MACH'S PRINCIPLE AND

WHITEHEAD’S RELATIONAL FORMULATION

OF SPECIAL RELATIVITY

enrico giannetto

Dipartimento di Fisica "A. Volta", Università 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 mathematical-logic 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 special-

relativistic 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 event-particle relations.7

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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 Théorie Relationniste

de L'Espace ,9 published in 1916. He then gave a complete solution to the problems of

relationism in 1919-1920 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

Whitehead’s 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 so-called “Mach’s principle” within the framework of general relativity.16

That is, by dealing with the misleading interpretation given by Einstein of Mach's idea of

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inertia (in Mach’s 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 pseudo-machian 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

Poincaré,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 Poincaré 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 non-metrical 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 chrono-

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geometry 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 matter-energy

tensor).25 That is, such an identification happens only at a level of high-order (non-

fundamental ) geometrical and physical constructions.

Indeed, even if, apart from the Einstein’s operational formulation, it was recognized only by

Poincaré 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

chrono-geometry , 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 so-called Mach's principle: it is only a problem of

the relation between two tensors, two non-fundamental 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 “Mach’s principle”.27

I would like to show that one must come back to Whitehead’s relational formulation of

relativity (which - it must be repeated - is completely independent from his special-

relativistic theory of gravitation as opposed to general relativity); then, also through the

4

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

meta-physics. 28 The fundamental concepts of physics like space and time cannot have any

mathematically or logically given a priori structure.

In Whitehead’s formulation of special relativity, physics not only defines the metrical

geometry, but it also defines non-metrical, descriptive or projective geometry, that is

geometry tout court from its “foundations”. Physics defines geometry not only a posteriori,

at the level of high-order constructions as field equations like in Eddington’s 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 high-level tensors

(matter), but in terms of relations among event-particles .29 From this point of view, only

Whitehead's relational chrono-geometry is an actual physical geometry, free from any logico-

mathematical (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 non-fundamental,

complex entities.30 The statement that the event-particle which one can coordinatizes by four

quantities (p1 , p2 , p3 , p4 ) occupies or happens in the point (p1 , p2 , p3 ) means only that the

event-particle is only one of the series of event-particles which is the point. That is, point is

only a series, a set of physical event-particles.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

5

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 :

33 See, for example, A. N. Whitehead, The Principle of Relativity..., op. cit.20 See, for example, some of the most important papers written by Poincaré in which he gave the conceptual

foundations and then the complete formulation of special relativity: H. Poincaré, La mesure de temps, Revue

de Métaphysique et Morale, v. 6 (1898), pp. 1-13; H. Poincaré, La théorie de Lorentz et le principe de

réaction, Arch. Néerl., v. 5 (1900), pp. 252-278 and also in Recueil de travaux offerts par les auteurs à H. A.

Lorentz, Nijhoff, The Hague 1900; H. Poincaré, La Science et l'Hypothèse, Flammarion, Paris 1902; H.

Poincaré, L'état actuel et l'avenir de la Physique mathématique , Bulletin des Sciences Mathematiques, v. 28

(1904), pp. 302-324; H. Poincaré, The Principles of Mathematical Physics, The Monist, v. 15 (1905), p. 1; H.

Poincaré, Sur la dynamique de l'électron, Comptes Rendus de l'Académie des Sciences, v. 140 (1905), pp.

1504-1508; H. Poincaré, Sur la dynamique de l'électron, Rendiconti del Circolo Matematico di Palermo, v. 21

(1906), pp. 129-175. The absolute priority of Poincaré 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 1920-1926, v. 2, Nelson & Sons, London 1953 and in particular the chapter entitled 'The Relativity

Theory of Poincaré and Lorentz', pp. 27-77. Poincaré was the ignored author of the four-dimensional space-

time formulation of special relativity: he was aknowledged by Minkowski (with six quotations of Poincaré)

only in the first paper (and then never quoted), published only after eight years: H. Minkowski, Das

Relativitätsprinzip, Lecture delivered on 5 November 1907, Annalen der Physik, IV Folge, v. 47 (1915), pp.

