3.), see Theorem 4.8 of [55]. 14The C1-regularity hypothesis is of fundamental importance. 15That is to say, a diagram of class C1. 490 G. Iurato Finally, we have functional dependence in the case in which rx < min{n,m} and, in general, it is no longer true that such a value rx remains constant in a neighborhood of x. Nevertheless, in such a case, if we suppose that such a value rx remains constant in, at least, one neighborhood of x, then we have the following Theorem 5. Let f : Ω → Rn be a C1-map defined on an open set Ω ⊆ Rm. Given a point x ∈ Ω, we suppose that rx = rank J(f)(x) < min{n,m} is constant in a neighborhood of x. Then, locally, the image of f is a regular Cartesian diagram, say Γf , with base an open subset of a coordinated r-dimensional subspace of Rn. Moreover, the inverse image of an arbitrary point of Γf , is a Cartesian diagram with base an open subset of a (m − r)dimensional coordinated space of Rm. At this point, the author introduces the notion of a differentiable manifold on Rm. Precisely, if we wish to introduce particular subsets of Rm that are locally like to some affine numerical space Rn, with n ≤ m, then the above mentioned theorems are a fundamental tool for this problematic context. The problem has, in general, only solutions of local nature: for example, it is well-known that a circle of R2 and a line, are locally homeomorphic but not globally; on the other hand, the intersection point of two distinct lines is not even locally homeomorphic to a point of a line. Thus, a (topological) n-dimensional manifold of Rm (with n ≤ m) is a subset Γ ⊆ Rm such that every point of it has a neighborhood homeomorphic to some open subset of Rn, namely, for each x ∈ Γ, there exist an open neighborhood U of x on Rn, an open set V of Rn and a bijective continuous map r : V → U ∩ Γ with continuous inverse; in such a case, we say that r is a local coordinate system (or a local chart) of x. In general, further properties are required to hold for such a map r: among these, we mainly require that it is continuously differentiable (or of class Ck, with k ∈ N or k = ∞), and in such a case, we speak of a differentiable chart of class C1 (or of class Ck). If every point x ∈ Γ has a differentiable chart of class Ck, then we say that Γ has the structure of a n-dimensional differentiable manifold of class Ck. We have the following16 Theorem 6. For a subset Γ ⊆ Rm be a n-dimensional differentiable manifold of class Ck, it is necessary and sufficient that, for every x ∈ Γ, there exists an open neighborhood U of x such that Γ ∩ U is a Cartesian diagram of class Ck, with base an open subset B of a n-dimensional coordinated space. For a proof (making use of the above mentioned Theorem 2.), see Theorem 6.4 of [55]. From the previous Theorems 2. and 6., it follows that any inverse local chart r−1 : Γ ∩ U → V can be factorized into r−1 = ∆ ◦ p where p is the canonical projection of the given Cartesian diagram (of Theorem 6.) over the base B, whereas ∆ : B → V is a Ck-bijective map with continuous inverse. At this point, it becomes a natural question to treat the case in which a same point x ∈ Γ is into two distinct local charts, say r1 and r2. Exactly, let ri : Vi → Γ ∩ U, i = 1, 2 be two local charts on the same open neighborhood U with x ∈ U ; then, it is possible to prove (see Theorem 6.6 of [55]) that r−12 ◦ r1 and r −1 1 ◦ r2 are real homeomorphisms of class Ck: the proof follows from the decomposition r−1 = ∆ ◦ p. The differentiability properties of a manifold rely just on the differentiability of its transition maps among allowable coordinate systems, and it is clear that these last properties do not subsist in the abstract case, that is to say, them must be explicitly postulated: from here, it 16The Theorem 6., among other things, is a useful criterion determining whether a subset Γ is a manifold. On the history of differentiable manifolds 491 follows the abstract (Weyl's) definition of a differentiable manifold. Nevertheless, the author himself (see Remark 6.7 of [55]) says that the degree of (syntactic) logical generality of the abstract theory of differentiable manifolds is no higher than that of the real differentiable manifold theory, because of the works of Whitney. However, the axiomatic approach has methodological and pragmatic advantages since, for instance, we may define such a structure over arbitrary mathematical objects (with a some predefined topology). Finally, we may define a differentiable manifold by means of Dini's Theorem 4. other than through the inverse function theorem (see the above Theorem 2) as made in Theorem 6., for instance, as follows Theorem 7. For a subset Γ ⊆ Rm be a n-dimensional differentiable manifold of class Ck (with n ≤ m), it is necessary and sufficiency that, for each x ∈ Γ, there exist an open neighborhood (of Rn) and a Ck-function ψ : U → Rm−n with maximum rank on U such that Γ ∩ U = {x;x ∈ U,ψ(x) = 0}. In other words, the latter says that there are m − n real Ck-functions ψ1, ..., ψm−n, defined on U and whose Jacobian matrix has rank m−n, such that17 Γ = ∩m−n i=1 Γi, having put Γi = {x;x ∈ U,ψi(x) = 0} for i = 1, ...,m− n. For a proof (making use of the above mentioned Dini's Theorem 4.), see Theorem 6.8 of [55]. This last theorem assures us that a n-dimensional differentiable manifold of class Ck is locally representable as the set of zeros of a certain multivalued function. In conclusion, from the viewpoint of [55], Chapter 4, the inverse function theorem is related with the problem of local injectivity of a regular function, whereas the Dini's theorem is related with the problem of local surjectivity of a regular function. From both these points of view, we may get a definition of a differentiable manifold (respectively, like Theorem 6. as regard the problem of local injectivity, and like Theorem 7. as regard the problem of local surjectivity) in Rn, so that it is evident the historical importance played by Dini's works on implicit function theorem regarding the foundations of modern differential geometry. However, it would be a historical mistake to think that Ulisse Dini had in mind such a manifold theory (although in Rn): in fact, he only settled (maybe unconsciously) the fundamental syntactic tools need for the next modern construction of an abstract affine manifold, although it may be probable that some problems of Rn-imbedded surfaces (as we have seen in the previous 6) were (maybe again unconsciously) at the basis of his work18. As we'll see later, there is no (explicit) semantic link between Dini's work on implicit functions and the theory of manifolds; there exist, instead, only strong links of syntactic nature (that, despite all, has a proper historical importance, as we shall see in the next sections, as regard the notion of syntactic rigid designator). We have already mentioned the possible role played by Algebraic Geometry (see, for instance, [31]) and Complex Analysis with regard to the mindset of the modern concept of a differentiable manifold. We wish to outline some a 17The maximum rank condition assures that such an intersection is non-degenerate. 