> M bjbj=="WWDTql8>\40MD"@@@@%@FL$tN PLLR=6LR=R=R=/@R=@R=R=n@n@+R<n@n@$M00Mn@Q<pQn@R=MACH'S PRINCIPLE AND
WHITEHEADS RELATIONAL FORMULATION
OF SPECIAL RELATIVITY
enrico giannetto
Dipartimento di Fisica "A. Volta", Universit di Pavia,
via A. Bassi 6, 27100 Pavia; GNSF/CNR, sezione di Pavia
Introduction
Alfred North Whitehead was born in 1861 and dead in 1947. He is very well known as the author, with Bertrand Russell, of the great mathematicallogic treatise Principia Mathematica 1 and for other mathematical and philosophical works.2 Regarding physics, Whitehead is practically known only among few general relativistic theorists for his sort of specialrelativistic theory of gravitation, formulated in opposition to general relativity.3 However, in my opinion, the greatest work of Whitehead is just a physical one, even if it has very important philosophical and mathematical implications. And it is not his theory of gravitation, which, here, I have no time to discuss, but it is his relational formulation of special relativity, that is completely independent of his gravitation theory and also of his trials to link the world of experience and of perceptual representation and the world of physics at a foundational level.4 The importance of this work is rather exceptional, and it is very absurde that no physicist, no historian or philosopher of physics has actually recognized and accepted it.5 Indeed, Whitehead has given a solution to Ockham, Al Ghazali and Kalam school, and Leibniz' major problem of constructing a relational theory of space, time and motion, and so of geometry,6 by defining all the fundamental concepts and formulating (special) relativity in terms of eventparticle relations.7
The work of Whitehead started in 1906 with the paper on On Mathematical Concepts of the Material World ,8 and one can also remember the relevant paper on La Thorie Relationniste de L'Espace ,9 published in 1916. He then gave a complete solution to the problems of relationism in 19191920 by the books An Enquiry on the Principles of Natural Knowledge and The Concept of Nature.10
In 1903, only few years before Whitehead's solution, his scholar Bertrand Russell wrote, in the book entitled The Principles of Mathematics that a relational theory of space and time should describe the principles of geometry in terms of sensible entities.11 Russell noted that indeed right lines and planes are not such entities, whereas, on the contrary, metrical (distance) relations are. Russell went on saying that indeed there is a very complicated method, invented by Leibniz and revised by Frischouf and Peano, by which only distance is fundamental, and the right line is defined from it, even if some of its properties can be introduced only by suitable axioms.12 The field of a given distance is the whole space, at variance with the field of the relation that gives rise to a right line which is only such right line itself. Such a relation generating the right line, hence, at variance with the former, makes an intrinsic distinction among space points, that is a distinction that a relational theory has to avoid. Pieri and others Peano's scholars have tried to formulate geometry starting from the fundamental concept of abstract motion, but they never create an entirely relational theory of geometry.13 This kind of approach to a relational theory of geometry did not start from actual physics and involved a change in the fundamental concepts of geometry, metrical geometry concepts replacing descriptive and projective geometry ones at the foundation level.14 Whiteheads approach actually overcome this latter abstract (mathematical) one.
However, after these works and Whitehead's answer, the relational question was almost completely hidden by the debate on general relativity, and specifically on the problem whether general relativity is actually a relational theory of space, time, and motion.15 And it was also believed that this latter problem could be reduced to the technical problem of the embedding of the socalled Machs principle within the framework of general relativity.16 That is, by dealing with the misleading interpretation given by Einstein of Mach's idea of inertia (in Machs perspective, it was due to the kinematical relation of every body to the remaining part of the universe, not to a dynamical (gravitational) effect).17
Indeed, even if one accepts the historical analysis given by Gereon Wolters that Mach did not really reject relativity,18 and even if one accepts the pseudomachian formulation of general relativity given by Sciama and others,19 a relational theory of space, time and motion is a more complex task than this reformulation of general relativity, a task which was realized for special relativity by Whitehead.
The Relational Theory of Space, Time and Motion:
a Brief Account
Beyond Leibniz, Huyghens and Mach, a relational conception of physics was at the roots of the theory of the actual creator of special relativity before Einstein, that is Jules Henry Poincar,20 but this kind of foundation was almost completely lost (with the relevant exception of Eddington)21 in the formulation accepted by the scientific community as given by Einstein. However, one can say that neither Mach nor Poincar himself have developed such a deep, relational, understanding of the foundations of relativity as Whitehead.
It is well kown that general relativity has turned upside down the hierarchy between kinematics (in some interpretation, dynamics) and geometry: the kind of geometry which enters in the construction of a physical theory is no longer given a priori, but it is defined by the kinematical, physical invariance group of transformations related to kinematized gravitodynamics.22 In this perspective, however, geometry has a foundation completely independent of physics at least at the nonmetrical level, that is at the affine or projective geometrical level.23 It is mathematically constructed in a platonist world of ideas, on its own specific axioms regarding abstract concepts as points, lines, etc., and only after this stage physics could individuate by a very problematic choice only the kind of metric, that is only the kind of metrical geometry to be understood and used only as a physical application of already given mathematical structures. And even if one understands this determination of metrical geometry by physics in a more radical way as the emergence of a physical chronogeometry as opposed to mathematical geometry,24 it is only the metrical structure, the superficial structure  one can say , of geometry that is physically determined, not the deep structure of geometry.
Only Eddington has had the idea to reduce tout court geometry to physics, in a relational perspective of geometry and of general relativity, but he has realized this reduction only a posteriori , by interpreting field equations of general relativity as an identity of metrical geometry functions (the Gmn Einstein tensor) with physical functions (the Tmn matterenergy tensor).25 That is, such an identification happens only at a level of highorder (nonfundamental ) geometrical and physical constructions.
Indeed, even if, apart from the Einsteins operational formulation, it was recognized only by Poincar and Eddington (beyond Whitehead, of course), also special relativity can be interpreted as involving the breakdown of the hierarchy between geometry and physics: here, the problem is the elimination of magnetic forces, and the definition of geometry is given by the kinematical invariance group of transformations related to partially kinematized electrodynamics.26 Hence, already special relativity physics replaces a priori geometry with chronogeometry , but also in this case it is only metrical geometry which is determined by physics.
