Mechanics

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” (Gr. 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 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. 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”; see Matter) 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. 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. 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, 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.

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. 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” 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. 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). 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 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.

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. 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, v. I [New York - Oxford: Oxford University Press - Clarendon Press, 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. 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.

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. 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 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 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. 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.