18Clearly, these last considerations have to be considered, only at a semi-intuitive level, as a sort of possible insight for Dini's work on implicit function theorems. 492 G. Iurato few words about these last aspects. The work of Weyl, as is seen in 6, was centered around the study of the geometrical representation of certain analytic functions. On the other hand, we also remember that, for instance, Salvatore Pincherle, in Chapter XI of [49], exposes the implicit functions theory in the complex context, following Dini's work in the real case. In Chapter XII, he applies what has been said in the previous one, to the algebraic functions theory, whereas, in Chapter XIII, he resumed Lagrange's work on inverse function theorem in view of its analytical applications. This plan is common to all major treatises on Analytic Function Theory of that time. From all that, it is possible to guess (as, for instance, made by [40]) some not negligible influences of the 19th century Algebraic Geometry in the developments of some aspects of the Theory of Differentiable Manifolds, because many algebraic geometry tools and methods are applied to the study of the so-called Riemannian surfaces of an algebraic function. A posteriori, these conjectures find some partial (syntactic) confirmations by the so-called Nash-Tognoli imbedding theorems of Algebraic Geometry (see [5], Chapter 14), a sort of algebraic geometry analogous of Whitney's theorems, proving that any compact smooth manifold is diffeomorphic with a well-defined nonsingular real algebraic manifold. Hence, also the works of 19th century algebraic geometers should be considered having had some influences on the possible origins of modern theory of differentiable manifolds. Nevertheless, the comparison with the Nash-Tognoli theorems mentioned above, does not have a great historical importance within the question related to the rise of modern theory of differentiable manifolds, differently by the case of Dini's and Whitney's works (see next 11). At this point, it is necessary to introduce the minimal Model Theory notions, which will be essential for the following critical remarks: indeed, we want to introduce these basic quantitative tools to clarify, in a rigorous manner, the previous historical review, as well as to highlight the historical relevance of the possible syntactic links among these theories. 9 Some notions of Model Theory According to [10], Model theory is, roughly speaking, Universal Algebra plus Logic. In this section, we recall some notions of Model Theory, need for the follows. Our main references are [12], [13], [14], [36], [21], [38]. On the history of differentiable manifolds 493 9.1 Syntactic and semantic models Every axiomatic scientific theory has a either syntactic component and semantic one, and often these two aspects are mixed into a concrete (that is, non-axiomatic, or else intuitive) scientific theory. Therefore, in general, the formalization process of a scientific theory is an axiomatization process working out over an initial structure of intuitive theory, toward an abstract (axiomatic) structure, called model. The Model Theory deals with problems and methods of such a construction. In this problematic context, syntactic and semantic questions arise: for instance, the works of K. Gödel and A. Tarski show the possible existence of a non-contradictory syntactically closed theory, and the non-existence of a non-contradictory semantically closed theory. Hence, there exist, so to say, limitative theorems on the syntactic and semantic capacity of an axiomatic theory. Nevertheless, from these limitations, it also follows the reciprocal inseparability of the syntactic and semantic components. 9.1.1 Syntactic models The formalization process leads to the so-called notion of formal system. It is composed by both syntactic and semantic components. In this section, we expose the syntactic aspects. An elementary (syntactic) formal system (or a syntactic theory) F is a tuple F = ⟨⟨L,D⟩⟩ with language scheme L = ⟨Al, Te,Wr,E⟩ and deductive scheme D = ⟨Ax,Ru⟩, where • there exist disjoint sets Co,Qu, Fu, Pr, V a,Au, in such a way that Al =∪ {Co,Qu, Fu, Pr, V a,Au} is the alphabet (or the set of symbols) of F, with Co the set of logic connectives, Qu the set of logic quantifiers, Fu the set of functors, Pr the set of predicates, V a the set of individual variables and Au the set of auxiliary symbols (with Co ∪ Qu the set of logic constants and Pr∪Fu the set of descriptive constants or vocabulary); • Wr is the set of words, Te (⊆ Wr) is the set of atomic terms, E (⊆ Wr) is the set of atomic expressions (with Te∪E the set of well-formed words, and Prop the set of propositions defined as a subset of E whose elements have no free variables); • Ax (⊆ E) is the set of (logic and specific) axioms, whereas Ru is the set of logic deduction rules (with respect to a given Logic). 494 G. Iurato Note. In this section, from now on, we speak only of a formal system (or theory), without specify the term 'syntactic'. L determines the set of (explicit and implicit) definitions (say De) of F, whereas D determines the set of proofs (say Pf), and the set of theorems (say Th), of F. Therefore, a formal system (a theory) is a tuple of the type (1) F = ⟨⟨L,D⟩⟩ = ⟨⟨ ⟨Al, Te,Wr,E,De⟩, ⟨Ax,Ru, Pf, Th⟩ ⟩⟩. We may think D as the predicative, or propositional, or enunciative calculus of a theory F. If α is a theorem of F, we write ⊢F α. If an expression α of F is a logical derivation by a set of expressions M of F, then we write M ⊢F α. If the set of axioms Ax is decidable, then F is said to be axiomatizable, whereas, if the set of specific axioms is finite, then F is said to be finitely axiomatizable. If L is a formal [not formal (or intuitive)] language, then we say that F is a formal [not formal] theory. We need for some clarifications about the elements of Fu and Pr. Fu is the class of all n-functor Fun = {fni }0≤i<j with 0 ≤ j ≤ ω and 0 ≤ n < ω, where Fu0 is the set of individual constants, with Fun = ∅ if j = 0. Pr is the class of all n-predicate Prn = {P ni }0≤i<j with 0 ≤ j ≤ ω and 0 < n < ω; Pr2 contains, at least, the element P 20 said to be the identity predicate, with Prn = ∅ if j = 0. Let F1,F2 be two theories of the type (1); we say that • F2 is a predicative linguistic extension of F1 when Pr1 ⊆ Pr2; • F2 is a functorial linguistic extension of F1 when Fu1 ⊆ Fu2; • F2 is a linguistic extension of F1 (and we write L1 ⊆ L2) when F2 is a predicative and functorial linguistic extension of F1; • F2 is a deductive extension of F1 when Ax1 ⊆ Th2; • F2 is a theoretical extension of F1 (or that F1 is a sub-theory of F2) when F2 is a deductive and