In this perspective, one can understand how the question of relationism in relativity has been reduced to the technical satisfaction of the socalled Mach's principle: it is only a problem of the relation between two tensors, two nonfundamental variables.
However, I would like to point out this conclusion: Mach's principle is not sufficient for a relational theory of space, time and motion. Furthermore, in some sense, it is not even necessary. Thus, we can have also a relational formulation of special relativity. On the other side, the general covariant formulation of special relativity (and indeed even of classical mechanics) satisfies some sort of Machs principle.27
I would like to show that one must come back to Whiteheads relational formulation of relativity (which  it must be repeated  is completely independent from his specialrelativistic theory of gravitation as opposed to general relativity); then, also through the general covariant formulation of special relativity, one can automatically extend the relational formulation to general frameworks like general relatity too.
Whitehead, indeed, has solved the greatest question left by Leibniz: relationism actually implies that every concept and every structure within a physical theory must be defined in terms of relations among physical elements; no mathematical or logical concept or structure can be given independently from physical relations. Every other option leads to metaphysics. 28 The fundamental concepts of physics like space and time cannot have any mathematically or logically given a priori structure.
In Whiteheads formulation of special relativity, physics not only defines the metrical geometry, but it also defines nonmetrical, descriptive or projective geometry, that is geometry tout court from its foundations. Physics defines geometry not only a posteriori, at the level of highorder constructions as field equations like in Eddingtons interpretation of general relativity, but physics defines points, lines, planes and so on, in terms of fundamental physical processes, that is not in terms of relations among bodies or highlevel tensors (matter), but in terms of relations among eventparticles .29 From this point of view, only Whitehead's relational chronogeometry is an actual physical geometry, free from any logicomathematical (platonist or kantian, any way idealistic) presuppositions.
Let us consider, first of all, relationism in respect to the fundamental concepts of geometry. Already in 1906 paper, Whitehead was pointing out that the simplicity of spatial points was in opposition to the relational theory of space: this requires points to be nonfundamental, complex entities.30 The statement that the eventparticle which one can coordinatizes by four quantities (p1 , p2 , p3 , p4 ) occupies or happens in the point (p1 , p2 , p3 ) means only that the eventparticle is only one of the series of eventparticles which is the point. That is, point is only a series, a set of physical eventparticles.31 Hence, a theory of space is not a theory of relations of objects, but of relations of events. Whitehead explained that in the orthodox theory events are described by means of objects which occupy a dominant position, and so events are considered as a mere play of relations among objects. In this way space theory becomes a theory of relations among objects instead of relations among events.32 The consequence is that, for objects are not related to the becoming of events, space as relations among objects is mantained as unconnected to time. But there cannot be space without time, or time without space, or space and time without event becoming.33 Thus, at variance with the major part of interpretations of relativity which speak about the spatialization of time, Whitehead obtained a complete temporalization of space, so overcoming all the philosophical criticism about that seeming feature of relativity.34 Whitehead wrote in The Principle of Relativity with applications to Physical Science :
...nature is stratified by time. In fact passage in time is of essence of nature, and a body is a merely the coherence of adjectives qualifying the same route through the fourdimensional spacetime of events. But as the result of modern observations we have to admit that there are an indefinite number of such modes of time stratification. However, this admission at once yields an explanation of the meaning of the istantaneous spatial extension of nature. For it explains this extension as merely the exhibition of the different ways in which simultaneous occurrences function in regard to other timesystems. I mean that occurrences which are simultaneous for one timesystem appear as spread out in three dimensions because they function diversely for other timesystems. The extended space of one timesystem is merely the expression of properties of other timesystems. According to this doctrine, a moment of time is nothing else than an istantaneous spread of nature. Thus let t1 , t2 , t3 be three moments of time according to one timesystem, and let T1 , T2 , T3 be three moments of time according to another timesystem. The intersection of pairs of moments in diverse timesystems are planes in each istantaneous threedimensional space... 35
In a more synthetic way, he had written in the introduction:
Position in space is merely the expression of diversity of relations to alternative timesystems. Order in space is merely the reflection into the space of one timesystem of the timeorders of alternative timesystems. A plane in space expresses the quality of the locus of intersection of a moment of the timesystem in question (call it 'timesystem A') with a moment of another timesystem (timesystem B). The parallelism of planes in the space of timesystem A means that these planes result from the intersections of moments A with moments of one other timesystem B. A straight line in the space of timesystem A perpendicular to the planes due to timesystem B is the track in the space of timesystem A of a body at rest in the space of timesystem B. Thus the uniform Euclidean geometry of spaces, planeness, parallelism, and perpendicularity are merely expressive of the relations to each other of alternative timesystems. The tracks which are the permanent points of the same timesystem are also reckoned as parallels. Congruence  and thence, spatial measurements  is defined in terms of the properties of parallelograms and the symmetry of perpendicularity. Accordingly, position, planes, straight lines, parallelism, perpendicularity, and congruence are expressive of the mutual relations of alternative timesystems.36
It is as if Leibniz came back into existence and completed his work in Whitehead.37
Let us consider now properly kinematics. Motion is another relation of events, that is a series of events (p1 , p2 , p3 , p4 ) linked to an object, conceived as placed in them, which is defined by its relation with the remaining part of the universe. If one consider another timesystem (reference frame), the same motion will appear as a relation of other events (q1 , q2 , q3 , q4 ), which in general are associated to other different objects. Hence, even if the motion of one object is relative to the particular considered timesystem, such a motion cannot be reduced to an overall rest in any other timesystem: that is, it will transform itself into the motion of the remaining part of the universe.38 Thus, one must say that Whitehead rejected only the einsteinian content of the socalled Mach's principle, not its kinematical actual (machian) meaning. Indeed, Whitehead kinematized the concept of physical field of an object: it is nothing else than the collection of modifications of event series related to that object: it is a kinematical relation among events and it does not involve any contact or atadistance action (his theory of gravitation was not conceived as an action atadistance theory as often stated).39
