U23: Popper's Paper on Boscovich
- U23: Popper's paper on Boscovich
Below is the full paper of Karl Popper in which he deals with Boscovich's theory (which is the Unified Field Theory).
Note: Karl Popper means by "Metaphysics" - the "interpretation of physics"; one can set up mathematical equations, but what those equations mean is subject to interpretation, in the schema of Popper this is "beyond physics" which the term "metaphysics" means. ("meta" means "beyond".) In the way that some people use the term "metaphysics" it is not what Popper would mean, instead they might mean the term to mean say the "supernatural"; which is definitely not what Popper means. So, what we have below is an article by Popper explaining the descriptive basis of modern physics as being derived from Boscovich.
(My comments are in square brackets. )
The influence on theoretical and experimental physics of some metaphysical speculations on the structure of matter
The following remarks are intended as an illustration of the important thesis that science is capable of solving philosophical problems and that modern science, at any rate, has something important to say to the philosopher about some of the classical problems of philosophy - especially about the old problem of matter. I intend to discuss certain aspects of the problem of matter since Descartes. And I intend to point out the interesting fact that some of these problems were solved, in collaboration, by speculative philosophers, such as Descartes, Leibniz, and Kant, who all helped by proposing important though tentative solutions and thus prepared the way for the work of experimental scientists and theorists of physics such as Faraday, Maxwell, Einstein, de Broglie, and Schrodinger.
The history of the problem of matter has been sketched before, notably by Maxwell.  But though Maxwell gives an outline of the history of the relevant philosophical and physical ideas, he does not give the history of the problem situation, and of how it changed under the impact of the attempted solutions. It is this lacuna that I am now trying to fill. 
Descartes based the whole of his physics upon an essentialist  or Aristotelian definition of body or matter: a body is, in its essence or substance, extended; and matter is, in its essence or substance, extension. (Thus matter is extended substance, as opposed to mind which, as thinking or experiencing substance, is in essence intensity.) Since body or matter is identical with extension, all extension, all space, is body or matter: the world is full, there is no void. This is Parmenides' theory, as Descartes understood it. But while Parmenides concluded that there can be no motion in a full world, Descartes accepted a suggestion from Plato's Timaeus according to which motion is possible in a full world, as it is in a bucket of water: things may move in a full world like tea-leaves in a teacup. 
In this Cartesian world, all causation is action by contact: it is push. In a plenum, an extended body can move only by pushing other bodies. All physical change must be explicable in terms of mechanisms that work like cogwheels in clocks, or like vortices: the various moving parts pushing one another along. Push is the principle of mechanical explanation, of causation. There can be no action at a distance. (Newton himself sometimes felt that action at a distance was absurd, and at other times that it was supernatural.)
[[Newton was an alchemist and had heretical religious beliefs, so it is difficult to say in what context he was thinking about things; with his alchemical mode of thinking he must have had some sort of belief in the supernatural (?). But basically a theory in general seems to have some strange ideas within it, that one just has to accept if one wants to use the theory.]]
This Cartesian system of speculative mechanics was criticized by Leibniz on purely speculative grounds. Leibniz accepted the fundamental Cartesian equation, body=extension. But while Descartes believed that his equation was irreducible, self-evident, 'clear and distinct', and that it entailed the principle of action by push, Leibniz questioned all this: if a body pushes another body along instead of penetrating it, then this can only be because they both resist penetration. So this resistance must be essential to matter (or to bodies) - for it enables matter or body to fill space, and thus be extended in the Cartesian sense.
According to Leibniz, we must explain this resistance as due to forces: a body has 'a force and an inclination, as it were, to retain its state, and ... resist the cause of change'.  There are forces that resist interpenetration: repelling or repulsive forces. Thus body, or matter, in Leibniz's theory, is space filled by repulsive forces.
This is a programme for a theory that explains both the Cartesian essential property of body - that is extension - and the Cartesian principle of causation by push.
