According to Duhem's analysis[1]
, and as many philosophers of science, Einstein and the mathematician Robert
D. Carmichael declared about this point, without the slightest hesitation I
would be willing to affirm that there is nowhere a scientific theory which is
perfect and ultimate: even though there may be such a theory, I think it is
a good philosophical postulate to suppose that there is none.
Given as the result of a methodical process, being dependent on the history
of inquiry and ever new findings, a theory in physics cannot be taken as an
absolutely certain explanation of sensible appearances ; moreover, scientific
activity[*] can be regarded
as a special type of social activity which is culture regulated. By means of
sensible appearances we refer to an underlying reality, the approach of which
is only possible through scientific, but also historically determined research
because epistemology becomes for us more and more historical. Theory cannot
let our senses accede to the supposed underlying reality which is presumed to
exist. From that specific point of view, scientific theories work only "as though"
reality is exactly what they are asserting it to be. Therefore such a theory
is always at first an hypothetical construction concerning the reality.
However, certainty of theories must be possible to scientists. That possibility
is even an epistemological condition of their research, and a necessary belief
without which most of them would not go on leading their own research. And Pierre
Duhem, who was suspicious of the certainty of theories, nevertheless gave acoustical
theories as an example of certainty[2].
I. MEASUREMENT
With the reservations above mentioned, we can recognize that Carl G. Hempel
has exactly defined at what precise time, in the particular process of research,
a theory may be considered as being effective. We may retain what he proclaimed
justly : "Theories are usually introduced when previous study of a class of
phenomena has revealed a system of uniformities that can be expressed in the
form of empirical laws"[3].
Hempel's position is not different from Auguste Comte's position which stated,
in the last century, that scientific discourse must be developed according to
the "law" and not to the "cause", given that only the law can be expressed in
mathematical terms. Ernst Mach required, like Comte, that every statement states
relations between observable quantities. Also, concerning scientific law, we
can make further distinctions and say that a law involves numerical relations
allowing experimental verification by measurement. Carmichael expressed it very
well when he compared law and principle in relation to measurement:
"By a law we shall mean a statement of phenomenal fact in terms involving
numerical relations subject to experimental verification by measurement.
By a principle we shall mean a statement of fact, relating to phenomena in such
a form as to require a transformation by the use of other facts before one can
arrive at numerical relations subject to experimental verification by measurement"[4].
If experiments have a generative role, what is clear and well underlined, in
both of these cases enounced by Carmichael, is that the experimental function
is the function of "measurement". And the function of measurement may be identified
with Hempel's idea of uniformities, as much as with Comte's statements about
quantitative laws. Indeed, Auguste Comte explained how, upon the three Keplerian
laws, Newton could inversely determine a priori the planetary movements resulting
from the dynamical, mathematical law of gravitation which proved the independence
and the leading function of astronomy in relation to the other sciences.
With Carmichael's formulas we can better understand why one can speak, on the
one hand, of the Principle of the Conservation of Energy,
and on the other hand, of Newton's Law of Gravitation. Indeed,
Newton's law of gravitation is like any law defined as above: "a statement of
phenomenal facts involving numerical relations subject to experimental verification
by measurement". Thus, the principle of conservation of energy, not subject
to direct test, needs to be transformed in order to let the scientist arrive
at those numerical relations subject to experimental verification by measurement,
as above. Like any other principle, the principle of conservation states a fact
in a form which requires a transformation by means of other scientific facts.
And only the results obtained by means of a principle prove its value. We can
see also that it is not easy to verify theoretical symbolism with respect to
phenomenal reality, but it can be verified only punctually through another symbolism:
that of numbers, and by means of a relative system of reference with respect
to which measurement is possible.
Measurement begins to become effective through the regulated establishment of
numbers: with two numbers, for instance, the position of a ship may be defined
by its latitude and its longitude. But, for defining the position of an aeroplane,
which requires a third dimension, the height above sea-level, measurements are
accomplished with three numbers. In all these cases, measurements concern positions.
