- 这三篇是我在当代哥白尼一线所提到的文章,讨论物理中“力”的由来。我细读了第一篇,粗读了第二篇,尚未读第三篇。放在这里做参考。
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Whence the Force of F = ma? I: Culture Shock
Frank Wilczek
When I was a student, the subject that gave me the most trouble was classical mechanics. That always struck me as peculiar, because I had no trouble learning more advanced subjects, which were supposed to be harder. Now I think I've figured it out. It was a case of culture shock. Coming from mathematics, I was expecting an algorithm. Instead I encountered something quite different— a sort of culture, in fact. Let me explain.
Problems with F = ma
Newton's second law of motion, F = ma, is the soul of classical mechanics. Like other souls, it is insubstantial. The right−hand side is the product of two terms with profound meanings. Acceleration is a purely kinematical concept, defined in terms of space and time. Mass quite directly reflects basic measurable properties of bodies (weights, recoil velocities). The left−hand side, on the other hand, has no independent meaning. Yet clearly Newton's second law is full of meaning, by the highest standard: It proves itself useful in demanding situations. Splendid, unlikely looking bridges, like the Erasmus Bridge (known as the Swan of Rotterdam), do bear their loads; spacecraft do reach Saturn.
The paradox deepens when we consider force from the perspective of modern physics. In fact, the concept of force is conspicuously absent from our most advanced formulations of the basic laws. It doesn't appear in Schrödinger's equation, or in any reasonable formulation of quantum field theory, or in the foundations of general relativity. Astute observers commented on this trend to eliminate force even before the emergence of relativity and quantum mechanics.
In his 1895 Dynamics, the prominent physicist Peter G. Tait, who was a close friend and collaborator of Lord Kelvin and James Clerk Maxwell, wrote
"In all methods and systems which involve the idea of force there is a leaven of artificiality. . . . there is no necessity for the introduction of the word "force" nor of the sense−suggested ideas on which it was originally based."1
Particularly striking, since it is so characteristic and so over−the−top, is what Bertrand Russell had to say in his 1925 popularization of relativity for serious intellectuals, The ABC of Relativity:
"If people were to learn to conceive the world in the new way, without the old notion of "force," it would alter not only their physical imagination, but probably also their morals and politics. . . . In the Newtonian theory of the solar system, the sun seems like a monarch whose behests the planets have to obey. In the Einsteinian world there is more individualism and less government than in the Newtonian."2
The 14th chapter of Russell's book is entitled "The Abolition of Force."
If F = ma is formally empty, microscopically obscure, and maybe even morally suspect, what's the source of its undeniable power?
The culture of force
To track that source down, let's consider how the formula gets used.
A popular class of problems specifies a force and asks about the motion, or vice versa. These problems look like physics, but they are exercises in differential equations and geometry, thinly disguised. To make contact with physical reality, we have to make assertions about the forces that actually occur in the world. All kinds of assumptions get snuck in, often tacitly.
The zeroth law of motion, so basic to classical mechanics that Newton did not spell it out explicitly, is that mass is conserved. The mass of a body is supposed to be independent of its velocity and of any forces imposed on it; also total mass is neither created nor destroyed, but only redistributed, when bodies interact. Nowadays, of course, we know that none of that is quite true.
Newton's third law states that for every action there's an equal and opposite reaction. Also, we generally assume that forces do not depend on velocity. Neither of those assumptions is quite true either; for example, they fail for magnetic forces between charged particles.
When most textbooks come to discuss angular momentum, they introduce a fourth law, that forces between bodies are directed along the line that connects them. It is introduced in order to "prove" the conservation of angular momentum. But this fourth law isn't true at all for molecular forces.
Other assumptions get introduced when we bring in forces of constraint, and friction.
I won't belabor the point further. To anyone who reflects on it, it soon becomes clear that F = ma by itself does not provide an algorithm for constructing the mechanics of the world. The equation is more like a common language, in which different useful insights about the mechanics of the world can be expressed. To put it another way, there is a whole culture involved in the interpretation of the symbols. When we learn mechanics, we have to see lots of worked examples to grasp properly what force really means. It is not just a matter of building up skill by practice; rather, we are imbibing a tacit culture of working assumptions. Failure to appreciate this is what got me in trouble.
The historical development of mechanics reflected a similar learning process. Isaac Newton scored his greatest and most complete success in planetary astronomy, when he discovered that a single force of quite a simple form dominates the story. His attempts to describe the mechanics of extended bodies and fluids in the second book of The Principia3 were path breaking but not definitive, and he hardly touched the more practical side of mechanics. Later physicists and mathematicians including notably Jean d'Alembert (constraint and contact forces), Charles Coulomb (friction), and Leonhard Euler (rigid, elastic, and fluid bodies) made fundamental contributions to what we now comprehend in the culture of force.
Physical, psychological origins
Many of the insights embedded in the culture of force, as we've seen, aren't completely correct. Moreover, what we now think are more correct versions of the laws of physics won't fit into its language easily, if at all. The situation begs for two probing questions: How can this culture continue to flourish? Why did it emerge in the first place?
For the behavior of matter, we now have extremely complete and accurate laws that in principle cover the range of phenomena addressed in classical mechanics and, of course, much more. Quantum electrodynamics (QED) and quantum chromodynamics (QCD) provide the basic laws for building up material bodies and the nongravitational forces between them, and general relativity gives us a magnificent account of gravity. Looking down from this exalted vantage point, we can get a clear perspective on the territory and boundaries of the culture of force.
Compared to earlier ideas, the modern theory of matter, which really only emerged during the 20th century, is much more specific and prescriptive. To put it plainly, you have much less freedom in interpreting the symbols. The equations of QED and QCD form a closed logical system: They inform you what bodies can be produced at the same time as they prescribe their behavior; they govern your measuring devices— and you, too!— thereby defining what questions are well posed physically; and they provide answers to such questions— or at least algorithms to arrive at the answers. (I'm well aware that QED + QCD is not a complete theory of nature, and that, in practice, we can't solve the equations very well.) Paradoxically, there is much less interpretation, less culture involved in the foundations of modern physics than in earlier, less complete syntheses. The equations really do speak for themselves: They are algorithmic.
By comparison to modern foundational physics, the culture of force is vaguely defined, limited in scope, and approximate. Nevertheless it survives the competition, and continues to flourish, for one overwhelmingly good reason: It is much easier to work with. We really do not want to be picking our way through a vast Hilbert space, regularizing and renormalizing ultraviolet divergences as we go, then analytically continuing Euclidean Green's functions defined by a limiting procedure, . . . working to discover nuclei that clothe themselves with electrons to make atoms that bind together to make solids, . . . all to describe the collision of two billiard balls. That would be lunacy similar in spirit to, but worse than, trying to do computer graphics from scratch, in machine code, without the benefit of an operating system. The analogy seems apt: Force is a flexible construct in a high−level language, which, by shielding us from irrelevant details, allows us to do elaborate applications relatively painlessly.
Why is it possible to encapsulate the complicated deep structure of matter? The answer is that matter ordinarily relaxes to a stable internal state, with high energetic or entropic barriers to excitation of all but a few degrees of freedom. We can focus our attention on those few effective degrees of freedom; the rest just supply the stage for the actors.
