poincare's conventionalism of applied geometry
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F . P . O GORM AN
POINCARl S CONVENTIONALISM OF
APPLIED GEOMETRY
1. Physical and Mathematical Spaces
MATHEMATICIANS
tend to look on pure geometry as an uninterpreted formal
system, or what Frege calls a formal theory. Applied mathematicians, on the
other hand, are often said to be concerned with the question of which
interpreted formal geometry is true of our world, or, to use Freges terminology,
which of the first-level geometrical propositions is true. From this latter point
of view, while the statement-forms of a purely formal system of geometry
are, as such, neither true nor false, the question of the truth of an interpreted
system does arise. Nicod criticizes Poincare for having overlooked this in
arguing for the geometrical conventionalism of applied geometryZ, and
Nagel holds a similar viewa.
Poincare, however, did not overlook this point; rather he held, at least
implicitly, that this kind of criticism does not apply in the case of the
interpretation of geometry within physics. Indeed Nagel himself in his
comments on Poincares conventionalism of applied geometry, seems to have
seen this, but he failed to realise its significance because of the dichotomy he
himself makes between pure and applied geometry. Thus he remarks that
PoincarC sometimes wrote as if the grounds for the conventional or definitional
status of applied geometry were identical with those for pure
geometrf
, but
he did not devote sufficient attention to this aspect of Poincarts thought and,
therefore, failed to see its true import. As we shall see, Poincare held that the
space studied in physics is the space of classical mechanics, and that this
space is the mathematical continuum.
In other words, the mathematical
continuum was, for Poincare, part of the model or interpretation of geometry
in physics, i.e. any assertion about physical space within classical mechanics
Strictly speaking, in Freges opinion, one should not speak of the interpretation of a for&
theory. However, since this manner of speaking is standard, I will retain it.
Cf . icod, Geometry and Induction. p. 17.
Cf. Nagel, The
Structure of Science,
p. 261.
Cf. Nagel, op. cit. p. 262.
Stud. Hist. Phil. Sci. 8, (1977), No. 4. Printed in Great Britain.
303
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presupposes that it is mathematically continuous. If Poincare was right in this,
then his argument for the conventionalism of pure geometry, (viz. the
metrical amorphousness of the real continuum5) holds also for applied
geometry, and despite Nagels views to the contrary, Poincare was perfectly
entitled to argue in this manner.
(a)
Poincar and
Lobachewsky s
Experiment
In his discussion of Lobachewskys parallax experiment, Poincare states
that, even if it were proved by experiment that the parallax of a distant star is
negative, one could choose either to abandon Euclidean geometry or to modify
the laws of optics by supposing that light waves are not strictly propagated in
a straight line, and that therefore Euclidean geometry has nothing to fear
from new experiments6.
Prescinding for the moment from the kind of change necessary in optics
for the rentention of Euclidean geometry in such a contingency, the question
may be asked whether Poincares statement above is intended to be an
argument for geometrical conventionalism in applied geometry. Despite
Nagels interpretation to the contrary, I believe that Poincare did not ground
his geometrical conventionalism on considerations of this kind. If he did so
his argument is invalid since, as Russell points out, the question as to
whether the metrical axioms of Euclidean geometry are conventions, or are
true or false, is logically distinct from the question as to whether we can
verify
whether they are true or false. If Poincares statement above is taken as his
only argument for geometrical conventionalism, all it proves is at most that
we cannot verify the truth of any geometry. It also implies that verifiability is
the ultimate ground of Poincares geometrical conventionalism. But this is to
read too much of logical positivism into Poincares works. If Poincare intended
to argue from verifiability, one would expect him to explicitly say so in his
reply to Russells criticism. But in this reply Poincare does not refer to
verifiability; on the contrary, he explicitly rejects Russells contention that
his arguments merely show that some one geometry is true but we cannot
verify which onelo.
Indeed Poincare, in his reply to Russell, expressly rules
out the possibility of arguing from verifiability to geometrical conventionalism,
and therefore his claim that one can retain Euclid despite apparently adverse
Poincare maintained that, topologically speaking, there is nothing in the nature of the real
continuum which singles out the Euclidean metric from the other possible metrics indicated by
the distance function d(x,y), and hence the real continuum is metrically amorphous.
Poincart,
Science and Hypothesis,
p. 73.
Cf. Nagel, op. cit. p. 262.
Cf. Russell, Sur les Axiomes de la Cikometrie, p. 685.
sin particular one would be reading too much of Reichenbach into Poincart, since there is no
doubt that Reichenbach rests his own thesis of geometrical conventionalism on testability (c$ The
Phitosophy of Space and Time, p. 16).
Cf.
Poincare, Sur les Principes de la Geometric, p. 74.
Cf. ibid.
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experimental evidence cannot be interpreted as an argument for the
conventionalism of applied geometry.
In making such a claim, Poincare is assuming that he has
already established
the thesis of geometrical conventionalism in the case of applied geometry,
and is merely spelling out its consequences. What he is saying is that, if one
were to discover, for instance, a negative parallax for a distant star it would
not be necessary to conclude that physical space is Riemannian, for, since
congruence is a matter of definition, it would still be possible to retain Euclid
provided one made the necessary modifications in our optics. If this
interpretation is correct, we have yet to discover Poincarts reason for
geometrical conventionalism in the case of applied geometry. His reason is the
following. Within classical mechanics, and hence within classical physics,
space is taken by convention to be mathematically continuous; and, because
such a continuum is metrically amorphous, congruence is a matter of
convention. We shall now discuss these points more in detail.
(b) Classical Space and the Mathematical Continuum
In his discussion of Russells geometrical empiricism, Poincare states that
the distance, for example, between London and Paris is not an absolute datum
of experiencej2. Poincare is here clearly talking about the material world,
and, to use Russells words, is contending that distance is not an absolute
datum preexisting measurement, as America, for instance, pre-existed its
discovery. In other words, Poincare is assuming that physical space is
metrically amorphous, and that therefore the choice of congruence is a
matter of definition13.
He makes the same assumption in Science and
Hypothesis, when he states that the question of the self-congruence of a
transported rod is a matter of definition14. The reason why Poincare held
physical space to be metrically amorphous is because he interpreted the
properties of physical space in the light of classical mechanics. While
Poincare held that classical mechanics is an experimental science, he also
maintained that it contains certain conventional elements, and among these
elements he mentions Euclidean geometry (which he caIls a kind of convention
of language) with its claim that space is continuous and therefore metrically
amorphous.5 In other words, Poincare accepted the physical space of classical
mechanics as being, by convention, mathematically continuous and as such
metrically amorphous.
In this context it is quite clear what Poincare meant when he said that
Euclidean geometry has nothing to fear from new experiments. Because the
Euclidean or continuous space of classical mechanics is conventional, it has,
Poincart, op. cit. p. 81.
He explicitly makes this claim in The
Value ofScience,
p. 37.
Cf. Poincart,
Science and H ypothesis,
p.
45.
Cf. Poincart, op. ci t. p. 89.
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according to Poincare, no causal, or any other empirical influence on physical
bodies and as such is not open to experimental verification. Thus Poincart
holds that there is an absolute dichotomy between physical space and physical
bodies: because of this dichotomy, experiments, which of their nature have to
do with physical bodies, cannot give us any information about the relations
between these bodies and space or, a fortiori, betweenthe points of space thus
understood6.
That Poincare did in fact hold this kind of dualism between
space and matter,
and that it is not merely a possible explanation of his
position, is clear from the fact that he explains how this dualism comes about*.
The relationships between bodies as given to us in experience are, he says,
highly complex and therefore, instead of directly considering the complex
relation of one body A to another body
B, we
introduce an intermediary, space,
thereby envisaging three distinct relations, viz. that of body A with the figure
A of space, that of body B with the figure B of the same space, and that of
the two figures A and B to each other. The advantage of this, according to
Poincart, is that the relations between A and B are simple in comparison with
the relations between A and
B,
the aggregate of principles governing the
former being expressed in some geometry.