But if we want to consider events, where there are not positions but things
which happen, then we need not only a portion of space (three dimensions or
three numbers) but also an interval of time: an extra dimension or number. Thus,
in order to locate an event we need four numbers because of its four dimensions.
Moreover, space and time are not independent in the theory of relativity for
which fundamental physical things are events. Einstein's "basic concept", the
concept of "event", corresponded at the beginning of Einstein's research to
Mach's concept of "element". Applying the Lorentz transformation, Einstein discovered
that events happening at different places cannot be absolutely simultaneous
because of what was later called the "conical time order". Thus, we must think
of two cones. The active present of an event is a four-dimensional cone stretching
backward in space and time. Its active future is a similar cone stretching forward
in space and time. And outside of both of these cones no part of the four-dimensional
world can be either in the active past or in the active future of that event.
But the time itself of an event was described in 1905 as being "given simultaneously
with the event by a stationary clock located at the place of the event"[5].
And he went further : as that time has meaning "only when it connects with our
consciousness through sense experience (that is, when it is subjected to measurement-in-principle
by means of a clock present at that same place), also is the place, or space
co-ordinate, of an event meaningful only if it enters our sensory experience
while being subjected to measurement-in-principle (that is, by means of meter
sticks present at the same time)"[6].
A re-reading of Einstein's paper with insight might let us distinguish between
"reality" and "events" ; but, for Einstein : "the ‘time’ of an event
is that which is given simultaneously with the event by a stationary clock located
at the place of the event"[7].
‘Time’' is measured time. In other words: outside the fourth dimension
is no event, that is no reality, observable to us.
The Restricted principle of Relativity holds in the general theory of relativity,
and may be expressed as follows:
"If S1 and S2 are two systems of reference in free space
having with respect to each other a uniform unaccelerated motion, then natural
phenomena run their course with respect to S2 in accordance with
precisely the same general laws as with respect to S1" [8]
The above statement means that if one system S1 is suitable, the
other system S2 is equally suitable in as much the requisite conditions
are satisfied. The required measurements may be brought into convenient relations
by means of a Principle of Correspondence of Units. When the relation
of the quantity L1 to the system S1is the same as the
relation of the quantity L2 to the system S2, the units
of S1 and S2 let us obtain the same numerical result by
measuring a quantity L1 with the units of S1 and a quantity
L2 with the units of S2. If we must understand the restricted
principle of relativity in agreement with the correspondence of units, then
it is in agreement with the facts of experiment, given that there is an intricate
interdependence of experiment and theory.
What Einstein wanted to achieve was a unified field theory in which there would
not be two separated "kinds of physical things"[9]:
"measuring rods and clocks" and "all other things". Measurement would be included
in the theory in as much as "the particles themselves would everywhere be
described as singularity-free solutions of the complete field-equations"[10].
Classical mechanics describes phenomena as existing in themselves : a force
is supposed to exist even if we cannot observe it directly. And even if Einstein
relativist classical mechanics gave the "observer" a particularly important
function, Einstein thought that the observed events existed in themselves :
each observer would have a view of the objective reality. Hence it is possible
to identify Einstein's philosophy with epistemological realism.
On the contrary, in quantum mechanics it cannot be so. Indeed, a measurement
brings an answer to the question put by the experimenters. But a measurement
model is described in quantum mechanics by coupling the system to be measured
with a measuring apparatus. And another description of quantum-mechanical measurement
is possible through the use of formalism. To be precise, as Samuel L. Braunstein
and Carlton M. Caves pointed out, the formalism of Effects and Operations in
the terms of a measurement model are all equivalent[11].
Obtaining the formalism of Effects and Operations is possible by applying the
standard rules accepted in quantum mechanics to the apparatus observation in
a measurement model. Formalism provides a convenient notation for a direct description
of the measurement of the system state. In Effects and Operations, quantum state
and type of apparatus, and quantum state and type of the interaction with the
system are incorporated together. Barchielli, Lanz and Prosperi developed the
formal description of a continuous measurement of position, for which they used
Effects and Operations; yet, they did not specify the measurement model corresponding
to the Effects and Operations they chose[12].
