Giuseppe Iurato
email: iurato@dmi.unict.itsommario

On the History of Dierentiable Manifolds

Keywords: Mathematics Subject Classication: 01A55, 01A60, 01A85, 03C07, 03C98, 03G27 dierentiable manifold, implicit functions, Whitney's theorem, Lindenbaum-Tarski algebra, semantics, syntax, principle of virtual works, rigid designator

We discuss central aspects of the history of the concept of an ane dierentiable manifold, as a proposal conrming 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.

Introduction

It is well-known (see, for instance, [START_REF] Maurin | The Riemann Legacy[END_REF], [START_REF] Scholz | The concept of manifold, 1850-1950[END_REF]) that the sources of the modern concept of an ane dierentiable manifold should be searched in the Weyl's work [START_REF] Weyl | Die Idee der Riemannschen Fläche[END_REF], 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, in modern terms, a real two-dimensional analytic dierentiable manifold).

Moreover, the well-known geometrical works of Gauss and Riemann are considered as prolegomena respectively of the topological and metric aspects Nevertheless, following what has been said in the Introduction to [START_REF] Moerdijk | Models for Smooth Innitesimal Analysis[END_REF], we may say that for a modern reader, it is very tempting to regard his [that is, of Riemann] eorts as an endeavor to dene a manifold, and it is precisely the clarication 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.

G. Iurato

of the structure of a dierentiable manifold respectively in R 3 and R n , n ≥ 3

(see [START_REF]A Century of Geometry[END_REF]).

All these common claims are well-established in the History of Mathematics, as witnessed, for example, by the crucial work of E. Scholz (see [START_REF] Scholz | The concept of manifold, 1850-1950[END_REF]).

Nevertheless, in this paper we would like to propose other possible viewpoints, about the same historical question, which are corroborated by some elementary methods of Model Theory applied to Mathematical Historiography. To be precise, we wish to show that Dini's work on implicit function theorems provides an essential syntactic tool, which was at the foundation of the modern theory of dierentiable manifolds (see Examples 5 and 6, Section 1.1 of [START_REF] Hicks | Notes on Dierential Geometry[END_REF]).

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 dierentiable manifolds, via the fundamental works of H. Whitney. For a modern treatment of the theory of dierentiable manifolds strictly related to Dini's and Whitney's theorems (and for other interesting imbedding results), see [START_REF] Narasimhan | Analysis on Real and Complex Manifolds[END_REF].

Furthermore, in this perspective, we wish to relate (logically) Dini's work with some arguments and statements of Lagrange's 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 dierentiable manifold (as we know it nowadays). At last (but not least), we prove that the geometric structure dierentiable manifold is a mathematical entity that should be understood as a syntactic rigid designator in the sense of Kripke-Putnam.

The papers of Hassler Whitney

With the papers [START_REF] Whitney | Analytic extensions of dierentiable functions dened in closed sets[END_REF], [START_REF] Whitney | Dierentiable manifolds[END_REF] and [START_REF] Whitney | The imbedding of manifolds in families of analytic manifolds[END_REF], Hassler Whitney begins a detailed study of the structure of a dierentiable manifold, mainly starting from the works of O. Veblen and J.H.C. Whitehead (see next 7).

Subsequently, he improves and extends part of these results: for instance, his celebrated imbedding theorem is rst stated in [START_REF] Whitney | Dierentiable manifolds[END_REF] for compact manifold, and extended to every paracompact manifold in [START_REF] Whitney | The self-intersections of a smooth manifold in 2n-space[END_REF].

In the Introduction to [START_REF] Whitney | Dierentiable manifolds[END_REF], he says that A dierentiable manifold may be dened in two ways: as a point set with neighborhoods homeomorphic with Euclidean spaces R n (hence, according to Weyl), or as a subset of R n dened near See footnote 2 of page 645 of [START_REF] Whitney | Dierentiable manifolds[END_REF].

each point by expressing some of the coordinates in terms of others by dierentiable functions (hence, according to Dini, as we will see).

The rst fundamental theorem is that the rst denition is no more general than the second; any dierentiable manifold may be imbedded in Euclidean space. In fact, it may be made into an analytic manifold in some R n .

In [START_REF] Whitney | Dierentiable manifolds[END_REF], Whitney uses many results of [START_REF] Whitney | Analytic extensions of dierentiable functions dened in closed sets[END_REF] and, especially, he uses some approximation theorems of the Weierstrass type (see I. [START_REF]A Century of Geometry[END_REF]. of [START_REF] Whitney | Dierentiable manifolds[END_REF]).

In II.8. of [START_REF] Whitney | Dierentiable manifolds[END_REF], he proves (a rst version of ) the following, celebrated imbedding theorem (of Whitney) Any C r -manifold of dimension m (with r ≥ 1 nite or innite) is C r -homeomorphic with an analytic manifold in Euclidean space R m+1 .

There is another fundamental theorem proved by Whitney in [START_REF] Whitney | Dierentiable manifolds[END_REF], namely the Theorem 2 (expounded 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 [START_REF] Whitney | Dierentiable manifolds[END_REF], many results of the theory of real analytic functions and their approximations, are used.

Finally, we recall what he says in I.1. of [START_REF] Whitney | The imbedding of manifolds in families of analytic manifolds[END_REF] Let f 1 , ..., f n-m be dierentiable functions dened in an open subset of R n . At each point p at which all f i vanish, let the gradients ∇f 1 , ..., ∇f n-m be independent. Then the vanishing of the f i determines a dierentiable manifold M of dimension m. Any such manifold we shall say is in regular position in R n . Only certain manifolds are in regular position [...]. The purpose of the paper is to show that any m-manifold M in regular position in R n may be imbedded in a (n -m)-parameter family of homeomorphic analytic manifold; these ll out a neighborhood of M in R n .

We may extend the above denition 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 of M near p. If M is dierentiable, the two denitions agree; the ∇f i 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 subsequently to the works of Whitney (see, for instance, 1.1 and Theorem 3.2. of [START_REF] Hirsch | Dierential Topology[END_REF]; see also [START_REF] Narasimhan | Analysis on Real and Complex Manifolds[END_REF]), the Theorem 2 of [START_REF] Whitney | Dierentiable manifolds[END_REF], G. Iurato nowadays called regular value theorem, may be re-expressed and simplied through the implicit function theorem, starting from the original Whitney's proof, with a few modications.

Furthermore, the implicit function theorems are at the basis of the important notion of transversality, a modern dierential topology tool (see [START_REF] Hirsch | Dierential Topology[END_REF]) that species 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 reason: we shall use this imbedding theorem for proving a certain logical (syntactical) equivalence between the theory of dierentiable manifolds according to Weyl (that is to say, the modern one) and that deducible by the work of U. Dini. [START_REF] Arnold | Metodi Matematici della Meccanica Classica[END_REF] The implicit function theorem: a brief history The most complete work on the history of implicit function theorem, is [START_REF] Krantz | The Implicit Function Theorem[END_REF], to which we refer for a deepening of the subject; but, for the chief aspects of this history, see also [START_REF] Scarpello | A Historical Outline of the Theorem of Implicit Functions[END_REF].

The germs of the idea for the implicit function theorem, can be retraced both in the works of I. Newton, G.W. Leibniz, J. Bernoulli and L. Euler on Innitesimal Analysis, and in the works of R. Descartes on algebraic geometry.

Later on, in the context of analytic functions, J.L. Lagrange nds a theorem that may be seen as a rst version of the present-day inverse function theorem (see also, [START_REF] Kripke | Naming and Necessity[END_REF], 2, for the limitations of this theorem). We shall return on this question in regard to the inuences of the Theory of Analytic Functions and Algebraic Geometry, in the birth of the modern notion of dierentiable manifold. Subsequently, A.L. Cauchy gives a rigorous formulation of the previous semi-theories of implicit functions assuming that they are expressible as power series, a restriction removed by Dini (see [START_REF] Cajori | A History of Mathematics[END_REF], p. 431). Indeed, from here on, the implicit function theorem evolves until up the denitive Dini's generalized real-variable version (see [START_REF] Dini | Lezioni di Analisi Innitesimale, parte I a : calcolo dierenziale, and parte II a : calcolo integrale[END_REF], [START_REF] Dini | Lezioni di Analisi Innitesimale[END_REF]), related to functions of any number of real variables.

But, only with Dini's works, we have a rst complete, general and organic theory of implicit functions (at least, from the syntactic viewpoint). In 1866, Dini comes back to Pisa, where he starts his academic teaching career, as a professor of Geodesy and Advanced Analysis.

Nearly seventy, Dini settles an important work on a rigorous revision of the mathematical foundations of Analysis, with his celebrated Lezioni di Analisi Innitesimale (see [START_REF] Dini | Lezioni di Analisi Innitesimale, parte I a : calcolo dierenziale, and parte II a : calcolo integrale[END_REF], [START_REF] Dini | Lezioni di Analisi Innitesimale[END_REF]) and the Fondamenti per la teorica delle funzioni di variabili reali (see [START_REF] Dini | Fondamenti per la teorica delle funzioni di variabili reali[END_REF]). In these works, many original results and contributions of the Author are inserted: among these, the (so-called Dini's) theory of implicit functions, in [START_REF] Dini | Lezioni di Analisi Innitesimale, parte I a : calcolo dierenziale, and parte II a : calcolo integrale[END_REF], [START_REF] Dini | Lezioni di Analisi Innitesimale[END_REF].