Here Poincare expressly adopts
the dualistic theory of matter and space mentioned above, and this precisely is
why he maintained that experiment cannot give us any information about the
relationship between bodies and space and the mutual relations between the
different parts of space.
Against this interpretation of Poincare, however, it could be objected that
Poincare himself clearly distinguishes between laws which he considered to
be experimental, and principles, which he considered to be conventionaPO,
and calls the expression of a relation between a body A and its corresponding
spatial figure A a law, and the expression of a relation between the spatial
figures A and
B
a principle. Hence, it could be argued, his dichotomy
between space and matter proves, at best, that experiment cannot give us any
information about the mutual relations of the various points of space, but
not that it cannot give us any information about the relations between bodies
and space.
In my opinion, however, Poincarts conception of the relation between a
body and its corresponding spatial figure does not bear out this objection.
Cf.
Poincare,
Science and Z-Z ypot hesis,
p. 79-84, and Sur les Principes de la Geometric,
pp. 79-86.
It is interesting to note that Quine also draws our attention to this distinction and refers to it
as the dualistic theory of spatio-temporal reality (cJ
Word und O bject,
p. 252). However,
Poincare, unlike Quine, was not concerned with the problem of the ontological status of the
points of such a space, since for him mathematical existence simply meant freedom from
contradiction.
Cf. Poincare, The Val ue f Science, pp. 125-126.
Cf.
ib id.
Cf . op. ci t .
pp. 123-124.
Cf. op. cit .
p. 125.
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The law expressing the relationship between, for example, a natural solid and
its corresponding spatial figure is that the natural solid moves approximately
according to the structures of the Euclidean group of transformations. But, in
Poincares view, this does not express a relationship between the natural solid
and some
actual
metrical space studied by physicists. He gives two reasons for
this. First, one may encounter the Euclidean group of transformations in
disciplines which have nothing at all to do with metrical spaces; hence one
does not have to know, or to assume, the metrical properties of physical
space in studying the structure of the group formed by the movements of
natural solids. Secondly, while it can be verified experimentally that solids
move approximately according to the structure of the Euclidean group of
transformations without making any assumption about the metrical structures
of space, it is logically possible to construct a non-Euclidean solid and to verify
experimentally that it moves approximately according to the structure of, for
instance, Lobachewskian geometry. In such a case the scientist would conclude,
not that physical space is both Euclidean and Lobachewskian, but rather that
such experiments give us information only about physical bodies, and not
about their relationship to the metrical space of classical mechanicsz3.
It has recently been claimed by Griinbaum that the theory that the continuity
of physical space is conventional is unfounded24. Though he admits the close
relationship between the theory that physical space is continuous and
Poincares theory of geometrical conventionalism, he maintains that
there is broad inductive evidence to support the former theory, and that
therefore this theory is not conventional. The principal reason he gives for
this view is the lack of any alternative convention which would express the
same total body of experimental findings, i.e. the fact that, in principle, no
mathematically discontinuous set of theories has been shown to be as
empirically viable as those based on the mathematical continuumz5. He fails,
however, to show that, in the case of classical physics, there is any inductive
evidence for the continuity of physical space. Indeed Grtinbaum himself
admits that this is not directly verifiable, but he offers no criteria for its indirect
Cf. SW les Principes de la Geometric, p. 82.
*Cf.
Science and H ypot hesis,
pp. 80-84. In these pages Poincare gives an account of how we
may experimentally verify that a natural solid moves according to the structure of the Euclidean
group of transformations and of the difference between its movements and the movements of a
non-Euclidean solid.
Grtinbaum,
Phi l osophi cal Probl ems of Space and Time,
pp. 334-337. Since he justifies this
claim prior to his discussion of Einsteins general theory of relativity, it is clear that he envisages it
to be applicable to pre-relativity physics, which is the domain of Poincares geometrical convention-
alism.
Cf. Grttnbaum,
op. cit .
p. 337. He concedes, however, that his argument merely shows the
unfoundedness, and not the falsity, of the conventionalist conception of the continuity of space;
indeed he appears to weaken his objection to the gratuitousness of the conventionalist conception
of continuity, by admitting that his argument shows merely that the advocates of the convention-
alism of continuity are merely offering a programme to be completed (c$ ibid).
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verificationz6,
so that it is difficult to see in what his broad inductive evidence
consists. Moreover, if in classical physics the postulate of the continuity of
space is, as Poincare maintained, simply logically conjoined to the basic
empirical propositions of physics, then, while it could be argued that the
propositions deduced from this combination are empirical, this empirical
character would not constitute broad inductive evidence for the continuity of
space. Hence the primary issue is whether Poincares account of the relationship
between space and physical objects in classical physics is correct, and
Grtinbaum does not discuss this. Finally, even if we agree with Grtinbaum
that noncontinuous spaces have not been shown, even in principle, to be as
empirically viable as continuous ones, this does not imply that the theory
that the continuity of space is conventional is gratuitous. For the continuum
used by classical physics is the real number continuum and, as Poincare
maintains, this continuum is merely one kind of continuum among others. In
other words, there are other kinds of continua than that of the real number
system, and any of these may be adequate for the construction of classical
physics*. Grtinbaums argument fails to take account of this.
(c) Models and I nterpretations of Formal Systems
We saw above the distinction made by Nicod and Nagel between the
statement-forms of geometry understood as a formal system, which are neither
true nor false, and these same statement-forms as interpreted, which are
geometrical propositions and as such either true or false. This distinction has
been used as an argument against Poincarts theory of (applied) geometrical
conventionalism but, in my opinion, it cannot be used against Poincarts
theory. In the first place, his argument in pure geometry from implicit
definitions to geometrical conventionalism has nothing to do with purely
formal systems. For example, the so-called axioms of congruence (as given by
Hilbert) combined with Euclids parallel postulate, used by Poincare to define
congruence, are not statement-forms; rather, to use Freges terminology, they
are defining characteristics of the concept equality of length. Secondly,
geometry, as a purely formal system, is not given a physical interpretation in
classical physics. In the case of three dimensional space, for instance, real
number co-ordinates are substituted for the point-variables of the formal
system, which is a mathematical and not a physical, interpretation. This is
clear from the following considerations. First, there is a rigid dichotomy
Ayer, for instance, in Language, Truth and Logic does offer us such a criterion but, as is well
known, any proposition whatsoever, according to this criterion, is indirectly verifiable.
Cf. Poincare,
Science and Hypothesis,
p. 29. He actually points out that the work of Du Bois
Reymond offers us an account of a continuum of higher order than the real number continuum.
Also Rogers argues that the continuum of the rational numbers may suffice for the construction of
a space adequate for classical physics (cJ Rogers, On Discrete Spaces, American Philosophical
Quurterly, (1968), 118-120).
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within classical physics between matter and space, and thus the nonempirical
kind of interpretation just mentioned is possible. Secondly, one of the basic
claims within classical physics is that space is continuous, and that each point
of space may be assigned a Cartesian co-ordinate. Thus the formal system of
geometry is given a non-empirical interpretation. Finally, since within classical
physics there is no causal or other empirical relation between space and matter,
one must interpret the point-variables in the above nonempirical fashion. It
would seem, therefore, that Nicods and Nagels distinction between
geometry as a formal system and as an interpreted empirical system is
inapplicable to classical physics, and thus misses the point involved in
Poincares theory of geometrical conventionalism.
The first reason mentioned above,
viz.
that in classical physics there is a
rigid dichotomy between space and matter may be stated in a more general
way as follows. Any empirical theory may be considered, from the point of
views of syntax, to be a formalized language. From this point of view, the
distinguishing characteristics of an empirical language are given by semantical
considerations, especially by the condition that experience must decide the
truth-value of some at least of its interpreted theorems, which presupposes
that we have a material model for an empirical language. Now, if we apply
these considerations to classical physics, which in Poincares view is an
empirical language, we find the situation is not as simple as it might seem at
first. Nicod and Nagel assume that there is one and only one material model (or
fragment of reality) of which the language of classical physics speaks. But
Poincares theory of the dichotomy between space and matter implies that
this assumption does not hold. The factors which decide what the language of
classical physics actually speaks about do not determine a unique model;
rather, to use Przeleckis phrase, they determine a family of models. This
family consists of at least two distinct models, namely the mathematical
continuum, with its nonempirical relations between its elements, and the
natural solids and their motionsSo. Hence, the formalized language of
geometry is not, contrary to Nicods and Nagels assumption, given a physical
interpretation within classical physics, and hence the continuity of the space
of classical physics is not an empirical issue.