Since Bernard d'Espagnat found a discrepancy between what he thought to be a
new theory conceived by Barchielli et al.[13]
and the standard quantum mechanics, Braunstein and Caves have explored that
discrepancy and may prove now, on the contrary, that there was another measurement
model than the one which d'Espagnat supposed to be chosen by Barchielli et
al. Braunstein and Caves are also able to conclude that Barchielli et
al.. did not use a new theory of measurement. By the way, the formalism
Effects and Operations remains valid in the examined case[14].
Braunstein and Caves underline the possibility of one Effect have more than
one Operation: many model measurements accomplish the same measurement statistics
; however after measurement they lead to different system states. The measuring
apparatus is an observer and it may be any macroscopic tool used by experimenters.
But in that which concerns the very small, a macroscopic tool is being used
to measure a microscopic system : under such conditions it may occur that the
tool disturbs the measured system and give different system states.
Therefore, as Bernard d'Espagnat explains it[15],
the result measurement is expressed by a number given by the tool through its
temporary interaction with the system according to the rules direction for use
of the measuring apparatus. But the number of the measured quantity does not
necessarily show the real quantity supposed to be there before the measurement.
Therefore Bohr used the term ‘phenomenon’ only in the proper meaning
of observations including the measuring apparatus. When velocity is at first
measured, then momentum may give the measurement of mass and permit the physicist
to identify a particle.
All forces in nature are generated by the exchange of messenger particles between
interacting particles : photons (quanta of electromagnetic energy with zero
mass) may extend the influence in the case of the electromagnetic force to be
identified.
II. PRINCIPLES
Since Auguste Comte's positivism, all scientists know that a theory lets them
link phenomena to some principles, and that without a theory the isolated observations
could not be combined, nor could the facts be noticed in the first place.
I began to speak about the essential function of measurement because physical
laws are always verified through measurement without which they simply would
not be laws. But in the meantime I have already referred to the notion of principle,
necessary from the outset for measurement. A kind of circularity is operative
between measurement and principles.
Indeed, in order to go all the way with measurement, scientists use the certainty
of various principles: for instance, a principle of correspondence of units.
But, before that principle, they suppose the use of systems of reference with
respect to which measurement can be made and principles can be formulated. Principles
are statements of facts relating to phenomena, which are to be transformed through
other facts before being verified by measurement. By the term ‘facts’
Auguste Comte meant either "particular facts" observed on the condition
of a theory, or "general facts" that are scientific laws or theories.
Likewise, for Einstein, ‘facts’ were, besides "particular
facts", either, for instance, the constancy of the velocity of light,
which is a law, or the equality of gravitational and inertial mass, which is
a principle called Einstein's principle of equivalence (E=mc2, where
m is the relativistic mass, E the sum of kinetic energy: associated with the
motion).
Whereas a law is subject to experimental verification by measurement, a principle
is not subject to a direct test : it only proves its value through the results
of measurement, after a transformation.
Thus, the principle of correspondence of units lets the
physicist get a transformation allowing the continuation of measurement,
with which he verifies another principle, the restricted principle
of relativity. And so, we can see that measurement is in the very
bosom of a physical theory, which includes laws the truth of which is dependent
on measurement and principles, on the basis of systems
of reference which imply also measurement and principles! Thus,
any scientific theory is to be tested by experiment through measurement. There
must be an unified theory which presents the relationship between
theory and experiment.
As to the way of setting up a theory, Einstein has stated against Hume's induction
that "there is no way from the experience to the setting up of a theory"
[16]. What is needed
at first is to have "sufficient strong formal conditions", such
as principles or other theories ; afterwards is only needed a "little
knowledge of facts"[17].
If he had been strictly getting along only with a "constructive theory"
built up by induction of generalization, on the basis of a logical ladder departing
from a set of individual observations, Einstein would never have created the
theory of relativity. Einstein was not searching for a constructive theory,
but, on the contrary, for a "principle theory"[18].