linguistic extension of F1; in such a case, we write F1 4 F2, and we say that 2 is the theoretical inclusion relation; • a theoretical extension F2 of F1 is a linguistically invariant extension when L1 = L2, that is to say, when F2 is an improper linguistic extension of F1; On the history of differentiable manifolds 495 • a theoretical extension F2 of F1 is an inessential19 extension when Th1 = Th2 ∩ E1. If F1 4 F2 and F2 4 F1, then we say that F1 is equivalent to F2, and we write F1 ≈ F2; we say that ≈ is the theoretical equivalence relation. We refer to [13], Capitolo 1, 3, Definizione 7, for the definition of the elements of De (the set of predicative and functorial definitions of a theory F). We say that F2 is a simple definitional extension of F1 if and only if there exists a predicative [functorial] definition δP n i [δf n i ] in F2, such that 1. P ni /∈ Pr1 [fni /∈ Fu1]; 2. Pr2 = Pr1 ∪ {P ni }, Fu2 = Fu1 [Fu2 = Fu1 ∪ {fni }, Pr2 = Pr1]; 3. Ax2 = Ax1 ∪ {δP n i } [Ax2 = Ax1 ∪ {δf n i }]. We say that F2 is a definitional extension of F1 when there exists a sequence of theories Fk1 , ...,Fkp (1 < p ≤ ω), such that: 1. F1 = Fk1 and F2 = Fkp ; 2. for each 1 ≤ i ≤ ω, Fi+1 is a simple definitional extension of Fi. In other words, F1 = Fk1 → ...→ Fki → ...→ Fkp = F2 1 < i < p, is a chain of simple definitional extensions. Every simple definitional extension is a (proper) deductive and linguistic extension as well. Moreover, we have the following Theorem 1. If F2 is a [simple] definitional extension of F1, then F2 is an inessential extension of F1. For a proof, see [13], Capitolo 1, 3, Teoremi 5, 6. Remark 1. Theorem 1. is the final result of a part of the works due to Giuseppe Peano, Alessandro Padoa and Mario Pieri on the logical analysis of formal systems; a consequence of the so-called (Peano-Padoa-Pieri) noncreativity principle20 of the logical definitions, is that the definitions (elements of De) of a formal theory F must not determine deductive novelties21 but only expressive novelties. From here, it follows why a [simple] definitional extension is proved to be "inessential". 19See also next Remark 1. 20See [38], Cap. III, 5, and Cap. VI, 2, or [4], Cap. I. 21That is, the definitions must not involve the demonstrability of new theorems, or rather it must not broaden or enlarge the deductive capacity of a theory. 496 G. Iurato If L is a pure syntactic [or not] language, then we say that F is a pure syntactic [not pure syntactic] theory. This last classification leads us to an extra-syntactic area, as we will see later, when we shall introduce the notion of semantic model. We now introduce the various notion of theoretical homomorphisms (for details, see [13], Capitolo 2). Let F1,F2 be two theories of the type (1). A theoretical representation of F1 into F2 is a map ρ : Wr1 → Wr2; hence, we write ρ : F1 → F2. Remembering that E, Th ⊆ Wr, we can say that a theoretical representation ρ : F1 → F2 is • an expressive homomorphism if ρ(E1) ⊆ E2; • a theorematical homomorphism if ρ is an expressive homomorphism and ρ(Th1) ⊆ Th2; • a deductive homomorphism if ρ(Pr1) ⊆ Pr2. A deductive homomorphism is a theorematical homomorphism as well. This last classification defines the so-called class of theoretical homomorphisms. A theoretical representation ρ : F1 → F2 is said to be • a version of F1 into F2, if there exists a map (called the base of this version) ψ : Fu1 ∪Pr1 → Th2 ∪E2, satisfying a certain set of compatibility properties (see [13], Cap. 2, 1, Def. 3, a)); • a quasi-relativization of F1 into F2, if there exists an expression α(v) ∈ E2 (v is a free variable) and a map ψ : Fu1 ∪ Pr1 → Th2 ∪ E2, verifying a set of compatibility properties (see [13], Cap. 2, 1, Def. 3, b)); we say Bρ =< α(v), ψ > to be the base of this quasi-relativization; • a relativization of F1 into F2, if there exists a quasi-relativization ρ′ of F1 into F2, with base Bρ′ =< α(v), ψ >, in such a way that ρ(β) ⇒ ρ′(β) for each β ∈ E1, and ρ(β) = ρ′(β) for each β ∈ Pr1. Versions, quasi-relativizations and relativizations, are expressive homomorphisms. A theorematical homomorphism ρ of F1 into F2 is said to be • a translation, if ρ(¬β) = ¬ρ(β) for each β ∈ Pr1; • an interpretation, if ρ is a version of F1 into F2; • a relative interpretation, if ρ is a relativization of F1 into F2; On the history of differentiable manifolds 497 • an isomorphism, if ρ : Wr1 →Wr2 is a bijection such that ρ(Ax1) = Ax2, and there exists a map ψ : Al1 → Al2, commuting with ρ, such that ψ(Fu1) ⊆ Fu2, ψ(Pr1) ⊆ Pr2, ψ(V a1) ⊆ V a2, ψ(Au1) ⊆ Au2, ψ(Co1 ∪ Qu1) ⊆ Co2 ∪Qu2. Therefore, we say that F1 is translatable, interpretable, and relatively interpretable into F2 if, respectively, there exists a translation, an interpretation, and a relative interpretation of F1 into F2. We say that F1 is isomorphic to F2 if there exists an isomorphism between F1 and F2, and we write F1 ∼ F2. An isomorphism is a deductive homomorphism as well, but not conversely, in general (see [13], Capitolo 2, 1, Teorema 5). It is also possible to prove (see [13], Cap. 2, 2, Teorema 1) the following Theorem 2. If F1 4 F2, then F1 is translatable, relatively interpretable and interpretable into F2; moreover, if it is also L1 = L2 and F1 is isomorphic to F2, then F1 ≈ F2, the converse being not true, in general. The relations of translatability, relative interpretability and interpretability, are pre-orders. We have the following chain of implications (see [13], Capitolo 2, 1, Teoremi 6, 7, 8, 9) Isomorphism ⇒ Interpretation ⇒ ⇒ Relative Interpretation ⇒ Traducibility. If a representation ρ, inducing a certain theoretical homomorphism [isomorphism], is computable, then we speak of an effective theoretical homomorphism [isomorphism]. If F1 is relatively interpretable into F2, then we say that F1 has a syntactic model into F2, and we write F1 F2. It is important the following Theorem 3. If F1 is [relatively] interpretable in F2, then F2 has a definitional extension F′2 containing a sub-theory F ′ 1 isomorphic to F1. For a proof, see [13], Capitolo 2, 2, Teoremi 10, 11. Among the theoretical homomorphisms defined above, for our historiographical purposes, we are interested in the interpretable and relatively interpretable ones. The adjective "interpretable" leads us towards the semantic context. To each formal theory F = ⟨⟨L,D⟩⟩ of the type (1), it is associable a particular universe U , that is to say, the set of truth values of its statements (propositions, theorems, expressions, and so on); its choice is independent22 of the syntactic structure of F. Therefore, the interpretability of F1 into F2, means that it is always possible to give an interpretation of the concepts of F1 in the terms of the concepts of 22We'll take again this argument in the semantic context. 