| > >
Copyright © Interdisciplinary Encyclopedia of Religion and Science ISSN: 2037-2329 and Alberto Strumia
No part of this article may be reproduced, stored in a retrievial system or transmitted without the prior permission of the Editors.
To refer to the content of this article, quote: INTERS – Interdisciplinary Encyclopedia of Religion and Science, edited by G. Tanzella-Nitti, P. Larrey and A. Strumia, http://www.inters.org
 Mechanics
Alberto Strumia
I. Mechanics as the Study of Motion - II.
Motion as a Philosophical-Theological Concept. 1. The Physical
Approach. 2. The Metaphysical Approach. 3. The
Mathematical Approach. 4. Change in Non-Corporeal Entities:
Philosophical-Theological Aspects. - III. Scientific Inquiry
into the Nature of Motion. 1. Newtonian Mechanics. 2. Relativistic
Mechanics. 3. Quantum Mechanics. 4. Instability
and Deterministic Chaos. - IV. Mechanism and Reductionism. 1.
What is Mechanism? 2. Mechanism and the Structure of
the Universe. 3. Reductionism and Mathematics. -
V. Mechanics and Causality 1. Mechanical Causality.
2. Mathematical Formal Causality. 3. Philosophical
Causality. - VI. Mechanics and Finalism. 1. Finalism in the
Formulation of Laws. 2. The Anthropic Principle.
3. Concluding Observations.
I. Mechanics as the Study of Motion
The term “mechanics” (gr. mechané, machine; mechanikós,
engineer; lat. mechanica) denotes that branch of physics which deals with the study
of “motion” or “movement” of material bodies (cf. e.g.
Goldstein, 2002). In Ancient Greece, under the influence of Platonic thought, the word was
often used in a negative, even pejorative sense, because mechanics, understood as an
experimental “art” (téchne) and not as a theoretical science,
involved a certain manual labor of a material nature which was considered in opposition to
the ascent of thought to the “world of ideas” (cf. Koyré, 1971). It is
known that although Archimedes (287-212 B.C.) was considered to be one of the greatest
mechanical geniuses of all time, he sincerely disdained his practical inventions (cf.
Bell, 1990). Nevertheless, his methods of research were judged masterly by Galileo who
considered him to be a master (cf. Koyré, 1978). Some of Archimedes’ results, such
as his famous principle according to which a heavy body immersed in a liquid is pushed up
with a force equal to the weight of the liquid displaced, are still recognized as valid.
Regarding the mechanics of the ancients, in the field which we call today
“celestial mechanics” (because it is related to astronomy), we are well
aware, among other things, that around the centuries 8th - 7th B.C. the Babylonians were
able to predict lunar eclipses (cf. Daumas, 1957).
Unlike pure mathematics, where new results were added to older ones without
substantially changing the latter and at most leading one to re-think their formulation,
mechanics has undergone, through the centuries, much evolutionary change. Such change has
led to the definition of new theories and has involved several radical conceptual
innovations. We only have to think of recent advances from Newtonian to relativistic
physics ( RELATIVITY, THEORY OF) and to quantum mechanics.
If it is true that a new theory must reproduce in first approximation the results of the
older theory, it is also true that the new theory requires, at the same time, a
far-reaching conceptual revision of the older one. Several authors have even maintained
that such conceptual changes are accompanied by the rise of new kinds of metaphysics which
are “incommensurate” with the preceding ones (cf. P. Feyerabend, Against
Method, Verso, London-New York 1993). Certainly, however, even in matters not related
to the problem of “discontinuity” and “continuity” in
scientific rationality — a problem which has caused some discussion between all
contemporary epistemologists — the problem of motion has proven, from its
origins, to be of great philosophical and not only scientific interest.
We will therefore try to analyze this issue as far as possible
from both the scientific and philosophical points of view, and to
take into consideration their interrelation. In addition, it will
be useful to keep in mind that this entry and what was presented
in the entry on matter
are in a certain sense complementary: in the former, the problem
of the “structure” and “constitutive elements”
of corporeal bodies is examined, while in the latter the problem
of their “evolution” in time and their “change”
is considered primarily; in the former, emphasis is placed on what
is “permanent” in a corporeal body as it undergoes change,
on what guarantees its identity, while in the latter, there is an
emphasis on what “changes” and on the way in which such
change takes place and on its causes.
II. Motion as a Philosophical-Theological concept
From the philosophical point of view, the problem of motion is
related most of all to the classical problem of “becoming”:
how is change possible in all its various forms? From antiquity,
the search for an explanation of the coexistence of something permanent
(the physicist of today would say “invariant”) with something
which changes in the same subject, has been the source of numerous
paradoxes whose solution has been a great challenge for physicists,
mathematicians, and philosophers. In this section, I will try to
take into consideration the “qualitative” or better still
the “metaphysical” differences between the different ways
of approaching the problem of motion and to indicate, among other
things, aspects more directly relevant to philosophy and theology.
1. The Physical Approach. The Ionian philosophers (6th
-5th centuries B.C.) such as Thales, Anaximen, Anaximander, etc.,
later called “physicists” because they studied nature
(gr. physis), posed the problem of what the “constitutive
elements” are of all that falls under our senses ( MATTER,
II.1). What we can gather from the information we have from these
philosophers and what their successors have said about them, suggest
that the Ionian philosophers were mostly concerned with the problem
of identifying the constitutive unitary principle of the entire
sensible world. In their view, there is no doubt that matter is
one and the same even if they choose a different constitutive principle,
because one witnesses all manner of things transforming into each
other in the great geographical and meteorological observatory of
the world. Unity is the result of a rational principle of permanence
which is implicitly accepted and by which the ultimate nature of
things remain the same under the appearance of change (cf. Enriques
and De Santillana, 1973). Change and motion are considered “primitive”
data which can be verified by experience and which have no need
for explanation, whereas the permanence of things behind the change
and motion which animates them is what needs to be explained. With
Heraclitus (530-470 B.C.) the point of view is reversed and change
itself becomes the fundamental principle of the universe, whereas
the permanence of things is only apparent. However, motion is not
explained in the physical sense of the word, but is instead described
as a continuous transition between opposites, without an understanding
in causal terms. Democritus of Abdera (460-360 B.C.), who formulated
the atomic theory of matter following an idea of Leucippus (5th
sec. B.C.), introduced the notion of the vacuum and conceived motion
as the displacement of atoms through such a vacuum.
2. The Metaphysical Approach. With Parmenides (5th
century B.C.), the decisive step from the physical to the metaphysical
approach is made. The question one asks regarding the problem of
understanding reality is no longer “what are the constituent
elements?” but “how is change possible in things?”,
that is, the question of becoming. The idea that becoming and motion
are merely factual data is no longer satisfactory, and then a response
to this question is sought by speculating about “principles”.
Parmenides wished to explain the process, or the world’s becoming,
as an effect of causes which explain the events, but in admitting
a unique constitutive principle of reality — the undifferentiated
being always identical to itself — he could not conceive
the reality of motion and of change since the passage from one mode
of being to another of the same thing is not possible. The “univocal”
being of Parmenides has only one mode of being. It is therefore
completely immobile and changeless. It can only remain identical
to itself since it has nothing different from itself in which to
change. One has to wait for the philosophy of Aristotle (384-322
B.C.) to have a metaphysical theory in which the possibility of
motion can be understood without logical contradictions. This theory
has to be inseparably linked to a theory of the nature of corporeal
material bodies (“hylemorphism”; MATTER,
II.3) in order to make even that particular kind of motion which
is their “local motion” comprehensible. In order to become
(and in particular, in order for corporeal objects to move), being
must be structured according to certain constitutive principles.
The negation of these principles makes the coexistence of being
and becoming and the becoming of being (that is motion), inconceivable
since it is contradictory.