We are interested in the Lezioni di Analisi Innitesimale. Finally, in the following Chapters (of [START_REF] Dini | Lezioni di Analisi Innitesimale, parte I a : calcolo dierenziale, and parte II a : calcolo integrale[END_REF]), Dini exposes some geometrical and analytical " applications of some theorems of the previous Chapters XIII and XV.

At the beginning of the 20 th century, Dini publishes a new revised and en- larged edition of the previous lessons [START_REF] Dini | Lezioni di Analisi Innitesimale, parte I a : calcolo dierenziale, and parte II a : calcolo integrale[END_REF], into two volumes (and each volume, into two parts). Nevertheless, as specied in the Preface to each volume, the new edition is dierent from the rst only in notations and terminology, but not in the contents: indeed, he notices that the Editorial publication of these lessons is motivated by the will to giving a historical evidence to his teaching of 1876-1877, with a lesser provisional publication.

For our purposes, we are interested in part 1 a and part 2 a of the vol. I of [START_REF] Dini | Lezioni di Analisi Innitesimale[END_REF]; the part 1

The paper of Henry Poincaré

Following E. Scholz ( [START_REF] Scholz | The concept of manifold, 1850-1950[END_REF]; see also [START_REF] Mccleary | A History of Algebraic Topology[END_REF]), in the paper [START_REF] Poincaré | Analysis Situs[END_REF] may be found another possible source of the concept of a manifold.

" Where, among other things, the Author introduces the famous Dini's numbers of Mathematical Analysis.

In fact, H. Poincaré, in 1 and 3 of [START_REF] Poincaré | Analysis Situs[END_REF], gives a constructive denition of (unilateral/bilateral # ) manifold as follows.

If x 1 , ..., x n are generic variables of R n (n ≥ 2), then he considers the following system of p equalities and q inequalities

(1)

               F 1 (x 1 , ..., x n ) = 0 .... F p (x 1 , ..., x n ) = 0 φ 1 (x 1 , ..., x n ) > 0 .... φ q (x 1 , ..., x n ) > 0,
with F i , φ j continuous and uniform functions, with continuous derivatives in such a way that J = ∂F i ∂x k ̸ = 0 in each point of the common denition domain of the F i . If p = 0, we have a domain.

The system (1) denes a manifold of dimension m = n -p, that, when $ q = 0, it is possible to prove (see 3 of [START_REF] Poincaré | Analysis Situs[END_REF]) to be equivalent to a manifold dened by a system of equations of the following type (2)

   x 1 = θ 1 (y 1 , ..., y m ) .... x n = θ n (y 1 , ..., y m ).
Again, the (syntactic) recalls to the implicit function theory, are evident. However, the main historical interest of the paper [START_REF] Poincaré | Analysis Situs[END_REF] is known to be related to the origins of Algebraic Topology, and not to the (possible) concept of dierentiable manifold (see [START_REF] Sarkaria | The topological work of Henri Poincaré[END_REF]). [START_REF]A Century of Geometry[END_REF] The work of Hermann Weyl The rst denition of a complex two-dimensional topological manifold, as we know it nowadays, is exposed in 4 of [START_REF] Weyl | Die Idee der Riemannschen Fläche[END_REF], while in 6 of [START_REF] Weyl | Die Idee der Riemannschen Fläche[END_REF], the Author gives the notion of a dierentiable structure on such a manifold type.

Weyl's analysis starts from a geometrical representation of an analytic form (according to K. Weierstrass and Riemann), and attaining to a particular structure of (Riemann) surface % , through the new topological developments achieved # The distinction between unilateral and bilateral manifolds is given in 8 of [START_REF] Poincaré | Analysis Situs[END_REF]. We refer to the bilateral case. $ Henceforth, if not otherwise specied, when we shall consider the equivalence between (1) and (2), it is understood that q = 0. In all these works, there are some rst results relative to a particular class of R n -imbedded surfaces, generated by analytic functions.

Klein also known other works on R n -imbedded surfaces as, for instance, those of A. Tonelli (Atti della R. Accademia Reale dei Lincei, ser. II, v. 2, 1875), W.K. Cliord (1876), F. Prym (1874) and P. Koebe.

As said by Weyl himself, these works of Klein seem to assume an important role in the (Weyl's) denition of a dierentiable structure on a manifold. Furthermore, Klein's Erlangen Program viewpoint seems to be at the base of Weyl's denition 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 xed the origin of R 2 ; such a group characterizes the manifold, and Weyl speaks of a surface of type Γ.

Later on, in [START_REF] Weyl | [END_REF] the author makes use of what is said in [START_REF] Weyl | Die Idee der Riemannschen Fläche[END_REF], with applications to General Relativity. & Entitled Über Riemann's Theorie der algebraischen Funktionen und ihrer Integrale, Leipzig, 1882. See also F. Klein, Neue Beiträge zur Riemannschen Funktionentheorie, Mathematische Annalen, 21 (1883). [START_REF] Th | Introduction to Dierential Topology[END_REF] The works of O. Veblen and J.H.C. Whitehead O. Veblen and J.H.C. Whitehead, in the paper [START_REF] Veblen | A Set of Axioms for Dierential Geometry[END_REF] (and, more extensively, in [START_REF] Veblen | Foundations of Dierential Geometry[END_REF]), introduce two denitions of a n-dimensional (regular) ane manifold through three groups of axioms.

In the Introduction, the Authors dene 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 p. 552 of [START_REF] Veblen | A Set of Axioms for Dierential Geometry[END_REF]). This notion is put at the foundation of the denition of regular manifold, through the further notion of pseudo-group of transformations (see [START_REF] Nomizu | Lie Groups and Dierential Geometry[END_REF], [START_REF] Kobayashi | Foundations of Dierential Geometry[END_REF] or [START_REF] Chern | Dierentiable Manifolds[END_REF]), via three groups of axioms that, on the whole, characterize the concept of manifold (see 5).

The next sections of [START_REF] Veblen | A Set of Axioms for Dierential Geometry[END_REF] are devoted to the consistency and independence of the previous groups of axioms, to some topological considerations and to a few analytic applications. Also in this case, Dini's implicit function theorems play a crucial role in the denition of manifold, since this is characterizable as an abstract entity locally dieomorphic to R n , via allowable through regular transformations local coordinate systems ' (see Examples 5. and 6., Section 1.1 of [START_REF] Hicks | Notes on Dierential Geometry[END_REF], and also the next paragraph). [START_REF] Cajori | A History of Mathematics[END_REF] The role of Dini's theory on implicit functions in dierential geometry

In this paragraph, we want to put in evidence the existence of important logical (and historical) links between the theory of implicit functions, as settled by Ulisse Dini, and the construction of the abstract theory of a (topological) ane manifolds.

It is possible to build up a theory of ane manifolds in R n , by means of the Dini's implicit function theorem and the inverse function theorem: see, for instance, [START_REF] Pini | Primo Corso di Algebra[END_REF], [START_REF] Pini | Primo Corso di Analisi Matematica[END_REF], [START_REF] Pini | Secondo Corso di Analisi Matematica, parte I e parte II[END_REF], [START_REF] Pini | Terzo Corso di Analisi Matematica, Cap. 1 e Cap. 2[END_REF] in particular [START_REF] Pini | Secondo Corso di Analisi Matematica, parte I e parte II[END_REF], parte I, Capitolo 2, 2 and parte II, Capitolo 7,3 and [15], secondo volume, Cap. V.

Implicit function theorem and inverse function theorem, characterize the local structure of any manifolds (see the parametrization technique of [START_REF] Sernesi | [END_REF],

' Besides, the Authors devote 2 of Chapter III of [START_REF] Veblen | A Set of Axioms for Dierential Geometry[END_REF], to explain the Implicit Function Theorem, as a fundamental tool that will be used in the remaining text.

G. Iurato

Cap. 5; see also Chapter 5 of [START_REF] Hermann | Dierential Geometry and the Calculus of Variations[END_REF]): to this end, see, for instance, Theorems 3 and 4, Chapter 5, of [START_REF] Singer | Lecture Notes on Elementary Topology and Geometry[END_REF].

Moreover, a manifold (in R n ) may be thought, in a certain sense, as the zero values of a given system of functions of the type (1) (equivalent to (2)), already discussed in the previous 5.

Here, we do not develop the detailed calculations connected with these claims, since we have other interests and aims. Nevertheless, it is necessary to recall the main denitions and theorems related to such a question, following, respectively, the expositions given by [START_REF] Gilardi | Analisi Due, 2 a edizione[END_REF] in Chapters VII and VIII, and by [START_REF] Prodi | Lezioni di Analisi Matematica, Parte II[END_REF] in Chapter 4.

According to the exposition given by [START_REF] Gilardi | Analisi Due, 2 a edizione[END_REF], the local character of the Dini's implicit function theorems led to important applications even having local character: among these, there are the inverse function theorems (or local invertibility theorems).

Roughly speaking, a dierentiable manifold is a subset Γ ⊆ R n that may be locally represented as a set of zeros of a many-variables function whose Jacobian matrix has maximum rank. For example, we may consider a surface Γ ⊆ R 3 given by g(x 1 , x 2 , x 3 ) = 0 with ∇ x g ̸ = 0 for each x = (x 1 , x 2 , x 3 ) ∈ Γ, or the geometric entity Γ given by the non-degenerate intersection of p (≥ 2) hyperplanes Γ 1 , ..., Γ p of R n . In this last case, if Γ i is represented by the linear function g i (x) = ∑ n j=1 a ij x j , x = (x 1 , ..., x n ), i = 1, ..., p, then Γ is represented by the zeros of the linear function g(x) = (g 1 (x), ..., g p (x)), so that, if Γ = Ker g, then dim Γ = dim Ker g = n -dim Im g = n -rank A where A = ∥a ij ∥; moreover, we suppose that det A ̸ = 0. Finally, if we wish that such a Γ have dimension m ∈ N with m < n, then we must impose that either p = n -m and rank A = n -m, or 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 have maximum rank, and since this is variable with the variation of the points of Γ, it follows that the representation of Γ as sets of zeros can only have a local nature.