498 G. Iurato F2, in such a way that what F1 says to be true with respect to its universe U1, is also true − by means of such an interpretation − with respect to the universe U2 of F2. Instead, the relative interpretation of F1 into F2, means that it is always possible to give an interpretation of the concepts of F1 in terms of the concepts of F2, but in such a way that what F1 says to be true with respect to its universe U1, is also true with respect to a particular sub-universe Uα of U2, determined by the relativization condition α(v) (of the base Bρ =< α(v), ψ > of the given representation ρ : F1 → F2). At this point, it is possible to apply these considerations to the historiographical context, as follows. Indeed, one of the central problems in the Historiography of Exact Sciences, is to determine the possible relationships among different theories, as, for instance, those holding among a concrete (or intuitive) theory and its formalizations23. A first rational (or quantitative) comparison of this last type, it is possible, for instance, taking into account the possible existence of a theoretical representation among the theories under comparison: for example, if there exists an interpretation, or a relative interpretation, of a theory F1 into a theory F2, then we can say that F1 is, in a certain sense, ¡¡included¿¿ into F2. Analogously, the possible determination of a syntactic model (and the possible theoretical connections that it may give) provides a useful criterion for the ¡¡reducibility¿¿ of a theory into another. These types of (syntactic) connections, provide "natural" interpretations of certain theories into others, also in the case in which their (historical) sources are very far off between them. Nevertheless, for methodological motivations, we should consider such a syntactic comparison criteria, with the suitable cautions. Anyway, at this point, we may do a simple historical application of what has been said so far. If FDini1 is the theory of differentiable manifolds in the Dini's sense, while FWeyl2 is the theory of differentiable manifolds in the Weyl's sense (that is, the modern one), then it is obvious that FDini1 is interpretable into FWeyl2 . On the other hand, by means of Whitney's theorems, we can say too that FWeyl2 is interpretable into FDini1 . From Theorem 3, it follows that F Dini 1 ∼ F′ Weyl 2 4 FWeyl2 and F Weyl 2 ∼ F′ Dini 1 4 FDini1 , for certain definitional extensions Fi of Fi i = 1, 2. Moreover, we may suppose the equality 24 between the languages 23Although, it would be more correct to consider such a type of logical comparison only among theories having almost the same syntactic degree of formalization. 24In fact, again by Whitney's works, it is no restrictive to think any abstract smooth n-manifold as the closed subset of some RN (with N = N(n) > n), locally representable (according to Dini) as intersection of the diagrams of a system of differentiable functions defined on some common open subset of Rn, with values into Rs, s = N − n. On the history of differentiable manifolds 499 of FDini1 and F ′Weyl 2 , and of F Dini 2 and F ′Weyl 1 , so that, by Theorem 2., we have FDini1 ≈ F′ Weyl 2 and F Weyl 2 ≈ F′ Dini 1 . From here, it does not follow the (syntactic) equivalence FDini1 ≈ F Weyl 2 , but rather a "minor" equivalence, as follows. If we take into account the notion of deductive equivalence, then we may say that F1 and F2 are deductively equivalents, and we write F1 ≃ F2, when F2 is a deductive extension of F1, and vice versa. Therefore, if we take into account what has been said in Remark 1, about the inessentiality of the definitional extensions, then we may set Fi ≃ Fi i = 1, 2. Thus, the relations FDini1 4 FWeyl2 ≃ FWeyl2 and FWeyl2 4 FDini1 ≃ FDini1 , implies the following deductive equivalence FDini1 ≃ F Weyl 2 . On the other hand, it is clear that this equivalence cannot be extended to the theoretical syntactic equivalence ≈, because there is no linguistic equivalence between FDini1 and F Weyl 2 : indeed, in F Dini 1 , there exists neither the explicit nor the implicit definition of manifold. In conclusion, FWeyl2 is a proper linguistic and inessential extension of FDini1 . Another almost equivalent way leading to the same conclusions (about the relationships between FDini1 and F Weyl 2 ), is centered around the (logic) immersion theorems (see [36], Cap. 2, 2.3), through which we have FDini1 ∼ F Weyl 2 . Let T be the class of all possible elementary theories, and T = T / ≈ the set of equivalence classes of T , with respect to the equivalence relation ≈. If 4ri is the relation of relative interpretability, then (T,4∗ri) is a pre-ordered set, putting [F1] 4∗ri [F2] if and only if F1 4ri F2 (this being a well-posed definition). We call rational power of a theory F, its equivalence class [F] ∈ (T,4∗ri): intuitively, [F] is the class of all theories F′ containing a sub-theory "that says the same things said" by F, whereas, in turn, F contain a sub-theory "that says the same things said" by F′. Analogously, if 4eri denotes the effective relative interpretation relation, we have that (T,4∗eri) is a pre-ordered set; [F] ∈ (T,4∗eri) is said to be the rational content of F, and, intuitively, it "contains everything said by F and, also, everything saids" by the weaker theories of F. Since it is possible to prove the existence of a (syntactic) isomorphism between (T,4∗ri) and (T,4∗eri), the unique formal entity they determined, is called a theoretical pre-order. Therefore, it is possible to consider this theoretical pre-order as a tool to determine a certain "scale of importance" among theories; further, it may turn out also useful in certain historical classifications of the "importance" of a theory identified with its rational contents. Moreover, such a pre-ordering may correspond to the historical development of the theories, so that it is evident the usefulness of the syntactic tools here exposed, for the possible 500 G. Iurato historical-critical comparison of theories. 