While force itself does not appear in the foundational equations of modern physics, energy and momentum certainly do, and force is very closely related to them: Roughly speaking, it's the space derivative of the former and the time derivative of the latter (and F = ma just states the consistency of those definitions!). So the concept of force is not quite so far removed from modern foundations as Tait and Russell insinuate: It may be gratuitous, but it is not bizarre. Without changing the content of classical mechanics, we can cast it in Lagrangian terms, wherein force no longer appears as a primary concept. But that's really a technicality; the deeper questions remains: What aspects of fundamentals does the culture of force reflect? What approximations lead to it?
Some kind of approximate, truncated description of the dynamics of matter is both desirable and feasible because it is easier to use and focuses on the relevant. To explain the rough validity and origin of specific concepts and idealizations that constitute the culture of force, however, we must consider their detailed content. A proper answer, like the culture of force itself, must be both complicated and open−ended. The molecular explanation of friction is still very much a research topic, for example. I'll discuss some of the simpler aspects, addressing the issues raised above, in my next column, before drawing some larger conclusions.
Here I conclude with some remarks on the psychological question, why force was— and usually still is— introduced in the foundations of mechanics, when from a logical point of view energy would serve at least equally well, and arguably better. The fact that changes in momentum— which correspond, by definition, to forces— are visible, whereas changes in energy often are not, is certainly a major factor. Another is that, as active participants in statics— for example, when we hold up a weight— we definitely feel we are doing something, even though no mechanical work is performed. Force is an abstraction of this sensory experience of exertion. D'Alembert's substitute, the virtual work done in response to small displacements, is harder to relate to. (Though ironically it is a sort of virtual work, continually made real, that explains our exertions. When we hold a weight steady, individual muscle fibers contract in response to feedback signals they get from spindles; the spindles sense small displacements, which must get compensated before they grow.4) Similar reasons may explain why Newton used force. A big part of the explanation for its continued use is no doubt (intellectual) inertia.
References 1. P. G. Tait, Dynamics, Adam & Charles Black, London (1895).
2. B. Russell, The ABC of Relativity, 5th rev. ed., Routledge, London (1997).
3. I. Newton, The Principia, I. B. Cohen, A. Whitman, trans., U. of Calif. Press, Berkeley (1999).
4. S. Vogel, Prime Mover: A Natural History of Muscle, Norton, New York (2001), p. 79.
Frank Wilczek is the Herman Feshbach Professor of Physics at the Massachusetts Institute of Technology in Cambridge.
- posted on 07/14/2005
Whence the Force of F = ma? II: Rationalizations
Frank Wilczek
In my previous column (Physics Today, October 2004, page 11), I discussed how assumptions about F and m give substance to the spirit of F = ma. I called this set of assumptions the culture of force. I mentioned that several elements of the culture, though often presented as "laws," appear rather strange from the perspective of modern physics. Here I discuss how, and under what circumstances, some of those assumptions emerge as consequences of modern fundamentals—or don't!
Critique of the zeroth law
Ironically, it is the most primitive element of the culture of force—the zeroth law, conservation of mass—that bears the subtlest relationship to modern fundamentals.
Is the conservation of mass as used in classical mechanics a consequence of the conservation of energy in special relativity? Superficially, the case might appear straightforward. In special relativity we learn that the mass of a body is its energy at rest divided by the speed of light squared (m = E/c2); and for slowly moving bodies, it is approximately that. Since energy is a conserved quantity, this equation appears to supply an adequate candidate, E/c2, to fill the role of mass in the culture of force.
That reasoning won't withstand scrutiny, however. The gap in its logic becomes evident when we consider how we routinely treat reactions or decays involving elementary particles.
To determine the possible motions, we must explicitly specify the mass of each particle coming in and of each particle going out. Mass is a property of isolated particles, whose masses are intrinsic properties—that is, all protons have one mass, all electrons have another, and so on. (For experts: "Mass" labels irreducible representations of the Poincaré group.) There is no separate principle of mass conservation. Rather, the energies and momenta of such particles are given in terms of their masses and velocities, by well−known formulas, and we constrain the motion by imposing conservation of energy and momentum. In general, it is simply not true that the sum of the masses of what goes in is the same as the sum of the masses of what goes out.
Of course when everything is slowly moving, then mass does reduce to approximately E/c2. It might therefore appear as if the problem, that mass as such is not conserved, can be swept under the rug, for only inconspicuous (small and slowly moving) bulges betray it. The trouble is that as we develop mechanics, we want to focus on those bulges. That is, we want to use conservation of energy again, subtracting off the mass−energy exactly (or rather, in practice, ignoring it) and keeping only the kinetic part E − mc2 ≅ 1/2 mv2. But you can't squeeze two conservation laws (for mass and nonrelativistic energy) out of one (for relativistic energy) honestly. Ascribing conservation of mass to its approximate equality with E/c2 begs an essential question: Why, in a wide variety of circumstances, is mass−energy accurately walled off, and not convertible into other forms of energy?
To illustrate the problem concretely and numerically, consider the reaction 2H + 3H → 4He + n, which is central for attempts to achieve controlled fusion. The total mass of the deuterium plus tritium exceeds that of the alpha plus neutron by 17.6 MeV. Suppose that the deuterium and tritium are initially at rest. Then the alpha emerges at .04 c; the neutron at .17 c.
In the (D,T) reaction, mass is not accurately conserved, and (nonrelativistic) kinetic energy has been produced from scratch, even though no particle is moving at a speed very close to the speed of light. Relativistic energy is conserved, of course, but there is no useful way to divide it up into two pieces that are separately conserved. In thought experiments, by adjusting the masses, we could make this problem appear in situations where the motion is arbitrarily slow. Another way to keep the motion slow is to allow the liberated mass−energy to be shared among many bodies.
Recovering the zeroth law
Thus, by licensing the conversion of mass into energy, special relativity nullifies the zeroth law, in principle. Why is Nature so circumspect about exploiting this freedom? How did Antoine Lavoisier, in the historic experiments that helped launch modern chemistry, manage to reinforce a central principle (conservation of mass) that isn't really true?
Proper justification of the zeroth law requires appeal to specific, profound facts about matter.
To explain why most of the energy of ordinary matter is accurately locked up as mass, we must first appeal to some basic properties of nuclei, where almost all the mass resides. The crucial properties of nuclei are persistence and dynamical isolation. The persistence of individual nuclei is a consequence of baryon number and electric charge conservation, and the properties of nuclear forces, which result in a spectrum of quasi−stable isotopes. The physical separation of nuclei and their mutual electrostatic repulsion—Coulomb barriers—guarantee their approximate dynamical isolation. That approximate dynamical isolation is rendered completely effective by the substantial energy gaps between the ground state of a nucleus and its excited states. Since the internal energy of a nucleus cannot change by a little bit, then in response to small perturbations, it doesn't change at all.
Because the overwhelming bulk of the mass−energy of ordinary matter is concentrated in nuclei, the isolation and integrity of nuclei—their persistence and lack of effective internal structure—go most of the way toward justifying the zeroth law. But note that to get this far, we needed to appeal to quantum theory and special aspects of nuclear phenomenology! For it is quantum theory that makes the concept of energy gaps available, and it is only particular aspects of nuclear forces that insure substantial gaps above the ground state. If it were possible for nuclei to be very much larger and less structured—like blobs of liquid or gas—the gaps would be small, and the mass−energy would not be locked up so completely.
Radioactivity is an exception to nuclear integrity, and more generally the assumption of dynamical isolation goes out the window in extreme conditions, such as we study in nuclear and particle physics. In those circumstances, conservation of mass simply fails. In the common decay π0 → γγ, for example, a massive π0 particle evolves into photons of zero mass.