Generalizing this, we have that Dini's theorem implies that a manifold may be locally thought either as non-degenerate intersection of diagrams of regular functions (denition 1) and as images of regular functions (denition 2), in both these cases the Jacobian matrices having maximum rank; furthermore, from the pointwise variability of the Jacobian matrix, it follows the possibility of introducing local coordinate systems.

Hence, a rst denition of manifold arises when we consider this latter 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 R n obtained intersecting a subspace (of R n ) with an open set (of R n ).
Precisely, we have the following

Denition 1. Let Γ ⊆ R n , m ∈ N with m < n, and k ∈ N or k = ∞. Then we say that Γ is a C k -manifold of R n with dimension m, when, for each x 0 ∈ Γ, there exists an open neighborhood I of x 0 and a function g ∈ C k (I, R n-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 denition of a manifold arises when we consider it locally identied as image of regular functions. Exactly, we have the following Denition 2. Let Γ ⊆ R n , m ∈ N with m < n, and k ∈ N or k = ∞. Then we say that Γ is locally the diagram of a m-variables C k -function when, for each x 0 = (x 10 , ..., x n0 ) ∈ Γ,

there exists two open neighborhood I ′ and I ′′ respectively of the points (x 10 , ..., x m0 ) and (x (m+1)0 , ..., x n0 ), and there is a C k -function h :

I ′ → I ′′ , such that, setting I = I ′ × I ′′ , we have Γ ∩ I = {(x 1 , ..., x n ); (x 1 , ..., x n ) ∈ I, (x m+1 , ..., x n ) = h(x 1 , ..., x m )},
unless unessential permutations of x 1 , ..., x n . In such a case, we say that Γ has a structure of a C k -manifold with dimension m.

The latter is the denition of a manifold via parametrizations, which are the result of a formalization of the geographical mapping that put into bijective correspondence a geographical chart C with a certain zone C ′ of Earth's surface; in such a way, it is evident that the tools and methods of the Geodesy have played a fundamental role in developing the intuitive idea of what a manifold can be . Indeed, from this last point of view, we reach the following

Denition 3. Let Γ ⊆ R n , m ∈ N with m < n, and k ∈ N or k = ∞. If x 0 ∈ Γ and Ω is an open set of R n , then a local m-chart of class C k of Γ at x 0 is an injective C k -function r : Ω → R n such that there exists an open neighborhood I of x 0 in such a way that Γ ∩ I = r(Ω) and rank J(r)(t) = m for any t ∈ Ω. An m-atlas of class C k of Γ is a family {r i } i∈Ξ (Ξ is a set of indices) of local m-charts of class C k such that the union of the related image sets is Γ. Finally, Γ is a C k -manifold of dimension m if it has a m-atlas of class C k . 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 t 1 , ..., t m 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 necessary in order that be possible to prove the equivalence among the above mentioned denitions of manifold.

Let n = 3, m = 2 and Γ be a plane of R 3 containing the origin of R 3 . Such a plane may be considered as the set of zeros of a suitable linear operator with rank n -m = 1; let a 1 x 1 + a 2 x 2 + a 3 x 3 = 0 such an operator with (a 1 , a 2 , a 3 =) ̸ = 0, and, for instance, let a 3 ̸ = 0. Hence we have x 3 = px 1 + qx 2 , so that such a plane is also the diagram of a linear operator from R 2 to R (that is to say, from R m to R n-m ); nally, the same plane has the following parametric equations x 1 = t 1 , x 2 = t 2 and x 3 = pt 1 + qt 2 , so that it is the image of the linear operator (t 1 , t 2 ) → (t 1 , t 2 , pt 1 + qt 2 ), operating from R 2 to R 3 (that is to say, from R m to R n ), with rank 2 (that is to say, m) whatever be p, q. From here, in the more general case in which Γ ⊆ R n be a subspace of dimension m(< n), it is possible to prove that Γ may be represented as the 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 R m to R n-m , and as the image of a certain linear operator from R m to R n with rank m.

Finally, if Γ is a manifold, then instead having to do with a linear operator (like in the previous examples, in which such a linear operator globally represents Γ), we shall have to do with regular non-linear operators providing (in general, only) a local representation of such a manifold. The fundamental tools which allow us to get such a local representation are the Dini's theorem and the inverse function theorem. Indeed, as seen in the above mentioned example concerning a plane of R 3 , it has been 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 out of its n variables, and the Till to the rst middle of the 20-th century, Geodesy was a common subject-matter of mathematical academic studies.

G. Iurato

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 the set of zeros of a unique function.

In Theorem 1.11 of Chapter VIII of [START_REF] Gilardi | Analisi Due, 2 a edizione[END_REF], it is proved the following, fundamental result: In the proof of implication 1 . ⇒ 2 . it is used the Dini's implicit function theorem, whereas in the proof of the implication 3 . ⇒ 1 . it is used the inverse function theorem. Among other things, the above theorem 1. provides a useful criterion to verify whether a certain subset of R n is a manifold or not.

Theorem 1. Let Γ ⊆ R n , m ∈ N with m < n, and k ∈ N or k = ∞.
Lastly, we observe that Dini's implicit function theorem and the inverse function theorem are strictly correlated of each other. The above exposition, drew from [START_REF] Gilardi | Analisi Due, 2 a edizione[END_REF], starts from Dini's theorems toward inverse function theorems. Instead, according to the exposition of [START_REF] Prodi | Lezioni di Analisi Matematica, Parte II[END_REF], it is possible to start from inverse function theorems toward Dini's theorems.

For instance, Chapter 4 of [START_REF] Prodi | Lezioni di Analisi Matematica, Parte II[END_REF] begins with problems concerning possible inversions of dierentiable functions between R n -type spaces, hence with problems of local inversion of functions of many variables. The rst historical methods related to this problem type concerns the class of dierentiable functions, as the dierential of a function is the rst, natural linear approximation tool for these functions, and we have a large class of results for linear applications (like the dierential map) suitable to answer to the above mentioned inversion problems. Therefore, the principle of the method consist in a generalization of what is known about linear maps to the more extended class of dierentiable maps. Because of the local nature of the dierential map, it is clear that the so obtained results from this generalization, have a local character as well.

In 2 of [START_REF] Prodi | Lezioni di Analisi Matematica, Parte II[END_REF], the author deals with some problems concerning the local inversion of maps. If Ω ⊆ R m and Λ ⊆ R n 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.

As regard the local inversion problems of maps, let us consider the following examples.

Given two open sets Ω, Λ ⊆ R n , for a C 1 -map f : Ω → Λ to be invertible in a point x ∈ Ω, it is necessary and sucient that its Jacobian matrix in x, say J(f )(x), is not singular when n = m); so, we obtain a characterization of the local invertibility of a C 1 -function in the case n = m.

The general case of arbitrary n, m ∈ N, is as follows.

Let f : Ω → R n be a C 1 -map dened on an open set Ω ⊆ R m ; if we want to study locally f in a neighborhood of a point x ∈ Ω, then we have to consider the rank of the Jacobian matrix J(f )(x) of f at x (that represents the dierential of f at x). If r x = rank J(f )(x), then r x ≤ min{n, m} for every n, m ∈ N and x ∈ Ω, the local invertibility of f in x being possible only when r x is the highest.

The C 1 -regularity hypothesis is a fundamental one. Therefore, we rst consider the case r x = min{n, m}, in such a way that it remains maximum in a neighborhood of x, since f ∈ C 1 (Ω, R n ).

Let r x = m < n. In such a case, we have the following inverse function theorem: Theorem 2. Let f : Ω → R n be a C 1 -map dened on an open set Ω ⊆ R m , and let r x = 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 regular Cartesian graph with base an open subset of R m .

For a proof, see Theorem 4.3 of [START_REF] Prodi | Lezioni di Analisi Matematica, Parte II[END_REF].

Instead, if r x = n < m, then f is locally surjective, so that it follows the problem of studying the inverse image of every point y ∈ R n that lies into a neighborhood of f (x). To this end, Dini's theorems are fundamental tools for the resolution of such a problem. For instance, if we consider the case-study m = 2 and n = r x = 1, then it holds the following Dini's implicit function theorem

Theorem 3. Let R 2 = R ′ × R ′′ with R ′ ∼ = R ′′ ∼ = R. Let f : Ω → R be a continuous function dened on an open set Ω ⊆ R 2 with
f y continuous on it; let P 0 = (x 0 , y 0 ) ∈ Ω be a point such that f (x 0 , y 0 ) = 0 and f y (x 0 , y 0 ) ̸ = 0. Then, f is locally surjective in P 0 . Moreover, there exists a neighborhood U of x 0 on R ′ and a neighborhood V of y 0 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, Γ f = {(x, y); (x, y) ∈ Ω, f (x, y) = 0} and there is a point (x, ȳ) ∈ Γ f such that (f x (x, ȳ), f y (x, ȳ)) ̸ = (0, 0), then Γ f , in a neighborhood of (x, ȳ), is of the form y = φ(x) or x = ψ(y), for certain C 1 -functions φ or ψ. This result fails into the degenerate case f x (x, ȳ) = f y (x, ȳ) = 0, that is to say, on the singular points of Γ f . In the general case, we have the following Dini's theorem Theorem 4.