9.1.2 Semantic models In this section, we should discuss the elementary semantic aspects of a (syntactic) formal system. The emergence of the semantic context has the following motivations. The above exposed syntactic methods, may turn out to be useful when we are mainly interested in the syntactic comparison of theories: for instance, with these methods, it is possible a comparison of theories with different languages. Nevertheless, the historical comparison is often oriented towards a language comparison, and the syntax shows its own limits25 with respect to this framework. A method to avoid these limits, consists in the introduction of the so-called Metamathematical Semantics. Roughly speaking, the Semantics studies the sets of possible meanings (or interpretations) associable to syntactic symbols. In [13], Capitolo 4, it is possible to find a purely abstract formalization of Semantics; instead, we are interested in a more extended setting, suitable to historical questions. To this end, we refer to [12], [14], [21] and [38]. We follow the algebraic viewpoint of the Semantics as developed by the Polish school. One of the central concepts of Algebraic Semantics is that of (Peirce-Schröder) logical matrix, built up on a syntactic system F = ⟨⟨L,D⟩⟩. Such a logical matrix is a tuple of the type M = ⟨⟨F,D⟩⟩, where D is the set of the so-called appointed (or designated) values, defined as follows. If C = Fu ∪ Pr is the set of descriptive constants26 of F, U is a possible world (or a universe of discourse) and v : C → U is a valuation, then R = (U , v) is said to be a (Frege) extensional interpretation of F. Therefore, we may define (extensively) D as follows: for each formula F of F, we have v(F) ∈ D if and only if F is true. F is a tautology if and only if v(F) ∈ D for every valuation v. If Ev(M) is the set of all formulas true under v (that is to say, such that v(F) ∈ D), then we set E(M) = ∩ v Ev(M). In such a way, the logical matrix generalizes the concept of (Tarski-Huntington-Bernstein) deductive system (or deductive theory); in general, F is a Boolean algebra and D is a filter on F (instead, the set of not true formulas, is an ideal of this algebra). We say that R is a semantic interpretation of the language L of F. We also may write 25There are further problematic limits of the syntactic context: for instance, there exist finiteness problems, connected with the attempts to avoid the impossible identification between mathematical truth and demonstrability, that led to the failure of the Hilbert's formalistic program. 26Descriptive constants (or atomic propositions) and specific axioms, characterize (syntactically) a formal theory. On the history of differentiable manifolds 501 M = ⟨⟨F,R⟩⟩ = ⟨⟨F, (U , v)⟩⟩, instead of M = ⟨⟨F,D⟩⟩. Now, we can introduce the fundamental notion of Lindenbaum-Tarski algebra. If M is a deductive theory (according to Tarski), we define the following pre-order φ ≤M ψ def.⇔ F ⊢M φ⇒ ψ. Its symmetrization gives the following equivalence relation27 φ ≡M ψ def.⇔ ( M ⊢M φ⇒ ψ ) ∧ ( M ⊢M ψ ⇒ φ ) , and it is immediate to prove that AM = M/ ≡M is a Boolean algebra with [φ] ∪ [ψ] = [φ ∨ ψ], [φ] ∩ [ψ] = [φ ∧ ψ], ¬[φ] = [¬φ], 0 = [(∀x)(x = x)], 1 = [(∀x)(x = x)]. Often, the above Lindenbaum-Tarski construction is made on F instead of the whole M, so that we obtain the following (syntactic) Lindenbaum-Tarski algebra AF = F/ ≡F. It is possible to prove that AM is a free algebra generated by C. By means of the Lindenbaum-Tarski algebra, it is possible to set a bijective correspondence between valuations and some particular homomorphisms of Boolean algebras, as follows. Let F(L) be the set of all formulas of L (in F) (as defined in [9], Appendice B, B.1.), and let AF(L) = F(L)/ ≡F(L) be the Lindenbaum-Tarski algebra of F(L); then, it is possible to prove that any valuation of M, bijectively corresponds to a well-determined homomorphism (of Boolean algebras) from AM to F, defined on the set of generators C. Moreover, if M is an arbitrary set of formulas of F (⊆ F(L)) and T (L,M) is the set of all theorems of the formal system having language L, and M as set of specific axioms (see [9], l.c.), then T (L,M) is a sub-theory of F, while T (L,M)/ ≡T (L,M) is a filter of AM. Thus, a Theory has a unique filter (on AM) as algebraic counterpart [precisely, a maximal filter for a (syntactically) complete Theory]: it is generated by the equivalence classes of the specific axioms M of T (L,M). On the other hand, following [36], if it is given a language L, a consistent set T of L-sentences is, roughly speaking, a Theory (see above T (L,M)), while a model of T (or a T -model) is a L-structure (see [36], Cap.1, 1.1, 1.2 and 1.3), 27There exist other equivalence relations leading to the so-called (Halmos) polyadic algebras, or to the so-called (Tarski) cylindric algebras. For simplicity, we restrict ourself to the Lindenbaum-Tarski algebras. 502 G. Iurato say S, such that every sentence in T is true into S. We say that a theory T proves a L-sentence ψ if T ⊢S ψ for every model S. Sometimes, the elements of T are called axioms, whereas the theorems (of T ) are the sentences proved in T , that is, the deductive closure of T (see also [60]). If we write, for simplicity's sake, S ⊢ ψ instead of T ⊢S ψ, then ⊢ sets up a Galois connection between the class of models of T and the set of all L-sentences of the deductive closure of T (see [12], Chapter 5, 4). Precisely, to each class C of T -models corresponds the set C∗ of all Lsentences true into every model of C, while, to each class S of L-sentences of T , corresponds the class S∗ of T -models with respect to which any sentence of S is true. Then, we have the following bijective correspondences C ξ→ C∗ and S ξ−1→ S∗, induced by the above Galois connection. On the other hand, if we consider the Lindenbaum-Tarski algebra associated (with the formal system corresponding) to the deductive closure of T , say AT , then the Galois connection, ξ, induces a Galois connection between AM and the space of models of T , say MT . Hence, we may write AT ξ∼= MT . The (logical) closure operators define (following Kuratowski) a well-determined topology on the model space MT , and the corresponding topological space is called the Boole space of T (see [12], Chapter 5, 6); it is a Stone space. If we want to apply these last considerations to the case related to History of Differentiable Manifolds, then we may deduce, via Whitney's theorems28, the existence of a Galois connection between MFDini1 and MFWeyl2 , hence between their corresponding Lindenbaum-Tarski algebras (computed with respect to the syntactic context, or with respect to the extensional semantic context). At this point, it is necessary to specify some above exposed semantic concepts. If B = (B,∨′,∧′,¬′, 0, 1) is any Boolean algebra, then a realization (or representation) of the language L into B, is a map ρ : F(L) → B, such that 1. ρ(¬α) = ¬′ρ(α), 2. ρ(α ∧ β) = ρ(α) ∧′ ρ(β), 3. ρ(α ∨ β) = ρ(α) ∨′ ρ(β), 4. ρ(α⇒ β) = ¬′ρ(α) ∨′ ρ′(β). 