927-938. An aknowledgement, among others, of Poincaré's work was present in: R. Marcolongo, Relatività,

Principato, Messina 1921, 19232, in which also Whitehead's works are quoted. Indeed, Marcolongo was the

second, after Poincaré and before Minkowski, to use a four-dimensional 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. 344-349. Fundamental works for

a historical inquiry about the rise of special relativity, exploring a different view, are: A. I. Miller, A Study of

Henry Poincaré's 'Sur la dynamique de l'électron, Archives for the History of Exact Sciences, v. 10 (1973),

pp. 207-328; A. I. Miller, Albert Einstein's Special Theory of Relativity: Emergence (1905) and Early

Interpretation (1905-1911), Addison-Wesley, 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 Poincaré, see: A. A. Tyapkin, Expression of the General Properties of Physical Processes in the

Space-Time Metric of the Special Theory of Relativity, Soviet Physics Uspekhi, v. 15 (1972), pp. 205-229. For

6

...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 four-dimensional

space-time 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

a complete historical and theoretical analysis, see: E. Giannetto, Poincaré 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 relatività 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. Non-Newtonian 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. 221-228; E. Giannetto, Note sull'interpretazione della relatività 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. 107-127; E.

Giannetto, The Epistemological and Physical Importance of Gödel's Theorems, in First International

Symposium on Gödel's Theorems, ed. by Z. W. Wolkowski, World Scientific, Singapore 1993, pp. 136-147.23 E. Giannetto, Il crollo..., op. cit.24 See note 22.34 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 relatività di spazio e tempo, aut-aut, v. 254-255 (1993), pp. 33-

48. It was also present in M. Heidegger, Sein und Zeit, Niemeyer, Tübingen 1927, § 70. It is not known wether

he knew Whitehead’s 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, durée et

7

it explains this extension as merely the exhibition of the different ways in which

simultaneous occurrences function in regard to other time-systems. I mean that occurrences

which are simultaneous for one time-system appear as spread out in three dimensions because

they function diversely for other time-systems. The extended space of one time-system is

merely the expression of properties of other time-systems. According to this doctrine, a

moment of time is nothing else than an istantaneous spread of nature. Thus let t1 , t2 , t3 be

simultanéité, 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 1959-1972) 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,

Tübingen 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 space-time. 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 Weiszäcker, Zeit und Wissen, Hanser

Verlag, München 1992; D. R. Mason, Time in Whitehead and Heidegger: a Comparison, in Process Studies

(1975), pp. 87-105; 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, Time-

Space rather than Space-Time, in Diogenes, v. 123 (1983), pp. 3O-49; M. Capek, The New Aspects of Time.

Its Continuity and Novelties, Boston Studies in the Philosophy of Science, v. 125, Reidel, Dordrecht 1991. 1 A. N. Whitehead & B. Russell (1910-1913), 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,

8

three moments of time according to one time-system, and let T1 , T2 , T3 be three moments

of time according to another time-system. The intersection of pairs of moments in diverse

time-systems are planes in each istantaneous three-dimensional space... 35

In a more synthetic way, he had written in the introduction:

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. 202-209; A. Grünbaum, Philosophical

Problems of Space and Time , Reidel, Dordrecht 1973, pp. 48-65 and 425-428; J. D. North, The Measure of

the Universe. A History of Modern Cosmology, Oxford University Press, Oxford 1965 & Dover, New York

1990, pp. 186-197 and references therein; J. L. Synge, Relativity: The Special Theory , North-Holland,

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, Montréal preprint 1967; M. Bunge & A. G. Maynez, A Relational Theory of

Physical Space , International Journal of Theoretical Physics, v. 15, n. 12 (1976), pp. 961-972; A. Grünbaum,

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. 707-711; G. Gerla,

Pointless Geometry, in Handbook of Incidence Geometry, ed. by F. Buekenhout and W. Kantor, North-

Holland, Amsterdam 1991 and references therein; G. Gerla, Pointless metric spaces, Journal of Symbolic

Logic, v. 55 (1990), pp. 207-219; 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. 204-218; B. L. Clarke, Individuals and points , Notre Dame Journal of Formal