Let f : Ω → R n be a C 1 -function dened on an open set Ω ⊆ R m , with n < m. If r x = rank J(f )(x) = n in a point x ∈ Ω, then f is locally surjective on x. Moreover, there exists a neighborhood V of ȳ = f (x), a neighborhood U of x and a (m -n)-dimensional open set V ′′ of R m 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 [START_REF] Prodi | Lezioni di Analisi Matematica, Parte II[END_REF].

Finally, we have functional dependence in the case in which r x < min{n, m} and, in general, it is no longer true that such a value r x remains constant in a neighborhood of x. Nevertheless, in such a case, if we suppose that such a value r x remains constant in, at least, one neighborhood of x, then we have the following Theorem 5. Let f :

Ω → R n be a C 1 -map dened on an open set Ω ⊆ R m . Given a point x ∈ Ω, we suppose that r x = rank J(f )(x) < min{n, m} is constant in a neighborhood of x.

G. Iurato

Precisely, if we wish to introduce particular subsets of R m that are locally like some ane numerical space R n , with n ≤ m, then the above mentioned theorems are fundamental tools for this problematic context. This problem has, in general, only solutions of local nature: for example, it is well-known that a circle of R 2 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 R m (with n ≤ m) is a subset Γ ⊆ R m such that every point of it, has a neighborhood homeomorphic to some open subset of R n , namely, for each x ∈ Γ, there exists an open neighborhood U of x on R n , an open set V of R n 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 holding for such a map r: among these, we mainly require that it is continuously dierentiable (or of class C k , with k ∈ N or k = ∞), and, in such a case, we speak of a dierentiable chart of class C 1 (or of class C k ).

If every point x ∈ Γ has a dierentiable chart of class C k , then we say that Γ has the structure of a n-dimensional dierentiable manifold of class C k .

We have the following ! Theorem 6. For a subset Γ ⊆ R m to be a n-dimensional dierentiable manifold of class C k , it is necessary and sucient that, for every x ∈ Γ, there exists an open neighborhood U

of x such that Γ ∩ U is a Cartesian diagram of class C k , 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 [START_REF] Prodi | Lezioni di Analisi Matematica, Parte II[END_REF].

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 C k -bijective map with continuous inverse.

At this point, a natural question is to treat the case in which a same point x ∈ Γ is into two distinct local charts, say r 1 and r 2 . Exactly, let r i : V i → 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 [START_REF] Prodi | Lezioni di Analisi Matematica, Parte II[END_REF]) that r -1 2 • r 1 and r -1 1 • r 2 are real homeomorphisms of class C k : the proof follows from the decomposition r -1 = ∆ • p.

The dierentiability properties of a manifold lies just in the dierentiability 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, these must be explicitly postulated: from here, it follows the abstract (Weyl's) denition of a dierentiable manifold. Nevertheless, the author himself (see Remark 6.7 of [START_REF] Prodi | Lezioni di Analisi Matematica, Parte II[END_REF]) says that the degree of (syntactic) logical generality of the abstract theory of dierentiable manifolds is no higher than that of the real dierentiable manifold theory, because of the works of Whitney. However, the axiomatic approach has methodological and pragmatic advantages since, for instance, we may dene such a structure over arbitrary mathematical objects (with a some predened topology).

Finally, we may dene a dierentiable manifold by means of Dini's Theorem 4. other than through the inverse function theorem (see the above Theorem 2) as done in Theorem 6., as follows Theorem 7. For a subset Γ ⊆ R m to be a n-dimensional dierentiable manifold of class C k (with n ≤ m) it is necessary and suciency that, for each x ∈ Γ, there exists an ! The Theorem 6., among other, is a useful criterion for determining whether a subset Γ

is a manifold or not. open neighborhood (of R n ) and a C k -function ψ : U → R m-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 C k -functions ψ 1 , ..., ψ m-n , dened on U and whose Jacobian matrix has rank m -n, such that " Γ =

∩ m-n i=1 Γ i , having put Γ i = {x; x ∈ U, ψ i (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 [START_REF] Prodi | Lezioni di Analisi Matematica, Parte II[END_REF].

This last theorem, assures us that a n-dimensional dierentiable manifold of class C k is locally representable as the set of zeros of a certain multivalued function.

In conclusion, from the viewpoint of the treatment given by [START_REF] Prodi | Lezioni di Analisi Matematica, Parte II[END_REF], Chapter 4, the inverse function theorem is related to the problem of local injectivity of a regular function, whereas Dini's theorem is related to the problem of local surjectivity of a regular function. From both these points of view, we may get a denition of a dierentiable manifold (respectively, like in Theorem 6. as regard the problem of local injectivity, and like in Theorem 7. as regard the problem of local surjectivity) in R n , so that it is evident the historical importance played by Dini's works on implicit function theorem regarding the foundations of modern dierential geometry.

However, it would be an historical mistake to think that Ulisse Dini had in mind such a manifold theory (although in R n ): in fact, he only settled the fundamental syntactic tools need for the next modern constructions of an abstract ane manifold, although it might be probable that some problems of R n -imbedded surfaces (as seen in the previous 6) had been (more or less unconsciously) at the base of his work # .

As we shall see later, there is no (explicit) semantic link between Dini's work on implicit functions and the theory of manifolds; there holds, instead, only strong links of syntactic nature (that, despite all, have a proper historical importance, as we shall see in the next paragraphes, as regard the notion of syntactic rigid designator).

We have already mentioned the possible role played by Algebraic Geometry (see, for instance, [START_REF] Krantz | The Implicit Function Theorem[END_REF]) and Complex Analysis in regard to the mind-setting of modern concept of a dierentiable manifold. We wish to outline some a few words about these last aspects.

The work of H. Weyl, as seen in 6, is 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 [START_REF] Pincherle | Gli Elementi della Teoria delle Funzioni Analitiche[END_REF], 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 resumes Lagrange's work on inverse function theorem in " The maximum rank condition assures that such an intersection is non-degenerate.

# This last consideration has to be considered valid only in an semi-intuitive context, or like a kind of insight, with respect to the Dini's work on implicit function theorems.

G. Iurato

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 [START_REF] Mccleary | A History of Algebraic Topology[END_REF]) some not negligible inuences of the 19-th century Algebraic Geometry, in the developments of some aspects of the Theory of Dierentiable 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 nd some partial (syntactic) conrmations by the so-called Nash-Tognoli imbedding theorems of Algebraic Geometry (see [START_REF] Bochnak | Real Algebraic Geometry[END_REF], Chapter 14), a sort of algebraic geometry analogous of the Whitney's theorems, proving that any compact smooth manifold is dieomorphics to a well-dened nonsingular real algebraic manifold.

Hence, the works of 19-th century algebraic geometers, it should be also considered have had some inuences on the possible sources of the modern theory of dierentiable manifolds. Nevertheless, just limited to the aim of the present paper, the comparison with the Nash-Tognoli theorems mentioned above, seems do not have had a great historical importance within the question related to the born of modern theory of dierentiable manifolds, dierently by the case of Dini's and Whitney's works (see next 11).

At this point, it is necessary formally 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 precise better, in a rigorous manner, the previous historical remarks, as well as to establish rigorously the historical relevance of the possible syntactic links among theories, in our case, between theory of functions and dierential geometry.

Some notions of Model Theory

According to [START_REF] Chang | Model Theory[END_REF], roughly speaking, Model theory is Universal Algebra plus Logic. In this section, we recall some notions of Model Theory, need for the following. Our main references are [START_REF] Cohn | Universal Algebra[END_REF], [START_REF] Dalla | Modelli sintattici e semantici delle teorie elementari[END_REF], [START_REF] Dalla Chiara | [END_REF], [START_REF] Lolli | Lezioni di Logica Matematica[END_REF], [START_REF] Freguglia | L'algebra della logica[END_REF], [START_REF] Mangione | Storia della logica[END_REF].

Syntactic and semantic models

Every scientic axiomatic theory has both a syntactic component and a semantic one, and, often, these two components are mixed of each other into a concrete (that is, non-axiomatic or intuitive) scientic theory.

$ This is an historical remark which should be thought back in regard to André Weil work on algebraic manifolds.

Therefore, in general, the formalization process of a scientic theory is an axiomatization process working out over an initial structure of intuitive theory, towards 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 theories, and the non-existence of a non-contradictory semantically closed theories. Hence, there exist limitative theorems on the syntactic and semantic capacities of an axiomatic theory.

Nevertheless, from these limitations, it also follows the reciprocal inseparability of the syntactic and semantic components.

Syntactic models

The formalization process leads to the so-called notion of formal system. It is composed both by syntactic components and semantic ones. 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, T e, W r, E⟩ and deductive scheme D = ⟨Ax, Ru⟩, where • there exists disjoint sets Co, Qu, F u, P r, V a, Au, in such a way that Al = ∪ {Co, Qu, F u, P r, 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 quantiers, F u the set of functors, P r 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 P r∪F u the set of descriptive constants or vocabulary);

• W r is the set of words, T e (⊆ W r) is the set of atomic terms, E (⊆ W r)

is the set of atomic expressions (with T e∪E the set of well-formed words, and P rop the set of propositions dened as a subset of E whose elements have no free variables);

• Ax (⊆ E) is the set of (logic and specic) axioms, whereas Ru is the set of logic deduction rules (with respect to a given Logic).

Note. In this section, from now on, we only speak of a formal system (or theory), without species the term syntactic.

L determines the set of (explicit and implicit) denitions (say De) of F, whereas D determines the set of proofs (say P f ), and the set of theorems (say T h), of F.

