Quantum Mechanics, Haldane, and Leibniz

N. Wiener
Philosophy of Science Vol. 1, No. 4 (Oct., 1934) pp. 479-482

Dear Sir:
There has recently appeared in this journal a very interesting paper by Professor J.B.S. Haldane, entitled, "Quantum Mechanics as a Basis for Philosophy." [Philosophy of Science I, 78-79, 1934.] The paper might indeed be called, "Leibniz as a basis for Quantum Mechanics." The name of Leibniz does not occur once in the article, yet it is permeated with his spirit, and is just such a document as an orthodox Leibnizian might write should he be confronted with the development of mathematics and experimental science in the two centuries since the death of the master.
It may seem a gross anachronism to speak of Leibniz and of quantum mechanics in one breath, but this anachronism disappears when we reflect on the historical background of Leibniz. Our classical phyics, the physics of our fathers and our text-books, is dominated by Newton and his dynamics of particles. We had come to regard this science as physics par excellence, and to demand a mechanical model for our optics, electrical theory, etc. This dictatorship of mechanics has only given way to a more democratic rule in the physics of our present generation. Thus we are inclinded to forget that this dictatorship is not coeval with physics itself, and that there were heros before Newton. Among the pre-Newtonian physicists was Huyghens, the prophet of wave optics and the master of Leibniz qua physicist.
The present state of physics represents a Hegelian synthesis between Newton's thesis of particle and Huyghens' antithesis of wave. Modern optics supplants the pure wave theory of Clerk-Maxwell by a theory in which the particle aspects of light are more strongly emphasized. In the field of mechanics, on the other hand, the well-intrenched particle theory has been superseded by one recognizing the wave aspects of matter. For the first time since Newton, wave and particle, field theory and a theory of collisions, share all branches of physics more or less evenly.
It is traditional to represent Leibniz' theory of monads as a pluralistic spiritualism. If we divest Leibniz' language of its protective layer of orthodox Christian phraseology, we shall see it might as well be called a pluralistic materialism. Only a small part of the monads are souls, the greater part being naked monads gifted with apperception rather than perception or consciousness. They mirror the universe, but do not integrate this activity of mirroring into a self-conscious consciousness. This mirroring is best to be understood as a parallelism, incomplete it is true, between the inner organization of the monad, and the organziation of the world as a whole. The structure of the microcosm runs parallel to that of the macrocosm.
In modern quantum theory, the electron or similar particle is a microcosm. It carries with it a set of proper coordinates embracing the entire universe. In terms of these coordinates we may express everything happening everywhere and "everywhen", but the picture we thus obtain is blurred because of our ignoration of the proper coordinates pertaining to all other particles, and we have no unique way, as far as physics has gone at present, of translating statements concerning the microcosm into statements concerning the macrocosm. The more organized a system of elementary particles is, the less "naked" its electrons and constituent elements, the better it mirrors the universe, and the better can we read back from our partial system to the universe at large.
Haldane assimilates the living organisms of the world to the particles of physics. He reduces the apparent randomness of their action to the indeterminacy of quantum theory and the ability which they show to make a choice on the basis of a representation of the future, to the ability of an electron to cross a potential barrier because of a lower potential on the other side of that barrier. In other words, although Haldane calls his system a materialism, and jsut because he calls it a materialism he reduces the living organism and the soul to something of the same nature as the proton or electron. I can see no essential difference between the materialism which includes soul as a complicated type of material particle and a spiritualism which includes material particles as a primitive type of soul.
With all these similarities between Haldane and Leibniz, there are important differences which cast a curious light on the changes both of physical and biological science since Leibniz' time. Haldane and Leibniz treat the problem of individuality somewhat differently. For Leibniz the monad is a completely closed entity over all eternity. It may rise to the rank of a soul or be degraded to that of a naked monad, but it never loses its identity or fuses it with that of another monad. On the other hand in deference to the exigencies of modern biology and modern physics, Haldane does not make the individual identity of his particles or living being absolute. The electrons of present-day physics do not retain a traceable identity after a collision, nor is it possible to trace the identity of a living organism back to the conjugation of germ cells in which it has its origin. These two types of indeterminate identitiy known to modern science, are identified by Haldane in an extremely interesting speculation. On the other hand, at the time of Leibniz, neither problem had arisen. No one doubted that the identity of two particles could be traced beyond their collision and the prevailing spermatist theory of heredity and conception made the individual find its sole origin in a single spermatozoon, to which at that time was supposed to have an existence continuable back to the Creation. Nevertheless, Leibniz' great principle of the identity of indiscernibles, to which he appeals in connection with the perfect individuality of the monad, is retained in the modern view. Since we may permute two electrons in such a way that they will be indiscernible from each other, we now say that two electrons cannot have complete separable individualities, but must be no more than two aspects of a complex containing two or more electrons. The modern philosopher is more subtle than Leibniz in distinguishing between a unity of indiscernibility and a multiplicity arrived at by counting. This unity in plurality is perfectly familiar to the mathematician: for example, the Möbius strip is locally endowed with distinguishable sides, but as a whole has only one side.
The other important issue on which Haldane and Leibniz take opposite sides concerns the nature of causality. With Leibniz, causality is still an unbroken chain, with no place for chance, while Haldane takes the modern standpoint according to which chance is a category prior to causality. It is thus all the more remarkable that the Leibnizian principle of sufficient reason is a direct ancestor of the mathematical methods of quantum mechanics. The principle of sufficient reason asserts that this world differs from the other possible self-consistent worlds by embodying some greatest perfection -- in other words, by satisfying some principle of minimization. This must have been suggested to Leibniz by his knowledge of optics and his acquintance with the principle of Fermat, in accordance with which rays of light minimize the time of passage between two near points. Leibniz calls this principle the principle of optimism, and in this form it may readily serve as a basis for a rather fatuous contentment with the world as it is. Voltaire ridiculed this contentment on the part of the Leibnizian Maupertuis, but the fact remains that Maupertuis' application of Leibnizian method to the expression of the laws of Newtonian mechanics as the minimization principle of least action, has proved to be anything but fatuous. It must always be considered as a mark of the greatness of Leibniz, that his idea of representing the laws of physics by principles of minimization is the source of the mechanics of Lagrange and Hamilton, from whom the line of inheritance to Heisenberg and Schrödinger is clear and direct.
Again, Leibniz may be regarded as the forerunner of field theory. For Leibniz, the world is a plenum, and there is no action at a distance. A monad can only mirror a distant monad through the intervention of intermediate monads. Thus Leibnizian physics only awaits the introduction of the partial derivative to become a field physics. This step is taken in the writings of Leibniz' immediate disciples, the Bernoullis.
Leibniz clairvoyance into the future is explicable only on the basis of his universal information, universal interests, and universal genius. It is scarcely an accident that a scholar of the broad knowledge and interests of Professor Haldane takes up what is essentially a Leibnizian position. A philosophy with sufficient vvitality to appeal to first rate scholars two centuries apart is surely worth more consideration than that generally granted to it by the intellectual public.

N. Wiener
Department of Mathematics, Massachusettes Institute of Technology, Cambridge, Mass.

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