Logic, v. 26 (1985), pp. 61-75. For the very interesting M. Capek’s epistemological works see note 34.6 Guillelmi de Ockham, Opera Philosophica, voll. IV-V-VI, St. Bonaventure University, New York 1984-5;

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,

9

Position in space is merely the expression of diversity of relations to alternative time-

systems. Order in space is merely the reflection into the space of one time-system of the

time-orders of alternative time-systems. A plane in space expresses the quality of the locus of

intersection of a moment of the time-system in question (call it 'time-system A') with a

moment of another time-system (time-system B). The parallelism of planes in the space of

time-system A means that these planes result from the intersections of moments A with

moments of one other time-system B. A straight line in the space of time-system A

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 1875-90 & Olms, Hildesheim 1960; G. W. Leibniz, Leibnizens

mathematische Schriften, ed. by C. G. Gerhardt, Halle 1850-63; H. G. Alexander, The Leibniz-Clarke

Correspondence, Barnes & Noble, New York 1984; C. Huyghens, Oeuvres complètes de Christian Huyghens,

Aja 1905, v. X (correspondence 1691-1695), p. 609; D. J. Korteweg & J. A. Schouten, Jahresbericht der

Deutschen Mathematiker-Vereinigung, v. 29 (1920) 136. Very interesting for the connection of the rise of

Poincaré's special relativity and Leibniz is the following paper: H. Poincaré, Note sur le principes de la

mécanique dans Descartes et dans Leibnitz, in G. W. Leibnitz, La Monadologie, ed. by E. Boutroux and with

a note by H. Poincaré, Delagrave, Paris 1880, pp. 225-231; see also: E. Giannetto, Poincaré 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. 416-438; 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 1911-1920; E. Cassirer, Leibniz' System in seinen wissenschaftlichen Grundlagen,

Elwert, Marburg 1902 & Wissenschaftliche Buchgesellschaft, Darmstadt 1962; E. Giannetto, Il crollo del

concetto di spazio-tempo negli sviluppi della fisica quantistica: l'impossibilità di una ricostruzione razionale

nomologica del mondo, in Aspetti epistemologici dello spazio e del tempo, ed. by G. Boniolo, Borla, Roma

1987, pp. 169-224; E. Giannetto, Lectures on Leibniz, mimeographed paper, University of Messina, Messina

1988; E. Giannetto, Note sull'interpretazione della relatività 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 Space-Time. Absolute versus Relational Theories of Space and Time, The MIT

Press, Cambridge, Mass. 1989; M. Serres, Le système de Leibniz et ses modèles mathématiques, 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. 103-112.

10

perpendicular to the planes due to time-system B is the track in the space of time-system A of

a body at rest in the space of time-system B. Thus the uniform Euclidean geometry of spaces,

planeness, parallelism, and perpendicularity are merely expressive of the relations to each

other of alternative time-systems. The tracks which are the permanent points of the same

time-system are also reckoned as parallels. Congruence - and thence, spatial measurements -

is defined in terms of the properties of parallelograms and the symmetry of perpendicularity.

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 (1915-16), pp. 104-129.8 A. N. Whitehead, On Mathematical Concepts of the Material World , Philosophical Transactions, Royal

Society of London A, v. 205 (1906), pp. 465-525: already here the relation with Leibniz' ideas is evident.9 A. N. Whitehead, La Théorie Relationniste de L'Espace , Revue de Métaphysique et de Morale, v. 23

(1916), pp. 423-454.10 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 (1902-03), pp. 6-10.13 See, for example, M. Pieri, Della geometria elementare come sistema ipotetico-deduttivo, 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 relazionalità della relatività generale, 1990,

unpublished; see also, for example: A. Grünbaum, The Philosophical Retention of Absolute Space in Einstein's

General Relativity, The Philosophical Review, v. LXVI (1957), pp. 525-534.16 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.

303-349; J. A. Wheeler, Geometrodynamic Steering Principle Reveals the Determiners of Inertia, Princeton

preprint 1988.