In connection with formal systems and their interpretation, it is interesting
to compare the view of Popper in The Logic of Scientif ic Discovery with that
of Poincare. Popper maintains that there are two distinct ways of viewing any
8Przelecki, The Logic of Empir icul Theories, p. 18. Przelecki maintains that this seems to be
true of all empirical languages.
AS we mentioned above, once we assert that space is continuous the argument for the
geometrical conventionalism of pure geometry holds also for applied geometry. Moreover, the
fact that these two models are so different is, as we shall see later, Poincares principal reason for
distinguishing between geometrical conventionalism and the conventionalism of the principles
of mechanics.
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system of axioms,
viz.
either as conventions or as hypothese?. Taken as
conventions, the axioms may be compared to algebraic equations (e.g.
x + y = 12), and when numbers are substituted for the variables in such
equations we sometimes get true, and sometimes false, propositions. For this
reason Popper calls the axioms statement-functions, and he maintains that if we
decide, with respect to some one statement-function, to admit only such
substitutional values as turn this function into a true statement, a definite class of
admissible value-systems is defined, and we may regard the axioms in question
as implicit definitions of this c1a.s~~~.ach class which satisfies a system of axioms
is a model of that system of axioms, and the substitution of such a model will
result in a system of analytic statements, since, as Popper points out, it will be
true by convention. According to Popper, Poincares conventionalism of applied
geometry is a particular instance of this kind of approach. In this Popper is
manifestly wrong. In the first place, Poincare explicitly maintains that it is not
always possible to view any system of axioms as either conventions or hypotheses.
He insists that, for example, Peanos postulates cannot be conventions (in his
sense of the word). This results from the fact that Poincart, unlike Popper,
imposes restrictions on the notion of an implicit definition: an axiomatic system
may be viewed as an implicit definition of a notion only if it is consistent and is the
one and only definition of the notion in questionJ3. Secondly, if as we maintained
above, Poincares implicit definition of congruence is in no way based on the
logical notion of a purely formal system it is unlikely that it is based on
Poppers statement-functions, assuming the latter to be distinct from the
former. Finally, while Poppers conventions may define a family of models,
it cannot, unlike Poincares conventions, define the predicates indicated by the
predicate variables of the statement functions. For instance, in the elementary
case where each member of the family of models defined by a formal axiomatic
system is isomorphic with every other member, the predicates indicated by the
predicate-variables of the system are, to use Przeleckis phrase, merely
determined up to an isomorphism, which amounts to saying that only their
structure is determined. In such a case, however, the models in question must
be finite34, and in a more complicated case, the models do not even determine
the structural properties of the predicates and hence it is impossible to say that
they define the predicates indicated by the predicate variables of the system. On
the other hand, the axioms of congruence combined with Euclids postulate
are used by Poincare to define distance which is a predicate indicated by a
predicate variable of the formalized language of Euclidean geometry.
3CJ The Logic of Scientif ic Di scovery, p. 12.
Cf. bid.
33Science nd Method,
pp.
151-154.
Przelecki,
op. cit.
p. 27.
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(d) The Empirical Element in Geometry
So
far we have argued that, for Poincare, the continuity of physical space is
a matter of convention and as such is not empirically verifiable, and that
within this continuum the choice of congruence is also conventional. Because
of this latter thesis Reichenbach claims that Poincare was an extreme conven-
tionalist, i.e. that he held that it is impossible to make objective metrical
statements about real space35.
The implications of this claim may be understood
if we consider Reichenbachs own theory of qualified geometrical convention-
alism. Reichenbach agrees with Poincart that the choice of congruence is a
definitional, and not an empirical issue, but he maintains that, once this
co-ordinative definition is fixed, the geometrical structure of the physical
space relative to it is an empirical one. As he himself puts it, once the definitions
have been formulated, it is determined through objective reality alone which is
the actual geometry36.
Thus Reichenbachs view of the the nature of physical
geometry emphasizes, on the one hand, the empirical character of this geometry
and, on the other, the limited, but important role of conventions in its actual
ascertainment. Poincare, according to Reichenbach, failed to recognise this
empirical character and he quotes the following passage from
Science and
Hypothesis in justification of this: To sum up, whichever way we look at it, it
is impossible to discover in geometrical empiricism a rational meaning37.
Carnap and Griinbaum, however, disagree with this interpretation of
Poincares geometrical conventionalism. According to Carnap, while Poincare
emphasized the conventional aspect of the structure of physical space in
stating that the rules of measurement should be adjusted in the light of
experimental results, he also clearly saw that, if the rules for the measurement
of length are defined, the geometrical structure of physical space is an empirical
issue38. He does not, however, produce any evidence to substantiate this claim,
and while it would seem that he is right in holding that the rules of measure-
ment of length were, for Poincare, a matter of definition, this does not imply
that Poincare held that the question of the structure of physical space is an
empirical one relative to ones rule of measurement. Unlike Carnap, Grtinbaum
discusses in detail Reichenbachs interpretation of Poincare as an extreme, or,
if one prefers, pure geometrical conventionalisF. He points out that
Reichenbach has taken Poincares statement quoted above out of its proper
35Cf.
Reichenbach,
The Phi l osophy of Space and Time,
p.
36.
Op. ci t .
p.
47.
Science and Hypot hesis,
p.
79.
Nagel (c$
op. ci t .,
p. 261) and Weyl
(cf. The Phi l osophy of
M at hemat i cs nd Nat ural Science, D. 34)
or examele. also interpret Poincart in this way.
Cf. Carnapsintroductory re&rks;o Reichenbachs;
The Phi l osophy of Space and Ti me,
p. 6.
J8Griinbaum,op.
cit.
pp. 127-131.
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context, namely Russells extreme geometrical empiricism40, and argues that
in this context, the statement does not imply that Poincart was a pure
geometrical coventionalist. On the positive side Grtinbaum quotes a long
passage from Poincarts reply to Russell as evidence for his own interpretation
of Poincare as a qualified geometrical empiricist. However, he does admit
that there are certain passages which do not fit in with this interpretation of
Poincart, notably Poincares statement in Science
and Hypothesis
that the
axioms of geometry are not experimental truths, and he suggests that there
is an apparent contradiction in Poincares works. In order to avoid this
contradiction he gives the following as a possible interpretation of Poincarts
position. There are practical rather than logical obstacles which frustrate the
complete elimination of perturbational distortions, and the resulting
vagueness as well as the finitude of the empirical data provide scope for the
exercise of a certain measure of convention in the determination of a metric
tensor43.
There is, however, no need for any such interpretation. Grtinbaum, as we
have already seen, does not take sufficient account of Poincares view of the
relationship between physical space and geometrical space, a view which
provides the solution to Griinbaums apparent contradiction. As we saw
above Poincare held for a rigid dichotomy between the relations between the
elements of the space of classical physics and the relations between material
bodies, and thus thought it possible to distinguish between the structural
relations of physical space and the metrical relations between the material
bodies of our universe. Therefore, since he held that physical space is the
mathematical continuum, he believed, on the one hand, that the question of
the geometrical structure of such a space is
not an
empirical issue4, and, on
the other, that the assertion (or denial) of a metrical relation between material
bodies
is an
empirical issue4. Thus in saying, for instance, that the distance
between London and Paris is not an absolute datum of experience, Poincart
did not mean that such statements are wholly conventional. Statements about
ORussell maintained that the ascertainment of the geometry of physical space is an immediate
empiricalssue in no way dependent on the prior stipulation of a co-ordinative definition. Similarly,
Gauss and Lobachewsky were also unaware of this Reichenbachian-type of qualified geometrical
empiricism.
