This is chapter 13 of the "The
Phenomenon of Science" by Valentin F.
WHEN THE FOUNDATIONS of the new mathematics were being constructed at the turn of the seventeenth century, the basic principles of experimental physics were also developed. Galileo (1564-1642) played a leading role in this process. He not only made numerous discoveries and inventions which constituted an epoch in themselves, but also--in his books, letters, and conversations--taught his contemporaries a new method of acquiring knowledge. Galileo's influence on the minds of others was enormous. Francis Bacon (1566-1626) was also important in establishing experimental science. He gave a philosophical analysis of scientific knowledge and the inductive method.
Unlike the ancient Greeks, the European scientists were by no means contemptuous of empirical knowledge and practical activity. At the same time they were full masters of the theoretical heritage of the Greeks and had already begun making their own discoveries. This combination engendered the new method. "Those who have treated of the sciences,'' Bacon writes.
"have been either empirics or dogmatical. The former like ants only heap up and use their store, the latter like spiders spin out their own webs. The bee, a mean between both, extracts matter from the flowers of the garden and the field, but works and fashions it by its own efforts. The true labor of philosophy resembles hers, for it neither relies entirely nor principally on the powers of the mind, nor yet lays up in the memory the matter afforded by the experiments of natural history and mechanics in its raw state, but changes and works it in the understanding. We have good reason, therefore, to derive hope from a closer and purer alliance of these faculties (the experimental and rational) than has yet been attempted.
THE CONCEPT of the experiment assumes the existence of a theory. Without a theory there is no experiment: there is only observation. From the cybernetic (systems) point of view the experiment is a controlled observation: the controlling system is the scientific method, which relies on theory and dictates the organization of the experiment. Thus, the transition from simple observation to the experiment is a metasystem transition in the realm of experience and it is the first aspect of the emergence of the scientific method. Its second aspect is awareness of the scientific method as something standing above the theory--in other words, mastering the general principle of describing reality by means of formalized language, which we discussed in the previous chapter. As a whole, the emergence of the scientific method is one metasystem transition which creates a new level of control, including control of observation (organization of the experiment) and control of language (development of theory). The new metasystem is what we mean by science in the modern sense of the word. Close direct and feedback ties are established between the experiment and the theory within this metasystem. Bacon describes them this way: ''Our course and method . . . are such as not to deduce effects from effects, nor experiments from experiments (as the empirics do), but in our capacity of legitimate interpreters of nature, to deduce causes and axioms from effects and experiments.''
We can now give a final answer to the question: what happened in Europe in the early seventeenth century? A very major metasystem transition took place, engulfing both linguistic and nonlinguistic activity. In the sphere of nonlinguistic activity it took shape as the experimental method. In the realm of linguistic activity it gave rise to the new mathematics, which has developed by metasystem transitions (the stairway effect) in the direction of ever-deeper self-awareness as a formalized language used to create models of reality. We described this process in the preceding chapter without going beyond mathematics. We can now complete this description by showing the system within which this process becomes possible. This system is science as a whole with the scientific method as its control device--that is, the aggregate of all human beings engaged in science who have mastered the scientific method together with all the objects used by them. When we were introducing the concept of the stairway effect in chapter 5 we pointed out that it takes place in the case where there is a metasystem Y which continues to be a metasystem in relation to systems of the series X, X', X",. . . , where each successive system is formed by a metasystem transition from the preceding one and, while remaining a metasystem, at the same time insures the possibility of metasystem transitions of smaller scale from X to X', from X" to X"', and so on. Such a system Y possesses inner potential for development: we called it an ultrametasystem. In the development of physical production ultrametasystem Y is the aggregate of human beings who have the ability to convert means of labor into objects of labor. In the development of the exact sciences ultrametasystem Y is the aggregate of people who have mastered the scientific method--that is, who have the ability to create models of reality using formalized language.
We have seen that in Descartes the scientific method, taken in its linguistic aspect, served as a lever for the reform of mathematics. But Descartes did not just reform mathematics; while developing the same aspect of the same scientific method he created a set of theoretical models or hypotheses to explain physical, cosmic, and biological phenomena. If Galileo may be called the founder of experimental physics and Bacon its ideologist, then Descartes was both the founder and ideologist of theoretical physics. It is true that Descartes' models were purely mechanical (there could be no other models at that time) and imperfect, and most of them soon became obsolete. But those imperfections are not so important as the fact that Descartes established the principle of constructing theoretical models. In the nineteenth century, when the first knowledge of physics was accumulated and the mathematical apparatus was refined. this principle demonstrated its full utility.
It will not be possible here to give even a brief survey of the evolution of the ideas of physics and its achievements or the ideas and achievements of the other natural sciences. We shall dwell on two aspects of the scientific method which are universally important, namely the role of general principles in science and the criteria for selecting scientific theories, and then we shall consider certain consequences of the advances of modern physics in light of their great importance for the entire system of science and for our overall view of the world. At the conclusion of this chapter we shall discuss some prospects for the development of the scientific method.