The mass of an individual electron is a universal constant, as is its charge.Electrons do not support internal excitations, and the number of electrons is conserved (if we ignore weak interactions and pair creation). These facts are ultimately rooted in quantum field theory. Together, they guarantee the integrity of electron mass−energy.
In assembling ordinary matter from nuclei and electrons, electrostatics plays the dominant role. We learn in quantum theory that the active, outer−shell electrons move with velocities of order αc = e2/4πħ ≈ .007 c. This indicates that the energies in play in chemistry are of order me(αc)2/mec2 = α2 ≈ 5 × 10-5 times the electron mass−energy, which in turn is a small fraction of the nuclear mass−energy. So chemical reactions change the mass−energy only at the level of parts per billion, and Lavoisier rules!
Note that inner−shell electrons of heavy elements, with velocities of order Zα, can be relativistic. But the inner core of a heavy atom—nucleus plus inner electron shells—ordinarily retains its integrity, because it is spatially isolated and has a large energy gap. So the mass−energy of the core is conserved, though it is not accurately equal to the sum of the mass−energy of its component electrons and nucleus.
Putting it all together, we justify Isaac Newton's zeroth law for ordinary matter by means of the integrity of nuclei, electrons, and heavy atom cores, together with the slowness of the motion of these building blocks. The principles of quantum theory, leading to large energy gaps, underlie the integrity; the smallness of α, the fine−structure constant, underlies the slow motion.
Newton defined mass as "quantity of matter," and assumed it to be conserved. The connotation of his phrase, which underlies his assumption, is that the building blocks of matter are rearranged, but neither created nor destroyed, in physical processes; and that the mass of a body is the sum of the masses of its building blocks. We've now seen, from the perspective of modern foundations, why ordinarily these assumptions form an excellent approximation, if we take the building blocks to be nuclei, heavy atom cores, and electrons.
It would be wrong to leave the story there, however. For with our next steps in analyzing matter, we depart from this familiar ground: first off a cliff, then into glorious flight. If we try to use more basic building blocks (protons and neutrons) instead of nuclei, then we discover that the masses don't add accurately. If we go further, to the level of quarks and gluons, we can largely derive the mass of nuclei from pure energy, as I've discussed in earlier columns.
Mass and gravity
On the face of it, this complex and approximate justification of the mass concept used in classical mechanics poses a paradox: How does this rickety construct manage to support stunningly precise and successful predictions in celestial mechanics? The answer is that it is bypassed. The forces of celestial mechanics are gravitational, and so proportional to mass, and m cancels from the two sides of F = ma. This cancellation in the equation for motion in response to gravity becomes a foundational principle in general relativity, where the path is identified as a geodesic in curved spacetime, with no mention of mass.
In contrast to a particle's response to gravity, the gravitational influence that the particle exerts is only approximately proportional to its mass; the rigorous version of Einstein's field equation relates spacetime curvature to energy−momentum density. As far as gravity is concerned, there is no separate measure of quantity of matter apart from energy; that the energy of ordinary matter is dominated by mass−energy is immaterial.
The third and fourth laws
The third and fourth laws are approximate versions of conservation of momentum and conservation of angular momentum, respectively. (Recall that the fourth law stated that all forces are two−body central forces.) In the modern foundations of physics these great conservation laws reflect the symmetry of physical laws under translation and rotation symmetry. Since these conservation laws are more accurate and profound than the assumptions about forces commonly used to "derive" them, those assumptions have truly become anachronisms. I believe that they should, with due honors, be retired.
Newton argued for his third law by observing that a system with unbalanced internal forces would begin to accelerate spontaneously, "which is never observed." But this argument really motivates the conservation of momentum directly. Similarly, one can "derive" conservation of angular momentum from the observation that bodies don't spin up spontaneously. Of course, as a matter of pedagogy, one would point out that action−reaction systems and two−body central forces provide especially simple ways to satisfy the conservation laws.
Tacit simplicities
Some tacit assumptions about the simplicity of F are so deeply embedded that we easily take them for granted. But they have profound roots.
In calculating the force, we take into account only nearby bodies. Why can we get away with that? Locality in quantum field theory, which deeply embodies basic requirements of special relativity and quantum mechanics, gives us expressions for energy and momentum at a point—and thereby for force—that depend only on the position of bodies near that point. Even so−called long−range electric and gravitational forces (actually 1/r2—still falling rapidly with distance) reflect the special properties of locally coupled gauge fields and their associated covariant derivatives.
Similarly, the absence of significant multibody forces is connected to the fact that sensible (renormalizable) quantum field theories can't support them.
In this column I've stressed, and maybe strained, the relationship between the culture of force and modern fundamentals. In the final column of this series, I'll discuss its importance both as a continuing, expanding endeavor and as a philosophical model.
Biography
Frank Wilczek is the Herman Feshbach Professor of Physics at the Massachusetts Institute of Technology in Cambridge. - posted on 07/14/2005
Whence the Force of F = ma?
III: Cultural Diversity
Frank Wilczek
The concept of force, as we have seen, defines a culture. In the previous columns of this series (PHYSICS TODAY, October 2004, page 11, and December 2004, page 10) I've indicated how F = ma acquires meaning through interpretation of—that is, additional assumptions about—F. This body of interpretation is a sort of folklore. It contains both approximations that we can derive, under appropriate conditions, from modern foundations, and also rough generalizations (such as "laws" of friction and of elastic behavior) abstracted from experience.
In the course of that discussion it became clear that there is also a smaller, but nontrivial, culture around m. Indeed, the conservation of m for ordinary matter provides an excellent, instructive example of an emergent law. It captures in a simple statement an important consequence of broad regularities whose basis in modern fundamentals is robust but complicated. In modern physics, the idea that mass is conserved is drastically false. A great triumph of modern quantum chromodynamics (QCD) is to build protons and neutrons, which contribute more than 99% of the mass of ordinary matter, from gluons that have exactly zero mass, and from u and d quarks that have very small masses. To explain from a modern perspective why conservation of mass is often a valid approximation, we need to invoke specific, deep properties of QCD and quantum electrodynamics (QED), including the dynamical emergence of large energy gaps in QCD and the smallness of the fine structure constant in QED.
Isaac Newton and Antoine Lavoisier knew nothing of all this, of course. They took conservation of mass as a fundamental principle. And they were right to do so, because by adopting that principle they were able to make brilliant progress in the analysis of motion and of chemical change. Despite its radical falsity, their principle was, and still is, an adequate basis for many quantitative applications. To discard it is unthinkable. It is an invaluable cultural artifact and a basic insight into the way the world works despite—indeed, in part, because of—its emergent character.
The culture of a
What about a? There's a culture attached to acceleration, as well. To obtain a, we are instructed to consider the change of the position of a body in space as a function of time, and to take the second derivative. This prescription, from a modern perspective, has severe problems.
In quantum mechanics, bodies don't have definite positions. In quantum field theory, they pop in and out of existence. In quantum gravity, space is fluctuating and time is hard to define. So evidently serious assumptions and approximations are involved even in making sense of a's definition
Nevertheless, we know very well where we're going to end up. We're going to have an emergent, approximate concept of what a body is. Physical space is going to be modeled mathematically as the Euclidean three-dimensional space R3 that supports Euclidean geometry. This tremendously successful model of space has been in continuous use for millennia, with applications in surveying and civil engineering that even predate Euclid's formalization.
Time is going to be modeled as the one-dimensional continuum R1 of real numbers. This model of time, at a topological level, goes into our primitive intuitions that divide the world into past and future. I believe that the metric structure of time—that is, the idea that time can be not only ordered but divided into intervals with definite numerical magnitude—is a much more recent innovation. That idea emerged clearly only with Galileo's use of pendulum clocks (and his pulse!)