Cf. Grtinbaum, op.
cit.
pp. 129-130, and Poincart, Sur les Principes de la G&rm&ie, pp. 85-
86.
The passage in question is the following. Should we conclude that the axioms of geometry are
experimental truths? - If geometry were an experimental science, it would be subject to continual
revision. Nay, it would from this very day be convicted of error, since we know that no rigorously
invariable solid exists. The axioms of geometry therefore are - conventions - Thus it is that the
postulates can remain rigorously true -
(Science andHypothesis,
pp. 49-50).
3Grtinbaum, op. cit.
p.
130.
Thus he can claim that no geometry is either true or false and that the axioms of geometry are
not experimental truths.
45PoincarC, Science and Hypothesis, p. 97.
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the metrical relations between material bodies contain a conventional and an
empirical element: once the choice of congruence is made, the ascertainment
of the distance is an empirical issue46. In this sense, Poincark may be called a
qualified geometrical empiricist, but, because of the manner in which he
conceived the relationship between physical space and geometrical space, this
does not imply that the ascertainment of the geometrical structure of physical
space was for him an empirical issue. In this way the apparent contradiction,
mentioned by Grtinbaum, disappears.
2. Geometrical Conventionalism and the Conventionalism of Classical
and Special Relativity Physics
We have seen that PoincarC understood physical space to mean the
mathematical continuous space of classical mechanics, and that he combined
this with a dualistic theory of space and matter, thereby implying that the
conventionalism of pure geometry also holds good of applied geometry. We
shall now develop PoincarCs thesis of applied geometrical conventionalism by
comparing and contrasting it with the conventionalism he attributed to the
principles of mechanics, to the measurement of time, and, more generally, to
the principles of physics. We shall see that, while PoincarC maintained that
the principles of physics share the conventional character of the geometrical
postulates, he held the former are more directly based on experience47. Thus
PoincarCs applied geometrical conventionalism is not, as Popper maintained48,
simply a particular instance of the general conventionalism of the principles or
laws of science. By way of conclusion, we shall examine PoincarCs interpretation
of the impact of the special theory of relativity on his thesis of geometrical
conventionalism.
(a)
Geometrical Conventionalism and the Conventionalism of the Laws of
Science
As
is well known, Poincark not only defended the thesis of the conventionalism
of geometry, but also developed the more general thesis of the conventionalism
of the principles of classical mechanics and of the other principles of classical
physics4g. Despite Poppers interpretation to the contrary, this latter
conventionalism cannot be identified with the conventionalism of the laws of
science, i.e. with the thesis that the laws of science are conventions or implicit
definitions. Indeed PoincarC himself expressly states that the laws of science
Cf. PoincarC, Sur Les Principes de la GComttrie, p. 81. This point is also borne out by the
fact that Poincare maintained that mechanics is an empirical science. If particular distance state-
ments are completely conventional then the concepts of velocity and acceleration, which are
essential to mechanics, are also completely conventional.
Poincart, Science and Hypothesis, p. 26.
Cf. Popper, The Logic of Scientific Discovery, pp. 78-84.
Cf. Poincart, op. cit. pp. 91-105. The classical mechanical principles in question are Newtons
laws of inertia, of the equality of action and reaction, and of acceleration.
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are not conventions or implicit definitions, and he goes so far as to contrast his
own thesis of geometrical conventionalism with a typical example of a
scientific law in order to bring out the differences between the two5.
For the present let us concentrate on this contrast. The typical example used
by Poincare is the law that phosphorus melts at 44 degrees centrigrade5. He
makes three points of distinction between this law and Euclids postulate which
he holds to be a genuine convention. First, he admits that both Euclids
postulate and the above law are known to be free from contradiction, and
that therefore both fulfil the mathematical formulation of the first condition
for the procedure of definition by postulates52. However, he insists on Mills
formulation of this condition as applied to the domain of material objects,
i.e.
that a definition must guarantee the existence of the object defiiedS3. In the
case of phosphorus, since existence means material existence, the statement
phosphorus melts at 44 degrees centigrade, however consistent, cannot guarantee
thiss4. His second point is that if a scientist accepts the above law as a definition
of phosphorus, he is contravening the second condition for the procedure
of definition by postulates, since, in using laboratory samples to define
phosphorus, he is using two different definitions of the same symbo155.
Cf. Poincart,
The Value of Science,
p. 125, and
Science and Method,
pp. 171-176. Also
Poincare makes it abundantly clear that his conception of a scientific principle is quite different
to that of a scientific law (cJ
Scienceand Hypothesis,
pp. xxvi, 150-153).
He borrows the law concerning phosphorus from Le Roy, who uses it as a typical illustration of
a law which functions as a disguised definition in order to justify his own thesis of nominalism
(cJ Poincart, The Value ofScience, p. 122).
5zCJ supra, p. 8.
Cf. Poincart, Science and Method, p. 172.
5While Poincares argument here is valid, its truth depends on the identification of mathematical
existence with freedom from contradiction and, in the opinion of the logisticians, this is too narrow
a view.
Cf. Poincare, op. cit. p. 174. Poincares point here appears to be vague and inconclusive. It is
not clear whether he understands the definition by means of laboratory samples to be a particular
instance of the procedure of ostensive definition, or to indicate some other kind of procedure. In
other words, he does not explain the kind of definition involved in the use of laboratory samples,
and such an explanation is crucial to his point. It could beargued that the samples play the following
role: the label marked phosphorus on the sample is used as, what Geach calls, a name (cf.
Reference and Generality) i.e. it is used to acknowledge the presence of the thing. In addition,
phosphorus is also what Geach calls a substantival term, i.e. the expression the same phosphorus
supplies a criterion of identity or, if one wishes, the term phosphorus conveys a nominal essence
(c$ Geach, op. cit. pp. 38-42). Moreover, if one holds that a statement of the nominal essence
required to identify a specific object, e.g. phosphorus, must contain one or more of what Locke
calls secondary qualities (c$ Swinburne, Space and Time, p. 17), it is possible that, even if the
above law (viz. that phosphorus melts at 44 degrees centigrade) does not express a real essential
property of phosphorus, it may, nevertheless, figure in the nominal essence and thus be required
as a part of the definition. In this connection Poincart makes the point that the above law is open to
verification in the following sense: all bodies which possess such and such properties in finite
number (namely, the properties of phosphorus given in chemistry books with the exception of its
melting point) melt at 44 degrees centigrade. But this point is more compatible with the nominal
essence interpretation of the naming of the laboratory samples than with an ostensive definition
of phosophorus in terms of these same samples. This beingthe case, Poincare should have explained
why the law in question is excluded from the defining characteristics of phosphorus, while other
laws are, at least implicitly, included in it. This he fails to do.
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Thirdly, Poincare contrasts the reaction of a scientist who, for instance,
discovered a negative parallax of a distant star, thereby apparently showing
that a light ray does not satisfy Euclids postulate with his reaction to the
discovery that phosphorus melts at 43.9 degrees centigrade, and not at 44
degrees centigrade, as stated in the above law. In the former case he might
conclude either that a straight line is, by definition, the path of a light ray and
thus does not satisfy Euclids postulate or, on the contrary, that, since a
straight line, by definition, satisfies Euclids postulate, the path of a light ray is
not rectilinear. Similarly, in the latter case, he might conclude either that
phosphorus melts at 43.9 degrees centigrade or, on the contrary, that, since
phosphorus is, by definition, that which melts at 44 degrees centigrade, the
substance called phosphorus and which melts at 43.9 degrees centigrade is not
really phosphorus. But in fact Poincare states that the scientist would adopt
the second alternative in the first case, and the first alternative in the second56.
The reason he gives is that the scientist does not (and cannot) change the name
of a substance every time he adds or subtracts a decimal to its melting points.