BACON SET FORTH a program of gradual introduction of more and more general statements (''causes and axioms'') beginning with unique empirical data. He called this process induction (that is to say, introduction) as distinguished from deduction of less general theoretical statements from more general principles. Bacon was a great opponent of general principles; he said that the mind does not need wings to raise it aloft, but lead to hold it on the ground. During the period of the ''initial accumulation'' of empirical facts and very simple empirical rules this conception still had some justification (it was also a counterbalance to Medieval Scholasticism), but it turned out later that the mind still needs wings more than lead. In any case, that is true in theoretical physics. To confirm this let us turn to Albert Einstein. In his article entitled ''The Principles of Theoretical Physics,'' he writes:
To apply his method the theoretician needs a foundation of certain general assumptions, so-called principles, from which he can deduce consequences. His activity thus breaks into two stages. In the first place he must search for the principles, and in the second place he must develop the consequences which follow from these principles. School has given him good weapons to perform the second task. Therefore, if the first task has been accomplished for a certain area, that is to say a certain aggregate of interdependencies, the consequences will not be long in coming. The first task mentioned, establishing the principles which can serve as the basis for deduction, is categorically different. Here there is no method which can be taught and systematically applied to achieve the goal. What the investigator must do is more like finding in nature precisely formulated general principles which reflect definite general characteristics of the set of experimentally determined facts.
In another article entitled ''Physics and Reality,'' Einstein speaks very categorically: ''Physics is a developing logical system of thinking whose foundations cannot be obtained by extraction from past experience according to some inductive methods, but come only by free fantasy.'' The words about "free fantasy" do not mean, of course, that general principles do not depend on experience at all but rather that they are not determined uniquely by experience. The example Einstein often gave is that Newton's celestial mechanics and Einstein's general theory of relativity were constructed from the same facts of experience. But they began from completely different (in a certain sense even diametrically opposed) general principles, which is also seen in their different mathematical apparatuses.
As long as the edifice of theoretical physics had just a few ''stories'' and the consequences of general principles could be deduced easily and unambiguously, people were not aware that they had a certain freedom in establishing the principles. The distance between the trial and the error (or the success) in the trial and error method was so slight that they did not notice that they were using this method, but rather thought that they were deducing (although it was called inducing, not deducing) principles directly from experience. Einstein writes: ''Newton, the creator of the first vast, productive system of theoretical physics still thought that the basic concepts and principles of his theory followed from experience. Apparently this is how his statement, 'Hypotheses non fingo' (I do not compose hypotheses) must be understood.'' With time, however, theoretical physics changed into a multistory construction and the deduction of consequences from general principles became a complex and not always unambiguous business, for it often proved necessary in the process of deduction to make additional assumptions, most frequently "unprincipled'' simplifications without which the reduction to numerical calculation would have been impossible. Then it became clear that between the general principles of the theory and the facts permitting direct testing in experience there is a profound difference: the former are free constructions of human reason, while the latter are the raw material reason receives from nature. True, we should not overestimate the profundity of this difference. If we abstract from human affairs and strivings it will appear that the difference between theories and facts disappears: both are certain reflections or models of the reality outside human beings. The difference lies in the level at which the models originate. The facts, if they are completely ''deideologized,'' are determined by the effect of the external world on the human nervous system which we are compelled (for the present) to consider a system that does not permit alteration, and therefore we relate to facts as the primary reality. Theories are models embodied in linguistic objects. They are entirely in our power and thus we can throw out one theory and replace it with another just as easily as we replace an obsolete tool with a more highly refined one.
Growth in the abstractness (construct quality) of the general principles of physical theories and their remoteness from the immediate facts of experience leads to a situation in which it becomes increasingly more difficult using the trial and error method to find a trial which has a chance of success. Reason begins to experience an acute need for wings to soar with, as Einstein too is saying. On the other hand, the increase in the distance between general principles and verifiable consequences makes the general principles invulnerable to experience within certain limits, which was also frequently pointed out by the classics of modern physics. Upon finding a discrepancy between the consequences of a theory and the experiment, the investigator faces two alternatives: look for the causes of the discrepancy in the general principles of the theory or look for them somewhere between the principles and the concrete consequences. In view of the great value of general principles and the significant expenditures required to revise the theory as a whole, the second path is always tried first. If the deduction of consequences from the general principles can be modified so that they agree with the experiment, and if this is done in a sufficiently elegant manner, everyone is appeased and the problem is considered solved. But sometimes the modification very clearly appears to be a patch, and sometimes patches are even placed on top of patches and the theory begins to tear open at the seams: nonetheless, its deductions are in agreement with the data of experience and continue to have their predictive force. Then these questions arise: what attitude should be taken toward the general principles of such a theory? Should we try to replace them with some other principles? What point in the ''patchwork'' process, how much ''patching,'' justifies discarding the old theory?
FIRST OF ALL let us note that a clear awareness of scientific theories as linguistic models of reality substantially lessens the impact of the competition between scientific theories and the naive point of view (related to Platonism) according to which the linguistic objects of a theory only express some certain reality, and therefore each theory is either ''really'' true if this reality actually exists or "really'' false if this reality is fabricated. This point of view is engendered by transferring the status of the language of concrete facts to the language of concept-constructs. When we compare two competing statements such as ''There is pure alcohol in this glass'' and ''There is pure water in this glass,'' we know that these statements permit an experimental check and that the one which is not confirmed loses all meaning as a model and all truth value. It is in fact false and only false. Things are entirely different with statements which express the general principles of scientific theories. Many verifiable consequences are deduced from them and if some of these prove false it is customary to say that the initial principles (or methods of deducing consequences) are not applicable to the given sphere of experience; it is usually possible to establish formal criteria of applicability. In a certain sense, therefore, general principles are ''always true'': to be more precise, the concepts of truth and falsehood are not applicable to them, but the concept of their greater or lesser utility for describing real facts is applicable. Like the axioms of mathematics, the general principles of physics are abstract forms into which we attempt to squeeze natural phenomena. Competing principles stand out by how well they permit this to be done. But what does ''well'' mean'?