Therefore, motion is a relation and is relative to a timesystem, but it has a real counterpart. This furnishes us with a leibnizian interpretation of relativity: the subject of motion is not an invariant, but overall motion is.40
Conclusions: Relativity as a Physical Hermeneutics
One can understand better Whitehead's work by schematizing and comparing in the following way the different kind of constructions of the physical theories:
Classical Mechanics Poincar's Special Relativity
 epistemology and  epistemology and ontology
ontology
 logic  logic
 set theory  set theory
 topology  topology
 nonmetrical geometry  nonmetrical geometry
 metrical geometry  electrodynamics
 kinematics  experimental defining operations
 dynamics  kinematics
 verification experiments  metrical chronogeometry
 general dynamics
 verification experiments
Whiteheads Special Relativity Einstein's General Relativity
 lifeworld experience  epistemology and ontology
 epistemology and ontology of  logic
interrelated events  set theory
 (electro)kinematics  topology
 logic of events  nonmetrical geometry
 set theory of events  gravitodynamics
 topology of events  electrodynamics
 nonmetrical chronogeometry  kinematics
 metrical chronogeometry  metrical chronogeometry
 (gravito)kinematics  general dynamics
 verification experiments  verification experiment
These four schemas represent very different followed hierarchies of steps from the top to the bottom in the construction of physical theories.41 Except Whitehead's case, in the other ones the steps from epistemology to nonmetrical geometry (and for classical theories indeed up to kinematics) are almost completely unquestioned presuppositions to a physical theory, that is metaphysical presuppositions.42 Even if Poincar (at variance with Einstein and Minkowski) had discussed (giving many contributes) practically all the problems related to such steps, he left these levels as untouched by relativistic physics (on the contrary he suggested more radical changes for quantum physics).43
Whitehead is the only one to derive all these levels (not only nonmetrical geometry) from the consideration of physical processes (events), trying to overcome the foundationalist paradigm of an epistemological or ontological (that is, subjectivistically or objectivistically metaphysical) ultimate ground for knowledge and physics.44 His starting point, as I have emphasized with the word lifeworld , is very similar (beyond Leibniz) to Husserl's concept of Lebenswelt with the reconsideration of the full experience in opposition to rationalizing function of experiments, but without any transcendental foundation; so, his position is indeed more similar to Heidegger's one by the consideration of a constitutive physical Dasein (the subject is a particular physical event, nonseparable from the world of all other events), and relationism is not a mere relativism or a particular epistemological option but a sort of a physical hermeneutics .45 In fact, Whitehead has given us the deepest conception of relativity: in his approach, the principle of relativity is first of all, actually, a principle of universal relatedness of nature, and it is this relatedness upon which the indeterminateness of the subject of motion is based, with all its epistemological implications. Nature is not a simple aggregate of independent and separate entities: the traditional mechanistic view has represented nature as an accidental system of contingent separate entities; however, relativity as relationism shows that events are nonseparable within the world as a whole.46
The appearance of an entirely physical (theoretical) practice represents an epochal change, an epochal departure from Western metaphysics .47 However, as it can be seen from his construction of special relativity, gravitokinematics was left by Whitehead into one of the bottom levels, at variance with general relativity: Whitehead recognized that a choice like the one operated in general relativity construction would lead to an actual hidden breakdown of the metrical geometry structure.48 Thus, Whitehead's choice in this respect was not good from a radical relationist point of view, just indipendently of the validity of general relativity.49 It should not be so difficult to elaborate, on one side, a complete relationist theory of gravitational processes too in an actual Whitehedian form, and, on the other side, in any case it is easy to give a relational construction to general relativity, by considering, just on the same level of matter eventparticles, gravitational eventparticles too.50
It is so clear that Whiteheads formulation of special relativity is not equivalent to Poincares or the other traditional version of the theory at least from an epistemological point of view; but indeed also from a mathematical point of view the structure of the theory is different until up to the metrical geometry level. Therefore, differing at an epistemic and mathematical level, and furthermore at the semantic level (for example, the idea of temporalization of space), Whiteheads special relativity seems to be a new, different physical theory more than a mere reformulation of Poincars or EinsteinMinkowskis special relativity.51 However, the observational consequences to be related to the metrical geometry structure are identical and Whitehead's special relativity indeed gives us a completely relational physical theory in which we no longer appear as having to include the world but as included in the world.
Notes and References
PAGE
PAGE 1
1 A. N. Whitehead & B. Russell (19101913), Principia Mathematica , Cambridge, Cambridge University Press, v. I, 1910, v. II, 1912, v. III, 1913.
2 The Philosophy of A. N. Whitehead , ed. by P. A. Schilpp, The Library of Living Philosophers, Northwestern University, Evanston and Chicago 1941 & Tudor, New York 1951: here a very good bibliography is available. Only to quote some of the most important books on mathematics and philosophy, published by A. N. Whitehead, see for instance: A. N. Whitehead, an Anthology , edited and selected by F. S. C. Nohrtrop & M. W. Gross, Cambridge University Press, Cambridge 1953; A. N. Whitehead, A Treatise on Universal Algebra, with Applications , Cambridge University Press, Cambridge 1898; A. N. Whitehead, The Axioms of Projective Geometry , Cambridge University Press, Cambridge 1906; A. N. Whitehead, The Axioms of Descriptive Geometry , Cambridge University Press, Cambridge 1907; A. N. Whitehead, An Introduction to Mathematics , Williams and Norgate, London & H. Holt and Co., New York 1911; A. N. Whitehead, Science and the Modern World , The Macmillan Co., New York 1925; A. N. Whitehead, Process and Reality. An essay in cosmology , The Macmillan Co., New York, 1929; A. N. Whitehead, Adventures of Ideas , The Macmillan Co., New York 1933; A. N. Whitehead, Nature and Life , The University of Chicago Press, Chicago 1934; A. N. Whitehead, Modes of Thought , The Macmillan Co., New York 1938; A. N. Whitehead, Essays in Science and Philosophy , Philosophical Library, New York 1947 .