This reason, however, is limited in scope and is not typical of all the laws of
science: while it is clearly applicable to laws concerning the melting points of
substances and the boiling and freezing points of liquids etc., its relevance to
other laws of science (for instance, laws which assert a relation of functional
dependence between two or more variable magnitudes associated with stated
properties or processes) is by no means apparent. It is possible that Poincare
himself recognized this limitation, for in
The Value of Science
he gives us a
more general account of the conventional and empirical aspects of the laws of
science, and it is to this account which we now turn5,
According to Poincare, when a law of science has received sufficient
experimental confirmation, one may adopt two attitudes towards it: one may
accept it as a law and, as such, open to future verification and revision, or
else one may elevate it into a principle, which by definition (and therefore by
convention) is not open to revision. For Poincare, however, the latter step is
not accomplished simply by stating that the law is a convention; rather, where
the original law expresses a relation between observable terms A and B, one
introduces a more or less fictitious and abstract term C. In this way one
obtains two relations: one between A and C which is assumed to be rigorous,
and this is the principle; the other between C and
B
which remains a law subiect
Cf. p. cit. pp. 174-176.
I this instance Poincare is, in my opinion, correct. Usually scientists do not change the name
of a
substance
simply because a decimal point has been added to its melting point, and if one assumes,
as indeed Poincare does, that generally speaking classical definitions are used within the positive
sciences (Max Black, for instance, argues against this view, cJ Problems of Anulysis, pp. g-14),
then this law does not form part of the definition of phosphorus.
Cf.Poincare, The Value of Science, pp. 122-127.
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to constant testing5. He illustrates this procedure as follows: the law the stars
obey Newtons law may be analysed into the principle gravitation obeys
Newtons law, and the experimental law
gravitation
is the only force acting on
the stars. In addition, he maintains that it is considerations of convenience
which decide whether or not one introduces these conventional principles into
the sciences. Hence, for Poincare, there appears to be no essential difference
between the conventional status of geometry and that of the principles of
science. Brunschvicg, for example, thus maintains that this kind of analysis of
scientific principles enables us to interpret without any fear of equivocation
Poincares remarks about the conventionalism of geometry in Poincares SW
les hypothkses fondamentales de la GkomktrieO, and Nagel makes the same
pointE . However, Brunschvicg (but not Nagel) draws attention to the
limitations, acknowledged by Poincare himself, in the analogy between the
principles of geometry and those of science, and especially those of mechanicssZ.
According to Poincare the limitations in question are due, first, to the fact
that, even though the principles of mechanics are conventional, they are
initially verified by mechanical experiments, and secondly, to the fact that,
despite the conventional nature of these principles, mechanics remains an
experimental science. The situation is different in the case of geometry: the
experiences which initially suggest Euclidean geometry are much more indirect,
consisting, for example, of physiological, kinematical and optical experiments;
and secondly, geometry itself is not an experimental science, but the study of
specific groups of transformationss3.
Hence, while in mechanics the separation
of principles from laws is artificial, in the case of geometry it is necessary to
recognize that it would have been difficult not to draw this distinction that is
pretended to be artificial.
And Poincare remarks that as one moves from
geometry to mechanics and from mechanics to physics, the radius of action,
so to speak, of principles diminishes, and thus in the latter two cases, unlike
the former, there is no reason either for separating the principles proper to these
sciences from the sciences themselves, or for considering these sciences to be
solely deductive.
Poincarts insight, expressed in this latter point, may be formulated in a
more technical fashion in the following way, not unlike what came to be called
the hypotheticodeductive method, and which has itself come to be the subject
of searching critiques in recent years. Generally speaking the extra-logical
5BPoincarks postulation of the continuous space of classical mechanics, Which we discussed
above, fits into this schema; indeed he introduced it in this same context.
Cf. Brunschvicg, Henri PoincarC: Le Philosophe, Revue de Mktaphysique et de Morale
(1913), 595. Poincarts article may be found in Bulletin de la Sock mathdmatiques de France
(1887), X3-216.
@CJNagel, The Structure of Science, pp. 260-261.
Cf.
Brunschvicg,
op. cit.
p.
596.
Cf. Poincark, op. cit. p. 216.
La Valeur de la Science, p.
242; my own translation.
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vocabulary of the empirical language of mechanics and physics may be divided
into two parts: an observational vocabulary, i.e. the predicates of the language
which are interpreted in a non-verbal (e.g., ostensive) way, and a theoretical
vocabulary, consisting of predicates which cannot be directly interpreted in
this way. For example, to use Poincares own illustration, the predicate
gravitational, while it directly applies to observable objects, does not directly
ascribe to them observable properties. However, since the interpretation of the
theoretical predicates, unlike that of the observational predicates, is not
direct, the question arises as to how exactly they are to be interpreted. Now,
Poincares point is that the languages of mechanics and physics are empirical,
and that consequently their theoretical predicates must be connected in some
way with their already directly interpreted observational predicates. According
to certain logicians this connection is effected by a set of statements called the
meaning postulates of the theoretical predicates. These postulates contain all
the theoretical predicates and all or some of the observational ones. In addition,
they must fulfil the following condition: the theoretical terms must be interpreted
in such a way that the meaning postulates be trues5. If this is so, then meaning
postulates, to use Poincares phrase, are removed from the fray, i.e. they are
not empirically verifiable, and as such conventional.
Poincares principles of mechanics and physics are meaning postulates of this
kind. But, in his opinion, while some of the geometrical predicates (for
example continuous) are theoretical,
i.e.
cannot be given a non-verbal
interpretation, they are not interpreted in the above manner,
i.e.
their
interpretation within classical mechanics does not connect them with
observational terms; rather their model is the mathematical continuum of the
type of the real number system as understood in pure mathematics. Hence
applied geometry, unlike mechanics, has no logical connection with observational
procedures, and so, to use Poincares terminology, its radius of action is
unlimited in this sense. In other words, while applied geometry, in so far as it
makes use of meaning postulates, shares in the general character of the
conventionalism of mechanics and physics, the kind of meaning postulate used
in applied geometry is essentially different to that of mechanics and physics.
For this reason, as we stated above, Poincares geometrical conventionalism is
not simply a particular instance of the general thesis of the conventionalism of
the principles of science.
(b) Geometr ical and Chronometr ical Conventionalism
In the last section we were concerned with the analogy between Poincares
geometrical conventionalism and the conventionalism of mechanics and
physics; in this section we shall discuss the analogy between his geometrical
and chronometrical conventionalism. For Poincare, this latter conventionalism
65Cf.Przelecki, The Logic of Empirical Theories, p. 48.
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was the outcome of his reflections on two problems: first, can we transform
psychological time, which is qualitative, into a quantitative time?, and,
secondly, can we reduce to one and the same measure facts which transpire
in different worlds?66
The manner in which Poincare poses these two problems is admittedly
psychological in tone and partially results from his belief that psychological
time is a datum of each individual consciousness. Nevertheless, under the first
problem he discusses the question of the equality of two intervals of physical
time, and under the second, the question of the simultaneity of events occurring
at different places, i.e. simultaneity at a distance. In connection with the former
question, he argues that psychological time is discontinuous, whereas physical
time is mathematically continuous. Hence, he implies that the continuum of
physical time, like that of physical space, is metrically amorphous, i.e. that it
lacks an intrinsic metric, and that therefore there is no unique standard of
equality of length imposed upon us by the nature of this continuum. Thus,
for instance, he says that we have not a direct intuition of the equality of
two intervals of time. However, unlike the case of geometry, there is no
distinction between pure and applied time; hence Poincare proceeds to discuss
the problem of the physical determination of the equality of two intervals of
time. In this discussion he points out that certain difficulties arises8 and, to use
a phrase coined by Putnam, he concludes that the concept of physical time is a
law-cluster or a multiple-criterion concepts, i.e. that there is a multiplicity
of compatible physical criteria, rather than one single criterion, by which
physical time can be measured7.
He maintains, however, that the choice of
these criteria is governed by considerations of convenience, and not of truth;
in other words, unlike Putnam, Poincare considered the multiple-criterion
nature of the concept of physical time to be irrelevant to the question of its
conventional or empirical status.