G. Iurato

Therefore, a formal system (a theory) is a tuple of the type (1) F = ⟨⟨L, D⟩⟩ = ⟨⟨ ⟨Al, T e, W r, E, De⟩, ⟨Ax, Ru, P f, T h⟩ ⟩⟩.

We may think to 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 specic axioms is nite, then F is said to be nitely axiomatizable.

If L is a formal [not formal (or intuitive)] language, then we say that F is a formal [not formal] theory.

We should make some clarications about the elements of F u and P r. F u is the class of all n-functor F u n = {f n i } 0≤i<j with 0 ≤ j ≤ ω and 0 ≤ n < ω, where F u 0 is the set of individual constants, with F u n = ∅ if j = 0. P r is the class of all n-predicate P r n = {P n i } 0≤i<j with 0 ≤ j ≤ ω and 0 < n < ω; P r 2 contains, at least, the element P 2 0 said to be the identity predicate, with P r n = ∅ if j = 0.

Let F 1 , F 2 be two theories of the type (1); we say that • F 2 is a predicative linguistic extension of F 1 when P r 1 ⊆ P r 2 ;

• F 2 is a functorial linguistic extension of F 1 when F u 1 ⊆ F u 2 ;

• F 2 is a linguistic extension of F 1 (and we write L 1 ⊆ L 2 ) when F 2 is a predicative and functorial linguistic extension of F 1 ;

• F 2 is a deductive extension of F 1 when Ax 1 ⊆ T h 2 ;

• F 2 is a theoretical extension of F 1 (or that F 1 is a sub-theory of F 2 ) when F 2 is a deductive and linguistic extension of F 1 ; in such a case, we write F 1 F 2 , and we say that is the theoretical inclusion relation;

• a theoretical extension F 2 of F 1 is a linguistically invariant extension when L 1 = L 2 , that is to say, when F 2 is an improper linguistic extension of F 1 ;

• a theoretical extension F 2 of F 1 is an inessential % extension when T h 1 = T h 2 ∩ E 1 . % See next Remark 1.
If F 1 F 2 and F 2 F 1 , then we say that F 1 is equivalent to F 2 , and we write F 1 ≈ F 2 ; we say that ≈ is the theoretical equivalence relation.

We refer to [START_REF] Dalla | Modelli sintattici e semantici delle teorie elementari[END_REF], Capitolo 1, 3, Denizione 7, for the denition of the elements of De (the set of predicative and functorial denitions of a theory F).

We say that F 2 is a simple denitional extension of F 1 if and only if there exists a predicative [functorial] denition δ

P n i [δ f n i ] in F 2 , such that 1. P n i / ∈ P r 1 [f n i / ∈ F u 1 ]; 2. P r 2 = P r 1 ∪ {P n i }, F u 2 = F u 1 [F u 2 = F u 1 ∪ {f n i }, P r 2 = P r 1 ]; 3. Ax 2 = Ax 1 ∪ {δ P n i } [Ax 2 = Ax 1 ∪ {δ f n i }].
We say that F 2 is a denitional extension of F 1 when there exists a sequence of theories F k 1 , ..., F kp (1 < p ≤ ω), such that:

1. F 1 = F k 1 and F 2 = F kp ; 2. for each 1 ≤ i ≤ ω, F i+1 is a simple denitional extension of F i .
In other words,

F 1 = F k 1 → ... → F k i → ... → F kp = F 2 1 < i < p,
is a chain of simple denitional extensions.

Every simple denitional extension is a (proper) deductive and linguistic extension as well. Moreover, we have the following

Theorem 1. If F 2 is a [simple] denitional extension of F 1 , then F 2 is an inessential extension of F 1 .
For a proof, see [START_REF] Dalla | Modelli sintattici e semantici delle teorie elementari[END_REF], Capitolo 1, 3, Teoremi 5, 6.

Remark

G. Iurato

We now introduce the notions of theoretical homomorphisms (for details, see [START_REF] Dalla | Modelli sintattici e semantici delle teorie elementari[END_REF], Capitolo 2).

Let F 1 , F 2 be two theories of the type (1). A theoretical representation of F 1 into F 2 is a map ρ : W r 1 → W r 2 ; hence, we write ρ :

F 1 → F 2 .
Remembering that E, T h ⊆ W r, we can say that a theoretical representation ρ :

F 1 → F 2 is • an expressive homomorphism if ρ(E 1 ) ⊆ E 2 ;
• a theorematical homomorphism if ρ is an expressive homomorphism and ρ(T h 1 ) ⊆ T h 2 ;

• a deductive homomorphism if ρ(P r 1 ) ⊆ P r 2 .

A deductive homomorphism is a theorematical homomorphism as well.

This last classication denes the so-called class of theoretical homomorphisms.

A theoretical representation ρ :

F 1 → F 2 is said to be • a version of F 1 into F 2 , if
there exists a map (called the base of this version) ψ : F u 1 ∪ P r 1 → T h 2 ∪ E 2 , satisfying a certain set of compatibility properties (see [START_REF] Dalla | Modelli sintattici e semantici delle teorie elementari[END_REF], Cap. 2, 1, Def. 3, a));

• a quasi-relativization of F 1 into F 2 , if there exists an expression α(v) ∈ E 2 (v is a free variable) and a map ψ : F u 1 ∪ P r 1 → T h 2 ∪ E 2 , verifying a set of compatibility properties (see [START_REF] Dalla | Modelli sintattici e semantici delle teorie elementari[END_REF], Cap. 2, 1, Def. 3, b)); we say B ρ =< α(v), ψ > to be the base of this quasi-relativization;

• a relativization of F 1 into F 2 , if there exists a quasi-relativization ρ ′ of F 1 into F 2 , with base B ρ ′ =< α(v), ψ >, in such a way that ρ(β) ⇒ ρ ′ (β) for each β ∈ E 1 , and ρ(β) = ρ ′ (β) for each β ∈ P r 1 .
Versions, quasi-relativizations and relativizations, are expressive homomorphisms.

A theorematical homomorphism ρ of F 1 into F 2 is said to be

• a translation, if ρ(¬β) = ¬ρ(β) for each β ∈ P r 1 ; • an interpretation, if ρ is a version of F 1 into F 2 ;
• a relative interpretation, if ρ is a relativization of F 1 into F 2 ;

• an isomorphism, if ρ : W r 1 → W r 2 is a bijection such that ρ(Ax 1 ) = Ax 2 , and there exists a map ψ :

Al 1 → Al 2 , commuting with ρ, such that ψ(F u 1 ) ⊆ F u 2 , ψ(P r 1 ) ⊆ P r 2 , ψ(V a 1 ) ⊆ V a 2 , ψ(Au 1 ) ⊆ Au 2 , ψ(Co 1 ∪ Qu 1 ) ⊆ Co 2 ∪ Qu 2 .
Therefore, we say that F 1 is translatable, interpretable, and relatively interpretable into F 2 if, respectively, there exists a translation, an interpretation, and a relative interpretation of F 1 into F 2 . We say that F 1 is isomorphic to F 2 if there exists an isomorphism between F 1 and F 2 , and we write

F 1 ∼ F 2 .
An isomorphism is a deductive homomorphism as well, but not conversely, in general (see [START_REF] Dalla | Modelli sintattici e semantici delle teorie elementari[END_REF], Capitolo 2, 1, Teorema 5).

It is possible to prove (see [START_REF] Dalla | Modelli sintattici e semantici delle teorie elementari[END_REF], Cap. 2, 2, Teorema 1) the following Theorem 2. If F 1 F 2 , then F 1 is translatable, relatively interpretable and interpretable into F 2 ; moreover, if it is also L 1 = L 2 and F 1 is isomorphic to F 2 , then F 1 ≈ F 2 , 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 [START_REF] Dalla | Modelli sintattici e semantici delle teorie elementari[END_REF], 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 eective theoretical homomorphism [isomorphism]. If F 1 is relatively interpretable into F 2 , then we say that F 1 has a syntactic model into F 2 , and we write F 1 F 2 .

It is important the following

Theorem 3. If F 1 is [relatively] interpretable in F 2 , then F 2 has a deni- tional extension F ′ 2 containing a sub-theory F ′ 1 isomorphic to F 1 .
For a proof, see [START_REF] Dalla | Modelli sintattici e semantici delle teorie elementari[END_REF], Capitolo 2, 2, Teoremi 10, 11.

Among the theoretical homomorphisms dened 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, the set of truth values of its statements (propositions, theorems, expressions, and so on); its choice is independent by the syntactic structure of F. Therefore, the interpretability of F 1 into F 2 , means that it is always possible to give an interpretation of the concepts of F 1 in the terms of the concepts of F 2 , in such a way that what F 1 says to be true with respect to its universe U 1 , is also trueby means of such an interpretationwith respect to the universe U 2 of F 2 .

We shall take again this argument in the semantic context.

Instead, the relative interpretation of F 1 into F 2 , means that it is always possible to give an interpretation of the concepts of F 1 in terms of the concepts of F 2 , but in such a way that what F 1 says to be true with respect to its universe U 1 , is also true with respect to a particular sub-universe U α of U 2 , determined by the relativization condition α(v) (of the base B ρ =< α(v), ψ > of the given representation ρ :

F 1 → F 2 ).
At this point, it is possible to apply these considerations to the historiographical context, as follows. Indeed, a central problem in the Historiography of Exact Sciences, is the determination of the possible relations among dierent theories, as, for instance, those that hold among a concrete (or intuitive) theory and its formalizations .