3 A. N. Whitehead, The Principle of Relativity with applications to Physical Science, Cambridge University Press, Cambridge 1922. See also for a discussion: A. Schild, On gravitational theories of Whitehead's type, Proceedings of the Royal Society of London, v. 235 (1956), pp. 202209; A. Grnbaum, Philosophical Problems of Space and Time , Reidel, Dordrecht 1973, pp. 4865 and 425428; J. D. North, The Measure of the Universe. A History of Modern Cosmology, Oxford University Press, Oxford 1965 & Dover, New York 1990, pp. 186197 and references therein; J. L. Synge, Relativity: The Special Theory , NorthHolland, Amsterdam 1956.
4 See, for instance, A. N. Whitehead, The Concept of Nature , Cambridge University Press, Cambridge 1920 and A. N. Whitehead, Process and Reality. An essay in cosmology , The Macmillan Co., New York, 1929; see also B. Russell, The Analysis of Matter , G. Allen & Unwin Ltd., London 1927 & Dover, New York 1954, where the problem of finding a link between the world of life experience and the world of physics is reconsidered.
5 Criticism about Whitehead's work is expressed in: M. Bunge, Physical Time: The Objective and Relational Theory, McGill University, Montral preprint 1967; M. Bunge & A. G. Maynez, A Relational Theory of Physical Space , International Journal of Theoretical Physics, v. 15, n. 12 (1976), pp. 961972; A. Grnbaum, Philosophical Problems..., op. cit.; G. J. Whitrow, The Natural Philosophy of Time , Nelson, 1961 & Oxford Univeristy Press, Oxford 1980. Some developments of Whitehead's perspectives are in: B. Russell, Our Knowledge of the External World , Open Court, La Salle 1914; B.Russell, The Analysis..., op. cit.; G. Gerla & R. Volpe, Geometry without points, The American Mathematical Monthly, v. 92, pp. 707711; G. Gerla, Pointless Geometry, in Handbook of Incidence Geometry, ed. by F. Buekenhout and W. Kantor, NorthHolland, Amsterdam 1991 and references therein; G. Gerla, Pointless metric spaces, Journal of Symbolic Logic, v. 55 (1990), pp. 207219; J. Nicod, Foundations of Geometry and Induction, Hartcourt, Brace and Co., New York 1930; B. L. Clarke, A calculus of individuals based on "connection" , Notre Dame Journal of Formal Logic, v. 22 (1981), pp. 204218; B. L. Clarke, Individuals and points , Notre Dame Journal of Formal Logic, v. 26 (1985), pp. 6175. For the very interesting M. Capeks epistemological works see note 34.
6 Guillelmi de Ockham, Opera Philosophica, voll. IVVVI, St. Bonaventure University, New York 19845; E. Giannetto, Ockham's Physics and Relational Theory of Motion, in Atti del XII Congresso Nazionale di Storia della Fisica, ed. by F. Bevilacqua, Goliardica Pavese, Pavia 1993 (in press). For Al Ghazali and Kalam school, see: M.Jammer, The History of Theories of Space in Physics, Harvard University Press, Cambridge, Mass. 1954. For Leibniz and relationism, see: M. Jammer, Concepts of force. A study in the foundations of dynamics, Harvard University Press, Cambridge, Mass. 1957; B. Russell, The Principles of Mathematics, Cambridge University Press, Cambridge 1903 and references therein; B. Russell, A Critical Exposition of the Philosophy of Leibniz, G. Allen & Unwin Ltd., London 1900; G. W. Leibniz, Die philosophischen Schriften von G. Leibniz, ed. by C. I. Gerhardt, Berlin 187590 & Olms, Hildesheim 1960; G. W. Leibniz, Leibnizens mathematische Schriften, ed. by C. G. Gerhardt, Halle 185063; H. G. Alexander, The LeibnizClarke Correspondence, Barnes & Noble, New York 1984; C. Huyghens, Oeuvres compltes de Christian Huyghens, Aja 1905, v. X (correspondence 16911695), p. 609; D. J. Korteweg & J. A. Schouten, Jahresbericht der Deutschen MathematikerVereinigung, v. 29 (1920) 136. Very interesting for the connection of the rise of Poincar's special relativity and Leibniz is the following paper: H. Poincar, Note sur le principes de la mcanique dans Descartes et dans Leibnitz, in G. W. Leibnitz, La Monadologie, ed. by E. Boutroux and with a note by H. Poincar, Delagrave, Paris 1880, pp. 225231; see also: E. Giannetto, Poincar and the Rise of Special Relativity, in Atti del XIII Congresso Nazionale di Storia della Fisica, ed. by F. Bevilacqua, Goliardica Pavese, Pavia 1993 (in press). And furthermore: M. Serres, Estime, in Politiques de la philosophie, ed. by D. Grisoni, Grasset & Fasquelle, Paris 1976; H. Reichenbach, Die Bewegungslehre bei Newton, Leibniz und Huyghens, Kantstudien, v. 29 (1924), pp. 416438; The Natural Philosophy of Leibniz, ed. by K. Okruhlik & J. R. Brown, Reidel, Dordrecht 1985; E. Cassirer, Erkenntnisproblem in der Philosophie und Wissenschaft der neuren Zeit, Berlin 19111920; E. Cassirer, Leibniz' System in seinen wissenschaftlichen Grundlagen, Elwert, Marburg 1902 & Wissenschaftliche Buchgesellschaft, Darmstadt 1962; E. Giannetto, Il crollo del concetto di spaziotempo negli sviluppi della fisica quantistica: l'impossibilit di una ricostruzione razionale nomologica del mondo, in Aspetti epistemologici dello spazio e del tempo, ed. by G. Boniolo, Borla, Roma 1987, pp. 169224; E. Giannetto, Lectures on Leibniz, mimeographed paper, University of Messina, Messina 1988; E. Giannetto, Note sull'interpretazione della relativit generale di A. S. Eddington, in Atti dell'XI Congresso Nazionale di Storia della Fisica, ed. by F. Bevilacqua, Goliardica Pavese, Pavia 1993; E. Giannetto, Relativity Theories and Leibniz' Dynamics , mimeographed paper, University of Pavia, Pavia 1992; J. Earman, World Enough and SpaceTime. Absolute versus Relational Theories of Space and Time, The MIT Press, Cambridge, Mass. 1989; M. Serres, Le systme de Leibniz et ses modles mathmatiques, P.U.F., Paris 1968. A very enlightening comparison between Leibniz and Whitehead is presented in: G. Deleuze, Le pli. Leibniz et le Baroque, Minuit, Paris 1988, pp. 103112.