Poincare is not as explicit about the physical criteria used in the measurement
of distance, but it is reasonable to assume that he also considered the concept
Poincare,
The Value
o Science (Chapter two), p. 27. This chapter consists of his article La
Mesure du Temps,
Revue deM aphysique et de Moral e (1898),
l-13.
Cf. Poincare,
op. cit.
p.
26.
He does not claim originality in pointing out these difficulties, but attributes them to Calinons
tudesur les diverses grandeur s, and Andrades Lefons de Mkhani quephysique.
Cf.
Putnam, The Analytic and Synthetic in
M innesota Studies in he Phi losophy ofScience,
I I I ,
pp. 376-381.
Cf. Poincare,
op. cit.
p.
30.
In
this connection we may say that Poincare would agree with the following remark made by
Griinbaum
..much as attention to the multiple-criterion character of concepts in physics may be
philosophically salutary in other contexts, it constitutes an intrusion of a pedantic irrelevancy in the
consideration of the consequences of alternative metrizability (of physical space and time)
(Griinbaum,
Phi losophical Problems of Space and Time,
p. 15).
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of physical distance to be a multiple-criterion one. Thus, entirely apart from
the distinction between pure and applied geometry and its consequences, it is
reasonable to say that Poincares thesis of the conventionalism of the measure-
ment of the equality of two spatial intervals is analogous to the conventionalism
of the measurement of the equality of two temporal intervals.
However, Poincare saw that not all the problems concerning physical time
are solved with the solution of the measurement of the equality of two intervals.
In his opinion, the logically prior and neglected problem of simultaneity
remains unsolved, i.e. the problem of the meaning of the statement that two
physical events which occur at a distance are simultaneous7. He held that we
have not a direct intuition of simultaneity at a distance; this is a matter of
definition or convention. However, though according to Poincare the
conventionalism of simultaneity at a distance is, not only in this sense, but also
fully analogous to the conventionalism of the measurement of the equality of
two temporal intervah?, the former, unlike the latter, is not based on
considerations of the metrical amorphousness of the continuum of time and
so, it could be argued, is not entirely analogous to the latter. More specifically,
while in the case of geometrical conventionalism the possible choices of
congruence standards are indicated by the function d x,y), there is no such
indication of the possible standards of simultaneity. Poincares point, however,
is that there is no inherent property in the continuum of time from which one
may abstract the concept of simultaneity at a distance and for this reason it is
analogous to the concept of equality of time. Moreover, we have no direct
intuition of either concept, and there is nothing in the nature of events in the
material world which imposes any set of criteria of either concept upon us.
Finally, both concepts are multiplecriterion concepts given by the application of
certain rules. Thus, according to Poincare, simultaneity at a distance, like the
equality of spatial and temporal intervals, is neither an absolute datum of the
mathematical continuum, nor of experience, and therefore the ascertainment
of the simultaneity of spatially separate events is not an immediate empirical
issue but depends on our prior conventions. However, once these conventions
are fixed the issue is an empirical one relative to them75. But, according to
Poincare, this is also e case with geometry. Hence his geometrical convention-
alism is analogous to his conventionalism of simultaneity.
Thus, for example, at times Poincare mentions intervals given by the coincidence behaviour of
unperturbed transported solid rods as a criterion of spatial congruence, while at other times he
mentions intervals for which light rays require equal transit times.
Cf. Poincare,
The Val ue f Science,
pp. 30-32.
Cf. op. cit .
p. 30.
However, if we examine more closely Poincares conventionalism of simultaneity, we shall
find indications of standards of simultaneity (cj. Poincare,
op. cit .
pp. 32-35). though these have
not as concise a form as those indicated by the distance function d(x,y).
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(c) Geometrical Conventionalism and the Special Theory of Relativity
The difficulty raised by the special theory of relativity against Poincares
geometrical conventionalism was stated succintly by Poincart himself: will
not the principle of relativity, as conceived by Lorentz, impose upon us an
entirely new conception of space and time and thus force us to abandon some
conclusions which might seem to have been established? Have we not said that
geometry was devised by the mind as a result of experience, but without having
been imposed upon us by experience, so that, once constituted, it is secure
from all revision and beyond the reach of new assaults from experience? And
yet do not the experiments on which the new mechanics is based seem to have
shaken it?76 To this question Poincare himself answered in the negative. His
conventionalism, he held, was not affected by the special theory of relativity,
since, in his opinion, scientists are not constrained by reality to adopt the
conventions of this theory. Rather, just as scientists in the past found the
conventions of classical mechanics to be the most convenient, so now other
scientists find the new conventions of the special theory of relativity to be the
most convenienF7.
To see Poincarts reasons for this, let us first consider his view of the
relationship between space and time in the special theory of relativity. According
to Poincare an essential element of this theory is that in it space and time are
no longer two entirely distinct entities which can be considered apart, but two
parts of the same whole (space-time), two parts which are so closely knit that
they cannot be easily separated.
Thus, as Poincare himself admits, if one
attributes an ontological or physical status to space-time one must maintain
that the connection between, or the unity of, space and time is not merely
conventional, but is, as Dingle puts it, an association that is more fundamental
than a mere ad hoc union to be recognized in some physical problems but
ePoincarC, Last &says, p, 15. In the above quotation Poincare mentions Lorentzs, and not
Einsteins, theory of relativity and, indeed, he continues this practice throughout his works. Thus
he appears not to credit Einstein with the discovery of the special theory of relativity. However,
there is no doubt that he was aware of Einsteins work
(cf.
M. Born,
Physics in my Generation,
p. 192). Also some commentators group together Poincares, Lorentzs and Einsteins works on
relativity, and consider them to be intimately connected (cJ Keswani, The Origin and Concept of
Relativity I and II, The British Joumalfor Philosophy o Science 15 (1965) 286306 and 16 (1966),
19-32), whereas others maintain that they are quite distinct (cJ H. Dingle, Note on Mr. Keswanis
Article, The Origin and Concept of Relativity, The British Journalfor the Philosophy of Science,
16 (1965), 242-246). From our point of view,
viz.
the conventionalism of geometry and of the
measurement of time, we can group Lorentzs and Einsteins theories together, and discuss their
implications vis d vis Poincarts thesis of geometrical conventionalism, and thus we shall simply
speak of the special theory of relativity.
Cf. Poincart, op. cit. p. 24.
81bid.
This point is also noted by other commentators. For example, Earman maintains that
space-time is the basic spatio-temporal entity (cJ Space-time, The Journal of Philosophy
(1970), 259). Similarly Russell maintains that events, and not particles, must be the stuff of physics
and that each event has to each other a relation called interval which could be analysed in various
ways into a time-element and a space-element (cf. History of Western Philosophy, pp. 860-861).
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abandoned in others if it becomes inconvenientv. From this point of view,
Earman argues that Poincares geometry of space, on which he grounds his
geometrical conventionalism, is taken out of its proper context, namely
space-time, whereas this question should be seen relative to the slicing off of
space from space-time, which contrary to Poincarts assumption, can be
accomplished in not just one, but in many, ways.
Poincare, however, did not agree that the space-time of the special theory
of relativity is a basic physical entity, in the sense that it is empirically or
otherwise necessary to view nature in this way, but regarded this unity merely
as a convention. He does not expressly state his reasons for this view, but these
reasons, I believe, are analogous to those he gives for the conventional nature
of geometry in classical mechanics, viz. the manner in which the concepts of
space and geometry are used in it. Thus he would agree, for example, with
Dingle that an examination of the way in which the concept of space-time has
appeared in physics shows that the implied association between space and time
has been chosen for a definite limited purpose, namely the derivation of the
laws of motion. Dingle illustrates this in a manner which would, in all
probability, be acceptable to Poincare. He points out that, in considering the
problem of, say, the derivation of the laws of radiation of energy by hot bodies,
one may choose to associate energy with time and to speak of energy-time in
connection with this problem, in exactly the same way as one chooses to
speak of space-time in connection with the problem of motion*. Poincare
goes further than Dingle in pointing out the limitation of the physicists
choice of space-time. He maintains that, even though space-time is used by
scientists to express the laws of motion, these same scientists still continue to
use classical mechanics, and thus space and time separately, in their investiga-
tions of the motions of terrestrial and other bodies whose velocities are
small relative to the velocity of lightaz. Moreover, when scientists do use space-
time in physics, they continue to maintain that space-time is mathematically
continuous, and therefore does not possess an intrinsic metric, which implies
(though Poincare does not expressly say so) that geometrical conventionalism
is not invalidated by the introduction of space-time into physics.
