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 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 event-particle relations.7
1
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
2
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-
3
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 (nonfundamental ) 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 general covariant
4
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 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 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 :
...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 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 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 timesystems are planes in each istantaneous three-dimensional 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 time-system of the timeorders 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
6
moments of one other time-system B. A straight line in the space of time-system A
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.
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
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 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
7
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
8
- (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
9
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
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.
10
Notes and References
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, 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,
11
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, 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. 103112.
12
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
15
B. Russell, An Essaay on the Foundations of Geometry, Cambridge University Press, Cambridge 1897.
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.
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
13
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 spacetime structure, op. cit. and references therein; D. J. Raine & M. Heller, The Science of Space-Time, Pachart
Publishing House, Tuscon 1981.
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. 302324; 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. 129175. 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
14
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 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.
25
See note 21
26
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
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,
15
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 BuraliForti, 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
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.
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 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
16
critical analysis, in particular, on the seeming spazialization of time: H. Bergson, durée et 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.
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.
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. 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.
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 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
17
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 Non-Separability, op. cit.
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.
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.
18