With Parmenides, motion cannot be conceived because of the univocity of being. With
Heraclitus, motion can be described at the expense of introducing a contradiction due to
the existence of non-being, nothingness (or “emptiness of being”) to
which the characteristics proper to being are attributed. With Aristotle, the explanation
of motion is possible in the theory of “potency” and
“act”, which allows one to introduce differentiated modes of being which
are intermediate between being in the sense of Parmenides, and non-being (or
“vacuum”) of Heraclitus and Democritus. Aristotle deals with this issue
in the third book of the Physics: «there are things which are only in
act, and others only in potency and in act: such a distinction must be applied to the
determined essence, to quantity, quality and, likewise, to all other categories of
being» (Physics, III, 1, 25). The basic idea behind the explanation of
the nature of things and their motion is, therefore, the diversification of being
according to a plurality of modes of being by which the being is actuated and referred to
in many different ways ( ANALOGY). In such a way, motion is conceived as the
act by which a being goes from one mode of being (in which it is in act) to another mode
of being (relative to which, it is in potency): «the act of what is in potency,
as such, is movement» (III, 1, 10). The phrase, «as such»,
— which Aristotle emphasizes «I insist on the expression “as
such”» — denotes the fact that the “final”
mode of being of the subject which moves is not completely actuated (if there is motion),
and if it were completely actuated, there would be no motion, just as if it had not been
actuated at all.
It is interesting to observe the following two points: a) the above definition is
ample enough to describe the motion of every kind of becoming or change, including: the
change of a subject into another subject (“substantial change”, as in
the case of a chemical reaction), the change of the qualities or properties of a subject
(“accidental change of quality”, as in the case of temperature change,
or change in color of the same corporeal object), the change of quantity of a subject
(“accidental change of quantity”, as the growth of living being), the
change of the position of a corporeal body with respect to another (“accidental
change according to place”, or local motion); b) in this definition, the
notion of time is not involved. Contrary to how we are used to thinking,
the concepts of space and time are derived and not primitive, like the Newtonian concept
of “absolute” space and time. Aristotle, in
fact, derives the concept of time from that of motion (as the “number”
associated with a certain “ordering relation” which characterizes
motion, cf. Physics IV, 11, 30), and the concept of space from the contact between
corporeal objects. This view is closer to that of Einstein’s (1879-1955) general
theory of relativity than to the physics of Newton (1643-1727): in fact,
«according to Einstein and Aristotle, time and space are in the universe and not
vice-versa» (Koyré, 1971, p. 269).
We observe that, as it often happens in Aristotelian physics, the
metaphysical principles of Aristotle are what strikes our interest
and are deemed useful even today from the epistemological point
of view. Whereas in the part which is more physical-mechanical (in
the modern sense) and where Aristotle seeks an explanation of the
constitution of corporeal objects (theory of four elements: air,
water, earth, and fire), or of physical or mechanical processes
through which motion occurs (distinction between natural motion
and forced motion, motive action of air, etc.), his physics is clearly
too qualitative and insufficient if not incorrect from the point
of view of quantitative modern science (cf. Sanguineti, 1992).
3. The Mathematical Approach. The mathematical approach
of the universe was first adopted by Pythagoras (6th century B.C.)
and his followers. In place of “atoms”, which were later
introduced by Democritus, we have “points”. This brings
us back to a geometrical description of physical space. The Pythagoreans
were not so much concerned with a mere description of the ponderable
or dynamical aspect of nature as with grasping order and harmony,
through numeric ratios. In this sense, they advanced from a materialist
description of the cosmos to an abstract or ideal one ( ASTRONOMY,
I). With their discovery of the correspondence between the points
of a line and numbers, their description became both geometrical
and arithmetic, or as one is oft to say, “arithmo-geometric”.
The crisis of “irrational” numbers, however, was not completely
resolved for many centuries after, and this mathematized way of
framing the problem, which had formed the basis of the entire Pythagorean
way of life and thought, reached a crisis and period of stagnation
for quite some time. Cartesian analytic geometry would later resume
in a certain sense and in a modern vein what the Pythagoreans had
begun. As to the problem of movement, motion was viewed by the Pythagoreans
more as a “state” than as change, especially in reference
to the perfect motion of heavenly bodies. Their vision was mostly
geometric and not dynamic. All of this seems to call to mind, in
a certain way, the unified space-time structure of relativity in
which even time is represented from a static and purely geometric
view.
The inquiry into the nature of motion led to another important problem in the history
of mathematics: the problem of the “continuum” and, related to it, the
problem of infinity and its paradoxes. As
Aristotle has observed, «It seems that movement belongs to the continuum; and
that infinity is manifested in the first place in the continuum. For this reason, it
happens often that he who sets out to define the continuum uses the concept of infinity,
because something is continuous if it is infinitely divisible» (Physics
III, 1, 15). The first famous paradoxes inherent to the problem of motion, which emerge
with the necessity of “crossing infinity” and the infinite possibility
to divide in parts the continuum, are associated with the name of Zeno (495-435 B.C.). A
disciple of Parmenides, Zeno pushed to the extreme limit the principles of his teacher and
deduced from them the contradictory nature, and conceptual and metaphysical impossibility
of motion. According to the “dichotomy paradox”, «movement is
impossible because before the mobile object has reached the point of arrival, it has to
have moved half the distance, and so on and so forth, up to infinity; this means, in
modern terms, that movement assumes that the sum, or synthesis, of an infinite number of
elements is possible» (Koyré, 1971, p. 10). In the “paradox of
Achilles and the tortoise”, «movement is impossible, because a fast
runner can never reach a slower one. In fact, if the latter has a head start, the former,
before reaching the other runner, has to arrive at a point in which the slower runner was
at the beginning. The head start is shortened in this way, but never reaches zero. In
modern terms, this means: 1) Every corporeal object must pass through an infinite
number of points, 2) Since every point of Achilles’ trajectory corresponds to a
point of the tortoise’s trajectory who is ahead of Achilles, and vice-versa, the
number of points must be the same. Therefore, it is impossible for the distance covered by
Achilles to be greater than that of the tortoise in the same interval of time» (ibidem).
The “arrow paradox” arises in a different manner, but the same
conclusion (i.e., that motion is impossible), is reached. «A flying arrow is
immobile in every instant and in every point of its trajectory. If, according to the
finitist hypothesis, we assume that every interval of time is composed of indivisible
elements (points and instants), then the arrow must be necessarily at rest in every point
and in every instant of time. In fact, motion cannot take place in indivisible points and
instants of time» (ibidem). Finally, we have the “stadium
paradox”, which is as follows: «three lines of equal length composed of
the same number of indivisible elements are placed in a stadium: one is immobile, and the
other two move in a direction parallel but opposite to the stationary one. According to
the finitist hypothesis, “the half must be equal to the whole”, as Zeno
says. This is so because in a fixed instant assumed indivisible, one and the same spatial
element must pass in front of one or two spatial elements and, consequently, must be equal
to one and two elements at the same time» (ibidem).
These and other paradoxes arise for two reasons: the first reason
is that the infinite number of parts (points or instants of time)
of the line, or the interval of time, are treated as if they “actually”
existed in the line, or in the interval of time, while, in reality,
they are only so “potentially”. Therefore, there can be
no “actual infinity” to cross (which would be impossible
to do). The second reason is that the points and instants of time,
which result from an operation of dividing infinitely, are treated
as if they were indivisible. But in the modern theory of the continuum,
the operation of division (of length and time) produces elements
of the same “kind” as those from which one started with,
that is, elements which though being arbitrarily small are still
divisible. In essence, the paradoxes arise when one identifies what
mathematicians call “infinitesimals” with size-less elements
which are, by the very fact of being size-less, not homogeneous
with the quantities one began with. If one treats the problem of
motion using a “discrete” approach instead of a continuous
one, then motion, as the continuous and gradual passage from one
position to another, would not even be thinkable and it would be
necessary to view motion instead in terms of discontinuous and instantaneous
“leaps” from one state of the mobile system to another,
as it so happens in the transitions from one atomic level to another
in modern quantum mechanics.