The mathematical structures involved are so familiar and fully developed that they can be, and are, used routinely in computer programs. This is not to say they are trivial. They most definitely aren't. The classical Greeks agonized over the concept of a continuum. Zeno's famous paradoxes reflect these struggles. Indeed, Greek mathematics never won through to comfortable algebraic treatment of real numbers. Continuum quantities were always represented as geometric intervals, even though that representation involved rather awkward constructions to implement simple algebraic operations
The founders of modern analysis (René Descartes, Newton, Gottfried Wilhelm Leibniz, Leonhard Euler, and others) were on the whole much more freewheeling, trusting their intuition while manipulating infinitesimals that lacked any rigorous definition. (In his Principia, Newton did operate geometrically, in the style of the Greeks. That is what makes the Principia so difficult for us to read today. The Principia also contains a sophisticated discussion of derivatives as limits. From that discussion I infer that Newton and possibly other early analysts had a pretty good idea about what it would take to make at least the simpler parts of their work rigorous, but they didn't want to slow down to do it.) Reasonable rigor, at the level commonly taught in mathematics courses today—the much-bemoaned epsilons and deltas—entered into the subject in the 19th century.
"Unreasonable" rigor entered in the early 20th century, when the fundamental notions from which real numbers and geometry are constructed were traced to the level of set theory and ultimately symbolic logic. In their Principia Mathematica Bertrand Russell and Alfred Whitehead develop 375pages of dense mathematics before proving 1 + 1 = 2. To be fair, their treatment could be slimmed down considerably if attaining that particular result were the ultimate goal. But in any case, an adequate definition of real numbers from symbolic logic involves some hard, complicated work. Having the integers in hand, you then have to define rational numbers and their ordering. Then you must complete them by filling in the holes so that any bounded increasing sequence has a limit. Then finally—this is the hardest part—you must demonstrate that the resulting system supports algebra and is consistent.
Perhaps all that complexity is a hint that the real-number model of space and time is an emergent concept that some day will be derived from physically motivated primitives that are logically simpler. Also, scrutiny of the construction of real numbers suggests natural variants, notably John Conway's surreal numbers, which include infinitesimals (smaller than any rational number!) as legitimate quantities.1 Might such quantities, whose formal properties seem no less natural and elegant than those of ordinary real numbers, help us to describe nature? Time will tell.
Even the unreasonable rigor of symbolic logic does not reach ideal strictness. Kurt Gödel demonstrated that this ideal is unattainable: No reasonably complex, consistent axiomatic system can be used to demonstrate its own consistency.
But all the esoteric shortcomings in defining and justifying the culture of a clearly arise on an entirely different level from the comparatively mundane, immediate difficulties we have in doing justice to the culture of F. We can translate the culture of a, without serious loss, into C or FORTRAN. That completeness and precision give us an inspiring benchmark.
The computational imperative
Before they tried to do it, most computer scientists anticipated that to teach a computer to play chess like a grand master would be much more challenging than to teach one to do mundane tasks like drive a car safely. Notoriously, experience has proved otherwise. A big reason for that surprise is that chess is algorithmic, whereas driving a car is not. In chess the rules are completely explicit; we know very concretely and unambiguously what the degrees of freedom are and how they behave. Car driving is quite different: Essential concepts like "other driver's expectations" and "pedestrian," when you start to analyze them, quickly burgeon into cultures. I wouldn't trust a computer driver in Boston's streets because it wouldn't know how to interpret the mixture of intimidation and deference that human drivers convey by gestures, maneuvers, and eye contact
The problem with teaching a computer classical mechanics is, of course, of more than academic interest: We'd like robots to get around and manipulate things; computer gamesters want realistic graphics; engineers and astronomers would welcome smart silicon collaborators—up to a point, I suppose.
The great logician and philosopher Rudolf Carnap made brave, pioneering attempts to make axiomatic systems for elementary mechanics, among many other things.2 Patrick Hayes issued an influential paper, "Naive Physics Manifesto," challenging artificial-intelligence researchers to codify intuitions about materials and forces in an explicit way.3 Physics-based computer graphics is a lively, rapidly advancing endeavor, as are several varieties of computer-assisted design. My MIT colleagues Gerald Sussman and Jack Wisdom have developed an intensely computational approach to mechanics,4 supported every step of the way with explicit programs. The time may be ripe for a powerful synthesis, incorporating empirical properties of specific materials, successful known designs of useful mechanisms, and general laws of mechanical behavior into a fully realized computational culture of F = ma. Functioning robots might not need to know a lot of mechanics explicitly, any more than most human soccer players do; but designing a functioning robotic soccer player may be a job that can best be accomplished by a very smart and knowledgeable man-machine team.
Blur and focus
An overarching theme of this series has been that the law F = ma, which is sometimes presented as the epitome of an algorithm describing nature, is actually not an algorithm that can be applied mechanically (pun intended). It is more like a language in which we can easily express important facts about the world. That's not to imply it is without content. The content is supplied, first of all, by some powerful general statements in that language—such as the zeroth law, the momentum conservation laws, the gravitational force law, the necessary association of forces with nearby sources—and then by the way in which phenomenological observations, including many (though not all) of the laws of material science can be expressed in it easily.
Another theme has been that F = ma is not in any sense an ultimate truth. We can understand, from modern foundational physics, how it arises as an approximation under wide but limited circumstances. Again, that does not prevent it from being extraordinarily useful; indeed, one of its primary virtues is to shield us from the unnecessary complexity of irrelevant accuracy
Viewed this way, the law of physics F = ma comes to appear a little softer than is commonly considered. It really does bear a family resemblance to other kinds of laws, like the laws of jurisprudence or of morality, wherein the meaning of the terms takes shape through their use. In those domains, claims of ultimate truth are wisely viewed with great suspicion; yet nonetheless we should actively aspire to the highest achievable level of coherence and explicitness. Our physics culture of force, properly understood, has this profoundly modest but practically ambitious character. And once it is no longer statuized, put on a pedestal, and seen in splendid isolation, it comes to appear as an inspiring model for intellectual endeavor more generally.
Frank Wilczek is the Herman Feshbach Professor of Physics at the Massachusetts Institute of Technology in Cambridge.
References
1. D. Knuth, Surreal Numbers, Addison-Wesley, Reading, MA (1974).
2. R. Carnap, Introduction to Symbolic Logic and Its Applications, Dover, New York (1958).
3. P. Hayes, in Expert Systems in the Microelectronic Age, D. Michie, ed., Edinburgh U. Press, Edinburgh, UK (1979).
4. G. Sussman, J. Wisdom, Structure and Interpretation of Classical Mechanics, MIT Press, Cambridge, MA (2001).
- Re: 力的由来 Whence the Force of F = ma?posted on 07/14/2005
I suppose I've been exerting. :)
This is a very interesting article on a perennial fascinating subject: the relationship between our cultural views and our understanding of the physical world. A perfunctory reading does us no good. - posted on 08/17/2005
Comments on the Culture of the Force
One of Frank Wilczek's main themes in "Whence the Force of F = ma? I: Culture Shock" (PHYSICS TODAY, October 2004, page 11) appears to be that although the force is, in Wilczek's words, "vaguely defined," it "continues to flourish" because the microscopic details it conceals are not really relevant for the scale of the phenomena it serves to describe. Further, it "survives the competition" because "it is much easier to work with." To this second point one might add that nothing succeeds like success. Let me explain.