In order to discover the laws that he was looking for, Einstein applied another
epistemological imperative, that is the reference to new universal principles.
And he explained that first necessity of discovering "a universal formal
principle (that) could lead us to assured results"[19],
even though always "a theoretical system can claim completeness only when the
relations of concepts and experiences facts are laid down firmly and unequivocally"[20].
Searching for "theories of principle", Einstein put forward his
postulates and drew the logical deductions in order to point the deductions
to experimental tests.
The fundamental start of the general theory of relativity is a four-dimensional
manifold of space-time: in the general theory of relativity, the restricted
principle of relativity is suitable under the condition that the phenomena involved
occur in free space, that is in the absence of a gravitational field or of any
disturbance. The restricted theory of relativity is also validly incorporated
in the general theory of relativity, in small portions of space-time within
the range of a vanishing difference due to macroscopic phenomena, since another
of Einstein's principles is the principle of covariance.
Let us review the conditional principles of the theory of relativity: 1. The
restricted principle of relativity; 2. the requirement that the first principle
shall hold as a limiting condition in a gravitational field; 3. the requirement
that, in an infinitesimal portion of space-time, the first principle shall hold
in an indefinitely close approximation ; 4. the requirement to express the laws
of nature in a form independent of the particular reference system used, because
there are no co-ordinates in nature.
A few laws, a few concepts and a few principles could succeed in the most complete
theory. But Einstein also had an operational approach to concepts which had
meaning for him only when they could refer, like Kant claimed, to objects, and,
besides, to the rules which were assigned to these objects. Einstein's ambition
was to both generalize and unify as much as possible: thus, he obtained a logical
simplicity which he thought was the essence of his theory of relativity. We
must remember now that, though suspecting any scientific theory of not being
perfect, Carmichael added: "But the theory of relativity comes nearer
to this ideal than any of its rivals"[21].
Maybe the reason for that near perfection is that Einstein knew the way he thought
himself: he claimed to his friend Solovine that it was necessary for the physicist
to know the way he usually thought[22].
Einstein asked elsewhere : "What is thinking?"[23].
Therefore Einstein gave a theory of "physical" thinking: a kind
of cognitive theory of his own. He established a reconstruction of the emergence
of the conceptual framework that had been necessary to make possible his theory.
Through a design with comments, Einstein showed that to the physicist is given
the E (Experience), which is a multiplicity of immediate sense experiences :
we philosophers must notice here the same beginning as Kant's in the Critique
of Pure Reason, referring to sense experience or to what Kant called,
in German, "Sinnlichkeit". But we need to let some A ("Axioms"
or fundamental Principles) which are based upon these sense experiences, and
from which consequences are going to be drawn, act upon them. Here, for Kant,
were working together the pure principles of understanding and transcendental
imagination. Now, for Einstein, between E and A, there would be no logical path,
but only an intuitive connection which is, as Einstein said, "subject
to revocation".
Thus, in Einstein's cognitive theory, the concept plays the part of a mental
connection between sense experiences, but it "is not identical with the
totality of sense impressions referred to"[24],
and it "cannot be gained from material given to us by the senses"[25].
If the choice of concepts was seen by him as being free, Einstein also supposed
working a theoretical structure of mind, permitting theoretical description
not "directly dependent upon acts of empirical assertions"[26].
Einstein saw concepts and systems of concepts as a human creation. From the
E (Experience) to the A (Axioms or Principles), there was for him a J (Jump);
and, from A to E (again Experience but the final experience), there were the
necessary consequences S, S', S", etc. Holton has recapitulated the complete
cycle : E-J-A-S-E, as being Einstein's process of scientific theory construction,
the criteria of which were an external validation along with an inner perfection[27].
III.QUESTIONS ABOUT INDUCTION AND EXPLANATION
In the same order of ideas about scientific imagination, Carl G. Hempel admits
that there are "no generally applicable ‘rules of induction’"[28],
by means of which both hypothesis and theory are mechanically inferred from
empirical data. He recognizes the function of a creative imagination and particularly
of the "method of hypothesis". Rules of induction and rules of deduction
are to be understood as "canons of validation rather than of discovery"[29].