4. Change in Non-Corporeal Entities: Philosophical-Theological
Aspects. In mechanics, which deals with the motion of material
bodies, one studies among all the various types of motion that which
is called “local motion”, a type of motion which can be
characterized by the change of positions and distances between corporeal
objects, or between the parts which comprise them. Nevertheless,
as we have already seen, motion and becoming have a wider meaning
than that of “local motion”, which is only a specific
case. This wider meaning has been the object of study of philosophers
and theologians. Philosophy and theology deal with non-corporeal
entities and the inquiry into the nature of change in such entities
has been one of the concerns of such disciplines. Clearly, a similar
inquiry cannot be conducted by means of direct or observational
experiment, but only on the basis of logic and metaphysics, as far
as philosophy is concerned, and with reference to truths revealed
by God, in the case of theology.
Basing himself on the Aristotelian analysis of motion (see above,
II.2), Thomas
Aquinas had pointed out that motion can occur in an immaterial entity
only if it is diversified in itself, and therefore, if it is by
no means simple. Consequently, one cannot speak of “motion”
in God (cf. Summa Theologiae, I, q. 9), who is a completely
simple being (ibidem, q. 3). He is Pure Act, in which everything
coincides with his very Being and in which there is nothing to reach
which is not already and completely actuated in Him. It is however
possible to speak, in an analogical sense, about motion in the angels
(cf. ibidem, q. 53) or in the human soul,
even when the latter is separated from the body. These, in fact,
reveal a certain degree of composition, even though they are of
an immaterial (spiritual) nature, since they are endowed with faculties
distinct from their essence (“potencies”) through which
they act and thereby change their state and their action on other
beings. In these beings, motion consists in their changing from
one state to another and their action on beings distinct from them
and on others. Since these immaterial beings (angels and souls separated
from the body) can act on other beings, it is possible to speak
in an analogical sense, not only of internal motion related to the
use of their cognitive and volitional faculties, but also to local
motion in so far as it makes sense to say that an immaterial being
is where it acts. Since God is present in every place, as he acts
on every being in creating it and keeping it in existence ( CREATION,
I), one can in no way attribute to God a local motion.
III. Scientific Inquiry into the Nature of Motion
Modern science, which is based on the Galilean method, abandons
the metaphysical approach in order to resume the “physical”
approach of the Ionian philosophers and the “mathematical”
approach of the Pythagoreans, approaches which modern science founds
anew and in a certain sense unifies. The aim of this section is
not so much to describe in an exhaustive way the various scientific
theories of motion, but rather to highlight the principal conceptual
changes in the field of mechanics which the paradigm shifts from
one conception to another have entailed (for a “classical”
concept of paradigm, cf. Kuhn, 1996).
1. Newtonian Mechanics. Newtonian mechanics was developed
through a gradual abandonment of qualitative concepts of Aristotelian
physics, which turned out not to be correct, in view of replacing
them with a quantitative-relational Archimedean approach, that is,
a mathematized one. This replacement of the more descriptive aspects
of Aristotelian physics was accompanied by the abandonment of Aristotle’s
theory of foundations which was no longer understood correctly especially
due to a gradually decreasing understanding of the notion of analogy.
The theoretical framework and application of the concept of analogy
reached its climax in the philosophy of Thomas Aquinas, and analogy
plays a decisive role in Aquinas’ thought. With the abandonment
of Aristotelian metaphysics one has witnessed a gradual, but decisive,
shift of scientific thought towards Platonism (cf. Koyré, Discovering
Plato, 1968). In light of the new scientific discoveries in
the last few decades, it seems as if the abandonment of Aristotle's
physics has brought an indisputable positive gain to scientific
thought, the abandonment of Aristotelian metaphysics has been, at
least in part, a loss, in the theory of foundations. Nowadays, the
sciences — and most of all the science of complexity —
seem to show, however, renewed interest in recovering Aristotelian
and Thomistic metaphysics which give important ideas with which
to overcome reductionism
present in the sciences.
However, at the time of Galileo, one dealt with what had become “corrupt
aristotelism”. «The concept of form at the basis of the hylemorphic
theory and of all of Aristotelian physics had been misunderstood by the scholastics of the
period of decline: “form”, which in the thought true to Aristotle and
St. Thomas Aquinas is an incomplete and partial reality, an ens quo, was described
as a complete substance, an ens quod, a description which led to a host of
contradictions» (Masi, 1957, p. 85). The quantitative approach to mechanics
was possible only if one took the road of abstract simplification. This required that one
isolate as much as possible that single factor considered the most relevant among the many
which contribute to the motion, and abstract from the other factors by making them
negligible. One only has to think, in this regard, of the painstaking work of Galileo in
which he tried to reduce as much as possible the effect of friction on his measurements. The mathematization of mechanics (and more generally of all
of science), already upheld in the thirteenth century by Roger Bacon as
the right path to take, led to two different roads: a) from the experimental point of
view, it entailed a large step from “a world of approximately to a world of
precision” (cf. Koyré, 1971, pp. 341-362), effected by the gradual
development of measuring devices and methods of ever increasing accuracy. In the field of
astronomy, the accurate measurements of Tycho Brahe (1546-1601) are well known, in which
Kepler (1571-1630) put so much trust that he began to question all of his preceding
theoretical work; b) from the theoretical point of view,
it required the unification of celestial and terrestrial mechanics on the basis of a
unified theory of matter which admitted a single type of “matter” common
to both heavenly and sub-lunar bodies. This theoretical way of thinking required a
two-fold program involving the development of a “kinematics” (that is,
an analytic and geometrical description of how motion happens in reality) and of a
“dynamics” (that is, an inquiry into the causes of motion) which worked
equally well for the motion of the planets and for earth-bound corporeal objects.
The most important conceptual advances made in the field of celestial kinematics were:
a) the discovery made by Copernicus (1473-1543) of the heliocentric theory which
simplified the description of planetary motion; b) the three laws of Kepler and, in
particular, the description of planetary motion by ellipses instead of circles. Regarding
terrestrial kinematics: a) the determination of the law of falling bodies by Galileo;
b) the principle of inertia, also discovered by Galileo, deduced from experiments
made using inclined planes, which constituted the point of contact with the later
Newtonian dynamics. In the field of dynamics, Newton's laws would be responsible for the
unification, in the strict sense of the word, of the sub-lunar and celestial worlds.
All of the subsequent advances in nineteenth-century rational mechanics were none other
than the fulfillment of the potential locked in the Newtonian paradigm brought about by
the development of new mathematical and geometrical tools of ever increasing
sophistication. The mechanics of Lagrange (1736-1813), of Euler (1707-1783) and of
Hamilton (1805-1865) would lead to subsequent formulations of Newtonian mechanics which
were equivalent under the proper analytic conditions and extended their application to
ever more structured mechanical systems (from the mechanics of point particles to the
mechanics of the rigid body, or to that of holonomic and anholonomic systems with any
number of degrees of freedom, to the mechanics of deformable continuous systems such as
fluids and solids, to force fields of all kinds). In this manner, one reaches three
analytical formulations of the same Newtonian mechanics, which are equivalent under the
proper conditions a) the formulation in terms of differential equations (equations of
Lagrange and Hamilton), b) the variational formulation (based on the principle of
least action of Maupertuis and Hamilton), c) Hamilton-Jacobi’s theory which
interprets motion as a special canonical transformation (that is a transformation through
which Hamilton’s equations remain invariant). The mathematical tools common to all of
these generalized formulations of Newtonian mechanics would be however differential and
integral calculus. The identification of constants of motion (first integrals and
invariants) and of the conservation laws related to symmetry (Noether’s theorem)
would permit a physical and mathematical understanding giving ever more interesting and
significant results ( LAWS OF NATURE, I-II).
Nevertheless, the forsaking of the old metaphysics, even if it
had been by then abandoned for several centuries, would be later
on, but inexorably felt. This was primarily because of an incurable
error within Newtonian mechanics related to the conception of absolute
space and time as “primitives” prior to the very notion
of motion, or even as, in the Kantian philosophical interpretation,
a priori categories belonging to the knowing subject.