28This correspondence is bijective since, by a fundamental theorem due to H. Grauert (see [23], and [43]), any abstract manifold corresponds to a unique real manifold, via the Whitney's imbedding. Hence, it follows the existence of a unique (Whitney) imbedded structure, for each assigned abstract manifold. On the history of differentiable manifolds 503 We say that ρ is a model of α, or that α is true with respect to ρ, if and only if ρ(α) = 1. α is said to be valid into B if and only if it is true with respect to any realization ρ into B; α is said to be valid if and only if it is valid into any Boole's algebra B. A semantic meaning may be defined with respect to the (Frege) extensional context (extensional semantic) or with respect to the intensional context (intensional semantics). We have seen that a possible extensional interpretation is given by R = (U , v), where U is a possible world (or a universe of discourse), while v is a map that assigns a meaning, into U , to the descriptive constants (∈ C) of L. Then, according to G. Frege, v should satisfy the following conditions: 1) v(a) ∈ U for each a ∈ C; 2) v(P ni ) ⊆ Un ∀n ∈ ω,∀P ni ∈ Pr. We say that R is an (extensional semantic) interpretation of L, or a (extensional) semantic structure associated to L. Then, we say that a proposition ψ ∈ Pr is true with respect to the interpretation R = (U , v) if and only if v(ψ) ⊆ Un, where n ∈ ω is the arity of ψ. In general, for an arbitrary L-sentence ψ, we say that ψ is true with respect to R, and we write |=R ψ, if a set of (Frege) conditions are fulfilled (see [14], Capitolo 2, 2.2.). These are the basic elements of the (Frege) extensional semantics in the modern formulation given by A. Tarski. Nevertheless, especially in the historical context, it is more important to consider an intensional semantic context, as, for example, the one given by Kripke Semantics (belonging to the general class of Modal Logic). The main limit of Tarski Semantics is due to the existence of only two possible cases: such a Semantics considers either one universe of discourse U or all possible universes of discourse. Instead, S. Kripke (see [32]) considers a suitable system of possible universes of discourse in dependence on the uses and purposes of the given formal system. So, we speak of a Kripke realization with respect to a particular set of universes of discourse, that is to say, those accessible. These universes of discourse are connected among them by the so-called accessibility relations. In such a way, we go towards the realm of Modal Logic (Temporal, Epistemic, etc) and the intensional theories of meaning (as, for example, the Carnap's one). The Modal Logic may play a very important role in some historical interpretation, as we will see in the next section. Finally, we may consider a Kripke deductive system as a tuple of the type MKripke = ⟨⟨F, (Ui, vi)i∈J⟩⟩, where Ri = (Ui, vi), i ∈ J , is the Kripke's set of realizations of MKripke (if J is a singleton, or an infinite set, then we obtain a Tarskian deductive system). Mutatis mutandis, what has been said above about Lindenbaum-Tarski methods, may be applied to MKripke as well. 504 G. Iurato Analogously to what has been said in section 9.1.1., the critical comparison between Lindenbaum-Tarski algebras built over a [Kripkian] deductive theories, may turn out to be useful for possible historical comparisons between the related theories (see next 11). 9.2 The work of S. Kripke Saul Kripke is considered one of the most important founders of the so-called Semantic Modal Logic, which gives a more extended semantic context to the Tarski's one (for the Classical Logic) and to the Gödel's one (for the Intuitionist Logic). The book [32] is a philosophical continuation of the first sixties Kripke's research on the semantic analysis of Modal Logic. This work has, among other things, a prominent role in the Historiography of Sciences, as we will see. In [32], among other things, it is discussed the historical role of the Factuals, Counterfactuals and of the so-called Historical Chains, in the framework of the so-called Possible Worlds; there is a deep critical analysis of the Aristotele's distinction between Essential and Accidental properties, and of some related metaphysical Kantian conceptions (as the "a priori", the "analytical" and "necessity" truth Categories, and so on). The kripkian logical and philosophical analysis, start from a critical study of the already known (philosophical) concepts and notions of Name, Necessity, Possibility, Essence, Analytical Truth, Referent, Meaning, Reference, Description, rigid and not rigid Designators, Cluster Concept, and so forth. He examines the Modalities of the relations that hold between Names and Things; besides, in his first January 20, 1970 lesson, the author discusses the role of the concept of Possible World in the mathematical definitions, as regard the importance of the Identity Criterion in time (hence, also concerning the historical viewpoint). From a critical re-examination of the previous Name Theories (as, for example, the Name Reference Theory of G. Frege and B. Russell), Kripke attains his semantic theory of Possible Worlds, with some possible its applications; among these, we recall those having usefulness in some epistemological questions: precisely, the author says that his theory is an essential tool to establish the existence, or not, of correct historical connections among historical facts. On the other hand, this is just what is necessary, for instance, for the historical comparison of the mathematical theories treated in this paper. Saul Krikpe with Hilary Putnam (see [56]), are the founders of the modern, new reference theory. On the history of differentiable manifolds 505 10 The role of the principle of virtual works in differential geometry In this section, we want briefly recall the important role took by principle of virtual works of Analytical Mechanics. [39] is the main reference for the History of Mechanics up to 1920. This principle has played a truly fundamental role in Lagrange's work (see [33]): in fact, it is at the basis of the analytical mechanics arguments29. There are many modern textbooks on Analytical Mechanics whose first chapters, devoted to the formulation of the celebrated Lagrange's equations, begin with the exposition of the so-called D'Alembert-Lagrange principle of virtual works. For instance, a modern historical exposition very similar to the original Lagrange's formulation, is given by [1], vol. I, Capitolo I: in it, once again, the reference to Dini's work on implicit function theorems is evident and this proves the essential syntactic necessity of these methods for the formal setting of Analytical Mechanics and, hence, for the subsequent formulation of Differential and Riemannian Geometry. For a brief, but rigorous, exposition of these arguments, see [46], [47] and the more complete treatment of [20]. We briefly recall the main results of [1], vol. I, Capitolo I. In 2 of Chapter I, it is expounded the so-called D'Alembert principle mia⃗i = F⃗i + R⃗i i = 1, ..., N , for a system of N point particles, each of which has mass mi, acceleration a⃗i, and is subject both to the total active forces F⃗i and to the total constraint forces R⃗i. This principle reduces every dynamical problem to a statical one, and in it underlies a well-defined equilibrium condition. For a smooth30 systems with holonomic constraints, this last equilibrium condition is equivalent to the so-called principle of virtual works, whose analytical formulation is based on the invertibility of the virtual displacements δPi of the point particle Pi, and it is ∑N i=1(F⃗i −mia⃗i) × δPi = 0, said to be the general (or symbolic) equation of dynamics. In 3 and 4, respectively, the [angular] momentum conservation theorems and the Lagrange's equations31, are deduced from this symbolic equation. In the following sections, many other possible formal expressions of the Lagrange's equations are deduced: δ-d Lagrange's formalism is the main an29This principle has also been used on some questions related to the constrained motion of a quantum particle (see, for instance, [28]). 