If a theory is a model of reality, then obviously it is better if its sphere of application is broader and if it can make more predictions. Thus, the criterion of the generality and predictive power of a theory is the primary one for comparing theories. A second criterion is simplicity; because theories are models intended for use by people they are obviously better when they are simpler to use.
If scientific theories were viewed as something stable, not subject to elaboration and improvement, it would perhaps be difficult to suggest any other criteria. But the human race is continuously elaborating and improving its theories, which gives rise to one more criterion, the dynamic criterion, which is also the decisive one. In The Philosophy of Science this criterion was well stated by Phillip Frank:
If we investigate which theories have actually been preferred because of their simplicity, we find that the decisive reason for acceptance has been neither economic nor esthetic, but rather what has often been called "dynamic.'' This means that the theory was preferred that proved to make science more "dynamic," i.e., more fit to expand into unknown territory. This can be made clear by using an example that we have invoked frequently in this book: the struggle between the Copernican and the Ptolemaic systems. In the period between Copernicus and Newton a great many reasons had been invoked on behalf of one or the other system. Eventually, however, Newton advanced his theory of motion, which accounted excellently for all motions of celestial bodies (e.g., comets), while Copernicus as well as Ptolemy had accounted for only the motions in our planetary system. Even in this restricted domain, they neglected the "perturbations'' that are due to the interactions between the planets. However, Newton's laws originated in generalizations of the Copernican theory, and we can hardly imagine how they could have been formulated if he had started with the Ptolemaic system. In this respect and in many others, the Copernican theory was the more ''dynamic'' one or, in other words, had the greater heuristic value. We can say that the Copernican theory was mathematically "simpler'' and also more dynamic than the Ptolemaic theory.
The esthetic criterion or the criterion of the beauty of a theory, which is mentioned by Frank, is difficult to defend as one independent of other criteria. But it becomes very important as an intuitive synthesis of all the above-mentioned criteria. To a scientist a theory seems beautiful if it is sufficiently general and simple and he feels that it will prove to be dynamic. Of course, he may be wrong in this too.
IN BOTH PHYSICS and pure mathematics, as the abstractness of the theories increased the understanding of their linguistic nature became solidly rooted. The decisive impetus was given to this process in the early twentieth century when physics entered the world of atoms and elementary particles, and quantum mechanics and the theory of relativity were created. Quantum mechanics played a particularly large part. This theory cannot be understood at all unless one constantly recalls that it is just a linguistic model of the microworld, not a representation of how it would "really" look if it were possible to see it through a microscope with monstrous powers of magnification; there is no such representation nor can there be one. Therefore the notion of the theory as a linguistic model of reality became a constituent part of modern physics, essential for successful work by physicists. Consequently their attitude toward the nature of their work also began to change. Formerly the theoretical physicist felt himself to be the discoverer of something which existed before him and was independent of him, like a navigator discovering new lands; now he feels he is more a creator of something new, like a master artisan who creates new buildings, machines, and tools and has complete mastery of his own tools. This change has even appeared in our way of talking. Traditionally, Newton is said to have ''discovered'' [otkryl] infinitesimal calculus and celestial mechanics; when we speak of a scientist today we say that he has ''created'' [sozdal], "proposed" [predlozhil], or ''worked out'' [razrabotal] a new theory. The expression ''discovered'' sounds archaic. Of course, this in no way diminishes the merits of the theoreticians, for creation is as honorable and inspiring an occupation as discovery.
But why did quantum mechanics require awareness of the "linguistic quality'' of theories?
According to the initial atomistic conception, atoms were simply very small particles of matter, small corpuscles which had, in particular, a definite color and shape which determined the color and physical properties of larger accumulations of atoms. The atomic physics of the early twenth century transferred the concept of indivisibility from the atom to elementary particles--the electron, the proton, and soon after the neutron. The word ''atom'' began to mean a construction consisting of an atomic nucleus (according to the initial hypothesis it had been an accumulation of protons and electrons) around which electrons revolved like planets around the sun. This representation of the structure of matter was considered hypothetical but extremely plausible. The hypothetical quality was understood in the sense discussed above: the planetary model of the atom must be either true or false. If it is true (and there was virtually no doubt of this) then the electrons ''really'' are small particles of matter which describe certain trajectories around a nucleus. Of course, in comparison with the atoms of the ancients, the elementary particles were already beginning to lose some properties which would seem to be absolutely essential for particles of matter. It became clear that the concept of color had absolutely no application to electrons and protons. It was not that we did not know what color they were; the question was simply meaningless, for color is the result of interaction with light by at least the whole atom, and more precisely by an accumulation of many atoms. Doubts also arose regarding the concepts of the shape and dimensions of electrons. But the most sacred element of the representation of the material particle, that the particle has a definite position in space at each moment, remained undoubted and taken for granted.
QUANTUM MECHANICS destroyed this notion, through the force of new experimental data. It turned out that under certain conditions elementary particles behave like waves, not particles; in this case they are not ''blurred'' over a large area of space, but keep their small dimensions and discreteness. The only thing that is blurred is the probability of finding them at a particular point in space.