Therefore Hempel is right to reject the presumed ideal of scientific inquiry
developed in four stages: "(1) Observation and recording of all facts,
(2) analysis and classification of these facts, (3) inductive derivation of
generalizations from them, and (4) further testing of the generalizations"[30].
Hempel calls that view "the narrow inductivist conception of scientific
inquiry"[31]. However,
he thinks that scientific inquiry may be said to be only "inductive in
a wider sense"[32].
Hempel conceives correlatively the rules of induction as stating, like the rules
of deduction, criteria for the soundest arguments. But we have seen that Einstein
went much further with his own epistemology of the theory of relativity, when
he "jumped" [*]
to an intuition which did not necessarily follow the rules of induction. In
order to be corroborated, theories must be already hypothetically imagined,
since, without any theory at all, the physicist could not project a set of observations,
for which he needs a basis of provisional hypotheses precisely belonging to
an imaginary theory.
That was indeed the original idea of Comte in his Cours de philosophie
positive (1830-1842). Already in 1825 - in Considérations sur
les sciences et les savants - Comte claimed that "one cannot
make observations without theories more than one can create a positive theory
without observations"[33].
It is as though a special kind of circularity were needed : without any research
theory there is no suitable observations, and, on the contrary, without suitable
observations there will be no final theory at all. But this presumed circularity
is only an appearance, because the first theory needed in order to make observations
is but hypothetical, and the second theory justifying the effective observations
has been, in principle, first corroborated. With the above reservations, there
may be hypotheses without any inductive-evidential support, but they have some
strong theoretical support, even if the first theory which elicited the observations
was false.
The principle of following an inductive way was abandoned by Pierre Duhem in
favor of holism: in his book concerning physical theory (1905), he gave preference
to a theory conceived as a whole yielding a representation rather than an explanation.
Duhem stated that no isolated hypothesis could be corroborated outside of the
field of physics[34].
Auguste Comte held exactly the same opinion, when he rejected whatever observations
and experimentations could be developed outside of a theory. And effectively
he did so in 1854, when he rejected, probably with exaggeration, the astronomical
prevision and prediction of the planet Neptune by Le Verrier in Paris (also
claimed by John Couch Adams in Cambridge and verified by Galle in Berlin). Indeed,
Comte asserted that it was an "astronomical illusion" to presume
a deduction concerning the existence of a planet simply from calculation –
while Le Verrier did what he had learned to do, that is: measuring angles and
counting time[35].
We could now suggest that scientific explanation is often suspicious as induction.
As did also Pierre Duhem, Carl G. Hempel justly emphasized the regularities
and the system of uniformities which he recognized to stand at the very bottom
of theories. Moreover, we must say that such an assertion might be right for
corroborated theories. In the above quoted essay, Comte proved Adam Smith's
History of Astronomy on the subject of regularities. In fact, Adam Smith
evoked not exactly the regularities, but, the contrary, the "magnificent
irregularities, whose grandeur cannot be overlooked" and which called
forth the amazement of mankind in the first ages of society. According Adam
Smith's observation, men ascribed all the irregular events of nature "to
the agency power of their gods"[36]
. Thus, explanations were looked for in the case of natural irregularities.
Beginning with the remote ‘theologian’ age, explanations went on
until the age of science.
Carl G. Hempel joined theories and regularities: "Theories then seek to
explain (...) regularities"[37].
Measurement would then need an explanation! But explanation is not always what
is necessarily procured by scientific theories. According to Comte, and despite
the motto "ramener l'inconnu au connu", scientific work is essentially
constituted by the fact of going out of the concrete reality in which we live,
toward its abstract and relative representation. For Pierre Duhem, it is even
more explicit: "A physical theory is not an explanation. It is a system
of mathematical propositions, deduced from a small number of principles, in
order to represent as simply, as completely and as exactly as possible, a set
of experimental laws"[38].