The concept of force had been under attack much before the comments of Peter Tait and Bertrand Russell. Even some of Isaac Newton's immediate successors, most notably Joseph-Louis Lagrange and Jean Le Rond d'Alembert, were critical of the concept. D'Alembert regarded it as "useless to mechanics" and said that it "ought therefore to be banished from it."1 However, the use of Newton's idea that force is a primary, nonderived concept, which was pursued steadfastly by Leonhard Euler,1 led to the greatest successes in continuum mechanics in the two centuries immediately following publication of Newton's Principia. That period culminated in the 1820s in Augustin-Louis Cauchy's stress principle,1 which unified the seemingly disparate fields of fluid mechanics and elasticity. This approach, commonly attributed to Newton rather than to Euler or Cauchy, is chosen over its main competitor, the variational formulation of Lagrange, to be taught in a typical fluid mechanics course. The stress has also been given a microscopic interpretation in kinetic theory and in more general statistical mechanics.Wilczek mentions some assumptions about forces. Newton regarded mechanics as "the science of motions that result from any forces whatever."1 Thus, he did not exclude contact forces, the dominating concept in continuum mechanics. Nor did he demand that all forces be central, which has particular relevance to the derivation of the angular momentum principle.
In 1776, Euler, guided by his research on elasticity, came to regard the balance of angular momentum as an independent, second principle of mechanics,1 the first principle being the balance of linear momentum. When Euler arrived at the rigid-body equations of motion in 1752 using the first principle, he had to invoke hypotheses about internal forces. However, once he saw the balance of the moments as an independent principle, he had no need of such hypotheses. In special cases such as that of a perfect fluid, the second principle follows from the first. In fact, the second principle leads to the symmetry of the stress tensor when all torques may be obtained as moments of forces.2 The status of the third law has been clarified by the work of Walter Noll, who gives a precise mathematical interpretation of Newton's verbal statement of the law.3Even in applications of quantum mechanics, Richard Feynman emphasized the importance of forces.4 He commented that "many of the problems of molecular structure are concerned essentially with forces," that "forces are almost as easy to calculate as energies are," and that "the quantities are quite as easy to interpret." Another application of the concept of force is found in nonequilibrium statistical mechanics. Just as the contact force had to be found as the appropriate force to describe the dynamics of the continuum, the physically realistic short-time force derived from the mean instantaneous potential had to be discovered as the force that describes typical chemical dynamics in liquids,5 in contrast to the traditional concept of the potential of mean force, which is more appropriate for slow or diffusion processes.
References
1. C. Truesdell, in Essays in the History of Mechanics, Springer-Verlag, New York (1968).
2. R. Aris, Vectors, Tensors and the Equations of Fluid Mechanics, Dover, New York (1962).
3. W. Noll, in The Axiomatic Method with Special Reference to Geometry and Physics, North-Holland, Amsterdam (1959), p. 266.4.R. P. Feynman, Phys. Rev. 56, 340 (1939).
5. S. Adelman, R. Ravi, Adv. Chem. Phys. 115, 181 (2000).
Ramamurthy Ravi(rravi@iitm.ac.inIndian Institute of TechnologyMadrasChennai, India
I think Frank Wilczek is too harsh when he implies that the use of equations like F = ma is a matter of intellectual inertia. In practical terms, in engineering, and even in the design of physics instruments, we are interested in the values taken by certain variables xi and the known dependence is in the form of differential equations dxi/dt =vi and dvi/dt = f(xi, vi). When xi is some position, the last is a form of F = ma. Many physicists—David Bohm and John Stewart Bell, for example—have argued that position is the fundamental variable . . . hence the importance of F = ma.
Brent Meeker(meekerdb@rain.org)Naval Air Warfare CenterPoint Mugu, California
Frank Wilczek's column exposes a delicate point in physics teaching. Good teachers avoid implanting misconceptions to be overwritten later. Yet Newtonian mechanics courses do just that! During 20 years teaching I've maintained that Newton's three laws are neither good laws nor independent. Students enjoy hearing the first law is just as circular as it seems. Textbook apologies that falsely limit physics to inertial frames contradict later teaching that physics can be used in any coordinates. Perhaps the first law was just a political device to start discussion, and to divide Newton's detractors. The third law is necessary for beginning physics of ropes and pulleys, but is wrong as "principle": Momentum conservation via translational symmetry has myriad solutions. The second law is okay, but it is indefensible to promote Newton's emphasis on "force" as primary, only later to revise it with Hamilton's equations of greater scope. Eventually I evolved a refreshing approach to non-calculus physics with energy and conservation laws as primary, and it works well.
Students happily accept that Newton sometimes guessed wrong. A timid teaching culture and careless textbook writing create the intellectual inertia Wilczek observes. Good physics teachers need to demonstrate critical thinking, distribute their own notes, and have the courage not to brainlessly repeat what is written in the book.
John RalstonUniversity of KansasLawrence
Wilczek replies: Ramamurthy Ravi's letter is an excellent, scholarly supplement to my October column, which emphasizes that some classical masters of mechanics had logical and aesthetic misgivings about the force concept, even before modern physics began to push us strongly toward different ones.Regarding Brent Meeker's letter, my critique was meant to be directed at foundational issues including, specifically, which principles should be regarded as primary, and which as derived. There are some significant problems with using F = ma as a primary principle, as I discussed. They could be avoided, perhaps advantageously, by focusing on momentum and energy. Of course, in that approach it would still be appropriate and extremely useful to have F= ma as a derived equation, with its limitations indicated. Some intellectual inertia isn't necessarily a bad thing, if it keeps you moving in the right direction and allows you to remain in sync with long-established flows.
Frank WilczekMassachusetts Institute of TechnologyCambridge - Re: 【物理】力的由来和误解posted on 08/17/2005
Well well well. Again it is an interesting subject for fuzzy guys like me. I still remember the old days when I read article after article about the concept of force in the journal called "Dialectics of Nature".
Old ghosts still lurch and huant modern thinking men. The concept of force in physics, like the concept of value in economics, is not going to go away. They may prove to be indispensable to their respective disciplines after a modern, ingenious reformulation. - Re: 【物理】力的由来和误解posted on 08/18/2005
终于读完了阿珊布置的作业。:-) 有意思。这类文章不是严格意义上的科普,因为它的读者至少应该受过大学教育并对自然科学有起码的兴趣。中学物理教师应该读这一类文章吧。普通人都只在中学了解一点经典物理学,然后就只有少数理科学生进一步去了解现代物理学。
关于经典力学与神和独裁的关系,而现代物理学,包括相对论和量子力学,同民主概念的一致性的讨论很有意思。
- Re: 【物理】力的由来和误解posted on 08/21/2005
- posted on 08/21/2005
Susan wrote:
美 国 地 心 引 力 最 新 研 究 动 态 , 摘 自 洋 葱 报 。 :)))))
赫赫,还是玛雅她们肯萨斯城的呢。:))))))
Evangelical Scientists Refute Gravity With New 'Intelligent Falling' Theory
KANSAS CITY, KS—As the debate over the teaching of evolution in public schools continues, a new controversy over the science curriculum arose Monday in this embattled Midwestern state. Scientists from the Evangelical Center For Faith-Based Reasoning are now asserting that the long-held "theory of gravity" is flawed, and they have responded to it with a new theory of Intelligent Falling.
"Things fall not because they are acted upon by some gravitational force, but because a higher intelligence, 'God' if you will, is pushing them down," said Gabriel Burdett, who holds degrees in education, applied Scripture, and physics from Oral Roberts University.
