Poincare's Conventionalism of Applied Geometry

7/25/2019 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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304 Studies in History and Philosophy of Science

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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Poincarb s Conventi onali sm of Appl ied Geometry

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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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St udi es in H i stor y and Phil osophy of Science

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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Poincarek Conventionalism

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Applied Geometry

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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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PoincarP s Conventi onali sm of Appli ed Geometry

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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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Studies in History and Philosophy of Science

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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Poincarb Conventionalism of Applied Geometry

313

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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314 Studies in History and Philosophy of Science

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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Poincarks Conventionalism of Applied Geometry

315

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.


14/38

316


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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Poincarks Conventionalism

of

Applied Geometry

317

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.


16/38

318

Studies in H istory and Phi losophy of Science

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).


17/38

Poi ncares Convent i onal i sm of Appl i ed Geometry 319

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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320


(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).


19/38

PoincarP s Conventi onali sm of Appli ed Geometry 321

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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322


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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Poincarks Conventionalism of Applied Geometry

325

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

327

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

Poincare's Conventionalism of Applied Geometry

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