2. Relativistic Mechanics. The incompatibility between
Newtonian mechanics and the theory of electromagnetism would reveal
in an unavoidable way the inexact nature of the Newtonian conception
of space and time with the famous Michelson-Morley experiment (1887).
By this experiment, it was discovered that the velocity of light
in the vacuum is the same with respect to any observer. As it is
known, taking this result as a postulate and combining it with the
“principle of relativity”, which was already formulated
by Galileo for mechanics alone and extended by Einstein to all of
physics and which assumes that physical laws are invariant under
uniform translations of the reference frame, Einstein deduced the
theory of special relativity ( RELATIVITY,
THEORY OF, I). Among the most conceptually revolutionary results
of this theory, are the Lorentz transformations together with their
correct interpretation. On the basis of such transformations, the
concepts of space and time can no longer be treated as absolute.
The general theory of relativity would push the principle of relativity
to the end by requiring that the invariance of physical laws (covariance)
be valid not only under uniform translations of the reference frame
but also under an arbitrary coordinate transformation. It involves
a sort of “principle of objectivity” which removes all
subjective elements, all dependence on the observer of the fundamental
laws of physics. Paradoxically, a theory originally called theory
of “relativity” (a name which has led a few to view, and
to view erroneously, the theory of relativity as a theory paving
the way for subjectivism in science) is a theory of invariants,
a theory of the objective formulation of the laws of mechanics and
of physics.
3. Quantum Mechanics. Quantum
mechanics brings completely new and unthinkable (from the Newtonian
point of view) elements whose philosophical interpretation has posed
and continues to pose a number of problems. The different paradoxes
such as the “Schrödinger’s cat” paradox and the Einstein-Podolski-Rosen
paradox, which are related to the duality of wave and particle representations
necessary for every physical object, to “non-locality”
and to the ensuing apparent “a-causality” of certain microscopic
systems, have been difficult to resolve. From the point of view
of the school of Copenhagen, these paradoxes can be resolved only
by giving up an understanding in terms of classical realism
and by interpreting them instead in the framework of philosophical
idealism.
The alternative interpretation given by Bohm (cf. Bohm, 1984), which
reformulates quantum mechanics in purely classical and deterministic
terms introducing the “quantum potential”, has been often
viewed as contrived and has not had a large following. Only recently
has it been readopted by various scientists. It seems that if one
wishes to give sufficient physical support to this interpretation,
one would have to explain the nature of the quantum potential in
terms of physical forces: that is, one would have to come up with
a field theory from which this potential can be obtained. One of
the possible directions of research seems to involve a non-linear
field theory of which the present-day quantum mechanics is a linear
approximation. From the philosophical point of view, the quantum-mechanical
paradoxes that manifest the “non-separability” of the
parts of a system, bring about once more the classic problem of
the relationship between the parts and the whole and of the non-reducibility
of the whole to the sum of the parts.
4. Instability and Deterministic Chaos. As we have
seen, the incompatibility between Newtonian mechanics and the electromagnetic
theory of Maxwell
(1831-1879) has shown the inadequacy of the absolute conception
of space and time. With the theory of relativity, the foundations
of mathematized physics and Aristotelian concepts have been brought
back together. However, even before the birth of Einsteinian relativity
and quantum mechanics, Newtonian mechanics has run up against another
large problem which has at the same time revealed the limits of
the mathematical tools used up to then to describe nature. It has
also shown the limits of the reductionist method (see below, IV)
which, up to then, have characterized science. This problem involves
the “stability” of the solutions of differential equations
and their sensitivity to small variations of the initial conditions.
This problem forms the basis of recent studies of the so-called
“deterministic chaos” ( DETERMINISM/INDETERMINISM,
II.4) and it has been treated systematically for the first time
by Poincaré (1854-1912). Qualitatively speaking, a solution of a
differential equation (which from the mechanical point of view represents
the motion of a physical system) is considered “stable”
if in changing by a small amount the initial conditions, the motion
changes only by a small amount for every successive instant of time.
If the motion changes a lot, the solution is called “unstable”.
It is clear that only systems having stable solutions are mathematically
“predictable” and that the high sensitivity to variations
of the initial conditions renders useless a mathematical description
in terms of differential equations of a mechanical, or more generally,
physical system. In fact, because of the instability, an initially
small error can become so large that after a certain time any description
of the motion is meaningless. On the other hand, the systems with
stable solutions turn out to belong to a restricted class of all
possible systems with which one works to find a description of the
motion. This situation has opened up new avenues of research.
From the mathematical point of view, “local” analysis
turns out to be insufficient. It requires, in so far as possible,
a global analysis with the aid of “topology”, a branch
of mathematics which analyzes the set-theoretical and geometric
structure of the parts of a whole in their mutual relations and
their relation to the whole and the irreducible properties of the
whole taken together. Consequently, one has understood, from the
epistemological and methodological point of view, that it is necessary
to overcome the reductionist view. This represents yet another step
towards the Aristotelian conception of the relationship between
the whole and the parts.
IV. Mechanism and Reductionism
1. What is Mechanism? What we mean by the word “mechanism”,
is a philosophical trend, which developed before the formulation
of Maxwell’s equations, whose adherents maintained that the
entire universe can be described and explained through mechanical
action of physical “contact” between material bodies alone.
This “strong” mechanism was not reconcilable even with
the very Newtonian mechanics which introduces the notion of “action
at a distance” to explain the “gravitational force”.
The mechanists (and especially the Cartesians who aspired to an
ideal of science completely describable by geometry) met the very
notion of force with suspicion «fearing that one might find
in this concept a residue of the much abhorred occult forces»
(Masi, p. 87; cf. also Koyré, Newtonian Studies, 1968). Following
the indisputable success of Newtonian mechanics and its application
to planetary motion, even the concept of force was, in the end,
accepted and one attempted to interpret it as an action, in a certain
sense of contact, even if indirect, realized by “fluxes”
of particles which interacting bodies exchange with each other.
This idea was inspired by an older conception of Pierre
Gassendi (1592-1655) and led to the gradual development of a “weaker”
mechanism which accepted the idea of “force” and action
at a distance.
In such a way, mechanism became more simply a philosophy of science which holds that
the entire universe can be described and explained by the laws of Newtonian mechanics
alone. The famous statement of Pierre-Simon de Laplace (1749-1827) is witness to this
position: «A mind which in a given instant knows all the forces which animate
nature and the corresponding situation of beings which comprise it and which is so vast
that it can submit these data to analysis, would encompass with one formula alone all of
the movements of the largest bodies of the universe as well as those of the lightest atom:
such a mind would not be uncertain of anything, either of the future, the past, and those
things which pass under its eyes» (Théorie analytique des probabilites,
Paris 1920, p. VII). The consequences of such a theory in philosophy and theology are
quite clear. Such a conception is based on a metaphysics of pure quantity and relation and
does not contain the fundamental concepts of the metaphysics of being. If mechanists of
that period did not immediately realize this, it was because their religious beliefs made
up for, in a fideistic way, the insufficiency at the rational basis of their metaphysics
( FIDEISM). The creative act of God was reduced to the initial
“setting-in-motion” of the machine of the world and His intervention was
no longer considered necessary to keep created things in existence (as causa essendi).
The more the success with which Newton succeeded in explaining natural phenomena from the
operations of natural forces which obey fixed and immutable laws was complete, the more it
became difficult to view the Creator of the world as conserver of the material universe.
He makes a weak attempt at demonstrating the need of His constant collaboration to prevent
and remedy the disturbances and irregularities which occur in the mechanism of the world,
but in so doing, he does nothing else but expose himself to the derision of Leibniz, who
asks him if perhaps the omnipotent creator had not produced an imperfect mechanism. The
mechanization of the world view lead to an irresistible coherence which supported the
conception of God as an engineer at rest, and, eventually, to His complete elimination
(cf. Dijksterhuis, 1969; NEWTON, IV and VI). He did not hesitate to add to
this elimination, «I have no need of this hypothesis», which was
Laplace’s reply to Napoleon who had asked him where was God’s place in his
system (cf. Koyré, Newtonian Studies, 1968).