Cf. Dingle, The Philosophical Significance of Space-Time, The
Proceedings of the
Ar istoteli an Society, (1947-Q,
155.
OCJ Earman, op.
cit.
pp. 261-262.
Cf. Dingle, op.
cit. p.
156.
Cf. Poincart, The Principles of Mathematical Physics, The
Monist
(l IOS), 24. While this
article, which consists of an address delivered by Poincare before the International Congress of
Arts and Science in St. Louis, in September 1904, was written before the publication of Einsteins
work, it, nevertheless, contains in germ some of Einsteins principles. Thus it is relevant to the
above discussion. Moreover, Poincarts point above is correct, as may be seen from an examination
of most texts on the special theory of relativity. For instance, Landau and Lifshitz point out that
in the limiting case when the velocities of moving bodies are small compared with the velocity of
light - relativistic mechanics goes over into the usual mechanics - which is called Newtonian or
classical (The Classical Theory of F ields p. 2).
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Another characteristic of the special theory of relativity, according to
Poincare, is the importance it attributes to the principle of the finiteness of the
velocity of the propagation of interactions for the notion of metrical time,
the maximum velocity being that of light.
While a detailed discussion of the
conventionalism of time is beyond the scope of this paper, it is both useful and
interesting to note the analogy between the chronometrical conventionalism of
the special theory of relativity and Poincares geometrical conventionalism. As
we mentioned above, in classical mechanics there is no functional or algebraic
way of indicating the possible choices of standards of simultaneity and in this
respect the conventionalism of geometry differs from that of time. In the
special theory of relativity, however, there is such a wag4. For this reason
Grtinbaum, for instance, maintains that Poincarts geometrical conventionalism
is fully analogous to Einsteins conventionalism of simultaneitf5, though in
all probability, Poincare himself would maintain that there is no essential
difference between the conventionalism of simultaneity in classical and
relativity mechanicsa6.
Poincare finally considers the importance of the Lorentz transformations of
the special theory of relativity, with particular reference to their bearing on his
geometrical conventionalism. In classical mechanics rigid bodies undergo
Galilean transformations, which is compatible with the claim that they move
(in any system whatsoever) approximately according to the structure of the
Euclidean group of transformations used by Poincart to define spaces7. In the
special theory of relativity, however, the motions of solid bodies are governed
by the Lorentz transformations which are incompatible with the Euclidean.
Hence, it would seem, the classical notion of space as defined by Poincare
must be changed in the light of the special theory of relativity. Thus, it would
appear, Poincares thesis of the conventionalism of geometry is false, since it
implies that the geometry of space should not be changed for any empirical
considerations.
Cf.Last Essay, pp. 23-24.
This functional indication of simultaneity may be explained as follows: let us consider an event
E, namely the departure of a light ray from a point A at a time t, measured by a clock at A, and
let us suppose that this light ray is reflected from a point B (an event we shall call
E,)
and that it
returns to A at a time Ismeasured by the same clock at A (an event we shall call E,). Finally, let us
consider any event E, at A between the times t, and t,. The problem is how can an observer at A
know whether E, is simultaneous or not with E,. This is not an empirical issue, since E, and E,
are
not connectible by interacting velocities less than, or equal to, the velocity of light; rather, as
Einstein says, it is a matter of definition (CA The
Principles ofRelativity,
p. 40). Its answer depends
on how one synchronizes the clocks at A and B and, as Reichenbach clearly points out (c$ The
Philosophy
of
Space and Time,
pp. 126-127), this may be done in an infinite number of ways.
The time t, assigned to the event
E2
s indicated by the function t2 = t, + E 1, t,), such that E is
greater than zero and less than one. This function also indicates the possible choices of simultaneity.
In other words, depending on the scientists choice of .r,
E2
s said to be simultaneous or not with
E,.
Cf.Grttnbaum, Philosophical Problems of Space and Time, p. 28.
Cf. upra p. 17.
8 Cf. Poincare,
op.
cit. p. 25.
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Poincark s Conventi onali sm of Appl ied Geometry 323
According to Poincare this objection does not hold*. First he says, it takes
the concept of space in a different, and more limited, sense than the sense it
has in his argument for geometrical conventionalism. In the objection space is
considered solely from the point of view of the principle of relativity, according
to which the laws of mechanics (which, in Poincares view are expressed by
means of differential equations) are identical in all inertial frames of references,
whereas in Poincares argument the concept of space is not limited to the
mechanics of inertial frames of reference. In other words the notion of space
defined by the group of Lorentz transformations, which is assumed to be the
correct notion of space in the above objection, is limited to considerations of
the invariance of the laws of mechanics vis d vis the Einstein principle of
relativity. But, if we look on space from the broader perspective of the
mathematical continuum, which is assumed both in the special theory of
relativity and in the other positive sciences, we see that the notion of space
defined by the group of Lorentz transformations is only one of the possible
metrical spaces compatible with the metrical amorphousness of this continuum.
Moreover, even within the limited point of view of the principle of relativity,
the above objection, according to Poincart, can be shown to be invalid. It
assumes that the metrical spaces of classical mechanics and of the special
theory of relativity are defined respectively by the groups of Galilean and
Lorentz transformations. But, in the first place, there is no essential difference
between these definitions, and secondly, these definitions result respectively
from the Galileo and Einstein principles of relativity understood as conventional
postulates, and not as experimental truths.
Poincares argument for the first
point may be summed up as follows. In classical mechanics we adopt the
convention that two figures are equal if the same solid body can be superimposed
on these figures such that it coincides first with one and then with the other. But
the solid body in question may be considered to be a mechancial system in
equilibrium under the influence of the various forces acting on its constituent
molecules, in which case the above convention is equivalent to an agreement
that the laws describing the equilibrium of the mechanical system of molecules
constituting the solid body remain invariant in all inertial systems. On the
other hand, in the special theory of relativity we agree to call two figures equal
if any mechancial system is placed in such a way that it coincides with these
figures in any inertial system, i.e. we agree that the laws describing the equili-
Cf . Last Essays, pp. 15, 18-22.
Cf
op cit.
pp.
22, 23.
One must distinguish, as indeed PoincarC himself does, between this.
principle of relativity and what may be called the Einstein principle of relativity, which consists
of the former principle combined with the principle of the finiteness of velocity of propagation of
interactions, and also between it and what may be called the Galileo principle of relativity of
classical mechanics, viz. the principle of relativity as explained above combined with the principle
of the infinity of the velocity of propagation of mechanical interactions.
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324 Studies in History and Philosophy of cience
brium of any mechanical system are invariant in all inertial systemsgo. But this
latter convention is merely a more general version of the former and hence there
is no essential difference between the definitions of space in terms of the
Galilean and Lorentz transformations. His second point, viz. that the
different definitions result from the Galileo and Einstein principles taken as
conventional principles, is more complex. As we saw above, Poincare held that,
if a scientist decides to elevate a corroborated general empirical proposition to
the status of a conventional principle, he must do so by transforming it into a
conventional principle and an experimental law. Thus the Einstein principle of
relativity can be transformed into the conventional principle that the differential
equations of dynamics satisfy the Lorentz group of transformations, and
experimental laws among which we have the law that the velocities of
propagation of mechanical interactions are finite. Now, according to Poincare,
it is the conventional principle (which includes the conventional principle of
relativity common to the Galileo and Einstein principles) that enables the
relativity physicist to define metrical space in terms of the Lorentz group of
transformations. Hence the definition of metrical space in terms of the
Lorentz group of transformations results from the conventional, and not the
experimental, dimension of the special theory of relativity.