As an illustration of this let us consider figure 13.1.
Figure 13.1. Diffraction of electrons.
The figure shows an electron gun which sends electrons at a certain velocity toward a diaphragm behind which stands a screen. The diaphragm is made of a material which is impervious to electrons, but it has two holes through which electrons pass to strike the screen. The screen is coated with a substance that fluoresces when acted upon by electrons, so that there is a flash at the place struck by an electron. The stream of electrons from the gun is sufficiently infrequent so that each electron passes through the diaphragm and is recorded on the screen independently of others. The distance between the holes in the diaphragm is many times greater than the dimensions of the electrons (according to any estimate of their size) but comparable with the quantity h/p where h is the Planck constant and p is the momentum of the electron--i.e., the product of its velocity and mass. These are the conditions of the experiment. The result of the experiment is a distribution of flashes on the screen. The first conclusion from analyzing the results of the experiment is the following: electrons strike different points of the screen and it is impossible to predict which point each electron will strike. The only thing that can be predicted is the probability that a particular electron will strike a particular point--that is, the average density of flashes after a very large number of electrons have struck the screen. But this is just half the trouble. One can imagine that different electrons pass through different parts of the hole in the diaphragm, experience effects of differing force from the edges of the holes, and therefore are deflected differently. The real troubles arise when we begin to investigate the average density of flashes on the screen and compare it with the results which are obtained when we close one of the holes in the diaphragm. If an electron is a small particle of matter, then when it reaches the region of the diaphragm it is either absorbed or passes through one of the holes. Because the holes in the diaphragm are set symmetrically relative to the electron gun, on the average half of the electrons pass through each hole. This means that if we close one hole and pass 1 million electrons through the diaphragm then close the second hole and open the first and pass 1 million more electrons through, we should receive the same average density of flashes as if we were to pass 2 million electrons through the diaphragm with two holes open. But it turns out that this is not the case! With two holes open the distribution is different; it contains maximums and minimums as is the case in diffraction of waves.
The average density of flashes can be calculated by means of quantum mechanics, relating the electrons to the so-called wave function, which is a certain imaginary field whose intensity is proportional to the probability of the observed events.
It would take too much space to describe all the attempts, none successful, which have been made to correlate the representation of the electron as a ''conventional" particle (such particles have come to be called classical as opposed to quantum particles) with the experimental data on electron behavior. There is a vast literature, both specialized and popular, devoted to this question. The following two things have become clear. In the first place. if we simultaneously measure the coordinate of a quantum particle (any such particle, not necessarily an electron) on a certain axis X and the momentum in this direction p, the errors of measurement, which we designate [Delta]x and [Delta]p respectively, comply with Heisenberg's uncertainty relation:
[Delta]x [Delta]p >= h
No clever tricks can get around this relation. When we try to measure coordinate X more exactly the spread of magnitudes of momentum p is larger, and vice versa. The uncertainty relation is a universally true law of nature, but because the Planck constant h is very small, the relation plays no part in measurements of bodies of macroscopic size.
In the second place, the notion that quantum particles really move along certain completely definite trajectories--which is to say at each moment they really have a completely definite coordinate and velocity (and therefore also momentum) which we are simply unable to measure exactly--runs up against insurmountable logical difficulties. On the other hand, the refusal on principle to ascribe a real trajectory to the quantum particle and adoption of the tenet that the most complete description of the state of a particle is an indication of its wave function yields a logically flawless, mathematically simple and elegant theory which fits brilliantly with experimental facts; specifically. the uncertainty relation follows from it immediately. This is the theory of quantum mechanics. The work of Niels Bohr (1885-1962) the greatest scientist-philosopher of our time, played the major part in clarifying the physical and logical foundations of quantum mechanics and interpreting it philosophically.
SO AN ELECTRON does not have a trajectory. The most that can be said of an electron is an indication of its wave function whose square will give us the probability of finding the electron in the proximity of a particular point in space. But at the same time we say that the electron is a material particle of definite (and very small) dimensions. Combining these two representations, as was demanded by observed facts, proved a very difficult matter and even today there are still people who reject the standard interpretation of quantum mechanics (which has been adopted by a large majority of physicists following the Bohr school) and want to give the quantum particles back their trajectories no matter what. Where does such persistence come from? After all, the expropriation of color from the electrons was completely painless and, from a logical point of view, recognizing that the concept of trajectory cannot apply to the electron is no different in principle from recognizing that the concept of color does not apply. The difference here is that when we reject the concept of color we are being a little bit hypocritical. We say that the electron has no color, but we ourselves picture it as a little greyish (or shiny, it is a matter of taste) sphere. We substitute an arbitrary color for the absence of color and this does not hinder us at all in using our model. But this trick does not work in relation to position in space. The notion of an electron which is located somewhere at every moment hinders understanding of quantum mechanics and comes into contradiction with experimental data. Here we are forced to reject completely the graphic geometric representation of particle movement. And this is what causes the painful reaction. We are so accustomed to associating the space-time picture with true reality, with what exists objectively and independently of us, that it is very difficult for us to believe in an objective reality which does not fit within this conception. And we ask ourselves again and again: after all, if the electron is not ''blurred'' in space, then it must really be somewhere, mustn't it?