Since body or matter or physical extension is to be explained as due to forces filling space, Leibniz's theory is a theory of the structure of matter, like atomism. But Leibniz rejected the theory of atoms (which he had believed in when he was very young). For atoms, at this time, were nothing but very small bodies, very small bits of matter, very small extensions. So the problem of their extension and impenetrability was precisely the same for atoms as for larger bodies: extended atoms could not help to explain extension, the most fundamental of all the properties of matter.
In what sense, however, can a part of space be said to be 'filled' by repulsive forces? Leibniz conceives these forces as emanating from unextended points and thus as located ('located' only in the sense that they emanated from them) in unextended points, the monads'. they are central forces whose centres are these unextended points. (Being an intensity attached to a point, a force may be compared, say, to the slope (or 'inclination') of a curve at a point, that is, to a 'differential': forces cannot be said to be 'extended' any more than differentials, though their intensities may of course be measurable and expressible by numbers; and being unextended intensities, forces cannot be 'material' in the Cartesian sense). Thus an extended piece of space - a body in the geometrical sense (a volume integral) - may be said to be 'filled' by these forces in the sense in which it is 'filled' by the geometrical points or 'monads' that fall within it.
For Leibniz, as for Descartes, there can be no void - empty space would be space free of repulsive forces, and since it would not resist occupation, it would at once be occupied by matter. One might describe this theory of the diplomat Leibniz as a political theory of matter: bodies, like sovereign states, have borders or limits which must be defended by repulsive forces; and a physical vacuum, like political power vacuum, cannot exist because it would at once be occupied by the surrounding bodies (or states). Thus we might say that there is a general pressure in the world resulting from the action of the repulsive forces, and that even where there is no movement there must be a dynamic equilibrium due to the equality of the forces present. While Descartes could not explain an equilibrium except as mere absence of movement, Leibniz can explain an equilibrium- and also the absence of movement - as dynamically maintained by equal and opposite forces (whose intensity may be very great).
So much for the doctrine of point-atomism (or of monads) which grew out of Leibniz's criticism of the Cartesian theory of matter. His doctrine is clearly metaphysical. And it gives rise to a metaphysical research programme: that of explaining the (Cartesian) extension of bodies with the help of a theory of forces.
This programme was carried out in detail by Boscovitch (who was anticipated by Kant).  The contributions of Kant and Boscovitch will perhaps be better appreciated if I first say a few words about atomism in its relation to Newton's dynamics.
The no-vacuum theory of the Eleatic-Platonic school and of Descartes and Leibniz has one great difficulty - the problem of the compressibility and elasticity of bodies. Yet Democritus' theory of 'atoms and the void' (this was the password of atomism) had been designed, very largely, to meet precisely this difficulty. The void between atoms, the porosity of matter, was to explain the possibility of compressing and expanding it. But Newton's (and Leibniz's) dynamics created a new and grave difficulty for the atomistic theory of elasticity. Atoms were small bits of matter, and if compressibility and elasticity were to be explained by the movement of atoms in the void, atoms could not, in their turn, be compressible or elastic. They had to be absolutely incompressible, absolutely hard, absolutely inelastic. (This is how Newton conceived of them.) On the other hand, there could be no push, no action by contact, between inelastic bodies according to any dynamic theory which - like that of Newton or Leibniz - explained forces as proportional to accelerations (in a finite unit of time). For a push given by an absolutely inelastic body to another such body would have to be instantaneous (and of finite magnitude in the instant), and an instantaneous finite acceleration would be an infinite acceleration (in the unit of time), involving infinitely large forces. 
Thus only an elastic push can be explained by finite forces. And this means that we have to assume that all push is elastic. Now if we wish to explain elastic push within a theory of inelastic atoms, we have to give up action by contact altogether. In its place we have to put short-distance repulsive forces between atoms, or, as it might be called, action at short distance, or action in the vicinity: the atoms must repel each other with forces which rapidly increase with decreasing distance (and which would become infinite when the distance became zero).