.... - posted on 08/22/2005
三个月前,堪萨斯州教育委员会开始了所谓有关进化论教学的辩论。当然,科学界是抵制那个所谓的辩论的。我一气之下写了一封长信去骂他们,并要挟起诉他们。有三个委员回信表示支持,有两个回信反对。
唉,我本来是很主张人们信教的,曾经不遗余力地推崇基督教,并坚信基督教可以救中国。远志明来我们这里演讲时,我去机场接送,并和他谈的很深。自从美国攻打伊拉克以来,我一天天看透了那些所谓基督教徒的卑鄙嘴脸。也许我中了撒旦的邪,也许基督教那一切根本就是一场骗局。总而言之,善良和罪恶就那么一线之差。无论人们信什么,做坏是的还是要做坏事。这个世界没救了。
阿姗 wrote:
Susan wrote:赫赫,还是玛雅她们肯萨斯城的呢。:))))))
美 国 地 心 引 力 最 新 研 究 动 态 , 摘 自 洋 葱 报 。 :)))))
Evangelical Scientists Refute Gravity With New 'Intelligent Falling' Theory
.... - Re: 【物理】力的由来和误解posted on 05/27/2007
两年前贴的,给大家“科普”一下关于力的概念。再提上来读读。 - Re: 【物理】力的由来和误解posted on 05/28/2007
赶明天我把史宾格勒的分析敲出来,与阿姗一和。
以前有不少条讨论线吧?
现在我对科学学渐渐有了新的认识。科学哲学对于科学分析与讨论是
很必要的。
马赫就说过,物理学的牛顿表述并不唯一。科学与哲学怎么能分开呢?
前两天读到,图灵还是罗素的学生,还精通生物学。
许多科学领域的创始人都是学神学的嘛。
- posted on 05/29/2007
今天正准备敲英文节版《浮士德与阿波罗的自然知识》的汉语翻译,
便在网上找到了全版的中文翻译,可以网上阅读了,免敲。
昨天找空把英文节版与全版都重读一遍,好歌德!
我把第一、二节转到这里,下面给一个LINK,也便以后参考。
====
第十一章浮士德式与阿波罗式的自然知识(1)
一
赫尔姆霍兹在1869年的一次著名演讲中指出,“自然科学的最终目标,就是去发现作为一切变化之基础的运动以及这些运动的动力;也就是,决心投身于力学。”这一投身于力学的决心意味着以定量化的基值(base-values)亦即广延和位置的变化来指涉一切质的印象。进一步说,它还意味着——如果我们记得生成与既成、形式与定律、意象与概念的对立——用一种在数字上和结构上可以度量的秩序的想象图象去指涉所见的自然图象。所有是西方力学皆有一特殊倾向,就是用度量来征服才智,因而,它势必要在一个具有某些恒定要素的体系中来寻找现象的本质,而体系中的这些恒定要素,很显然,是需要用度量来加以充分的和总括的鉴别的,赫尔姆霍兹从这些要素中区分出了运动(在日常意义上使用这个词),将其视作最重要的要素。
对于物理学家来说,这一界定是明确而全面的,但对于深究这一科学信念之历史的怀疑主义者来说,情况远非如此。对物理学家来说,现今的力学是一个逻辑体系,里面有明晰、意义独特的概念以及简单、必然的关系;而对怀疑主义者来说,它乃是西欧精神结构所特有的一种图象,尽管他承认这图象有着最高程度和最令人信服的说服力。不言而喻的是,没有任何实践结果和发现能证明那种理论、那个图象的“真理性”。实际上,对于绝大多数的人来说,“力学”似乎是各种自然印象不言而喻的综合。但是,它仅仅是似乎如此。什么是运动?一切质的东西可还原为永远相似的无数点的运动,这一先决条件本质上难道不是浮士德式的吗,不是其人性所共有的吗?例如,阿基米德并不觉得自己理应把他所看到的力学置换为运动的心理图象。运动一般地是一种纯粹的机械的量吗?它是一个表示视觉经验的词吗,或者说它是自那一经验中得出的一个概念吗?它是由经验地产生的事实之度量所发现的数字吗,或者说是从属于那一数字、由那一数字表示出来的图象吗?如果某一天物理学真的成功地达到了它意想的目标,成功地设计出了一个由定律所支配的“运动”的体系,一个有关那些运动背后的作用力的体系,但凡可以为感官所理解的东西皆可纳入其中——它由此就算是对所发生的事获得了“知识”吗,甚或说就算是向这一成就迈进了一步吗?然而,力学的形式语言因此就不再只是一种教条吗?相反,它不就是像根词一样的神话的容器吗,这容器不是来自于经验,而是要构建经验,并且在这一情形中是要尽可能严密地构建经验?什么是力?什么是原因?什么是过程?还有,甚至在它自身的定义的基础上,物理学果真有特殊的问题吗?它有对于所有世纪都同等有效的对象吗?它甚至有一个无懈可击的想象单位,使它能以此为参照来表达它的结论吗?
答案可以预想。近代物理学作为一门科学,是一个庞大的指涉(indices)系统,里面有各种名称和各种数字的形式,我们由此可像探究一架机器一样去探究自然。同样地,它还有一个可确切界定的目标。但是,若作为历史的一个片断来看,则物理学完全是由致力于物理研究的人的生命中的命运和偶然、以及研究过程本身所集合而成的,故此,就对象、方法和结论等方面而论,物理学就像是一种文化的表现和实现,是那一文化之本质中的一个有机的和内含的要素,而它的每一个结论本身就是一个象征。物理学——它仅仅存在于文化人的醒觉意识中——认为,它在它的方法和结论中所发现的东西,业已在那里,那是它研究的基础,亦是那研究所固有的,不论是其对象的选择还是其研究的方式。物理学的发现,单就其想象性的内涵(必须与它们的可形诸书面的公式区分开来)来看,颇具一种纯粹神话的性质,甚至在诸如J.R.迈耶(Mayer)、法拉第和赫兹(Hertz)这些缜密严谨的物理学家的心目中,也这么认为。故而,在每一自然定律中,即便它在物理的意义上是真确的,我们还是要在无名数与数字的命名之间、在清晰划定的范围与对范围的理论解释之间作出区分。公式只代表通用的逻辑数值,代表纯粹的数字——也就是说,只代表客观的空间——以及有边界的要素。但是,公式本身是不会说话的。例如S=1/2gt2这个表达式,除非某人能在心理上把这些字母跟某些特殊的词汇及其象征意义联系起来,否则便毫无意义。但是,每当我们用词汇来说明这些死的符号,赋予它们血肉、形体和生命,简言之,使它们在世上有一种可感知的意义,我们就跨越了单纯科学秩序的范围。θεωρια(观视、形象)有意象、幻象的意思,正是这种东西使人们能从一个图形-字母的公式中了解到一个自然定律。一切真确的东西,本身是没有意义的,每一种物理观察的构想,全是为了替一些想象的预设找出其实际的基础;若能成功地从中得出结果,就可以使这些预设比以前更有说服力。若是没有这些预设,则物理学的结论就只是一堆空洞的图形符号。但事实上,我们不会、也不可能摈弃这些预设。即便一个研究者能把他所知的每一个假设全置之脑后,可一当他开动他的思维,去投入他自以为明确的研究工作时,便不是由他在控制那工作的无意识形式,而是由工作的无意识形式在控制他,因为,在活生生的生命活动中,他永远是一个属于他的文化、他的时代、他的学派、他的传统中的人。信仰和“知识”,是人类仅有的两种内在的确定事物,但是,在这二者中,信仰更为古老,它支配着一切认识的条件,即便那些条件从不会如此确切地呈现。因而,作为一切自然科学之支撑的,是理论而不是纯粹的数字。文化人的潜意识中,总有着对真正科学的渴望,因为这真正的科学(再重复一遍),乃是文化人的精神所特有的,而有了这渴望,那一精神才能够在它所掌握的自然的世界意象内去理解、深入和综合。单纯苦苦地为度量而度量,永远只不过是一些偏狭的心智的可怜嗜好而已。数字仅仅是秘密的钥匙,且仅此而已。无知无识的人永远只会为数字自身之故而在上面浪费时间。