From the point of view of the scientific method, mechanism is the
first shocking example of reductionism.
In fact, it assumes that the entire universe is a large machine
governed by the laws of Newton and that, consequently, all of physics
and even all of science is reducible to a series of applications
of mechanical laws. After the success of physics in reducing thermodynamics
to mechanics with the kinetic theory of gases and statistical mechanics,
mechanism was in a certain sense confirmed. Following this reductionist
line of thought, one has later attempted to interpret chemistry
as a chapter of physics and the very biological sciences ( BIOLOGY)
in mechanical terms, including a mechanistic vision not only of
the processes of “growth” and “corruption” of
organic and living bodies, but also of the cognitive processes ( MIND-BODY
RELATIONSHIP).
2. Mechanism and the Structure of the Universe. From
the point of view of the “structure” of the universe,
support for mechanism could come only from materialism
since mechanics necessarily assumes the existence of material bodies
whose motion it describes. Reducing everything to mechanics meant
reducing everything to matter; the success of the atomic-molecular
theory seemed to confirm this line of thought. But the real crisis
of mechanism came about towards the middle of the nineteenth century
with the dawn of Maxwell’s electromagnetic theory, a theory
that did not lend itself to a reduction to mechanics. This irreducibility
appeared so radical that it gave rise to a current of thought which
moved in the opposite direction, that is, towards energetism ( MATTER,
V). It proposed to interpret matter itself as a concentrated form
of field energy.
In light of modern physics, this irreducibility of electromagnetic
theory to mechanics is understandable for two reasons: the first
reason is the incompatibility of Newtonian mechanics with the invariance
of Maxwell’s equations under Galilean transformations; and
Einstein’s theory of special relativity would bring the appropriate
modifications to mechanics to solve this problem. The other reason
is related to the different respective natures of radiation and
matter. In light of quantum mechanics, this difference can be explained
by the irreducibility of the behavior of particles comprising matter
(“fermions”) endowed with the property of impenetrability
(related to the Pauli exclusion principle) to the behavior of the
particles comprising the electromagnetic field (photons, which are
“bosons”) which do not have this property of impenetrability.
The end of mechanism did not mean the immediate end of reductionism.
Classical physics in the beginning of the twentieth century gave
two parallel and consistent syntheses: the Newtonian synthesis for
mechanics which can be extended to thermodynamics as well by means
of statistical mechanics, and the Maxwellian synthesis for electromagnetism.
Both were reconciled by Einstein's correction to mechanics. In the
twentieth century, quantum mechanics and quantum electrodynamics
radically reformulated all of physics, making it adequate for the
study of the atom and the microscopic world in general, but continued
to legitimize the reductionist method. Only from the beginning of
the second half of the century, with the gradual reprise of research
in non-linear mechanics initiated by Poincaré, a field which was
later abandoned for several decades thereafter, reductionism reached
a serious crisis. In the same period, the study of complexity
made progress to some extent in all of the sciences.
3. Reductionism and Mathematics. From the mathematical
point of view, reductionism would seem intimately related to differential
and integral calculus: these two mathematical tools are in fact
reductionist in their very methodology. Differential calculus is
by its very nature a “local” calculus. It defines its
quantities and works with infinitesimal quantities which vary in
the neighborhood of a point, ignoring what happens outside such
a neighborhood. It therefore takes into consideration an infinitesimal
part of the whole. From the geometrical point of view, this means
that it locally approximates a curve as a tangent, a plane as a
surface, etc. Integrating a differential equation requires the integral
calculus: the latter consists in effecting, with a limit process,
the “sum” of infinite infinitesimal elements (integral).
In this way, such a calculus reconstructs the whole, on the basis
of infinite information about the local character of the object
in question, as the sum of the parts. From the geometric point of
view, this means reconstructing a curve from a knowledge of the
tangents at every point. The study of complexity probably requires
an examination of the properties of the “whole” which
cannot be reconstructed in this way. That is, the “global”
properties which cannot be deduced from information about the “local”
character. But this still remains an open problem.
V. Mechanics and Causality
Another problem at the crossroads of science and philosophy which
is of great relevance in the interpretation of scientific theories,
in general, and mechanical ones, in particular, is the problem of
“causality”. Closely related to it is the problem of determinism/indeterminism
which has become important with the advent of quantum mechanics.
It often happens that the same words are used in the area of science
and in that of philosophy with different meanings and are transferred
from one discipline to another without due attention. Terms such
as “cause” and “causality” can be used in such
an improper way, as Schrödinger (1887-1961), the father of the wave
formulation of quantum mechanics, observed (cf. Schrödinger, 1932).
It seems necessary, however, to clarify as far as possible the
use of such a term: to fix our ideas, let us call “mechanical
causality” the concept of causality that is used by physicists
in the interpretation of their theories and let us distinguish it
from “philosophical causality”. We will attempt to identify
the similarities and differences between these two terms.
1. Mechanical Causality. The word “cause”
is not so much used in a technical sense in scientific language
(cf. Nagel, 1979), in particular, in the area of mechanics, as in
the philosophical paradigm on the basis of which a scientific theory
is interpreted. It is taken from common language and applied without
much epistemological thought. Often one speaks of “principle
of causality”, meaning that, in the area of experiment, one
encounters regular associations between the occurrence of certain
phenomena and those which succeed them, in a more or less successive
time. The former are interpreted as causes of the latter. In this
case, one says that the principle of causality cannot be violated
because experience shows that the order of temporal succession in
which two distinct phenomena occur cannot be inverted.
Mechanical causality in the area of Newtonian mechanics.
Mechanical causality appears first in the consideration of the force
as “cause of the acceleration”. In the mechanistic interpretation
of Newtonian mechanics, one tends to identify force with the cause
of acceleration through Newton’s second law. The Newtonian
paradigm, nevertheless, keeps separate the “law of motion”,
that is, the causal relation relating the force of acceleration
and the “law of force”, which describes the behavior of
the specific force acting on a certain body, as for example the
law of universal gravitation of Newton. It does not give a causal
explanation of gravity, that is, it is not concerned with the characterization
of the nature of gravity so much as the description of the variation
of the gravitational force as a function of the masses of and distance
between the interacting bodies. This scheme was criticized by Einstein
who gave, by the general theory of relativity, a unified description
of the laws of motion and the gravitational force law by means of
the geometry of space-time. In so doing, he gives also an explanation
of the nature of gravity in terms of the curvature of space-time.
A second point concerns the cause of inertia. The Newtonian paradigm does not
even put forward the problem of assigning a cause to the inertia of corporeal bodies, that
is, of motion in the absence of forces: in other words, it is assumed as law that a body,
in absence of forces, will remain at rest, or moves uniformly along a straight line with
respect to an inertial reference frame. No causal
explanation is given for this fact. Instead, one tends to say that rectilinear, uniform
motion has no cause. Ancient physics had assigned a cause to motion, including
rectilinear, uniform motion, since motion, the “becoming” of a being,
required a fitting cause which kept it in existence. Even the mechanics of the Middle Ages
and the Renaissance had sought to give a response in the mechanical sense to this
question, for example, with the theory of the impetus (cf. Dijksterhuis, 1969).
The problem was resumed in modern terms by Mach (1838-1916) who
considered insufficient a theory of mechanics which did not give
a causal explanation of inertia. He proposed that such an explanation
was of global (holistic) character and that it could not be described
with a local (reductionist) theory. According to “Mach’s
principle”, inertia must be an effect of the interaction between
all bodies present in the universe. Mach, however, was not in the
position to describe this interaction quantitatively (cf. Sciama,
1959, and also Nagel, 1979). The research program of a scientist
who deals with mechanics is the institution of a principle from
which all accelerated and inertial motions derive (cf. Mach, 1977).