A rst rational (or quantitative) comparison of this last type, it is possible, for instance, when one takes 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 F 1 into a theory F 2 , then we can say that F 1 is, in a certain sense, included into F 2 .

Analogously, the possible determination of a syntactic model (and the possible theoretical connections that it may provide) gives 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 o.

Nevertheless, for methodological motivations, we should consider such syntactic comparison criteria, with the suitable cautions. However, at this point, we may do a simple historical application of what has been said above. If F Dini 1 is the theory of dierentiable manifolds in the Dini's sense, while F W eyl 2 is the theory of dierentiable manifolds in the Weyl's sense (that is, the modern one), then it is obvious that F Dini

1 is interpretable into F W eyl 2 .
On the other hand, by means of Whitney's theorems, we can say too that F W eyl

2 is interpretable into F Dini 1 . By Theorem 3, it follows that F Dini 1 ∼ F ′ W eyl 2 FWeyl 2 and F W eyl 2 ∼ F ′ Dini 1 FDini 1
, for certain denitional extensions Fi of F i i = 1, 2. Moreover, we may suppose the equality between the languages of F Dini , so that, by Theorem 2., we have

F Dini 1 ≈ F ′ W eyl 2 and F W eyl 2 ≈ F ′ Dini 1
. From here, it does not follow the (syntactic

) equivalence F Dini 1 ≈ F W eyl 2
, but a minor equivalence, as follows. If we take

Although, it would be more correct to consider such a type of logical comparison only among theories having almost the same syntactic degree of formalization.

In fact, even by Whitney's works, it is no restrictive to think any abstract smooth n-manifold as a closed subset of some R N (with N = N (n) > n), locally representable (according to Dini) as an intersection of the diagrams of a system of dierentiable functions dened on some common open subset of R n , with values into R s , s = N -n.

into account the notion of deductive equivalence, then we may say that F 1 and F 2 are deductively equivalents, and we write F 1 ≃ F 2 , when F 2 is a deductive extension of F 1 , and vice versa. Therefore, if we take into account what has been said in Remark 1, about the inessentiality of the denitional extensions, then we may set F i ≃ Fi i = 1, 2. Thus, the relations

F Dini 1 FWeyl 2 ≃ F W eyl 2 and F W eyl 2 FDini 1 ≃ F Dini 1 , implies the following deductive equivalence F Dini 1 ≃ F W eyl 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 F Dini ), is centered around the (logic) immersion theorems (see [START_REF] Lolli | Lezioni di Logica Matematica[END_REF], Cap. 2, 2.3), through which we have F Dini

1 ∼ F W eyl 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 ri is the relation of relative interpretability, then (T, * ri ) is a pre-ordered set, putting [F 1 ] * ri [F 2 ] if and only if F 1 ri F 2 (this is a well-posed denition).
We call rational power of a theory F, its equivalence class [F] ∈ (T, * ri ): intuitively, [F] is the class of all theories F ′ containing a sub-theory 'which says the same things said' by F, whereas, on its turn, F contains a sub-theory 'which says the same things said' by F ′ .

Analogously, if

eri denotes the eective relative interpretation relation, we have that (T, * eri ) is a pre-ordered set; [F] ∈ (T, * eri ) is said to be the rational content of F, and, intuitively, it 'contains everything said by F and, also, everything said' by the weaker theories of F.

Since it is possible to prove the existence of a (syntactic) isomorphism between (T, * ri ) and (T, * 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 axiological scale of importance among theories; further, it may turn out to be also useful in certain historical classications of the 'importance' of a theory identied by its rational content. 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, in the possible historic-critical comparison between theories.

Semantic models

In this section, we 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 dierent languages.

Nevertheless, the historical comparison is often oriented towards a language comparison, and the syntax shows its own limits ! 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 [START_REF] Dalla | Modelli sintattici e semantici delle teorie elementari[END_REF], Capitolo 4, it is possible to nd a purely abstract formalization of the Semantics; instead, we are interested in a more extended setting, suitable to historical questions. To this end, we refer to [START_REF] Cohn | Universal Algebra[END_REF], [START_REF] Dalla Chiara | [END_REF], [START_REF] Freguglia | L'algebra della logica[END_REF] and [START_REF] Mangione | Storia della logica[END_REF].

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, dened as follows. If C = F u ∪ P r is the set of descriptive constants " 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 dene (extensively) D as follows: for each formula

F of F, it is 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 E v (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 E v (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 lter 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 may also write M = ⟨⟨F, R⟩⟩ = ⟨⟨F, (U, v)⟩⟩, instead of M = ⟨⟨F, D⟩⟩.

! There are further problematic limits of the syntactic context: for instance, there exists niteness problems, connected with the attempts to avoid the impossible identication between mathematical truth and demonstrability, that led to the failure of the Hilbert's formalistic program.

" Descriptive constants (or atomic propositions) and specic axioms, characterize (syntactically) a formal theory.

We can now introduce the fundamental notion of Lindenbaum-Tarski algebra.

If M is a deductive theory (according to Tarski), we dene the following pre-order

ϕ ≤ M ψ def. ⇔ F ⊢ M ϕ ⇒ ψ.
Its symmetrization gives the following equivalence relation

# ϕ ≡ M ψ def. ⇔ ( M ⊢ M ϕ ⇒ ψ ) ∧ ( M ⊢ M ψ ⇒ ϕ ) ,
and it is immediate to prove that A M = 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)].
Frequently, the above Lindenbaum-Tarski construction is made on F instead of the whole M, so that we obtain the following (syntactic) Lindenbaum-Tarski algebra A F = F/ ≡ F . It is possible to prove that A M is a free algebra generated by C.

By means of the Lindenbaum-Tarski algebra, it is possible to set up 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 dened in [START_REF] Casari | La Matematica della Verità (strumenti matematici della semantica logica)[END_REF], Appendice B, B.1.), and let A F(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 correspond to a well-determined homomorphism (of Boolean algebras) from A M to F, dened 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 the set of specic axioms (see [START_REF] Casari | La Matematica della Verità (strumenti matematici della semantica logica)[END_REF], l.c.), then T (L, M ) is a sub-theory of F, while T (L, M )/ ≡ T (L,M ) is a lter of A M . Thus, a Theory has a unique lter (on A M ) as an algebraic counterpart [precisely, a maximal lter for a (syntactically) complete Theory]: it is generated by the equivalence classes of the specic axioms M of T (L, M ).

On the other hand, following [START_REF] Lolli | Lezioni di Logica Matematica[END_REF], if it is given a language L, a consistent set T of L-sentences is, roughly speaking, a Theory (see the above T (L, M )), while a model of T (or a T -model) is a L-structure (see [START_REF] Lolli | Lezioni di Logica Matematica[END_REF], Cap. 1, 1.1, 1.2 and 1.3), say S, such that every sentence in T is true into S. We say that a # There exists other equivalence relations leading to the so-called (Halmos) polyadic al- gebras, or to the so-called (Tarski) cylindric algebras. For simplicity, we restrict ourself to consider Lindenbaum-Tarski algebras.

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 [START_REF] Sernesi | [END_REF]).

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 [START_REF] Cohn | Universal Algebra[END_REF], Chapter 5,[START_REF][END_REF]. On the other hand, if we consider the Lindenbaum-Tarski algebra associated (to the formal system corresponding) to the deductive closure of T , say A T , then the Galois connection, ξ, induces a Galois connection between A M and the space of models of T , say M T . Hence, we may write A T ξ ∼ = M T . The (logical) closure operators denes (following Kuratowski) a well-determined topology on the model space M T , and the corresponding topological space is called the Boole space of T (see [START_REF] Cohn | Universal Algebra[END_REF], Chapter 5, 6); it is a Stone space.

If we want to apply these last considerations to the case related to the History of Dierentiable Manifolds, then we may deduce, via Whitney's theorems $ , the existence of a Galois connection between M F Dini At this point, it is necessary to specify some of the above expounded 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).
$ This correspondence is bijective since, by a fundamental theorem due to H. Grauert (see [START_REF] Grauert | On Levi's problem and the imbedding of real analytical manifolds[END_REF], and [START_REF] Narasimhan | Analysis on Real and Complex Manifolds[END_REF]), any abstract manifold corresponds to a unique real manifold, via Whitney's imbedding. Hence, it follows the existence of a unique (Whitney) imbedded structure, for each assigned abstract manifold.

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 algebra B.

A semantic meaning may be dened 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 n i ) ⊆ U n ∀n ∈ ω, ∀P n i ∈ P r. 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 ∈ P r is true with respect to the interpretation R = (U, v) if and only if v(ψ) ⊆ U n , 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 fullled (see [START_REF] Dalla Chiara | [END_REF], 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, that given by Kripke's Semantics (of the general class of Modal Logics).

The main limit of Tarski Semantics is due to the fact that it comprehends only two possible cases: indeed, such a Semantics considers either one universe of discourse U or all possible universes of discourse. Instead, S. Kripke (see [START_REF] Kripke | Naming and Necessity[END_REF]) 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, 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 Logics (Temporal, Epistemic, etc.) and the intensional theories of meaning (as, for example, the Carnap's one). The Modal Logics 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 M Kripke = ⟨⟨F, (U i , v i ) i∈J ⟩⟩, where R i = (U i , v i ), i ∈ J, is the Kripke's set of realizations of M Kripke (if J is a singleton, or an innite set, then we obtain a Tarskian deductive system). Mutatis mutandis, what has been said above about the Lindenbaum-Tarski methods, may be applied to M Kripke as well. [START_REF] Chang | Model Theory[END_REF] The role of the principle of virtual works in dierential geometry

In this section, we wish briey to recall the important role played by the principle of virtual works of Analytical Mechanics. [START_REF] Maurin | The Riemann Legacy[END_REF] is the main reference for the History of Mechanics up to 1920.