7 See, for instance, A. N. Whitehead, An Enquiry on the Principles of Natural Knowledge , Cambridge University Press, Cambridge 1919; A. N. Whitehead, The Concept of Nature , op. cit.; A. N. Whitehead, Essays in Science and Philosophy , op. cit.; A. N. Whitehead, The Principle of Relativity with applications to Physical Science, op. cit.; A. N. Whitehead, Space, Time, and Relativity, Proceedings of the Aristotelian Society, v. 16 (191516), pp. 104129.
8 A. N. Whitehead, On Mathematical Concepts of the Material World , Philosophical Transactions, Royal Society of London A, v. 205 (1906), pp. 465525: already here the relation with Leibniz' ideas is evident.
9 A. N. Whitehead, La Thorie Relationniste de L'Espace , Revue de Mtaphysique et de Morale, v. 23 (1916), pp. 423454.
10 A. N. Whitehead, An Enquiry on the Principles of Natural Knowledge , op. cit.; A. N. Whitehead,The Concept of Nature , op. cit.; in A. N. Whitehead, The Principle of Relativity with applications to Physical Science, op. cit., in which he formulated also his new theory of gravitation, there is a new overview of his relationistic approach to special relativity.
11 B. Russell, The Principles of Mathematics, op. cit. .
12 L. Couturat, La Logique de Leibniz, Paris 1901, p. 420; Frischauf, Absolute Geometrie nach Johann Bolyai, Anhang, quoted by Russell; G. Peano, La geometria basata sulle idee di punto e distanza, Atti della Reale Accademia delle Scienze di Torino, v. XXXVIII (190203), pp. 610.
13 See, for example, M. Pieri, Della geometria elementare come sistema ipoteticodeduttivo, Memorie della Reale Accademia delle Scienze di Torino, v. XLIX (1899), p. 176.
14 B. Russell, An Essaay on the Foundations of Geometry, Cambridge University Press, Cambridge 1897.
15 E. Giannetto, Il crollo..., op. cit.; E. Giannetto, Note sulla relazionalit della relativit generale, 1990, unpublished; see also, for example: A. Grnbaum, The Philosophical Retention of Absolute Space in Einstein's General Relativity, The Philosophical Review, v. LXVI (1957), pp. 525534.
16 See, for instance: D. W. Sciama, The Unity of the Universe, Faber and Faber, London 1959; D. W. Sciama, The Physical Foundations of General Relativity, Heinemann, London 1969; Cosmology now, ed. by J. Laurie, BBC Publications, London 1973; J. A. Wheeler, Mach's principle as boundary condition for Einstein's equations, in Gravitation and Relativity, ed. by H. Y. Chiu & W. F. Hoffman, Benjamin, New York 1964, p. 303349; J. A. Wheeler, Geometrodynamic Steering Principle Reveals the Determiners of Inertia, Princeton preprint 1988.
17 E. Mach, Die Mechanik in ihrer Entwickelung historischkritisch dargestellt, Brockhaus, Leipzig 1883; see, for example: G. Boniolo, Mach e Einstein, Armando, Roma 1988; J. B. Barbour, Relational Concepts of Space and Time, The British Journal for the Philosophy of Science, v. 33 (1982), pp. 251274; J. B. Barbour, Forceless Machian dynamics, il Nuovo Cimento, v. 26 B (1975), pp. 1622; J. B. Barbour and B. Bertotti, Gravity and Inertia in a Machian Framework, il Nuovo Cimento, v. 38 B (1977), pp. 127; J. B. Barbour and B. Bertotti, Mach's principle and the structure of dynamical theories, Proceedings of the Royal Society of London, v. A 382 (1982), pp. 295306; D. J. Raine, Mach's principle and spacetime structure, Report on Progress in Physics, v. 44 (1981), pp. 11511195 and references therein; F. Hoyle & J. Narlikar, Action at a distance in physics and cosmology, Freeman, San Francisco 1974. For the relations between Mach and Leibniz, see: E. Giannetto, Relativity Theories and Leibniz' Dynamics , mimeographed paper, University of Pavia, Pavia 1992.
18 G. Wolters, Mach I, Mach II, Einstein und die Relativittstheorie. Eine Flschung und ihre Folgen, BerlinNew York 1987.
19 See notes 16 and 17 and D. W. Sciama, On the origin of inertia, Monthly Notices of the Royal Astronomical Society, v. 113 (1953), p. 34; D. W. Sciama, The Physical Structure of General Relativity, Reviews of Modern Physics, v. 36 (1964), pp. 463469; D. W. Sciama, P. C. Waylen & R. C. Gilman, Generally Covariant Integral Formulation of Einstein's Field Equations, Physical Review, v. 187 (1969), p. 17621766; R. C. Gilman, Physical Review D, v. 2 (1970), p. 1400; D. LyndenBell, On the Origins of SpaceTime and Inertia, Monthly Notices of the Royal Astronomical Society, v. 135 (1967), pp. 413428; H. Goenner, Mach's Principle and Einstein's Theory of Gravitation, in Boston Studies in the Philosophy of Science, v. 6, Reidel, Dordrecht 1970; M. Reinhardt, Mach's Principle. A critical Review, Zeitschrift fr Naturforschung, v. 28 A (1972), 529537; B. L. Altshuler, Mach's Principle. Part 1. Initial State of the Universe, International Journal of Theoretical Physics, v. 24 (1985), pp. 99118; D. J. Raine, Mach's Principle in General Relativity, Monthly Notices of the Royal Astronomical Society, v. 171 (1975), pp. 507528; D. J. Raine & E. G. Thomas, Mach's Principle and the Microwave Background, Astrophysics Letters, v. 23 (1982), pp. 3745; D. J. Raine, Mach's principle and spacetime structure, op. cit. and references therein; D. J. Raine & M. Heller, The Science of SpaceTime, Pachart Publishing House, Tuscon 1981.