General relativity would resume, in a certain sense, Mach’s
idea through the “equivalence principle” which states
that a gravitational field is locally indistinguishable from an
apparent force field due to the non-inertiality of the frame of
reference, reducing the two to the curvature of space-time. Einstein
had hoped that Mach’s principle could be deduced from his equations
(cf. K. Gödel, Collected Works, vol. I, Oxford University
Press & Clarendon Press, New York - Oxford 1986). Nevertheless,
since such a theory uses differential geometry, which is based on
a local description of space-time, it cannot exhaustively translate
Mach’s principle which on the other hand represents a holistic
(or global) vision.
Mechanical causality in relativistic mechanics. The philosophical
paradigm of Einstein’s special and general theories of relativity
does not modify the idea of causality common to Newtonian mechanics
and electromagnetism. Such an idea of causality requires the temporal
priority of “phenomenon-cause” over the “phenomenon-effect”
and adds to this the principle that the speed of light, being invariant
with respect to every observer, is the maximum speed of propagation
of any signal. A force field cannot propagate at a higher velocity
and therefore instantaneous interactions between distant bodies
are not possible. Combining this condition with temporal priority,
one arrives at the conclusion that a phenomenon cannot be the cause
of another which occurs at a distance “before” the light
signal has travelled the entire distance between the two places
in which the two phenomena occur. This law of physics is usually
called by physicists who deal with relativity, “principle of
causality”: it forbids the instantaneous action at a distance
which was considered possible with Newtonian mechanics. In fact,
Newtonian mechanics allowed a signal (or flux) to travel at infinite
velocity. Therefore causality came to be called “locality”,
that is, the fact that the interaction does not occur at a distance,
but in the place where the signal (field) is present. Theories which
violate this principle are called “non-local”, or “a-causal”,
and are considered valid only as a non-relativistic approximation.
Mechanical causality in quantum mechanics. In this third
field, the manner of understanding causality depends on the interpretative
paradigm adopted ( QUANTUM
MECHANICS, IV). If the “Copenhagen interpretation” is
adopted, it is assumed that the principle of causality in the Newtonian
sense is no longer valid and the cause-effect connection on the
microscopic level is not deterministic, but only probabilistic,
probabilistic in the sense that from a certain cause follow effects
with a certain probability distribution. Moreover, in quantum mechanics,
causality is violated even in the sense of a violation of non-locality
since instantaneous effects at a distance between non-separable
systems are possible (in quantum mechanics). If a realist interpretation
is adopted, like that of Bohm, one assumes that the principle of
causality is valid in the classical sense and that quantum mechanics
is not indeterministic. In this interpretation, with the introduction
of the “quantum potential”, the particle trajectories
are identified in the classical sense, but they remain, nevertheless,
deterministically non-observable ( DETERMINISM/INDETERMINISM,
II. 3).
Mechanical causality in non-linear mechanics. In non-linear
mechanics, in the presence of solution instability, the time evolution
of a solution of differential equations which govern the system
cannot be determined without amplifying the error on the initial
conditions. One can interpret this as the impossibility of mathematics
to describe in a univocal manner the causes and effects of a given
process considered by its very nature to be causal. The description
of cause is codified through a law which is expressed as a differential
equation, together with the appropriate initial conditions. The
knowledge of the latter is indispensable for determining the solution
of the problem. The solution describes the time evolution of the
phenomenon and therefore permits an identification of the effect.
Nevertheless, if the solution is unstable, a small error in the
assignment of the initial conditions can lead to a large error in
the determination of the evolution of the system. Since it is not
practically possible to know the initial conditions with infinite
precision (one would need to know numbers with infinite digits and
perform calculations with such numbers) one cannot, in practice,
determine, after a sufficiently long time, the effect. Causality
in this case is not violated in so far as the system is deterministic,
but one does not have the sufficient tools to describe it mathematically.
2. Mathematical Formal Causality. The empiricist conception
of causality of David
Hume (1711-1776), with its emphasis on the constant conjunction
of two phenomena which occur consecutively in experience, is only
part of scientific explanation and does not fully describe the causality
implicit in the Galilean science. Thus it does not seem possible
to conclude that, from the empirical point of view, the ascertainment
of correlations is sufficient to establish a complete equivalence
between mechanical causality and “efficient causality”
encountered in philosophy (cf. Artigas and Sanguineti, 1989). To
this end, it is necessary to show that there exists an ontological
structure adequate enough to take into account such observed correlations
in causal terms: this can be obtained only by completing the analysis
in metaphysical terms. But Galilean science is a “middle science”
(scientia media). It is materially “physical” and
empirical and formally “mathematical”, but not formally
“metaphysical”. For this reason, explanations are given,
on the mathematical side, through formal causality, which allows
one to perform a demonstration starting from the essential definitions
of mathematical entities. One may think of the way of proceeding
typical of mathematical physics, which is not concerned with the
experimental aspects of the problem. Mathematical physics treats
the mathematically-formulated laws of physics, and the definitions
of the mathematical entities it works with, as axiomatic assumptions.
It then formally deduces from such definitions and such laws, in
the form of theorems, those results which experimental physics then
verifies and technology applies.
3. Philosophical Causality. It is known how Aristotelian
science, basing itself on metaphysics, includes in its method of
explanation the four causes: material, formal, efficient, and final
( METAPHYSICS,
I.1). Thomas Aquinas then worked out the distinction between “principal
cause” and “instrumental cause”, which he used especially
in theology to explain the idea of inspiration in Holy Scripture
and its principal Author, and to show the salvific action by Christ’s
humanity and by the sacraments of the Church.
If one wishes to understand the theory of the four causes in an unambiguous way, one
needs to keep in mind the two metaphysical theories it presupposes, that is, the
“hylemorphic” theory ( MATTER, II.3) and the theory of
“potency-act” (see above, II.2). Now the material and formal causes are
placed on the same footing as these two metaphysical principles. The “material
cause” gives the constitutive basis of a physical object making it capable of
assuming one “form” or another. The “formal cause”
makes the object assume the form it has now instead of another. Concerning the two other
causes: the “efficient cause” makes a physical object assume a form
(and/or properties) that are different from those that characterize it in its present
state. It is therefore responsible for change and hence for local motion. The
“final cause” resides in the final state to be reached at the end of a
certain motion.
In terms of the final cause, the end of change is interpreted as
inscribed in the very law which governs it. In this perspective,
the final cause is more important than the other three which in
a certain sense depend on it. The end that is to be reached determines
the material constitution of a physical object, its essential characteristics
(form), and requires an adequate efficient cause to effect the change
from a certain initial state towards the final one. Newtonian mechanics,
and the mechanics which succeeded it, abandoned Aristotelian language.
Or when they do use it, they change its interpretation. For this
reason, science in the proper sense of the word does not use terms
such as “cause”, which goes beyond mathematical language.
Nevertheless, in the mechanistic interpretation given to this theory,
one usually asserts that, in nature, no cause is necessary to explain
(local) motion of corporeal objects other than the efficient cause.
In fact, in treating motion as a “state” analogous to
the state of rest, one claimed it was sufficient to assign a cause
which allows a body to go from one “state of motion” to
another, or a cause of acceleration. Since the causes that can make
the state of a body change are the “efficient” ones, the
consequence is readily apparent: this efficient cause, in virtue
of the second law of mechanics (F = m a), can be none
other than “force” in the Newtonian sense. In reality,
physics, and for better reason, the other sciences, have always
gone well beyond the mechanistic scheme, showing in this way that
it makes tacit recourse to the three other causes.
The material cause. The material cause is present in the
physical sciences whenever one studies the constituent elements
of the physical universe, whether in the form of radiation or matter.