In this way we are compelled, by the internal logic of the dynamic theory of matter, to admit central repulsive forces into mechanics. But if we admit these, then one of the two fundamental assumptions of atomism - the assumption that atoms are small extended bodies
- becomes redundant. And since we have to replace the atoms by Leibnizian centres of repulsive forces, we might just as well replace them by Leibnizian unextended points: we can identify the atoms with Leibnizian monads which are nothing but repulsive forces. It seems, however, that we must retain the other fundamental assumption of atomism: the void. Since the repulsive forces tend towards infinity if the distance between the atoms or monads tends to zero, it is clear that there have to be finite distances between monads: matter consists of a void in which there are discrete centres of force.
The steps here described were taken by Kant and by Boscovitch. They may be said to give a synthesis of the ideas of Leibniz, of Democritus, and of Newton. The theory, like that of Leibniz, is a theory of the structure of matter, and thus a theory of matter. Extended matter is here explained, and by something that is not matter: by unextended entities such as forces and monads, the unextended points from which the forces emanate. The Cartesian extension of matter, more especially, is explained by this theory in a highly satisfactory way. Indeed, the theory does more: it is a dynamic theory of extension which explains not only equilibrium extension - the extension of a body when all the forces, attractive and repulsive, are in equilibrium - but also extension changing under external pressure, or impact, or push. 
There is another development, almost equally important, of the Cartesian theory of matter and of Leibniz's programme of a dynamic explanation of matter: while the Kant-Boscovitch theory anticipates in rough outline the modern theory of extended matter as composed of elementary particles invested with repulsive and attractive forces, this second development is the direct forerunner of the Faraday-Maxwell theory of fields.
The decisive step in this development is to be found in Kant's Metaphysical Foundations of Natural Science in which he repudiates  the doctrine that matter is discontinuous, which he had himself upheld in his Monadology. He now replaces this doctrine by that of the dynamic continuity of matter. His argument may be put as follows.
The presence of (extended) matter in a certain region of space is a phenomenon consisting of the presence of repulsive forces in that region, forces capable of stopping penetration (or forces which are at least equal to the attractive forces plus the pressure at that place). It is, accordingly, absurd to assume that matter consists of monads from which repulsive forces radiate. For matter would be present at places where these monads are not present, but where the forces emanating from them are strong enough to stop other matter. Moreover, it would be present for the same reason at any point between any two monads belonging to (and allegedly constituting) the piece of matter in question.
Now whatever the merits of this argument may be,  there is at any rate great merit in the proposal to try out (and perhaps make more specific) the vague idea of a continuous (and elastic) something - of an entity consisting in the presence of forces. For this is simply the idea of a continuous field of forces in the guise of the idea of continuous matter. It seems to me an interesting fact that this second dynamic explanation of (Cartesian) extended matter and of elasticity was mathematically developed by Poisson and Cauchy, and that the mathematical form of Faraday's idea of a field of forces, due to Maxwell, might be described as a development of Cauchy's form of Kant's continuity theory.
Thus the theory of Boscovitch and the two theories of Kant may be described as the two most important attempts to carry further Leibniz's programme for a dynamic theory that explains Cartesian extended matter. They may be described as the joint ancestors of all modern theories of the structure of matter; the theories of Faraday and Maxwell, of Einstein, deBroglie and Schrodinger, and also of the 'dualism of matter and field'. (This dualism, if seen in this light, is perhaps not so deep as it may appear to those who, in thinking of matter, cannot get away from a crude Cartesian and non-dynamical model.) It may be mentioned that another important influence deriving from the Cartesian tradition - and from the Kantian tradition via Helmholtz - was the idea of explaining atoms as vortices of the ether - an idea that led to Lord Kelvin's and to JJ. Thomson's models of the atom. Its experimental refutation by Rutherford marks the beginning of what may be described as model atomic theory.
[[Last part refers to experimental refutation of Kelvin's and Thomson's models; but they were only approximations. Mathematical modelling process works by making approximations that can then be refined. Boscovich's theory is a theory from which approximate mathematical models can be derived which fit to experiment as well as one wants to go in refining those models.]]
One of the most interesting aspects of the development which I have sketched is its purely speculative character, together with the fact that these metaphysical speculations proved susceptible to criticism : that they could be critically discussed. This discussion was inspired by the wish to understand the world, and by the hope, the conviction, that the human intellect could at least make the attempt in understand it, and could perhaps get somewhere. And an experimental refutation of a speculative solution to one of its problems led to its turning into nuclear science.