This principle has played a fundamental role in Lagrange's work (see [START_REF] Lagrange | Mécanique Analitique[END_REF]):

in fact, it is at the base of the analytical mechanics arguments % . There are many modern texts on Analytical Mechanics whose rst chapters, devoted to the formulations of the celebrated Lagrange's equations, just begin with the exposition of D'Alembert-Lagrange principle of virtual works. For instance, a modern historical exposition very similar to the original Lagrange's formulation, is given by [START_REF] Agostinelli | Meccanica Analitica, 2 voll[END_REF], vol. I, Capitolo I: here, 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 Dierential and Riemannian Geometry. For a brief, but rigorous, exposition of these arguments, see [START_REF] Pastori | Visioni geometriche in meccanica analitica[END_REF], [START_REF] Pastori | Vincoli e riferimenti mobili in meccanica analitica[END_REF] and the more complete treatment of [START_REF] Finzi | Calcolo Tensoriale e Applicazioni[END_REF].

We briey recall the main results of [START_REF] Agostinelli | Meccanica Analitica, 2 voll[END_REF], vol. I, Capitolo I.

In 2 of Chapter I, it is expounded the so-called D'Alembert principle

m i ⃗ a i = ⃗ F i + ⃗ R i i = 1, .
.., N , for a system of N point particles, each of which has mass m i , acceleration ⃗ a i , and subjected to both the total active forces ⃗ F i and the total constraint forces ⃗ R i . This principle reduces every dynamical problem to a statical one; it provides a well-dened equilibrium condition. For a smooth & systems with holonomic constraints, this last equilibrium condition being equivalent to the so-called principle of virtual works, whose analytical formulation is based on the invertibility of the virtual displacements δP i of the point particle P i , and it is ∑ N i=1 ( ⃗ F i -m i ⃗ a i ) × δP i = 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 equations ' , are deduced from this symbolic equation.

In the following sections, many possible formal expressions of Lagrange's equations are deduced: the δ-d Lagrange's formalism is the main analytical % In passing, this principle has also been used on some questions related to the constrained motion of a quantum particle (see, for instance, [START_REF] Jensen | Quantum Mechanics with Constraints[END_REF]).

& Here, the term 'smooth' means constraints without friction.

' There exist various forms of Lagrange's equations exposed in [START_REF] Agostinelli | Meccanica Analitica, 2 voll[END_REF], vol. I, Capitolo I. In particular, in the subsection 2 of 4., the authors expose a rst form of Lagrange's equations using the Lagrange's multipliers rule, that plays a fundamental role in the extremum theory with side conditions (so that, again, we go back to Dini's works).

G. Iurato

tool for deducing many formal dynamical properties of a holonomic smooth constrained systems, given in a (Hertz) form similar to (1) of our 5, with q = 0 (equivalent to [START_REF] Amerio | Analisi Matematica[END_REF], where the y i are replaced by the lagrangian coordinates q i );

these properties have both metrical nature (assuming it is assigned a certain metric given by the kinetic energy, according to Jacobi) and ane nature, and it is much probable that they have played a fundamental role in the subsequent development of Dierential Geometry.

For instance, to this purpose, it is important to remember that the rst dierential topology tool explaining the basic dierential geometry local concepts, is that of tangent space in a point of a manifold: historically, the rst denitions of tangent space have been the result of a generalization of the main basic concepts and methods of Analytical Mechanics concerning the constrained motion of a particle over a smooth holonomic system (see the so-called physicist's denition and the geometer's denition, of a tangent space, equivalent between them and to another, called the algebraic denition given in Chap. 2 of [START_REF] Th | Introduction to Dierential Topology[END_REF]; see also [START_REF] Maurin | The Riemann Legacy[END_REF]).

However, for our purposes, we follow the modern exposition given by V.I.

Arnold in [START_REF] Arnold | Metodi Matematici della Meccanica Classica[END_REF].

In Chapter IV, he gives a rst modern denition of smooth holonomic con- The holonomy of such constraints has physical motivations, and, therefore, it is evident the mathematical physics sources of the concept of a smooth holonomic constraint, hence of the dierentiable manifold: indeed, the principle of virtual works provides the local characterization of a manifold, locally like to R n , likewise to Dini's implicit function theorems.

! For a deduction of Lagrange's equation from this Leontovic's point of view, see [START_REF] Fasano | Meccanica Analitica[END_REF], and reference therein.

On the other hand, it is well-known that the sources of Lagrange's inverse function theorem (already mentioned), should be traced in the Lagrange's works on some static problems, where, among other, he introduced the nowadays well-known Lagrange's multipliers (see [START_REF] Marco | Analisi Due[END_REF], Capitolo V., 5., footnote 5

of page 382). The latter, in turn, results to be related to the principle of virtual works as well, hence to the local structure of a dierentiable manifold (via the connection of the Lagrange's multipliers with the inverse function theorem).

To this end, we briey recall the problem.

Almost 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 by a set of zeros of functions, then, by means of Lagrange's multipliers, an extremum problem on Γ is reduced to a local extremum problem related to the functions (locally) describing such a manifold Γ. To this end, we remember that, if Γ is a manifold of R n , f : Γ → R, x 0 ∈ Γ and r is a chart of Γ (see 8) containing x 0 , then we say that x 0 is a relative maximum/minimum extremum for f if and only if r -1 (x 0 ) is a relative maximum/minimum extremum for f • r.

Since, in general, it is a dicult 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 x 0 . Hence, we have the following Lagrange's Multipliers Theorem. Let Γ ⊆ R n be a manifold of dimension m(< n), and x 0 ∈ Γ; let I be an open neighborhood of x 0 and g(x 0 ) = 0 the local equation of Γ in x 0 with g ∈ C 1 (I) and rank J(g)(x) = n -m for each x ∈ Γ ∩ I. Let f : Γ → R with f ∈ C 1 (I). If x 0 is a relative extremum for f | Γ∩I then ∇f (x 0 ) ∈ N x0 Γ (normal space to Γ at x 0 ), that is, there exist n -m real numbers λ 1 , ..., λ n-m such that ∇f (x 0 ) = ∑ n-m i=1 λ i ∇g i (x 0 ) with λ 1 , ..., λ n-m uniquely determined by x 0 .

If, for each x 0 ∈ Γ, we have ∇f (x 0 ) ∈ N x0 Γ, then we say that x 0 is a stationary (or a critical) point of f ; such points are in bijective correspondence with the solutions of the system of equations g(x) = 0 and ∇f (x) =

∑ n-m i=1 λ i ∇g i (x), whose solutions are of the type (x 1 , ..., x n , λ 1 , ..., λ n-m ) ∈ R n × R n-m , with λ 1 , ..., λ n-m said to be Lagrange's multipliers.

For instance, in the case n = 3, if R 3 is a model of the physical space and Γ(⊆ R 3 ) represents a bilateral smooth holonomic constraint for the material point x 0 subjected to the force eld ∇f , then the above Theorem says that the force acting over a critical point x 0 is orthogonal to the constraint Γ, whereas the values of the Lagrange's multipliers are connected with the intensity of the constraint reactions. From here, it arises evident 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 dicult task to do sure historical claims about these suppositions, but, at most, having only probability nature.

The only certainty concerns the syntactic comparison among the previous arguments, whereas their possible semantic comparison may be conducted wi-G. Iurato thin the Kripkian context (or, more generally, into the Modal Logics context), if we choose the suitable Kripke's set of realizations upon which to interpret 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 conguration space of a moving particle, subsequently formalized by both D'Alembert-Lagrange principle and 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 Classical Analytical Mechanics). The just mentioned historical connections are, however, rather probably (see next 11).

We conclude this section, with an unusual remark on the work of Tullio Levi-Civita on his parallel displacement ! , from which it is possible to infer another prove of the importance played by the principle of virtual works in the foundations of Dierential Geometry.

In fact, it is almost always armed (in the current relative literature) that Levi-Civita parallel displacement was motivated by the attempt of giving a geometrical interpretation to the so-called covariant derivative of Absolute Dierential Calculus. Indeed, if one carefully reads the paper [START_REF] Levi-Civita | Nozione di parallelismo in una varietà qualunque e conseguente specicazione geometrica della curvatura riemanniana[END_REF], it is clear that the historical verity is quite dierent. Levi-Civita was motivated by the attempts to simplify the computation of the curvature of a manifold through the Riemann symbols, as he says in the Introduction of his paper.

Then, once introduced a generic metric structure on a manifold dened 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 as a smooth holonomic system subjected to (invertible) virtual displacements. From it, the author deduces an equivalent equation, the (8) of the same paragraph, hence another equivalent form, the (I a ) of 3, from which he deduces the analytical conditions characterizing his celebrated notion of parallel displacement. In the remaining paragraphes, the author does not make any explicit mention to the covariant derivative, except a secondary application relative to Ricci's rotation coecients (see 13 of [START_REF] Levi-Civita | Nozione di parallelismo in una varietà qualunque e conseguente specicazione geometrica della curvatura riemanniana[END_REF]).

Conclusions

Albeit it is surely erroneous to think that the concept of a dierentiable manifold (as we know it nowadays) is already present in the works of Dini on ! As regard the historical importance of Levi-Civita parallel displacement in Physics (as, for example, in Gauge Theories), see [START_REF]A Century of Geometry[END_REF]. implicit function, or in the foundations of Analytical Mechanics, nevertheless we may state, out of doubt, what follows.