It requires real mental effort to recognize and feel the meaninglessness of this question. First we must be aware that all our knowledge and theories are secondary models of reality, that is, models of the primary models which are the data of sensory experience. These data bear the ineradicable imprint of the organization of our nervous system and because space-time concepts are set in the very lowest levels of the nervous system, none of our perceptions and representations, none of the products of our imagination, can go outside the framework of space-time pictures. But this framework can still be broadened to some extent. This must be done, however, not by an illusory movement ''downward,'' toward objective reality ''as it is, independent of our sense organs,'' but rather by a movement "upward," that is, by constructing secondary symbolic models of reality. Needless to say, the symbols of the theory preserve their continuous space-time existence just as the primary data of experience do. But in the relations between the one and the other, which is to say in the semantics of the theory, we can allow ourselves significant freedom if we are guided by the logic of new experimental facts, and not by our customary space-time intuition. And we can construct a sign system whose functioning is in no way related to graphic representations but is entirely appropriate to the condition of adequately describing reality. Quantum mechanics is such a system. In this system the quantum particle is neither a little greyish sphere nor a shiny one, and it is not a geometric point; it is a certain concept, a functional node of the system which, together with the other nodes, ensures description and anticipation of the real facts of experience: flashes on the screen, instrument readings, and the like.
Let us return to the question of how the electron ''really'' moves. We have seen that, owing to the uncertainty relation, the experiment cannot in principle give an answer to this question. This question is therefore meaningless as an '"external part'' of the physical model of reality. All that we can do is to ascribe a purely theoretical meaning to it. But then it loses its direct linkage with observed phenomena and the expression "really" becomes pure deception! When we go outside the sphere of perception and declare that such-and-such ''really'' takes place we are always moving upward, not downward; we are constructing a pyramid of linguistic objects and it is only because of the optical illusion that it seems to us we are going deeper into the realm which lies beneath sensory experience. To put it metaphorically, the plane that separates sensory experience from reality is absolutely impervious; and when we attempt to discern what is going on beneath it we see only the upside-down reflection of the pyramid of theories. This does not mean that true reality is unknowable and our theories are not correct models of it; one must remember, however, that all these models lie on this side of sensory experience and it is meaningless to correlate distinct elements of theories with the illusory ''realities'' on the other side, as was done by Plato for example. The representation of the electron as a little sphere moving along a trajectory is just as much a construction as is the interlinking of the symbols of quantum theory. It differs only in that it includes a space-time picture to which, following convention, we ascribe illusory reality by using the expression ''really,'' which is meaningless in this case.
The transition to conscious construction of symbolic models of reality that do not rely on any graphic representations of physical objects is the great philosophical achievement of quantum mechanics. In fact physics has been a symbolic model since Newton's time and it owes its successes (numerical calculations) to precisely this symbolic nature; but graphic representations were present as an essential element. Now they are not essential and this has broadened the class of possible models. Those who want to bring back the graphic quality no matter what, although they see that the theory works better without it, are in fact asking that the class of models be narrowed. They will hardly be successful. They are like the odd fellow who hitched his horse to a steam locomotive for, although he could see that the train moved without a horse, it was beyond his powers to recognize such a situation as normal. Symbolic models are a steam engine which has no need to be harnessed to the horse of graphic representations for each and every concept.
THE SECOND IMPORTANT result of quantum mechanics, the collapse of determinism, was significant in general philosophy. Determinism is a philosophical concept. It is the name used for the view which holds that all events occurring in the world have definite causes and necessarily occur; that is, they cannot not occur. Attempts to make this definition more precise reveal the logical defects in it which hinder precise formulation of this viewpoint as a scientific proposition without introducing any additional representations about objective reality. In fact, what does ''events have causes'' mean? Can it really be possible to indicate some finite number of ''causes'' of a given event and say that there are no others? And what does it mean that the event "cannot not occur?'' If this means only that it has occurred then the statement becomes a tautology.
Philosophical determinism can, however, obtain a more precise interpretation within the framework of a scientific theory which claims to be a universal description of reality. It actually did receive such an interpretation within the framework of mechanism (mechanical philosophy), the philosophical-scientific conception which emerged on the basis of the advances of classical mechanics in application to the motions of the celestial bodies. According to the mechanistic conception the world is three-dimensional Euclidean space filled with a multitude of elementary particles which move along certain trajectories. Forces operate among the particles depending on their arrangement relative to one another and the movement of particles follows the laws of Newton's mechanics. With this representation of the world, its exact state (that is, the coordinates and velocities of all particles) at a certain fixed moment in time uniquely determines the exact state of the world at any other moment. The famous French mathematician and astronomer P. Laplace (1749-1827) expressed this proposition in the following words:
Given for one instance an intelligence which could comprehend all the forces by which nature is animated and the respective situation of the beings who compose it--an intelligence sufficiently vast to submit these data to analysis--it would embrace in the same formula the movements of the greatest bodies of the universe and those of the lightest atom; for it, nothing would be uncertain and the future, as the past, would be present to its eyes.