Positivism, from Berkeley to Mach, was always opposed to these speculations. And it is most interesting to see that Mach still upheld the view that there could be no physical theory of matter (which for was nothing but a metaphysical 'substance' and as such redundant if not meaningless) at a time (after 1905) when the metaphysical theory of the atomic structure of matter had turned into a testable physical theory as a result of Einstein's theory of Brownian motion. It is perhaps somewhat ironical, and certainly more interesting, that these views of Mach's reached the peak of their influence when the atomic theory was no longer seriously doubted by anybody, and that it is still most influential among the leaders of atomic physics, especially Bohr, Heisenberg, and Pauli. 
Yet the wonderful theories of these great physicists are the result of attempts to understand the structure of the physical world, and to criticize the outcome of these attempts. Thus their theories may well be contrasted with what they, and other positivists, try to tell us today: that we cannot, in principle, hope ever to understand anything about the structure of matter; that the theory of matter must forever remain the private affair of the expert, the specialist -a mystery shrouded in technicalities, in mathematical techniques, and in 'semantics'; that science is nothing but an instrument, void of any philosophical or theoretical interest, and only of 'technological' or 'pragmatic' or 'operational' significance. I do not believe a word of this 'post-rationalist' teaching. Nothing could be more impressive than the progress made in our attempts - and especially the attempts of these great physicists - to understand the physical world. No doubt, we shall modify and even discard our theories many times. But it seems that we have at last found a way towards the understanding of the physical world.
 See Maxwell's masterly article Atom in the ninth edition of the Encyclopedia, Britannica.
 For many years I have been in the habit of giving an outline of the story (which begins with Hesiod) in my lectures.
 I have criticized (Aristotelian) essentialism and also the essential theory of definitions in my books The Open Society and Its Enemies and The Poverty of Historirism; see the Indices, under 'essence' and 'essentialism .
 Descartes, Principia Philosophiae, Elzevir, Amsterdam, 1644, part II, point 33f. By asserting the infinite divisibility of matter, Descartes prepared the way for Leibniz's non-extended monads. (Monad=point. A point is unextended and therefore immaterial.) In II, 36 Descartes asserts the conservation of the 'quantity of motion' (quantitas motus): God Himself 'who in the beginning created matter together with motion and rest, conserves in its totality as much of motion and rest he originally put into it'. Note that this 'quantity of motion' is neither our 'momentum', which has a definite direction and which is indeed conserved, nor our 'angular momentum' but, rather, mass times the (non-vectorial) amount of velocity which, as Leibniz showed (Mathematische Schriften, edited by C.I. Gerhardt, Weidmann, Berlin and Halle, 1849-63, volume VI, pp. 117ff.), is not conserved. (On the other hand, 'force' - which Leibniz thought was conserved - is not conserved either, not even vis viva, (mv2 / 2), that is, kinetic energy. The fact is that both Descartes and Leibniz had an intuitive idea of the conservation laws, and though Leibniz came nearer to the truth than Descartes, neither got very near.)
 Leibniz , Philosophische Schriften, edited by C.I. Gerhardt, Weidmann, Berlin, 1875-90, volume II, p. 170, lines 27f. J.W.N. Watkins has developed this argument in some detail, showing that for these ideas Leibniz was essentially dependent on Hobbes, whose term 'conatus' (translated into English by 'endeavour') Leibniz adopted, and which he identified with force. See J.W.N. Watkins, Hobbes's System of Ideas, Hutchinson, London, 1965, pp. 122-32; 2nd edition, 1973, pp. 85-94.