Nevertheless, the framework of modern physics differs radically
from the framework of Aristotelian metaphysics in that modern physics
investigates the nature of elementary particles as “things”,
that is, as entities homogeneous with the physical objects that
they compose. The reductionist approach is here rather evident:
one has thought of the whole as the sum of the parts. Aristotelian
metaphysics, instead, seeks the fundamental constituents on different
levels. Such constituents are inhomogeneous with respect to the
bodies they comprise and, like matter and form, with respect to
each other. What is interesting is the fact that complexity seems
to introduce the need for differentiated levels in the constituents
of physical, chemical, and biological objects. It must be noted
that even Aristotelian physics used components homogeneous with
the things they comprise, such as the four elements (air, water,
earth, and fire), which in essence played a similar role to that
of the chemical elements of the periodic table. However, Aristotelian
physics also introduced more fundamental principles, principles
operating on different levels as those of matter and form, to explain
the very possibility for the existence of lower-level components.
Modern physics has not reached this point, but it comes closer to
it than mechanism because it hypothesizes some differentiated and
inhomogeneous levels following the ideas of the science of complexity.
In order to continue and develop a mathematized physics, one would
need a mathematical theory capable of treating this hierarchization
of levels.
The formal cause. The formal cause comes into play in physics
and therefore, tacitly, in mechanics, since modern physics is a
kind of “mathematical physics” in that it uses mathematical
and not metaphysical definitions and demonstrations to describe
and explain objects of experience. Now, from the “definition”
(the logical connotation of form) of a mathematical object, which
in a scientific theory represents a physical object which identifies
its quantitative and relational properties, the physical theory
(and in particular, mechanics) deduces the behavior of the object.
The efficient cause. The efficient cause, whose role is
more evident in mechanistic epistemology, determines changes of
state which are none other than accidental changes, or substantial
ones of the object in consideration. Nevertheless, the current conception
in the interpretation of scientific theories is reductive in that,
being conditioned by empiricism, it relates causality to the constant
conjunction of two phenomena and to their occurrence in a temporal
succession by which the cause precedes the effect. From a metaphysical
point of view, this is not always true since regular temporal successions
which have no causal relation are possible and since a cause can
also be supertemporal ( TIME,
II.3) and can cause an entire being with its time without being
involved in time. Concerning the final cause, this seems
to be completely excluded from science and mechanics, but not always.
I will devote the next section to this problem.
VI. Mechanics and Finalism
Among the different new problems which have emerged in the area
of modern science —which, in reality, are not new because they
are related to ancient philosophical problems, and are new only
in the context and the way in which they emerge today— the
question as to the possibility of a finalistic explanation of the
facts of experience within a scientific theory is certainly one
of the most philosophical, and therefore, one of the most subtle
in the framework of the scientific methodology ( FINALITY).
It does not seem possible to maintain that final causality has never
been present in modern science. One should say, instead, the opposite:
the problem consists of identifying the ways in which the final
cause is “legitimately” present together with the other
causes. On the level of scientific analysis, it can be recognized
as a kind of “immanent” finality, or “low-level”
finality: it is not a question (nor would it be thinkable) of introducing
into physics a kind of transcendent finalism, but instead, simply
to point out that principles of finalistic character can act as
“principles of understanding” the evolution of phenomena.
For example, when the analysis of complex phenomena requires us
to introduce hierarchized levels, a finalistic explanation can be
used on every level without having necessity to call into question
the existence of a “final end” towards which the whole
process tends.
1. Finalism in the Formulation of Laws. One can observe,
first of all, that finality comes into play in a legitimate and
valid way in the formulation of the laws of scientific, physical,
and in particular, mechanical theories and that this has been true
for quite some time. There are in fact several ways of formulating
laws (and not only physical ones). One can identify two of them
for the purposes of our discussion: a) the first way does assign
a law in a “direct” manner which is not finalistic; b) the
second way, on the contrary, does not assign a law in a direct manner,
but identifies it “indirectly”, assigning an “end”
that, through such a law, can be realized in a physical world. Examples
of the first category are all the laws formulated in terms of differential
(or algebraic) equations which govern the time evolution of physical
systems, materials properties, etc. Examples of the second category
of laws are those of thermodynamics and variational principles.
It is important to emphasize that while a directly-formulated law
of evolution generally admits an indirect, and therefore finalistic,
formulation — as it so happens in Lagrangian and Hamiltonian
mechanics — it can happen that one can formulate the laws
in a finalistic way without a knowledge of the direct formulation
of the theory. This means that one can know the final causes before
knowing the efficient ones. When one has obtained both formulations,
one can claim to know both final and efficient causes. It is important
to emphasize that a finalistic explanation does not contradict one
that makes recourse to other causes and even requires the latter’s
explanation, to a certain extent, so as to understand the processes
through which a certain end is reached.
Thermodynamics. Historically, a significant example of such
a situation came about with the discovery of thermodynamics. Since
thermodynamics is a macroscopic theory, it formulates its laws in
finalistic terms and cannot give a direct description of the microscopic
“mechanisms” realized in processes. The processes which
nature realizes are those which reach two ends a) the conservation
of energy (the first law), b) the increase of entropy (second
law). For this reason, the mechanists did not like thermodynamics.
They sought an explanation of thermodynamics in terms of efficient
and mechanical causes by means of the kinetic theory of gases and
statistical mechanics. The latter gave “direct” laws according
to which ends prescribed by thermodynamics in its macroscopic formulation
are reached.
Quantum Mechanics. This kind of formulation of laws is found
not only in classical physics, but also in quantum mechanics where
some laws are formulated in a prescriptive way without describing
the mechanism which allows one to carry out the prescription. The
first example of this can be found as early as the initial phase
of quantum mechanics in Bohr’s (1885-1962) quantization scheme.
This scheme prescribed the trajectories of the electrons of atoms
which could be physically realized to be those which reach the following
end: to make the action equal to an integer multiple of Planck’s
constant. A direct explanation of the mechanism through which such
an end could be realized was not found until De Broglie (1892-1987)
formulated his wave theory of matter. The second example is given
by Pauli’s exclusion principle which prescribes two electrons
in an atom not to occupy the same quantum state. Therefore, they
must position themselves so as to realize this prescription. The
direct explanation of this finalistic prescription would be only
found later with the discovery of quantum statistics and the odd
or even symmetry of the wavefunctions under the exchange of particles.
Conservation Laws. In mechanics and physics in general,
“conservation laws” can be interpreted finalistically:
in certain conditions, motion “tends” to keep a certain
quantity (linear momentum, mechanical energy, angular momentum,
etc.) constant. One can also say: among all of the kinematically
conceivable motions, those which are realized in nature under certain
conditions are those which reach the end of conserving certain determined
physical quantities.
Variational Principles. Even the most powerful mathematical
formulation of the mechanical, and in general, physical laws is
given by “variational principles” of finalistic type.
In fact, variational principles state that nature behaves in such
a way so as to reach the goal of making minimum (or stationary)
a certain action integral. Among all the possible processes which
lead a system from state A to state B, the one chosen
in nature reaches the goal of minimizing a certain quantity. It
is interesting to observe how, from the historical point of view,
the “finalistic formulation” obtained through Hamilton's
principle, to determine the equations of motion, and through the
variational principle of Maupertuis, to obtain the equation of a
single trajectory of motion in the case of conservative systems,
has instead followed preceding the “direct formulation”
of Newton's laws. This can be easily understood from the fact that
the variational formulation requires mathematical techniques discovered
only later.
States which do not depend on Initial Conditions. In the
field of the mechanics of dynamical systems, another kind of low-level
finalism appears related to stable regimes which are independent
of the initial conditions. The system eventually reaches this low-level
finalism in which it remains. Such examples of these stable regimes
are the stable “limit cycles”, stable “equilibrium
points”, or more generally, stable “attractors”.
In these cases, the initial conditions are not determining factors
(which can extend to the inside of an entire “basin of attraction”),
but rather, the final conditions. The most well-known example is
that of forced oscillations, which after a certain time stabilize
and oscillate with the same period with which they are externally
driven.