康德有一句众所周知的话:“我认为,在每一种自然哲学的学科中,唯一可能的就是在其中像发现数学那样去发现真正的科学。”他说的是对的。在此,康德心中的意思,是想在既成物的领域作一些纯粹的限定,直至只在那个领域能够(在任何特殊阶段)看到定律和公式、数字和体系。但是,一个没有文字的定律,一个仅仅由一系列图形符号构成的定律,只能看作是一个工具,甚至不能看作是在这一纯粹状态中完全有效的一种理智运作。每个学者的实验,不论什么样的实验,同时都是主宰该学者思路的那种象征主义的一个例证。一切用文字语言表述出来的定律都是已被激活、被赋予活力、充满那一——且只是那一——文化的本质的秩序。至于作为一切精确研究的先决条件的“必然性”,在此我们也必须考虑其中的两种,亦即精神的和活生生的东西所固有的必然性(因为它就是每一个体研究行为的历史何时、何地且如何开始其行程的命运)以及认识对象所固有的必然性(西方描述它的流行名称就是因果律)。如果说一个物理公式的纯粹数字代表一种因果必然性,那么,一个理论的生存、诞生和生命绵延就是一种命运。
每个科学事实,甚至最简单的事实,从一开始就包含一种理论。一个科学事实乃是一个独特发生的事件在醒觉存在那里留下的印象,一切都取决于那一存在、取决于该事实为之或曾经为之而发生的存在,究竟是、或曾经究竟是古典的还是西方的,是哥特式的还是巴罗克式的。比较一下闪电在一只麻雀身上跟在一个机敏的物理研究者身上产生的效果之间的差异,想一想那个观察者看到的“事实”比那只麻雀看到的“事实”所包含的东西要多多少。近代物理学家动不动就忘记了:甚至像量、位置、过程、状态和物体的变化这些词都特别地代表着西方的意象。这些词总能激发一种有意义的感受,这些意象则是一种有意义的感受的反映,它们对于语言描述来说太微妙了,难以跟古典的、或麻葛的、或其他文化的人类进行沟通,就像这些人类的微妙的思想和感受难以跟我们沟通一样。科学事实的此种特征——亦即它们成为认识对象的方式——完全受这一感受的支配;并且,果真这样的话,那么,诸如作用、张力、能量、热量、或然性这类复杂的理智概念就更有理由如此了,它们中每一个都包含一个有关其自身的纯科学的神话。我们认为这些理论意象是出自毫无偏见的研究,且从属于某些确然有效的条件。但是,阿基米德时代第一流的科学家在对我们近代的理论物理学做一番彻底研究以后,必定会说自己根本无法理解人们怎么能断言如此武断花哨、纠缠不清的概念是科学,尤其是怎么能说它们是来自实际事实的必然结果。他必定会说:“所谓科学地证明的结论,其实应当是如此这般”;随即他还会在他的肉眼和他的心智借以确定“事实”的相同要素的基础上来提出各种我们的物理学家听起来荒谬绝伦的理论。
那么,在我们的物理学的领域,以逻辑的内在确定性发展出来的那些基本概念究竟是为何而存在?极性化的光射线,飘忽不定的离子,飞驰和碰撞的气流,磁场,电流和电波:它们不就是浮士德式的幻象吗?——跟罗马风格的装饰、哥特建筑的垂直上升、北欧海盗向未知海域的航行、哥伦布和哥白尼的渴望何其相似。这个形式和图象的世界不就是跟同时代的透视法的油画艺术、器乐艺术完全和谐一致地成长起来的吗?简言之,它们不就是我们的热情的方向感、我们的第三向度的激情,在想象的自然图象以及心灵意象中所获得的象征性的表现吗?
二
由此可知,一切有关自然的“认识”,甚至最精确的认识,都是基于一种宗教信仰。物理学家给自己设定作为目标形式——即认为自己的任务(也是所有这种想象机器的目标)就是还原自然——的纯力学,是以一个教义,亦即哥特时代的宗教的世界图象为前提的。因为西方才智所特有的物理学,是从这一世界图象中发展出来的。但凡科学,没有不存在这种无意识的前提的,这前提研究者无法控制的,它甚至可以追溯到醒觉文化的最初时期。但凡自然科学,没有不存在一种先行的宗教的。在这一点上,天主教的世界观与唯物主义的世界观之间没有任何区别——二者是以不同的语言说同样的事。甚至无神论者的科学也有它的宗教;近代力学完全就是沉思性的信仰的复现。
当爱奥尼亚风格在泰勒斯那里或巴罗克风格在培根那里达到其顶峰的时候,当人类已经发展到其都市阶段的时候,他的自信使他开始关注与乡村的更原始的宗教相反的批判科学,将其视作是看待事物的一种优先态度,并强调他进行思考时的唯一焦点就是真知,就是要去经验地和心理地解释宗教本身——换句话说,就是要以其他的东西去“征服”它。现在,高级文化的历史表明,“科学”是一种暂时的景观(transitory spectacle),仅仅属于那种文化的生命历程的秋季和冬季,并且在古典思想、印度思想、中国思想、阿拉伯思想的情形中,要完全耗尽它们的可能性,需要几个世纪的时间。古典科学在坎尼战役和亚克兴战役之间的时期里消失了,让位于“第二信仰”的世界观。由此可以预见到我们的西方科学思想抵达其演进极限的日期。
没有任何理由可以指派此一理智的形式世界优越于其他的形式世界。每种批判的科学,跟每种神话和每种宗教信仰一样,皆取决于一种内在的确信。不论这确信的种类,无论在结构上还是在名称上,如何的繁多,其在基本原则上皆没有区别。因此,任何借自然科学之名对宗教的指责,都是搬石头砸自己的脚的行为。我们太过自以为是,总以为自己能建立永久的“真理”以取代那些“人神同形的”概念,其实,所有的概念,都是“人神同形的”,因为这有这种概念才是真正存在的。每一确实可能的观念,其实都是其作者的存在的反映。所谓“人是照自己的形象创造了上帝”的说法——这对每一历史的宗教都是有效的——对每一物理理论也同样有效,不论其想当然的事实基础如何的稳固。古典科学家认为光存在于从光源传播到观者的肉眼中的有形粒子中。对于阿拉伯思想而言,甚至在以得撒(Edessa)、累西那(Resaïna)、庞巴迪西亚(Pombaditha)的犹太-波斯人的学园的阶段(对于波菲利来说也是这样),事物的色彩与形式的呈现根本不需要某个媒介的干预,而是以一种魔幻的、“精神的”方式被带到视力(seeing-power)面前,并且认为这视力是实体性的,就居存于眼球中。这便是伊本·海丹(Ibn-al-Haitan)、阿维森纳(Avicenna)及“忠诚兄弟会”(Brothers of Sincerity)所传授的教义。