20 See, for example, some of the most important papers written by Poincar in which he gave the conceptual foundations and then the complete formulation of special relativity: H. Poincar, La mesure de temps, Revue de Mtaphysique et Morale, v. 6 (1898), pp. 113; H. Poincar, La thorie de Lorentz et le principe de raction, Arch. Nerl., v. 5 (1900), pp. 252278 and also in Recueil de travaux offerts par les auteurs H. A. Lorentz, Nijhoff, The Hague 1900; H. Poincar, La Science et l'Hypothse, Flammarion, Paris 1902; H. Poincar, L'tat actuel et l'avenir de la Physique mathmatique, Bulletin des Sciences Mathematiques, v. 28 (1904), pp. 302324; H. Poincar, The Principles of Mathematical Physics, The Monist, v. 15 (1905), p. 1; H. Poincar, Sur la dynamique de l'lectron, Comptes Rendus de l'Acadmie des Sciences, v. 140 (1905), pp. 15041508; H. Poincar, Sur la dynamique de l'lectron, Rendiconti del Circolo Matematico di Palermo, v. 21 (1906), pp. 129175. The absolute priority of Poincar as the creator of special relativity was strongly pointed out for the first time in: E. T. Whittaker, History of the Theories of Aether and Electricity. The Modern Theories 19201926, v. 2, Nelson & Sons, London 1953 and in particular the chapter entitled 'The Relativity Theory of Poincar and Lorentz', pp. 2777. Poincar was the ignored author of the fourdimensional spacetime formulation of special relativity: he was aknowledged by Minkowski (with six quotations of Poincar) only in the first paper (and then never quoted), published only after eight years: H. Minkowski, Das Relativittsprinzip, Lecture delivered on 5 November 1907, Annalen der Physik, IV Folge, v. 47 (1915), pp. 927938. An aknowledgement, among others, of Poincar's work was present in: R. Marcolongo, Relativit, Principato, Messina 1921, 19232, in which also Whitehead's works are quoted. Indeed, Marcolongo was the second, after Poincar and before Minkowski, to use a fourdimensional formulation and to develop a covariant formulation of special relativity: R. Marcolongo, Sugli integrali dell'equazione dell'elettrodinamica, Rendiconti della Regia Accademia dei Lincei, s. 5, v. 15 (I sem. 1906), pp. 344349. Fundamental works for a historical inquiry about the rise of special relativity, exploring a different view, are: A. I. Miller, A Study of Henry Poincar's 'Sur la dynamique de l'lectron, Archives for the History of Exact Sciences, v. 10 (1973), pp. 207328; A. I. Miller, Albert Einstein's Special Theory of Relativity: Emergence (1905) and Early Interpretation (19051911), AddisonWesley, Reading Mass. 1981; E. Zahar, Einstein's Revolution. A Study in Heuristics, Open Court, La Salle Ill. 1989; A. Pais, 'Subtle is the Lord...'. The Science and the Life of Albert Einstein, Oxford University Press, Oxford 1982. For a theoretical interpretation of special relativity in the spirit of Poincar, see: A. A. Tyapkin, Expression of the General Properties of Physical Processes in the SpaceTime Metric of the Special Theory of Relativity, Soviet Physics Uspekhi, v. 15 (1972), pp. 205229. For a complete historical and theoretical analysis, see: E. Giannetto, Poincar and the Rise of Special Relativity, in Atti del XIII Congresso Nazionale di Storia della Fisica, ed. by F. Bevilacqua, Goliardica Pavese, Pavia 1993 (in press).
21 A. S. Eddington, Space, Time and Gravitation. An Outline of the General Relativity Theory, Cambridge University Press, Cambridge 1920; A. S. Eddington, The Mathematical Theory of Relativity, Cambridge University Press, Cambridge 1923; A. S. Eddington, The Nature of the Physical World, Cambridge University Press, Cambridge 1928; A. S. Eddington, The Expanding Universe, Cambridge University Press, Cambridge 1933; A. S. Eddington, New Pathways in Science, Cambridge University Press, Cambridge 1935; A. S. Eddington, The Philosophy of Physical Science, Cambridge University Press, Cambridge 1938; A. S. Eddington, Fundamental Theory, Cambridge University Press, Cambridge 1946. See also: E. Giannetto, Note sull'interpretazione della relativit generale di A. S. Eddington, in Atti dell'XI Congresso Nazionale di Storia della Fisica, ed. by F. Bevilacqua, Goliardica Pavese, Pavia 1993.
22 See, for example: A. O. Barut, Geometry and Physics. NonNewtonian Forms of Dynamics, Bibliopolis, Napoli 1989 and references therein. For the epistemological relevance of this step in the construction of the physical theory, see: D. Finkelstein, Matter, Space and Logic, Boston Studies in the Philosophy of Science, v. 5 (1969), p. 199; E. Giannetto, On Truth: A Physical Inquiry, in Atti del Congresso Internazionale "Nuovi Problemi della Logica e della Filosofia della Scienza", ed. by C. Cellucci & M. Dalla Chiara, Clueb, Bologna 1991, v. I, pp. 221228; E. Giannetto, Note sull'interpretazione della relativit generale di A. S. Eddington, in Atti dell'XI Congresso Nazionale di Storia della Fisica, ed. by F. Bevilacqua, Goliardica Pavese, Pavia 1993; E. Giannetto, La logica quantistica tra fondamenti della matematica e della fisica, in Foundations of Mathematics & Physics, ed. by U. Bartocci & J. P. Wesley, Wesley, Blumberg 1990, pp. 107127; E. Giannetto, The Epistemological and Physical Importance of Gdel's Theorems, in First International Symposium on Gdel's Theorems, ed. by Z. W. Wolkowski, World Scientific, Singapore 1993, pp. 136147.