11

Accordingly, position, planes, straight lines, parallelism, perpendicularity, and congruence

are expressive of the mutual relations of alternative time-systems.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

17 E. Mach, Die Mechanik in ihrer Entwickelung historisch-kritisch 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. 251-274; J. B. Barbour,

Forceless Machian dynamics, il Nuovo Cimento, v. 26 B (1975), pp. 16-22; J. B. Barbour and B. Bertotti,

Gravity and Inertia in a Machian Framework, il Nuovo Cimento, v. 38 B (1977), pp. 1-27; 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. 295-306; D. J. Raine, Mach's principle and space-time structure, Report on

Progress in Physics, v. 44 (1981), pp. 1151-1195 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 Relativitätstheorie. Eine Fälschung und ihre Folgen , Berlin-

New 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. 463-469; D. W. Sciama, P. C. Waylen & R. C. Gilman,

Generally Covariant Integral Formulation of Einstein's Field Equations, Physical Review, v. 187 (1969), p.

1762-1766; R. C. Gilman, Physical Review D, v. 2 (1970), p. 1400; D. Lynden-Bell, On the Origins of Space-

Time and Inertia, Monthly Notices of the Royal Astronomical Society, v. 135 (1967), pp. 413-428; 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 für

Naturforschung, v. 28 A (1972), 529-537; B. L. Altshuler, Mach's Principle. Part 1. Initial State of the

Universe, International Journal of Theoretical Physics, v. 24 (1985), pp. 99-118; D. J. Raine, Mach's

Principle in General Relativity, Monthly Notices of the Royal Astronomical Society, v. 171 (1975), pp. 507-

528; D. J. Raine & E. G. Thomas, Mach's Principle and the Microwave Background, Astrophysics Letters, v.

23 (1982), pp. 37-45; D. J. Raine, Mach's principle and space-time structure, op. cit. and references therein;

D. J. Raine & M. Heller, The Science of Space-Time, Pachart Publishing House, Tuscon 1981.25 See note 2126 See notes 20 and 21 and in particular: H. Poincaré, Sur la dynamique de l'électron, 1906, op. cit.; A. S.

Eddington, The Nature of..., op. cit.; E. Giannetto, Poincaré and the Rise of Special Relativity, in Atti del XIII

12

by its relation with the remaining part of the universe. If one consider another time-system

(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 time-system, such a motion cannot be

reduced to an overall rest in any other time-system: that is, it will transform itself into the

motion of the remaining part of the universe.38 Thus, one must say that Whitehead rejected

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 Körper, Annalen der Physik, s. 4, v. 17 (1905), pp. 891-921; A. N. Whitehead,

An Enquiry..., op. cit.; A. N. Whitehead, Space, Time and Relativity, Proceedings of the Aristotelian Society,

v. 16 (1915-1916), pp. 104-129.27 For a modern general-covariant 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, Four-Dimensional Formulations of Newtonian Mechanics and Their Relation to the Special and the

General Theory of Relativity, Reviews of Modern Physics, v. 36 (1964), pp. 938-965; P. Havas, Simultaneity,

conventionalism, general covariance, and the special theory of relativity , General Relativity & Gravitation, v.

19 (1987), pp. 435-453; C. Giannoni, Relativistic Mechanics and Electrodynamics without One-Way Velocity

Assumptions, Philosophy of Science, v. 45 (1978), pp. 17-46. However, these general-covariant 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 Levi-Civita. These general-covariant vectorial

formulations of classical mechanics, special and general relativity are given in: C. Burali-Forti & T. Boggio,

Meccanica Razionale, Lattes, Torino 1921; C. Burali-Forti & R. Marcolongo, Analyse Vectorielle Générale, 2

voll., Mattei, Pavia 1912-1913; C. Burali-Forti & T. Boggio, Espaces Coubes. Critique de la Relativité, Sten,

Torino 1924; R. Marcolongo, Relatività, op. cit.; see also: A. Pagano, Su di un'opera dimenticata di fisica di

Boggio e Burali-Forti, Mondotre/Quaderni, suppl. al n. 4/5 (1989), Laboratorio, Siracusa, pp. 107-132; G.