2. The Anthropic Principle. The above considerations
and examples, have helped us to illustrate how the finalistic explanation
has entered into the realm of sciences, in physics, and mechanics
in particular, even if it is not defined as such. It must be said
that such a procedure has succeeded because it was possible to formulate
mathematically the finalistic prescriptions once they have been
introduced. The resistance of the “mechanists” to thermodynamics
remains significant as well as their satisfaction at the discovery
of a mechanical model based on the kinetic theory of gases and statistical
mechanics. Nevertheless, this model has not supplanted thermodynamics
which has never been abandoned. With the advent of Maxwell’s
electromagnetic theory and field theory, the possibility of a physics
which was not reducible to mechanics was definitively confirmed.
Today, there seems to be a certain resistance towards those forms of finalism which
cannot have, or do not yet have, a complete mathematical formulation, such as, at the
moment, the Anthropic Principle. Such a principle is used in cosmology to
deduce several properties of the physical universe from the idea that such properties must
be compatible with the appearance of life (weak formulation), or, alternatively, that life
itself, and in particular, the appearance of human beings, acts as a general principle
with which one can understand the necessary presence of those properties (strong
formulation).
It must not be forgotten that the use of the final cause on higher
(intentional) levels in the study of being is not the task of physical-mathematical
sciences, but that of metaphysics (final cause of being and the
nature of things) and of theology (questions of meaning).
3. Concluding Observations. To conclude, it is interesting
to examine, or at least to hint at, several criteria of a general
character with which to judge, in the context of a scientific theory,
a principle or behavior of finalistic character. I have already
examined the first criterion which was used in regards to thermodynamics
and variational principles. Such a criterion, which has survived
the test of time, can be formulated as: «a physical law can
be stated in finalistic form if such a formulation can be expressed
in mathematical form». In this regard, one can add that future
and more advanced (“wider”) mathematics, can uncover new
terrain for a kind of finalistic explanation which now might seem
unacceptable from the scientific point of view. Another significant
case which I have alluded to is that of stable attractors, which
concerns not so much the physical laws as the individual evolutionary
behavior of physical systems, i.e. a single solution of the laws.
This does not pose any problems as long as the behavior of the system,
which does not depend on the initial conditions, is represented
by particular solutions of differential equations, and as such,
arises directly from the mathematics which govern the physical system.
All of this is perfectly scientific and whether or not it involves
finalistic behavior is a question of philosophical interpretation
of a scientific fact. The third case which I have referred to, involves
the Anthropic Principle, and is by its nature more subtle, since
it deals with a finalistic principle for which no mathematical formulation
exists as of yet. Is it valid to accept a principle formulated this
way in a scientific theory? Generally, in science, one accepts a
hypothesis or theory when a comparison with experiment is possible
in the following two senses: a) in accounting for the known
experimental data, within the error of measurements and limits which
define the domain of validity of the theory; b) in being able
to predict new phenomena which are experimentally verifiable. As
a rule, verification and predictions of a quantitative nature are
required, that is, on the level of measurements. One, therefore,
asks the following two questions: «Is a philosophical principle
possible that allows one to deduce information about the values
of certain quantities?». And also: «Is it possible or
appropriate to work out a demonstrative but not mathematized scientific
theory which allows one to describe and make predictions on non-quantitative
data?».
Obviously, these are deep and completely open questions which are fascinating to the
researcher. Perhaps we are living in a very important moment in the development of
scientific thought, a moment witnessing the rise of philosophical questions arising from
the theory of the very foundations of science.
Alberto Strumia
(translated by Eric Chang)
See also: COMPLEXITY;
LAWS OF NATURE; FINALISM; MATTER; METAPHYSICS; QUANTUM MECHANICS; REDUCTIONISM;
.
RELATIVITY, THEORY OF
Bibliography
Scientific Aspects: E. SCHRÖDINGER, Über
indeterminismus in der Physik, J.A. Barth, Leipzig 1932; A. EINSTEIN
e L. INFELD, The Evolution of Physics. The Growth of Ideas from
Early Concepts to Relativity and Quanta (1938), Cambridge University
Press, Cambridge 1971; D. BOHM, Causality and Chance in
Modern Physics (1957), Routledge, London 1984; P.A.M. DIRAC,
Principles of Quantum Physics, Clarendon Press, Oxford 1958;
D.W.SCIAMA, The Unity of the Universe, Faber & Faber,
London 1959; P.A. SCHILPP (ed.), Albert Einstein. Philosopher-scientist
(1938), Open Court, LaSalle (IL) 1970; R. COURANT e H. ROBBINS,
What is Mathematics? An Elementary Approach to Ideas and Methods,
Oxford Univ. Press, New York - Oxford 1996; V.I. ARNOLD, Mathematical
Aspects of Classical and Celestial Mechanics, Springer, Berlin
- London 1997; V.I. ARNOLD, Geometrical Methods in the Theory
of Ordinary Differential Equations, Springer, New York 1988;
J. VON NEUMANN, Mathematical Foundations of Quantum Mechanics
(1932), Princeton Univ. Press, Princeton (NJ) 1955; H. GOLDSTEIN,
Classical Mechanics, Addison-Wesley, San Francisco - London
2002.
History and Philosophy of Science: A. EINSTEIN,
The World as I See It, J. Lane, London 1955; M. DAUMAS
(ed.), Histoire de la science, Gallimard, Paris 1957; A. KOYRÉ,
From the Closed World to the Infinite Universe, Johns Hopkins
Univ. Press, Baltimore 1957; R. MASI, Struttura della materia.
Essenza metafisica e costituzione fisica, Morcelliana, Brescia
1957; E. MACH, The Science of Mechanics. A Critical and
Historical Account of its Development (1883), Open Court, LaSalle
(IL) 1960; M. CLAGETT, The Science of Mechanics in the Middle
Ages, Univ. of Wisconsin Press - Oxford Univ. Press, Madison
- London 1961; M. BUNGE, Causality. The Place of the Causal
Principle in Modern Science, Meridian Books, Cleveland 1963;
A. KOYRÉ, Discovering Plato, Columbia Univ. Press,
New York 1968; A. KOYRÉ, Newtonian Studies, Univ.
of Chicago Press, Chicago 1968; E.J. DIJKSTERHUIS, The Mechanization
of the World Picture, Oxford Univ. Press, Oxford 1969; M. BOAS,
The Scientific Renaissance, 1450-1630, Fontana, London 1970;
A. KOYRÉ, Études d’histoire de la penseé philosophique,
Gallimard, Paris 1971; F. ENRIQUES, G. DE SANTILLANA, Compendio
di storia del pensiero scientifico (1936), Zanichelli,
Bologna 1973; A. KOYRÉ, The Astronomical Revolution.
Copernicus, Kepler, Borelli, Methuen, London 1973; A. KOYRÉ,
Études d’histoire de la penseé scientifique, Gallimard,
Paris 1973; A. KOYRÉ, Galileo Studies, Humanities
Press, Highlands (NJ) 1978; E. NAGEL, The Structure of Science.
Problems in the Logic of Scientific Explanation, Routledge &
Kegan Paul, London 1979; E.T. BELL, Men of Mathematics
(1937), Simon & Schuster, New York 1986; M. ARTIGAS, J.J.
SANGUINETI, Filosofia della natura, Le Monnier, Firenze 1989;
T. and I. ARECCHI, I simboli e la realtà, Jaca Book,
Milano 1990; J.J. SANGUINETI, Scienza aristotelica e scienza
moderna, Armando, Roma 1992; A. STRUMIA, Introduzione
alla filosofia delle scienze, Edizioni Studio Domenicano, Bologna
1992; A. CROMBIE, Styles of Scientific Thinking in the European
Tradition, 3 vols., Duckworth, London 1994; T.S. KUHN,
The Structure of Scientific Revolutions (1964), Univ. of
Chicago Press, Chicago - London 1996; R. COGGI, La filosofia
della natura? Ciò che la scienza non dice, Edizioni Studio Domenicano,
Bologna 1997; F. BERTELÈ, A. OLMI, A. SALUCCI e A. STRUMIA,
Scienza, analogia, astrazione. Tommaso d’Aquino e le scienze
della complessità, Il Poligrafo, Padova 1999. |