30Here, the term 'smooth' means constraints without friction. 31There exist various forms of the Lagrange's equations exposed in [1], vol. I, Capitolo I. In particular, in the subsection 2 of 4., the authors expose a first form of Lagrange's equations using Lagrange's multipliers rule, which plays a fundamental role in the extremum theory with side conditions (so that, again, we return to Dini's works). 506 G. Iurato alytical tool for deducing many formal dynamical properties of a holonomic smooth constrained system, given in a (Hertz) form similar to (1) of our 5, with q = 0 (equivalent to (2), where yi are replaced by lagrangian coordinates qi); these properties are both of metrical nature (assuming assigned a certain metric given by the kinetic energy, according to Jacobi) and affine nature, and it is much probable that they have played a fundamental role in the subsequent development of Differential Geometry. For instance, to this end it is important to remember that the first differential topology tool explaining the basic differential geometry local concepts, is that of tangent space in a point of a manifold: historically, the first definitions of tangent space have been the result of a generalization of the main basic concepts and methods of Analytical Mechanics concerning constrained motion of a particle over a smooth holonomic system (see the so-called physicist's definition as well as the geometer's definition, of a tangent space, equivalent between them and to another, called the algebraic definition all given in Chap. 2 of [7]; see also [39]). However, for our purposes, we follow the modern exposition given by V.I. Arnold in [3]. In Chapter IV, he gives a first modern definition of smooth holonomic constraint suggested32 by M.A. Leontovic (see 17, A.), with a second definition (see 17, B.) where it is substantially defined a manifold in the Dini's sense (see (1) of our 5); he goes back to the definition of smooth holonomic constraint in B., Example 10 of 18, where it is introduced the modern (Weyl's) definition of a differentiable manifold. In 21, Arnold introduces D'Alembert principle, and at the point B. of the same section, he proves the (syntactic) equivalence between the D'AlembertLagrange principle (of virtual works) and the definition of smooth holonomic constraint given at the point B. of 17, by means of the use (see point C. of 21) of a variational calculus argument already known to Lagrange (see point C. of 21, where the author also exposes the original Lagrange's static formulation). A similar exposition may be found in [1], vol. I, Capitolo I. The holonomy of such constraints has physical motivations, and, therefore, it is evident the mathematical physics origins of the concept of smooth holonomic constraint, hence of the differentiable manifold: indeed, the principle of virtual works provides the local characterization of a manifold, locally like to Rn, likewise to Dini's implicit function theorems. On the other hand, it is well-known that the sources of the Lagrange's inverse function theorem (already mentioned above), should be retraced into the 32For a deduction of Lagrange's equations from this Leontovic's standpoint, see [19] and reference therein. On the history of differentiable manifolds 507 Lagrange's works on some static problems, where, among other, he introduced the today known "Lagrange's multipliers" (see [15], Capitolo V., 5., footnote 5 of page 382). The latter, in turn, turns out to be related with the principle of virtual works as well, hence with the local structure of a differentiable manifold (via the connection of the Lagrange's multipliers with the inverse function theorem). To this end, we briefly recall the problem. Every extremum problem with side conditions historically started from questions of mechanics of constrained systems. If Γ is a smooth constraint, hence a manifold described as a set of zeros of functions, then, by means of Lagrange's multipliers, the extremum problems on Γ is reduced to local extremum problems related with the functions (locally) describing such a manifold Γ. To this end, we remember that, if Γ is a manifold of Rn, f : Γ → R, x0 ∈ Γ and r is a chart of Γ (see 8) containing x0, then we say that x0 is a relative maximum/minimum extremum for f if and only if r−1(x0) is a relative maximum/minimum extremum for f ◦ r. Since, in general, it is a difficult task to determine the charts of a manifold, because of the local nature of the question, for such an extremum problem it is enough to consider the same extremum problem related to a restriction of f on Γ′ = Γ∩I where I is a neighborhood of x0. Whence, we have the following Lagrange's Multipliers Theorem. Let Γ ⊆ Rn be a manifold of dimension m(< n), and x0 ∈ Γ; let I be an open neighborhood of x0 and g(x0) = 0 the local equation of Γ in x0 with g ∈ C1(I) and rank J(g)(x) = n − m for each x ∈ Γ ∩ I. Let f : Γ → R with f ∈ C1(I). If x0 is a relative extremum for f|Γ∩I then ∇f(x0) ∈ Nx0Γ (normal space to Γ at x0), i.e., there exist n−m real numbers λ1, ..., λn−m such that ∇f(x0) = ∑n−m i=1 λi∇gi(x0) with λ1, ..., λn−m uniquely determined by x0. If, for each x0 ∈ Γ, we have ∇f(x0) ∈ Nx0Γ, then we say that x0 is a stationary (or a critical) point of f ; such points are into bijective correspondence with the solutions of the system of equations g(x) = 0 and ∇f(x) = ∑n−m i=1 λi∇gi(x), whose solutions are of the type (x1, ..., xn, λ1, ..., λn−m) ∈ Rn × Rn−m, with λ1, ..., λn−m said to be Lagrange's multipliers. For instance, in the case n = 3, if R3 is a model of the physical space and Γ(⊆ R3) represents a bilateral smooth holonomic constraint for the material point x0 undergoes the force field ∇f , then the above theorem says that the force acting over a critical point x0 is orthogonal to the constraint Γ, whereas the values of Lagrange's multipliers are connected with the intensity of the constraint reactions. From here, it follows clear links with the principle of virtual works. In short, it is evident the existence of syntactic links between these analytical mechanics arguments and the basic formulations of the theory of differentiable manifolds, although it is a very difficult task to do sure historical claims about these suppositions. The only certainty concerns the syntactic comparison among the previous arguments, whereas their possible semantic comparison may be conducted within the Kripkian context (or, more generally, into the Modal Logic context), if we choose suitable Kripke's set of