This conception became called Laplacian determinism. It is a proper and inevitable consequence of the mechanistic conception of the world. It is true that Laplace's formulation requires a certain refinement from a modern point of view because we cannot recognize as proper the concepts of an all-knowing reason or absolute precision of measurement. But it can be modernized easily, almost without changing its meaning. We say that if the coordinates and velocities of all particles in a sufficiently large volume of space are known with adequate precision then it is possible to calculate the behavior of any system in any given time interval with any given precision. The conclusion that all future states of the universe are predetermined can be drawn from this formulation just as from Laplace's initial formulation. By unrestrictedly increasing the precision and scope of measurements we unrestrictedly extend prediction periods. Because there are no restrictions in principle on the precision and range of measurements (that is, restrictions which follow not from the limitations of human capabilities but from the nature of the objects of measurement) we can picture the extreme case and say that really the entire future of the world is already absolutely and uniquely determined today. In this case the expression ''really'' acquires a perfectly clear meaning: our intuition easily recognizes that this ''really'' is proper and we object to its discrediting.
Thus, the mechanistic conception of the world leads to the notion of the complete determinism of phenomena. But this contradicts our own subjective feeling of free choice. There are two ways out of this: to recognize the feeling of freedom of choice as ''illusory'' or to recognize the mechanistic conception as unsuitable as a universal picture of the world. It is already difficult today to say how thinking people of the "pre-quantum'' age were divided between these two points of view. If we approach the question from a modern standpoint, even knowing nothing of quantum mechanics, we must firmly adhere to the second point of view. We now understand that the mechanistic conception, like any other conception, is only a secondary model of the world in relation to the primary data of experience; therefore the immediate data of experience always have priority over any theory. The feeling of freedom of choice is a primary fact of experience just like other primary facts of spiritual and sensory experience. A theory cannot refute this fact; it can only correlate new facts with it, a procedure which, where certain conditions are met, we call explanation of the fact. To declare freedom of choice ''illusory'' is just as meaningless as telling a person with a toothache that his feeling is ''illusory.'' The tooth may be entirely healthy and the feeling of pain may be a result of stimulation of a certain segment of the brain, but this does not make it "illusory.''
Quantum mechanics destroyed determinism. Above all the representation of elementary particles as little corpuscles moving along definite trajectories proved false, and as a consequence the entire mechanistic picture of the world--which was so understandable, customary, and seemingly absolutely beyond doubt--also collapsed. Twentieth-century physicists can no longer tell people what the world in which they live is really like, as nineteenth-century physicists could. But determinism collapsed not only as a part of the mechanistic conception, but also as a part of any picture of the world. In principle one could conceive of a complete description (picture) of the world that would include only really observed phenomena but would give unambiguous predictions of all phenomena that will ever be observed. We now know that this is impossible. We know situations exist in which it is impossible in principle to predict which of the sets of conceivable phenomena will actually occur. Moreover, according to quantum mechanics these situations are not the exception; they are the general rule. Strictly determined outcomes are the exception to the rule. The quantum mechanics description of reality is a fundamentally probabilistic description and includes unequivocal predictions only as the extreme case.
As an example let us again consider the experiment with electron diffraction depicted in figure 13.1. The conditions of the experiment are completely determined when all geometric parameters of the device and the initial momentum of the electrons released by the gun are given. All the electrons propelled from the gun and striking the screen are operating under the same conditions and are described by the same wave function. However, they are absorbed (produce flashes) at different points of the screen, and it is impossible to predict beforehand at what point an electron will produce a flash. It is even impossible to predict whether the electron will be deflected upward or downward in our picture; all that can be done is to indicate the probability of striking different segments of the screen.
It is permissible, however, to ask the following question: why are we confident that if quantum mechanics cannot predict the point which an electron will strike no other future theory will be able to do this?
We shall give two answers to this question. The first answer can be called formal. Quantum mechanics is based on the principle that description by means of the wave function is a maximally complete description of the state of the quantum particle. This principle, in the form of the uncertainty relation that follows from it, has been confirmed by an enormous number of experiments whose interpretation contains nothing but concepts of the lowest level, directly linked to observed quantities. The conclusions of quantum mechanics, including the more complex mathematical calculations, have been confirmed by an even larger number of experiments. And there are absolutely no signs that we should doubt this principle. But this is equivalent to the impossibility of predicting the exact outcome of an experiment. For example, to indicate what point on the screen an electron will strike one must have more knowledge about it than the wave function provides.
The second answer requires an understanding of why we are so disinclined to agree that it is impossible to predict the point the electron will strike. Centuries of development in physics have accustomed people to the thought that the movement of inanimate bodies is controlled exclusively by causes external to them and that these causes can always be discovered by sufficiently precise investigation. This statement was completely justified as long as it was considered possible to watch a system without affecting it, which held true for experiments with macroscopic bodies. Imagine that figure 13.1 shows the distribution of cannonballs instead of electrons. and that we are studying their movement. We see that in one case the ball is deflected upward while in another it goes downward; we do not want to believe that this happens by itself, but are convinced that the difference in the behavior of the cannonballs can be explained by some real cause. We photograph the flight of the ball, do some other things, and finally find phenomena A1 and A2, which are linked to the flight of the cannonball in such a way that where A1 is present the ball is deflected upward and where A2 is present it goes downward. We therefore say that A1 is the cause of deflection upward while A2 is the cause of deflection downward. Possibly our experimental area will prove inadequate or we shall simply get tired of investigating and not find the sought-for cause. We shall still remain convinced that a cause really exists, and that if we had looked harder we would have found phenomena A1 and A2.