此种光作为一种力、一种推动力的观念,甚至自1300年左右就在巴黎奥卡姆主义者——以萨克森的阿尔伯特(Albert of Saxony)、布里丹(Buridan)和坐标几何的发现者奥里斯梅为中心——当中广为流行。每种文化都提出了自己的一套过程意象,那意象只对那文化本身是真确的,只有当那一文化本身是鲜活的且能实现自身的可能性的时候,那意象才是鲜活的。当一种文化走到了它的终点的时候,当那些创造性的要素——想象力、象征主义——消亡的时候,剩下的就只是“空洞的”公式,是死体系的骨架,另一文化的人就只能在字面上来读解它,就会觉得它毫无意义或价值,或是机械地把它保存起来,再不就干脆蔑视和忘记它。数字、公式、定律并不能意味什么,它们只是空无。它们必须有一个形体,只有活生生的人类——为了内在地创造它们,这人类把他的生命投射到它们之中,且通过它们来投射自身,通过它们来表现自身——能赋予它们一定的意义。因而,根本没有绝对的物理科学,而只有各别的物理科学,它们在各别文化内部产生、繁荣和消亡。
古典人的“自然”的最高艺术表征在于裸体雕像,从那里逻辑地生长出了一种身体的静力学,一种指向切近的物理学。阿拉伯文化拥有阿拉伯风格的图案和清真寺的洞穴穹隆,从这种世界感中,产生出了炼金术及其神秘有效的实体观念,诸如“哲学水银”(philosophical mercury)之类,这既不是一种物质,也不一种属性,而是某种经过金属的生命变形可以把一种金属转变成另一种金属的东西。至于浮士德式的人的自然观念所引发的后果,则是一种广度无限的动力学,一种指向远处的物理学。因此,属于古典文化的,是物质和形式的概念,属于阿拉伯文化的,是具有可见或神秘属性的实体观念(与斯宾诺莎的观念十分接近),而属于浮士德文化的,则是力和质量的观念。阿波罗式的理论是一种宁静的冥思,麻葛式的理论则是作为恩宠手段的静默的炼金术知识(甚至在这里也可以觉察到力学的宗教源泉),而浮士德式的理论从一开始就是一种运作的假设。希腊人问:什么是可见的存在的本质?我们问:支配生成的不可见的动因有何样的可能性?对希腊人而言,是心满意足地专注于可见物;对我们来说,则是满足于操控性地对自然和按部就班的实验进行提问。
不但问题的表述和处理问题的方法各有不同,而且连基本的概念也是这样。它们在各自那种文化且只在那种文化的情形中才是象征。古典的一些根词απειρον(虚无)、αρχη(始基)、μορφη(形象)、υλη(质料)等都无法译成我们的语言。用“prime-stuff”(基本质素)来移译αρχη,便失去了其阿波罗式的内涵,使那个词成为了一个空洞的外壳,听起来十分陌生。古典人看到他眼前的空间中的“运动”,他便以αλλοιωσιs(位移)——物体位置的变动——来加以理解;而我们西方人,则从我们经验运动的方式中演绎出一种过程的概念,一种“进行”的概念,以此来表达和强调我们的思想在自然进程中必须要设定的要素——方向能量。自然的古典批评家把可见的、并置在一起的状态视作是原本就迥异的,恩培多克勒著名的“四根说”就是例证——亦即,固态且有形的土,非固态但有形的水,无形的气,以及古典精神无疑会因为其形体性而将其视作是能产生最强烈的视觉印象的火。相反,阿拉伯的批评家所讲的“要素”是理想的,暗含着秘密的构成和组合,故而能为肉眼界定事物的现象。如果我们能更近地贴近这种情感,我们就会发现,固态和液态的对立对于叙利亚人来说所意味的东西完全不同于它对于亚里士多德式的希腊人来说所意味的东西,后者看到的是形体性的程度差异,前者看到的是神奇属性的差异。因此,这一对立对于前者,产生出来的是化学要素作为一种神奇实体的意象,一种秘密的因果关系可使那要素从事物中浮现出来(也可使其再消失于事物之中),那一要素甚至会受到星象的影响。在炼金术中,对于事物的雕塑式的现实性——希腊数学家、物理学家和诗人所谓的“体质”(somata)——以及它的为图发现自己的本质而消融和摧毁体质一事,有一种深刻的科学怀疑。它跟伊斯兰和拜占廷的鲍格米勒派(Bogomils)的怀疑一样,是一种真正的捣毁圣像运动。它揭示了对现象之自然的可感知形象的一种深刻的不信任,可那一形象对于希腊人来说是神圣不可侵犯的。在所有早期宗教会议中出现、并导致了聂斯脱利派和一性论的分离运动的有关基督位格的冲突,即是一种炼金术的问题。这类事决不会发生在古典世界,因为古典物理学家决不会一面探究事物,而同时却又否定或消灭事物的可感知的形式。也正是因此,根本就没有一种古典化学,也不会有对有违阿波罗式的表征的实体的任何理论化。
阿拉伯风格的化学方法的出现,展开了一种全新的世界感。它的出现,一举终结了阿波罗式的自然科学,终结了力学中的静力学。这个发现,与一个神秘的名字赫耳墨斯·特里斯美吉斯托斯(Hermes Trismegistus)联系在一起,一般认为此人生活在亚历山大里亚,跟普罗提诺和丢番图属于同时代。同样地,正当西方数学经由牛顿和莱布尼茨获得确定的解放的时候,西方的化学也经由斯塔耳(Stahl)和他的“燃素说”(Phlogiston theory)而摆脱了阿拉伯的形式。化学与数学一样,成为纯粹的分析。帕拉塞尔苏斯(Paracelsus)(1493~1541年)早就已经把提炼黄金的麻葛式努力转变成一种药剂科学——在这一转变中,人们不过是在揣测一种被改变的世界感。接着,罗伯特·波义耳(Robert Boyle)(1626~1691年)发明了化学解析的方法,从而出现了西方的元素概念。但是,不要误会了接下来的变化。所谓近代化学的创立,不过就是建立一种“化学”观念,以区别于意指炼金术士的自然观的化学观念。斯塔耳和拉瓦锡(Lavoisier)已经处在这一创立的转折点。事实上,这也是真正化学的终结,它消融于纯粹动态的理解系统中,被巴罗克时代由伽利略和牛顿所奠定的力学观所吸收同化。恩培多克勒的四根说是说明形体性的状态的,而拉瓦锡的元素说——他的燃烧理论的提出恰逢1771年对氧的成功离析——是说明人的意志可以理解的能量系统的,“固态”和“液态”成为描述分子间的张力关系的单纯术语。通过我们的分析和综合,自然不仅被追问或被制服,而且被施暴。近代化学是有关行为(Deed)的近代物理学的一个篇章。
我们所谓的静力学、化学和动力学——近代科学使用的这些词仅仅是传统的区分,但却没有更深刻的意义——实际上是阿波罗式的心灵、麻葛式的心灵、浮士德式的心灵各自的物理体系,每一种都是在自身的文化中生长起来的,且其有效性只限于同一种文化。与这些科学一一对应,我们在数学上有欧几里得几何、代数学和高等解析几何;在艺术上则有雕塑、阿拉伯风格图案和赋格曲。我们可以通过各自对待运动问题的立场来区分这三种物理学(当然要记住,其他文化可能且事实上产生的是其他类型的物理学),并分别称它们是有关“状态”、“秘密的力”以及“过程”的力学研究。
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