23 E. Giannetto, Il crollo..., op. cit.
24 See note 22.
25 See note 21
26 See notes 20 and 21 and in particular: H. Poincar, Sur la dynamique de l'lectron, 1906, op. cit.; A. S. Eddington, The Nature of..., op. cit.; E. Giannetto, Poincar and the Rise of Special Relativity, in Atti del XIII Congresso Nazionale di Storia della Fisica, ed. by F. Bevilacqua, Goliardica Pavese, Pavia 1993 (in press). See also: A. O. Barut, Geometry and Physics..., op. cit.; A. A. Tyapkin, Expression of the..., op. cit.; E. Giannetto, Lectures on Special relativity, mimeographed paper, University of Pavia, Pavia 1993; A. Einstein, Zur Elektrodynamik bewegter Krper, Annalen der Physik, s. 4, v. 17 (1905), pp. 891921; A. N. Whitehead, An Enquiry..., op. cit.; A. N. Whitehead, Space, Time and Relativity, Proceedings of the Aristotelian Society, v. 16 (19151916), pp. 104129.
27 For a modern generalcovariant formulation of special relativity and classical mechanics, see: A. Logunov, Lectures in Relativity and Gravitation. A Modern Look, Nauka, Moscow & Pergamon Press, Oxford 1990; P. Havas, FourDimensional Formulations of Newtonian Mechanics and Their Relation to the Special and the General Theory of Relativity, Reviews of Modern Physics, v. 36 (1964), pp. 938965; P. Havas, Simultaneity, conventionalism, general covariance, and the special theory of relativity, General Relativity & Gravitation, v. 19 (1987), pp. 435453; C. Giannoni, Relativistic Mechanics and Electrodynamics without OneWay Velocity Assumptions, Philosophy of Science, v. 45 (1978), pp. 1746. However, these generalcovariant formulations of physical theories are translations, in the language of coordinates, of a vectorial language developed in Italy by Peano's school as an "absolute omographic calculus without coordinates", opposed to the "absolute calculus with coordinates" of Christoffel, Riemann, Ricci and LeviCivita. These generalcovariant vectorial formulations of classical mechanics, special and general relativity are given in: C. BuraliForti & T. Boggio, Meccanica Razionale, Lattes, Torino 1921; C. BuraliForti & R. Marcolongo, Analyse Vectorielle Gnrale, 2 voll., Mattei, Pavia 19121913; C. BuraliForti & T. Boggio, Espaces Coubes. Critique de la Relativit, Sten, Torino 1924; R. Marcolongo, Relativit, op. cit.; see also: A. Pagano, Su di un'opera dimenticata di fisica di Boggio e BuraliForti, Mondotre/Quaderni, suppl. al n. 4/5 (1989), Laboratorio, Siracusa, pp. 107132; G. Boscarino, S. Notarrigo, A. Pagano, Geometria e Fisica, Mondotre/Quaderni, n. 8 (1992), Laboratorio, Siracusa, pp. 61123; S. Notarrigo, Applicazioni fisiche del calcolo geometrico di Peano, Catania preprint 1993 and references therein; in these two last papers it is shown how a sort of Mach's principle is embedded in classical mechanics. A critical and historical analysis of these generalcovariant vectorial formulations of physical theories can be found in: E. Giannetto, Classical Mechanics, Relativity and the Peano's School, mimeographed paper, University of Pavia, Pavia 1993.
28 E. Giannetto, The Epistemological and the Physical..., op. cit.; E. Giannetto, Lectures on Relativity, op. cit.
29 For eventparticles Whitehead means infinitesimal events; see, for instance, A. N. Whitehead, The Concept of Nature, op. cit.; F. S. C. Northrop, Whitehead's Philosophy of Science, in The Philosophy of Alfred North Whitehead, op. cit., pp. 165207.
30 See, for example, A. N. Whitehead, Einstein's Theory. An Alternative Suggestion, The London Times Educational Supplement, Feb. 12 (1920), p. 83, reprinted in A. N. Whitehead, Essays in Science..., op. cit.
31 See notes 29 and 30.
32 See notes 29 and 30.
33 See, for example, A. N. Whitehead, The Principle of Relativity..., op. cit.
34 An idea of temporalization of space was already present in Leibniz: G. W. Leibnitz, Leibnizens..., op. cit.; see also H. Poser, La teoria leibniziana della relativit di spazio e tempo, autaut, v. 254255 (1993), pp. 3348. It was also present in M. Heidegger, Sein und Zeit, Niemeyer, Tbingen 1927, 70. It is not known wether he knew Whiteheads formulation of special relativity, but he certainly was aware of relativity (and of Bergson's critical analysis, in particular, on the seeming spazialization of time: H. Bergson, dure et simultanit, P.U.F., Paris 1922, 1968; indeed, also Whitehead was influenced by Bergson's philosophy of temporal evolution and becoming: H. Bergson, Oeuvres, P.U.F., Paris 19591972) which affected his ideas of space and time. He elaborated philosophically the conceptual core of relativity, criticizing its scientific, operational and reductionistic dress. This is also much more evident in his idea of the being conceived as event (Ereignis ), developed in M. Heidegger, Zeit und Sein, in Zur Sache des Denkens, Niemeyer, Tbingen 1969, where however he rejected the idea of a fundamental temporalization of space as given in the 70 of Sein und Zeit, stressing the priority of events beyond spacetime. This temporalization, heideggerian, idea has been recently reconsidered by J. Derrida, without quoting Whitehead: J. Derrida, De la grammatologie, Minuit, Paris 1967, chh. II & III. See also: C. F. von Weiszcker, Zeit und Wissen, Hanser Verlag, Mnchen 1992; D. R. Mason, Time in Whitehead and Heidegger: a Comparison, in Process Studies (1975), pp. 87105; The Concepts of Space and Time: their Structure and their Development, ed. by M. Capek, Boston Studies in the Philosophy of Science, v. XXII, Reidel, Dordrecht 1976; M. Capek, Bergson and Modern Physics., Boston Studies in the Philosophy of Science, v. 7, Reidel, Dordrecht 1971; M. Capek, TimeSpace rather than SpaceTime, in Diogenes, v. 123 (1983), pp. 3O49; M. Capek, The New Aspects of Time. Its Continuity and Novelties, Boston Studies in the Philosophy of Science, v. 125, Reidel, Dordrecht 1991.