In the experiment with electrons, once again we see that the electron is deflected upward in some cases and downward in others and in the search for the cause we try to follow its movement, to peek behind it. But it turns out here that we cannot peek behind the electron without having a most catastrophic effect on its destiny. A stream of light must be directed at the electron if we are to ''see'' it. But the light interacts with the substance in portions, quanta, which obey the same uncertainty relation as do electrons and other particles. Therefore it is not possible to go beyond the uncertainty relation by means of light or by any other investigative means. In attempting to determine the coordinate of the electron more precisely by means of photons we either transfer such a large and indeterminate momentum to it that it spoils the entire experiment or we measure the coordinate so crudely that we do not find out anything new about it. Thus, phenomena A1 and A2 (the causes according to which the electron is deflected upward in some cases and downward in others) do not exist in reality. And the statement that there "really'' is some cause loses any scientific meaning.
Thus, there are phenomena that have no causes, or more precisely, there are series of possibilities from which one is realized without any cause. This does not mean that the principle of causality should be entirely discarded: in the same experiment, by turning off the electron gun we cause the flashes on the screen to completely disappear, and turning off the gun does cause this. But this does mean that the principle must be narrowed considerably in comparison with the way it was understood in classical mechanics and the way it is still understood in the ordinary consciousness. Some phenomena have no causes; they must be accepted simply as something given. That is the kind of world we live in.
The second answer to the question about the reasons for our confidence that unpredictable phenomena exist is that the uncertainty relation assists us in clarifying not only a mass of new facts but also the nature of the break regarding causality and predictability that occurs when we enter the microworld. We see that belief in absolute causality originated from an unstated assumption that there are infinitely subtle means of watching and investigating, of ''peeking'' behind the object. But when they came to elementary particles physicists found that there is a minimum quantum of action measurable by the Planck constant h, and this creates a vicious circle in attempts to make the description of one particle by means of another detailed beyond measure. So absolute causality collapsed, and with it went determinism. From a general philosophical point of view it is entirely natural that if matter is not infinitely divisible then description cannot be infinitely detailed so that the collapse of determinism is more natural than its survival would have been.
THE ABOVEMENTIONED SUCCESSES of quantum mechanics refer primarily to the description of nonrelativistic particles--that is, particles moving at velocities much slower than the velocity of light, so that effects related to relativity theory (relativistic effects) can be neglected. We had nonrelativistic quantum mechanics in mind when we spoke of its completeness and logical harmony. Nonrelativistic quantum mechanics is adequate to describe phenomena at the atomic level, but the physics of elementary high-energy particles demands the creation of a theory combining the ideas of quantum mechanics with the theory of relativity. Only partial successes have been achieved thus far on this path; no single, consistent theory of elementary particles which explains the enormous material accumulated by experimenters exists. Attempts to construct a new theory by superficial modifications of the old theory do not yield significant results. Creation of a satisfactory theory of elementary particles runs up against the uniqueness of this realm of phenomena, phenomena which seem to take place in a completely different world and demand for their explanation completely unconventional concepts which differ fundamentally from our customary scheme of concepts.
In the late 1950s Heisenberg proposed a new theory of elementary particles. Upon becoming familiar with it Bohr said that it could hardly prove true because it was ''not crazy enough.'' The theory was not in fact recognized, but Bohr's pointed remark became known to all physicists and even entered popular writing. The word "crazy" [Russian sumasshedshaya, literally ''gone out of the mind''] was naturally associated with the epithet ''strange,'' which was applied to the world of elementary particles. But does ''crazy'' mean just ''strange,'' ''unusual"? Probably if Bohr had said "not unusual enough,'' it would not have become an aphorism. The word ''crazy'' has a connotation of ''unreasoned,'' ''coming from an unknown place,'' and brilliantly characterizes the current situation of the theory of elementary particles, in which everyone recognizes that the theory must be fundamentally revised, but no one knows how to do it.
The question arises: does the ''strangeness'' of the world of elementary particles--the fact that our intuition, developed in the macroworld, does not apply to it--doom us to wander eternally in the darkness?
Let us look into the nature of the difficulties which have arisen. The principle of creating formalized linguistic models of reality did not suffer in the transition to study of the microworld. But if the wheels of these models, the physical concepts, came basically from our everyday macroscopic experience and were only refined by formalization, then for the new, ''strange" world we need new, "strange'' concepts. But we have nowhere to take them from; they will have to be constructed and also combined properly into a whole scheme. In the first stage of study of the microworld the wave function of nonrelativistic quantum mechanics was constructed quite easily by relying on the already existing mathematical apparatus used to describe macroscopic phenomena (the mechanics of the material point, the mechanics of continuous media, and matrix theory). Physicists were simply lucky. They found prototypes of what they needed in two (completely different) concepts of macroscopic physics and they used them to make a ''centaur,'' the quantum concept of the wave-particle. But we cannot count on luck all the time. The more deeply we go into the microworld the greater are the differences between the wanted concept-constructs and the ordinary concepts of our macroscopic experience: it thus becomes less and less probable that we shall be able to improvise them, without any tools, without any theory. Therefore we must subject the very task of constructing scientific concepts and theories to scientific analysis, that is, we must make the next metasystem transition. In order to construct a definite physical theory in a qualified manner we need a general theory of the construction of physical theories (a metatheory) in the light of which the way to solve our specific problem will become clear.