 Boscovitch's Philosophiae Naturalis Theoria Redacta ad Unicam Legem Virium in Natura Existentium was first published in 1758 in Vienna (the second, improved, edition was translated by J.M. Child as Theory of Natural Philosophy and published in London in 1922), Kant's Metaphysicae cum geometria iunctae usus in philosophia naturali, cuius specimen I. continet Monadologiam physicam (referred to in English as 'Monadology') in 1756 in Konigsberg. Thirty years later Kant repudiated part of his Monadology in his Metaphysische Anfangsgrunde der Naturwissenschaft, published in Riga in 1786 (translated by James W. Ellington as Metaphysical Foundations of Natural Science, Bobbs-Merrill, Indianapolis, 1970). Though the essential idea of Boscovitch's monadology is to be found in Kant (see Kant, proposition V for the finite number of discrete monads present in finite bodies, and proposition X for the central forces which are attractive over long distances and repulsive over short distances, and for Kant's explanation of extension), Kant's work is rather sketchy as compared with Boscovitch's. (Added 1973.) Restrictions on the size of this paper when it was originally presented prevented me from discussing Faraday. While Boscovitch developed the Newtonian research programme which treated physical events as due to central forces (acting, however, at infinitely small distances - from one point to the next, as it were), it was Faraday's revolutionary innovation that he broke with the dogma of central forces. Although Maxwell, with his models, still hoped, like Ampere, to reduce non-central forces to central forces, his theory in effect also broke with that dogma. It thereby achieved a generalization which, I suggest, inaugurated modern field theory, and so led to special and general relativity.
 The argument is clearly stated by Kant in his Metaphysical Foundations of Natural Science, 'General Observation on Mechanics', pp. 115-17. See also his Monadology, proposition XIII, and his 'Neuer Lehrbegriff der Bewegung und Ruhe', 1758 (the section on the Principle of Continuity). Similar arguments may be found in Leibniz (see his Mathematische Schriften, volume II, p. 145), who says that it seems that only elasticity 'makes bodies rebound'. The best statement of the argument is to be found in the work of Boscovitch.
 It is important to realize that Boscovitch's forces are not to be identified with Newtonian forces: they are not equal to acceleration multiplied by mass, but equal to acceleration multiplied by a pure number (the number of monads). This point has been clarified by L.L. Whyte (in a very interesting note in Nature, 179,1957, pp. 284f.). Whyte stresses the 'kinematic' aspects of Boscovitch's theory (as opposed to its 'dynamical' aspects, in the sense of Newton's dynamics). It seems to me that Whyte's reply to Maxwell is correct. I may perhaps express this by saying that Boscovitch not only gives a theory of extension and gravity but also of Newtonian inertial mass. On the other hand, although Boscovitch's forces are, as Whyte rightly stresses, from a formal or dimensional point of view accelerations, they are, from a physical and a metaphysical point of view, forces, very much like Newton's: they are dispositions, existing in their own right; they are the causes that determine accelerations. Kant, on the other hand, thinks in purely Newtonian terms, and attributes inertia to his monads; see his Monadology, proposition XL
 See the second chapter, theorem 4, especially the first paragraph of note 1, and note 2. Kant's repudiation is the result of that doctrine which he-calls (in the Critique of Pure Reason) his 'transcendental idealism': he rejects the monadology as a doctrine of the spatial structure of things in themselves. (This way of speaking would be for him a mixture of spheres - something like a 'category mistake'.)
 Like all such proofs Kant's proof is invalid, even in the form here given, which is an attempt to improve a little on Kant's own version. Kant illicitly identifies 'moving', in the sense of a moving (repelling) force, and 'movable'; cp. the penultimate paragraph of his note 1 to theorem 4. The ambiguity is bad, but it brings out the fact that he wishes to identify the presence of a moving force with that of movable matter. The logical situation is, in brief, as follows. In this post-critical work, Kant's transcendental idealism is used to remove - by a valid argument, incidentally - his original objections to the doctrine of continuous matter. But he now thinks, mistakenly, that he can prove continuity -by an argument which, though invalid, is interesting and important because it compelled him to push his dynamics to its very limits (and far beyond those limits which he anticipated in his definitions).
 Niels Bohr, Werner Heisenberg, and Wolfgang Pauli were all alive when this was written.
The Myth of the Framework, Karl R Popper, ed. M A Notturno, Routledge, London and NY, UK 1994 isbn 0-415-11320-2, p 112 - 120
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