realizations upon which to interpret 508 G. Iurato the syntactic contents of the previous statements. From this point of view, it is perhaps possible to think that the work of Lagrange (and others, as C.G.J. Jacobi, L. Euler, and so on) on Analytical Mechanics, were intuitively oriented towards a study of the (local) geometry of configuration space of a moving particle, subsequently formalized both by the D'Alembert-Lagrange principle and by a mathematical structure described by a system of the type (1) of our 5 (with q = 0), by means of a large use of the so-called δ-d symbolism (typical of initial Analytical Mechanics, perhaps also little studied in the History of Differential Geometry). The just mentioned historical connections are however rather probable to hold (see next 11). We conclude this section with an unusual remark on the work of Tullio Levi-Civita on his parallel displacement33, from which it is possible to infer another prove of the importance played by the principle of virtual works, for the foundation of Differential Geometry. In fact, it is almost always affirmed (in the current related literature) that Levi-Civita parallel displacement was mainly motivated by the tentative of giving a geometrical interpretation to the so-called "covariant derivative" of Absolute Differential Calculus. Indeed, if one carefully read the paper [35], it is clear that the historical verity is quite different. Indeed, Levi-Civita was motivated by attempts to simplify the computation of the curvature of a manifold through Riemann symbols, as he says in the Introduction to his paper. Then, once introduced a generic metric structure on a manifold defined by a system of the type (2) of our 5, the author establishes a fundamental equation, the (I) of 2. The latter is nothing but the principle of virtual works applied to such a manifold, thought like a smooth holonomic system undergo (invertible) virtual displacements. From it, the author deduces an equivalent equation, the (8) of the same section, hence another equivalent form, the (Ia) of 3, from which he deduces the analytical conditions characterizing his celebrated notion of parallel displacement. In the remaining sections, the author does not make any explicit mention to the covariant derivative, except a secondary application related to Ricci's rotation coefficients (see 13 of [35]). 11 Conclusions Albeit it is surely erroneous to think that the concept of a differentiable manifold (as we know it today) was already present in the works of Dini on implicit 33As regard the historical relevance of Levi-Civita parallel displacement in Physics (as, for example, in Gauge Theories), see [6]. On the history of differentiable manifolds 509 functions, as well as in the foundations of Analytical Mechanics, nevertheless we may affirm, without doubts, what follows. A mathematical theory does not born from nothing, but instead it starts from some previous ones34: to be precise, it begins from those having, at least, a some syntactic link with it35. Hence, from here, it is evident the importance of the notion of syntactic model in searching such syntactic links, in order to we are able to determine a possible chain of syntactic models which may remember the above Kripkian "historical chains" of 9.2. We have exposed a case study of this historiographical methodology, precisely, that related to the origins of the concept of a differentiable manifold. Beyond such a first epistemological theory comparison, further researches are possible concerning the semantic context, for instance, brought into the Modal Logic framework. Through this last perspective, it is subsequently possible to make suitable "interpretations" (on the basis of the previous syntactic comparison) which are more proper for a historical setting. For instance, since we have seen that certain filters algebraically correspond to theories, then it is possible to compare two theories comparing their corresponding filters, and so on. Likewise, it may be compared the corresponding (syntactic or semantic) Lindenbaum-Tarski algebras between them. In these last two cases, the resulting chains of filters, or the corresponding algebras, may be considered as an 'algebraic formalization' of the so-called "historical chains" of Historiography (already mentioned above). We have so sketched such a line of historiographical methodology in relation to a particular case related to the History of Differential Geometry. Thus, within this context, it is very likely that Dini's works on implicit functions as well as the basis of Analytical Mechanics, have played a considerable moreor-less tacit role in the formulation of the modern theory of Differentiable Manifolds, both from the syntactic and semantic viewpoint. 34On the other hand, this statement finds a further confirmation on a certain, not casual epistemological "evolution" of a mathematical structure along historical time (see [48]). As a concrete example of this, we recall the work of G. Peano on the axiomatization of natural numbers, which started from the previous work of R. Dedekind on the same argument (such a question, besides, is also treated, from the Modal Logic viewpoint, in the January 22, 1970 lesson of S. Kripke − see [32]). 35The further, not trivial question concerning the awareness, or not, of the existence of these theories by the author under historical examination, may be analyzed from suitable philosophical (as well as psychoanalytical, whenever possible) viewpoints. However, certain contemporaneous but independent (between them) mathematical discoveries/constructions (like those mentioned above in the footnote 12) prove that the previous syntactical capacity of a certain theoretical context reaches a certain degree which will allow, in turn, a subsequent discovery/construction. 510 G. Iurato Furthermore, from what has been said so far, it is also clear that the geometric structure called differentiable manifold is a syntactic36 rigid designator in the sense of the new reference theory due to S. Kripke and H. Putnam: in fact, the same syntactic structure (or mathematical entity), has been identified in, at least, two different semantical contexts (or in two discourse's worlds), that of the theory of implicit functions and that of Lagrange's Analytical Mechanics. Moreover, following H. Putnam, we might say that the mechanisms by which the names engage the entities, is a collective mechanism and not an individual one, that is to say, there exists a historical chain (see above), external to every individual, or a series of 'reference rings' transmitted through time, in which it is possible to identify a certain constancy of the discourse's terms (rigidity of the reference) leading to a given entity (rigid designator): in our case, it deals with the syntactic structure of 'differentiable manifold'. 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