The metaphor of the graphic models of the old physics as a horse and the abstract symbolic models as a steam engine can be elaborated as follows. Horses were put at our disposal by nature. They grow and reproduce by themselves and it is not necessary to know their internal organization to make use of them. But we ourselves must build the steam engine. To do this we must understand the principles of its organization and the physical laws on which they are based and furthermore we must have certain tools for the work. In attempting to construct a theory of the ''strange'' world without a metatheory of physical theories we are like a person who has decided to build a steam engine with his bare hands or to build an airplane without having any idea of the laws of aerodynamics.
And so the time has come for the next metasystem transition. Physics needs . . . I want to say ''metaphysics," but, fortunately for our terminology, the metatheory we need is a metatheory in relation to any natural science theory which has a high degree of formalization and therefore it is more correct to call it a metascience. This term has the shortcoming of creating the impression that a metascience is something fundamentally outside of science whereas in fact the new level of the hierarchy created by this metasystem transition must, of course, be included in the general body of science, thereby broadening it. The situation here is similar to the situation with the term metamathematics: after all, metamathematics is also a part of mathematics. Inasmuch as the term ''metamathematics was acceptable nonetheless, the term "metascience'' may also be considered acceptable. But because a very important part of metascientific investigation is investigation of the concepts of a theory, the term conceptology may also be suggested.
The basic task of metascience can be formulated as follows. A certain aggregate of facts or a certain generator of facts is given. How can one construct a theory that describes these facts effectively and makes correct predictions?
If we want metascience to go beyond general statements it must be constructed as a full-fledged mathematical theory and its object the natural science theory, must be presented in a formalized (albeit simplified: such is the price of formalization) manner, subject to mathematics. Represented in this form the scientific theory is a formalized linguistic model whose mechanism is the hierarchical system of concepts, a point of view we have carried through the entire book. From it, the creation of a mathematical metascience is the next natural metasystem transition, and when we make this transition we make our objects of study formalized languages as a whole--not just their syntax but also, and primarily, their semantics, their application to description of reality. The entire course of development of physico-mathematical science leads us to this step.
But in our reasoning thus far we have been basing ourselves on the needs of physics. How do things stand from the point of view of pure mathematics?
Whereas theoretical physicists know what they need but can do little, "pure" mathematicians might rather be reproached for doing a great deal but not knowing what they need. There is no question that many pure mathematical works are needed to give cohesion and harmony to the entire edifice of mathematics, and it would be silly to demand immediate ''practical'' application from every work. All the same, mathematics is created to learn about reality, not for esthetic or sporting purposes like chess, and even the highest stages of mathematics are in the last analysis needed only to the extent that they promote achievement of this goal.
Apparently, upward growth of the edifice of mathematics is always necessary and unquestionably valuable. But mathematics is also growing in breadth and it is becoming increasingly difficult to determine what is needed and what is not and, if it is needed, to what extent. Mathematical technique has now developed to the point where the construction of a few new mathematical objects within the framework of the axiomatic method and investigation of their characteristics has become almost as common, although not always as easy, a matter as computations with fractions were for the Ancient Egyptian scribes. But who knows whether these objects will prove necessary? The need is emerging for a theory of the application of mathematics, and this is actually a metascience. Therefore, the development of metascience is a guiding and organizing task in relation to the more concrete problems of mathematics.
The creation of an effective metascience is still far distant. It is difficult today to even picture its general outlines. Much more preparatory work must be done to clarify them. Physicists must master "Bourbakism" and develop a "feel" for the play of mathematical structures, which leads to the emergence of rich axiomatic theories suitable for detailed description of reality. Together with mathematicians they must learn to break symbolic models down into their individual elements of construction in order to compose the necessary blocks from them. And of course, there must be development of the technique of making formal computations with arbitrary symbolic expressions (and not just numbers) using computers. Just as the transition from arithmetic to algebra takes place only after complete assimilation of the technique of arithmetic computations, so also the transition to the theory of creating arbitrary sign systems demands highly sophisticated techniques for operations on symbolic expressions and a practical answer to the problem of carrying out cumbersome formal computations. Whether the new method will contribute to a resolution of the specific difficulties that now face the theory of elementary particles or whether they will be resolved earlier by ''oldtime'' manual methods we do not know and in the end it is not important because new difficulties will undoubtedly arise. One way or another, the creation of a metascience is on the agenda. Sooner or later it will be solved, and then people will receive a new weapon for conquering the strangest and most fantastic worlds.
 Francis Bacon, Novum Organum, Great books of the Western World, Encyclopedia Britannica, 1955, Aphorism 95, p 126.
 Bacon, Aphorism 117, p 131.
 See the collection A. Einstein, Fizika i real'nost (Phyics and Reality). Moscow: Nauka Publishing House, 1965. The quotations below are also taken from this Russian work [Original article available in Mein Weltbild, Amsterdam, 1934 -trans.]
 Original article in Franklin Institute Journal, vol 221, 1936, pp. 313-57 - trans.
 Phillip Frank, The Philosophy of Science (Englewood Cliffs, New York: Prentice Hall, 1957).
 Laplace. P., Opyt filosofi terertt veroyatnostei, Moscow, 1908. p 9 [Original Essai philosophique des probabilités, 1814. English translation, A Philosophical Essay on Probabilities, New York: Dover Publications. 1951--trans.]
 This section is written on the motifs of the author's article published under the same title in the journal Voprosy filosofi (Questions of Philosophy), No 5, 1968.