Against this, it could be argued that the definition of metrical space cannot
be isolated from the context of the special theory of relativity, and that in this
context it is imposed by the total empirical theory, and as such is not
conventional. In other words, since Einsteins law of relativity, unlike the
classical law, is an experimental truth, and since the definition of metrical
space in terms of the Lorentz group of transformations is associated with the
former, and not the latter, this definition of space is based on empirical, as
well as conventional, grounds. In Poincarts view, however, this objection does
not hold. First, since the dualism of metrical space and matter holds both in
the special theory of relativity and in classical mechanics, the definition of
metrical space in both cases is, logically speaking, prior to empirical
investigations, and thus cannot be influenced by the empirical results. It is true,
of course, that the special theory of relativity and classical mechanics are
mutually inconsistent and that the former is better corroborated on empirical
grounds than the latter, but this implies merely that the special theory of
relativity as a whole is, in some sense, more probable than classical mechanics;
it does not imply that the definition of metrical space in terms of the Lorentz
Cf. Poincare, op. cit . p. 22.
BLandau and Lifshitz maintain that the finiteness of the velocity of propagation of mechanical
interactions is an empirical truth (cJ The
CIossicul Theory
of
Fields,
p. #I).
It is worth noting, as Giedymin points out, that Poincare, in opposition to Le Roy, denied the
truth of the incommensurability thesis in general (cJ Giedymin, Logical Comparability and
Conceptual Disparity between Newtonian and Relativistic Mechanics, The Br iti sh Journal or the
Phi losophy of Science, (1973), 271).
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transformations is more probable than that used in classical mechanics.
Secondly, despite the fact that the law of the special theory of relativity,
(namely, that the velocities of mechanical interactions are finite) is better
corroborated than the laws of classical mechanics, nevertheless, if one takes into
account Einsteins assumption that the maximum velocity of interaction is
equal to the velocity of light in vacua then one can maintain that the choice
between the definitions of metrical space in terms of the Galilean or of the
Lorentz transformations is solely a matter of convenience. For Einsteins
assumption contains a conventional element, which can be highlighted by
Reichenbachs view of simultaneity mentioned above, namely that the
constancy of the velocity of light in vacua depends on ones choice of e in the
equation
t2 = t, + E -t,)
and that this choice is a matter of conventionB3. It follows that, for certain
purposes, the velocity of light in vacua may be taken to exceed its normal value.
For this reason it is legitimate to interpret the velocity of light in the Lorentz
transformations as tending towards infinity, in which case the Lorentz
transformations degenerate into those of GalileoB. Thus is would seem that the
choice of either set of transformations as a basis for ones definition of
metrical space is a matter of convenience.
3.
The Retention of Euclid
Poincare held that the scientist, in his empirical descriptions of the physical
world, could retain Euclidean geometry in the face of apparently adverse
experimental evidence. Poincart, however, went further than this: he held that
the scientist not only could, but should, retain Euclid in such an hypothesis.
We shall now examine his reasons for holding this, and his view of the required
modification of physics necessary for its accomplishment. As regard the first
point (the retention of Euclid), it is necessary to distinguish between Poincares
view prior to, and following upon, the development of the special theory of
relativity. In his earlier works he maintained that scientists always do, as a
matter of fact, find it more convenient to retain Euclid, but in his later
works he conceded that scientists might find it more convenient to adopt some
other geometry, but only for specialised purposes. As regards the kind of
change he thought necessary in physics for the retention of Euclid in the
hypothesis of adverse experimental evidence, we shall see that Poincart did not,
contrary to one widely held opinion, conceive this change as the introduction
of
ad hoc
hypotheses, but simply as a change in the (geometrical) language of
physics. Such a change, however, is not simply a matter of semantics, but is
Cf. supra note 84.
Cf. Landau and Lifshitz op. cit. p. 13.
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326
St udies in H i st ory and Phil osophy
of Science
based on the metrically amorphousness of space, and on the peculiar kind of
relationship he held to exist between geometry and mechanics mentioned
aboves5.
(a) The Reasons for Retaining Euclid
In the passage from Science and Hypothesis quoted above Poincart states
that, if the scientist were to discover a negative parallax for a distant star, he
could choose either to give up Euclidean for Riemannian geometry, or retain
Euclid by modifying the laws of optics. In both
Science and Hypothesis
and
Science and Method he states that this choice is a matter of convenience, and
that it is more convenient to retain Euclid, even at the expense of modifying
the laws of physics.
In Science and Hypothesis he gives two reasons for this, viz. that Euclidean
geometry is the simplest in itself, just as a polynomial of the first degree is
simpler than a polynomial of the second degree, and that it is the best
approximation to those properties of natural solids, which we can compare
and measure by means of our senses96.
As regards the first reason, Poincare
has been accused of giving too much weight to the analytical simplicity of
Euclidean geometry, and of neglecting the physics in which that geometry is
usedg7. Hence an argument frequently used against Poincare here is the overall
simplicity of the general theory of relativity, which demands the use of
Riemannian, rather than of Euclidean geometry. This however, cannot be
used as an argument against Poincare. As we saw above, Poincare held a
dualistic theory of space and matter in classical mechanics, in the sense that
there is no empirical relationship between them, and therefore his choice of
Euclidean geometry on the grounds that it is analytically simpler than the other
metrical geometries does not, in this particular case, conflict with the overall
simplicity of classical mechanics. In other words, Poincart could argue that,
since space and matter are empirically independent, the question of the overall
simplicity of classical physics entails two independent questions,
viz.
the
question of which metrical geometry is the simplest and the question of which
formulation of the physical laws is the simplest. From this point of view the
reference to the general theory of relativity as a counter argument to Poincares
argument for retaining Euclid is irrelevant, since in this latter theory the
dualism of space and matter no longer holds, and hence it is impossible to
divide the overall simplicity of this theory into the simplicity of independent
constituent element?. As regards his second reason, viz. that the Euclidean
Cf. lqxYl p.
34.
g6Science nd Hypot hesi s, p. 50.
Cf.
Grtinbaum,
op. cit .
pp.
21-22, 121.
The dualism of space and matter of classical mechanics no longer holds in the general theory
of relativity, since in this latter theory it is legitimate to enquire about the influence of a gravitational
field upon the metrical properties of space. Also one should note that Poincare died before the
development of this theory and he had not foreseen this kind of development.
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Poincarb Conventionalism of Applied Geometry
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metric is the best approximation to the natural solids used in measurements,
Poincare here is clearly correct. We do continue to use rulers, chains etc. in
our ordinary terrestrial measurements of length, and we do retain the
Euclidean metric in accomplishing these measurement@.
In Science and Method Poincare gives a more detailed account of the choices
open to the physicist in the event of the discovery of negative parallaxesoo. In
such an event the physicist, according to Poincare, may choose between the
following two positions: a straight line is, by definition, the path of a light ray,
and therefore a straight line does not satisfy Euclids postulate; or a straight
line is, by definition, that which satisfies Euclids postulate, and therefore the
path of a light ray is not a straight line. Poincart maintained that it would be
foolish, though not false, to opt for the former alternative. He gives
three reasons for this. He argues, first, that a light ray probably satisfies in a
most imperfect way not only Euclids postulate but the other properties of the
straight line. Poincare here implies that the choice of the Euclidean geodesics
as straight lines is a matter of definition and, all things considered, is the best
definition
O
Poincares second reason is more difficult to interpret. He states simply that
it would be foolish to adopt the path of a light ray as the definition of a straight
line, because it not only deviates from the Euclidean straight but also from the
axis of rotation of solid bodies which is another imperfect image of the straight
line. As I see it Poincare is arguing here that the physicist has a choice between
the Euclidean definition of a straight line and its definition as a cluster or
multiple criterion concept, and that, if he chooses the second definition, the
multiple criteria in question should render approximately the same results02.
But in fact this is not the case, since the paths of a light ray and some other
criteria of a straight line (e.g. the axis of rotation of a solid body ) do not give
the same resultio3.
Against this, however, it could be argued that, while
Poincare is correct in noting the possibility of what may be called a range
definitionO o
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