Any mathematical theory does not born from nothing, but, instead, it starts from some previous ones ! : precisely, it begins from those having, at least, a some syntactic link with it !! . Hence, from here, it is evident the importance of the notion of syntactic model in searching such syntactic links, in such a way that it is possible to determine a chain of syntactic models which may remember the Kripkian historical chains of 9.2.

So. we have exposed a case-study of such an historiographical methodology, precisely, that relative to the origins of the concept of a dierentiable manifold.

Beyond such a rst comparison among theories, further researches are possible concerning the semantic context, for instance, in 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 lters algebraically correspond to theories, then it is possible to compare two theories comparing their lters, and so on, to be then historically interpreted.

Analogously, it may be compared the corresponding (syntactic or semantic) Lindenbaum-Tarski algebras between them. In these last two cases, the resulting chains of lters, or algebras, may be considered as an algebraic formalization of the so-called historical chains of Historiography (already mentioned when we have discussed on Kripke's work).

We have therefore sketched such a line of historiographical methodology in relation to a particular case related to the History of Dierential Geometry. Thus, in this context, it is very likely that Dini's works on implicit functions and the bases of Analytical Mechanics, have played a considerable role in the formulation of the modern theory of Dierentiable Manifolds, both from syntactic and semantic viewpoint. ! On the other hand, this statement nds a further conrmation on a certain, not casual epistemological evolution of a mathematical structure along the historical time (see [START_REF] Piaget | Le Structuralisme[END_REF]).

As a concrete example of that, we recall the work of G. Peano on the axiomatization of Natural Numbers, that started from the previous works of R. Dedekind on the same argument (such a question, besides, it is treated, from the Modal Logic viewpoint, in the January 22, 1970 lesson of S. Kripke see [START_REF] Kripke | Naming and Necessity[END_REF]). !! The further, not trivial question concerning the awareness, or not, of the existence of these theories on behalf of the author under historical examination, may be analyzed from suitable philosophical viewpoints. However, certain contemporaneously but independent (between them) mathematical discoveries/construction (like those mentioned above in the footnote 12 ) prove that the previous syntactical capacity of a certain theoretical context reaches a certain degree in allowing a subsequent discovery/construction.

G. Iurato

Furthermore, from what has been said so far, it is clear too that the geometric structure, called dierentiable manifold, is a syntactic !" rigid designator in the sense of the new reference theory of S. Kripke and H. Putnam; in fact, the same syntactic structure (or mathematical entity), i.e., that of dierentiable manifold, has been identied in, at least, two dierent semantical contexts (or in two discourse worlds): that of the Theory of Implicit Functions and that of Lagrange's Analytical Mechanics. Moreover, following H. Putnam, we could say that it may exists a collective (not individual) historical chain (see above), so external to every individual, or else a series of reference transmitted through the 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 dierentiable manifold. Hence, the quantitative methods used in this paper for such a particular case-study, may turn out to be of some usefulness also in regard to the nature of other mathematical entities (in the context of Mathematical Philosophy).

!" Examples of semantic rigid designators are the physical entities (as, for instance, an elementary particle, a physical eld, etc.).

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 former was a physicist and a mathematician, deeply inuenced by the works of J.L. Lagrange ! , who taught Geodesy at the University of Pisa when Dini was a student. The latter was professor of Mathematical Physics at the University of Pisa and supervisor of the Dini's thesis. In 1864, Dini publishes a paper on an argument of his thesis suggested by Betti. This rst paper is followed by many other works on dierential geometry and geodesy. In that period, Dini is in a scientic correspondence with E. Beltrami who took Geodesy chair left by the late Mossotti. At the same time, Dini is into touch with B. Riemann, at the time visiting professor at Pisa under Betti's interests. In 1865, Dini spends one year of specialization in Paris under the supervision of J. Bertrand, where he continues his thesis arguments, with further researches in dierential geometry, geodesy, algebra and analysis.

  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-work Bertini, and the edition published by the printing-work Gozani. Both editions are in a unique volume, but divided into two parts: the rst devoted to the Dierential Calculus (with Chapters I-XXXII), the second devoted to the Integral Calculus (with Chapters I-XXIII).

  Dini's theory on implicit functions is expounded in the following Chapter (of [16]) XIII. Derivate e dierenziali dei vari ordini di funzioni implicite di una o più variabili indipendenti, ! On the other hand, O.F. Mossotti was a close colleague and collaborator of G.A.A. Plana at Torino, and the latter was a pupil of J.L. Lagrange at the Paris École Polytechnique. G. Iurato whereas, in the Chapter (of [16]) XV. Cangiamento delle variabili indipendenti, Dini deals with certain forms of the so-called inverse function theorem.

&

  on the concept of a Riemann sur- face, more general than the formulation used by Riemann in his studies on the theory of analytic functions. Klein based his work on previous Riemann's studies on Abelian functions, on the fundamental 1870 paper of H.A. Schwarz on the integration of the bidimensional Laplace equation ∆u = 0, and on a 1877 paper of R. Dedekind.

  Then, the following conditions are equivalent: 1. Γ is a C k -manifold of dimension m, according to the Denition 1; 2. Γ is locally the diagram of a m-variables function of class C k , according to the Denition 2; 3. Γ is a manifold having a m-atlas of class C k , according to the Denition 3.

  Then, locally, the image of f is a regular Cartesian diagram, say Γ f , with base an open subset of a coordinated r-dimensional subspace of R n . Moreover, the inverse image of an arbitrary point of Γ f , is a Cartesian diagram with base an open subset of a (mr)-dimensional coordinated space of R m . At this point, the author introduces the notion of a dierentiable manifold on R m . That is to say, a diagram of class C 1 .

1 and F ′ W eyl 2 ,

 12 and of F Dini

2 andF

 2 ′ W eyl 1

1 and F W eyl 2 :

 12 indeed, in F Dini 1 , there exists neither the explicit nor the implicit denition of manifold. In conclusion, F W eyl 2 is a proper linguistic and inessential extension of F Dini 1 . Another, almost equivalent way leading to the same conclusions (about the relations between F Dini 1 and F W eyl

2

 2

1 →

 1 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 -S * , induced by the above Galois connection.

1 and M F W eyl 2 ,

 12 hence between their corresponding Lindenbaum-Tarski algebras (computed with respect to the syntactic context, or with respect to the extensional semantic context).

  straint suggested ! by M.A. Leontovic (see 17, A.), with a second denition (see 17, B.) where it is dened, substantially, a manifold in the Dini's sense (see (1) of our 5); he returns on the denition of smooth holonomic constraint in B., Example 10 of 18, where it is introduced the modern (Weyl's) denition of dierentiable manifold. In 21, Arnold introduces D'Alembert principle, and, at the point B. of the same section, he proves the (syntactic) equivalence between D'Alembert-Lagrange principle (of virtual works) and the denition of smooth holonomic constraint given at the point B. of 17, by means of the use (see the point C. of 21) of a variational calculus arguments already known to Lagrange (see the 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.

  % This is not a surface, in the sense of Analysis Situs.

	G. Iurato
	by D. Hilbert and others. In particular, the local Hausdor 's concept of nei-
	ghborhood of a point, has played a crucial role in the Weyl's construction of
	a topological manifold.
	Moreover, some geometrical aspects of Complex Analysis of that time, have
	also played a fundamental (syntactic) role in the Weyl's work (as we shall see
	later).
	The central Weyl's idea is that of local homeomorphism of a manifold with
	R n .
	Subsequently, Weyl introduces a dierentiable structure on a topological
	manifold by means of such a local homeomorphism of this manifold with R n ,
	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.
	Klein wrote a fundamental monograph

  1. The Theorem 1. is the nal result of a part of the works of

	Giuseppe Peano, Alessandro Padoa and Mario Pieri on the logical analysis
	of formal systems; a consequence of the so-called (Peano-Padoa-Pieri) non-
	creativity principle & of the logical denitions, is that the denitions (elements
	of De) of a formal theory F, should not determine deductive novelties ' but
	only expressive novelties. From here, it follows why a [simple] denitional
	extension is proved to be inessential.
	If L is a pure syntactic [or not syntactic] language, then we say that F is
	a pure syntactic [not pure syntactic] theory. This last classication leads us
	to an extra-syntactic area, as we will see later, when we shall introduce the
	notion of semantic model.

& See

[START_REF] Mangione | Storia della logica[END_REF]

, Cap. III, 5, and Cap. VI, 2, or

[START_REF][END_REF]

, Cap. I.

' That is, the denitions should not involve the demonstrability of new theorems, or rather, it should not widen the deductive capacity of a theory.

G. Iurato

Analogously to what has been said in section 9.1.1., the critical comparison between the Lindenbaum-Tarski algebras built up on a [Kripkian] deductive theory, may turn out to be useful for possible historical comparisons between the relative theories (see next 11).

9.2

The work of Saul 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 than that of Tarski's one (for the Classical Logic) and of Gödel's one (for the Intuitionist Logic).

The book [START_REF] Kripke | Naming and Necessity[END_REF] is a philosophical continuation of the rst sixty Kripke's researches 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 [START_REF] Kripke | Naming and Necessity[END_REF], among other, 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, starts 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 which hold between Names and Things; besides, in his rst January 20, 1970 lesson, the author discusses the role of the concept of Possible World in the mathematical denitions, as regard the importance of the Identity Criterion along time (hence, from 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 reaches 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 [START_REF] Putnam | Philosophical Papers[END_REF]), are the founders of the modern new reference theory.