We find here the two principal characteristics of physical theory, that we emphasized
: measurement and principles. Comte and Duhem thought like, later, Philipp Frank,
that "scientific explanations" may often prove to be metaphysical.
And Philipp Frank himself developed the idea that a particular type of equation
being the only legitimate basis of the scientific explanation of physical phenomena,
could not be justified by science unless the attempts of justification were
based upon metaphysics.
From that point of view, the distinction between a description and a causal
theory was for Philipp Frank purely metaphysical[39],
For instance, to say that Einstein's theory of relativity is "descriptive",
or that Newton's theory is "causal" and "explanatory",
for both of these cases the similarity of the traditional attitude is noted
by Frank : the difference is only metaphysical. What was supposed to explain
in the sixteenth century was what could be derived from the Aristotelian philosophy
; in the nineteenth century, it was what could derived from Newton's philosophy.
We can say that throughout time science follows practices which are regulated
by culture. The reference of scientific theories to the different schools has
been pointed out by Kuhn in 1962, but it has also been explicitly noted by Duhem
in 1905. When Duhem distinguished four schools in succession :1) the Aristotelian,
2) the Newtonian, 3) the atomistic, 4) the Cartesian, he did not report these
schools to a banished metaphysics, and he even thought that these philosophic
schools had been also helpful in the creation of explanations in scientific
theories considered as good. And we know that, in the Structure of
Scientific Revolutions, Thomas S. Kuhn also referred to what he
called the "paradigms" which came from such schools, and which helped
the scientists at their own time : "the assimilation of Galileo’s
and Newton's mechanics gave rise to a particular famous debates with Aristotelians,
Cartesians, and Leibnizians about the standards legitimate to science"[40].
More recently, Hempel did not restrict the distinctive characteristics of a
good scientific theory to the explanatory import nor the testability-in-principle
because they were for him "only minimal necessary conditions that a
scientific theory must satisfy"[41].
For explanation is to Hempel a necessary condition of scientific theory. Indeed,
for Hempel, and according to the views of Comte, Duhem and Einstein, it seems
that the best scientific theory would be the one which could best unify the
greater number of diverse phenomena. For Hempel too, theory had to trace all
the diverse phenomena back to the same underlying processes : there was then
also a unification of views. Like Newton's theories of gravitation and of motion,
like the kinetic theory of gases, and also like Bohr's theory of the hydrogen
atom, for Hempel the best theory is the one which could best unify by saving
the phenomena.
Therefore, Bas C. Van Fraassen has proposed a new way to a theory : "to
specify a family of structures, its models ; and secondly to specify
certain parts of those models (the empirical structures) as candidates
for the direct representation of observable phenomena"[42].
Then, what is called ‘axioms of quantum energy’ in treatises on
quantum mechanics is exactly what B. C. Van Fraassen calls a "family of
models", with the indication of their "empirical substructures".
In that presentation, what is actually becoming possible is to choose between
two epistemic attitudes : either the assertion that the theory is "true"
and deserves belief, or the assertion of its empirical adequacy with the consequence
of its acceptance. Against any "metaphysical baggage" B. C. Van
Fraassen thinks it necessary to advocate "empirical minimality",
for pragmatic reasons. The assertion of empirical adequacy appears to B. C.
Van Fraassen weaker than the assertion of truth. The scientist must save the
phenomena by delivering himself from metaphysics, but we know that thematic
analysis shows how the scientific discovery is really working : scientific imagination
uses what Holton calls themata[43]
persisting metaphysical issues in scientific change.
CONCLUSION
Theories in physics are based upon principles and measurement. Rather than
the explanation of natural laws what the best theories are looking for is a
greater unification at a higher level of axioms. There is a cognitive function
of theories which has been well understood by Einstein, analyzing the way his
proper mind proceeded in the process of learning about reality. We may now raise
a question : is it possible be neither realist nor nominalist neither rationalist
nor empiricist ? If any of these characteristics remains unavoidable then one
could doubt that "science" or the human mind are able to avoid completely
all metaphysics.
Université d’Amiens.