Boscarino, S. Notarrigo, A. Pagano, Geometria e Fisica, Mondotre/Quaderni, n. 8 (1992), Laboratorio,

Siracusa, pp. 61-123; 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 general-covariant 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 event-particles 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

13

only the einsteinian content of the so-called 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 at-a-

distance action (his theory of gravitation was not conceived as an action at-a-distance theory

as often stated).39

Therefore, motion is a relation and is relative to a time-system, 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

Whitehead, op. cit., pp. 165-207.30 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 30.35 A. N. Whitehead, The Principle of Relativity..., op. cit., pp. 54-55.36 A. N. Whitehead, The Principle of Relativity..., op. cit., pp. 8-9.37 For a comparison, see, for instance: G. Deleuze, Le pli, op. cit.39 A. N. Whitehead, The Concept of Nature, op. cit.38 A. N. Whitehead, The Concept of Nature, op. cit.40 E. Giannetto, Lectures on Leibniz, op. cit.; E. Giannetto, Lectures on Relativity, op. cit.

14

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 Poincaré's Special Relativity

- epistemology and - epistemology and ontology

ontology

- logic - logic

- set theory - set theory

- topology - topology

- non-metrical geometry - non-metrical geometry

- metrical geometry - electrodynamics

- kinematics - experimental defining operations

- dynamics - kinematics

- verification experiments - metrical chrono-geometry

- general dynamics

- verification experiments

Whitehead’s Special Relativity Einstein's General Relativity

- lifeworld experience - epistemology and ontology

- epistemology and ontology of - logic

interrelated events - set theory

15

- (electro-)kinematics - topology

- logic of events - non-metrical geometry

- set theory of events - gravitodynamics

- topology of events - electrodynamics

- non-metrical chrono-geometry - kinematics

- metrical chrono-geometry - metrical chrono-geometry

- (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 non-metrical geometry (and for classical theories indeed up to

kinematics) are almost completely unquestioned presuppositions to a physical theory, that is

meta-physical presuppositions.42 Even if Poincaré (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 non-metrical geometry) from

the consideration of physical processes (events), trying to overcome the foundationalist

paradigm of an epistemological or ontological (that is, subjectivistically or objectivistically

meta-physical) 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, non-separable 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

16

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 non-separable within the world as a whole.46

The appearance of an entirely physical (theoretical) practice represents an epochal change,

an epochal departure from Western meta-physics .47 However, as it can be seen from his

construction of special relativity, gravito-kinematics 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 event-particles, gravitational event-particles too.50

It is so clear that Whitehead’s formulation of special relativity is not equivalent to Poincare’s

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

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. 121-124; E.

Giannetto, L'epistemologia quantistica come metafora antifondazionistica, in Immagini Linguaggi Concetti,

ed. by S. Petruccioli, Theoria, Roma 1991, pp. 303-322; 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, Poincaré 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.46 A. N. Whitehead, The Concept of Nature, op. cit.; A. N. Whitehead, The Principle of Relativity..., op. cit.;

E. Giannetto, Notes on Non-Separability, op. cit.

17

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), Whitehead’s special relativity seems to be a new, different physical theory more than

a mere reformulation of Poincaré’s or Einstein-Minkowski’s 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.

45 E. Husserl, Die Krisis der europäischen Wissenschaften und die transzendentale Phänomenologie , 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

Non-Separability, 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. 1966.47 E. Giannetto, Heidegger and the Question of Physics, conference delivered in Veszprém, 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.

18

Notes and References

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: Philosopher-Scientist, 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, München 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. 15-124; E. B. McGilvary, Space-Time, Simple Location, and Prehension, in The

Philosophy of Alfred North Whitehead, op. cit., pp.209-240. 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. 44-

57; A. N. Whitehead, Discussion: The Idealistic Interpretation of Einstein's Theory, Proceedings of

Aristotelian Society, v. 22 (1921-1922), pp. 130-134; A. N. Whitehead, The Philosophical Aspects of the

Principle of Relativity, Proceedings of Aristotelian Society, v. 22 (1921-1922), pp. 215-223. The holistic,

philosophical perspective of Whitehead is indeed appealed in the debate about non-separability 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 Non-Separability, op. cit. See also note 34.

19