35 A. N. Whitehead, The Principle of Relativity..., op. cit., pp. 5455.
36 A. N. Whitehead, The Principle of Relativity..., op. cit., pp. 89.
37 For a comparison, see, for instance: G. Deleuze, Le pli, op. cit.
38 A. N. Whitehead, The Concept of Nature, op. cit.
39 A. N. Whitehead, The Concept of Nature, op. cit.
40 E. Giannetto, Lectures on Leibniz, op. cit.; E. Giannetto, Lectures on Relativity, op. cit.
41 D. Finkelstein & E. Rodriguez, Quantum Simplicial Topology, GIT preprint 1983; E. Giannetto, Toward a Quantum Epistemology, in Atti del Congresso 'Temi e prospettive della logica e della filosofia della scienza contemporanee', v. II, ed. by M. Dalla Chiara e M. C. Galavotti, Clueb, Bologna 1988, pp. 121124; E. Giannetto, L'epistemologia quantistica come metafora antifondazionistica, in Immagini Linguaggi Concetti, ed. by S. Petruccioli, Theoria, Roma 1991, pp. 303322; E. Giannetto, La logica quantistica..., op. cit.
42 E. Giannetto, Lectures on Relativity, op. cit.; E. Giannetto, The Epistemological and Physical..., op. cit.
43 E. Giannetto, Poincar and the Rise..., op. cit.
44 R. Rorty, Philosophy and the Mirror of Nature, Princeton University Press, Princeton 1979; E. Giannetto, L'epistemologia quantistica..., op. cit. and references therein.
45 E. Husserl, Die Krisis der europischen Wissenschaften und die transzendentale Phnomenologie, in Husserliana, Gesammelte Werke, v. VI, Den Haag 1959; M. Heidegger, Sein und Zeit, op. cit.; for the idea of a physical hermeneutics, see E. Giannetto, L'epistemologia quantistica..., op. cit., and E. Giannetto, Notes on NonSeparability, conference delivered in Lecce, October 1993, in preparation. Here, we use the expression physical Dasein without any heideggerian, ontological and humanistic connotation. The idea to link Whitehead's and Husserl's philosophies in a phenomenological relationism was indeed explored by E. Paci: E. Paci, La filosofia di Whitehead e i problemi del tempo e della struttura, La Goliardica, Milano 1965; E. Paci, Relazioni e significati, v. 1, Lampugnani Nigri, Milano 1965; E. Paci, Dall'esistenzialismo al relazionismo, D'Anna, Messina 1967; see also C. Sini, Whitehead e la funzione della filosofia, Marsilio, Padova 1965; P. A. Rovatti, La dialettica del processo. Saggio su Whitehead, il Saggiatore, Milano 1969; J. J. Kockelmans, Phenomenology and Physical Science. An Introduction to the Philosophy of Physical Science, Duquesne University Press, Pittsburgh, Pa. 1966.
46 A. N. Whitehead, The Concept of Nature, op. cit.; A. N. Whitehead, The Principle of Relativity..., op. cit.; E. Giannetto, Notes on NonSeparability, op. cit.
47 E. Giannetto, Heidegger and the Question of Physics, conference delivered in Veszprm, September 1993, in press; E. Giannetto, The Epistemological..., op. cit.; E. Giannetto, Physical Theories: From Foundationalism to Practices, conference delivered in Como, September 1992, in press.
48 A. N. Whitehead, The Concept of Nature, op. cit.; A. N. Whitehead, The Principle of Relativity..., op. cit.; E. Giannetto, Note sull'interpretazione..., op. cit.; E. Giannetto, Lectures on Relativity, op. cit. and references therein.
49 E. Giannetto, Note sull'interpretazione..., op. cit.; E. Giannetto, Lectures on Relativity, op. cit. and references therein.
50 See note 48.
51 A brief comparison of Whitehead's and Einstein's positions is explored in F. S. C. Northrop, Whitehead's Philosophy ..., op. cit. and in A. P. Ushenko, Einstein's Influence on Contemporary Philosophy, in Albert Einstein: PhilosopherScientist, ed. by P. A. Schilpp, The Library of Living Philosophers, Evanston & Open Court, La Salle 1949 (here is interesting to point out that Einstein did not reply to whitehedian and russellian arguments presented by Ushenko); see also P. Franck, Einstein. Sein Leben und seine Zeit, Mnchen 1949; A. P. Ushenko, The Logic of Events, Berkeley 1929 and A. P. Ushenko, Power and Events, Princeton University Press, Princeton 1946; V. Lowe, The Development of Whitehead's Philosophy, in The Philosophy of Alfred North Whitehead, op. cit., pp. 15124; E. B. McGilvary, SpaceTime, Simple Location, and Prehension, in The Philosophy of Alfred North Whitehead, op. cit., pp.209240. Other philosophical, relevant papers written by Whitehead on this subject are: A. N. Whitehead, Symposium  Time, Space and Material: Are They, and If So in What Sense, the Ultimate Data of Science?, Proceedings of Aristotelian Society, suppl. v. 2 (1919), pp. 4457; A. N. Whitehead, Discussion: The Idealistic Interpretation of Einstein's Theory, Proceedings of Aristotelian Society, v. 22 (19211922), pp. 130134; A. N. Whitehead, The Philosophical Aspects of the Principle of Relativity, Proceedings of Aristotelian Society, v. 22 (19211922), pp. 215223. The holistic, philosophical perspective of Whitehead is indeed appealed in the debate about nonseparability in quantum mechanics: The World View of Contemporary Physics, ed. by R. F. Kitchener, State University of New York Press, Albany 1988; E. Giannetto, Notes on NonSeparability, op. cit. See also note 34.
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