Dear William,

Leibniz pointed out that particles, or fundamental dynamic units might be a better term, cannot exchange anything. The Scholastic view up until the seventeenth century was that causation was due to 'accidents' being passed between substances. But if particle P acquires accident A from particle Q then we have P+A, not P. So we no longer have a particle; we have an aggregate. Worse still if particle P were to lose an accident A it would have the strange nature of being a not quite P, or P-A.

Exchanging is the language of aggregates - it can only be. Leibniz showed that the dynamic rules of fundamental units must be radically different from the aggregates we are used to in daily life. Causation at the real fundamental level does not do exchanging. The idea of 'exchanging momentum' is merely a poetic metaphor. What is different after the collision is the relation of the dynamic unit to the universe - if anything it is a new property of the universe's organisation, not of the particle, I would suggest. The dynamic unit is a packet of disposition and to be the same unit that disposition must remain the same, as indeed it does in physics. If a new disposition is present then a new particle P' must have replaced P, as perhaps in a Feynman diagram. That replacement, as Leibniz says, can only occur by creation or annihilation, not by addition or subtraction. Otherwise we are dealing at the aggregate level.

Dear Jonathan,

Thanks for your comments.

Rather than accepting Leibniz’s thoughts as a preferred reference position from which to evaluate observations, would it not be better to make observations, examine them for interrelationships of sequential patterns, and then apply methods of theories to interpretations? An argument for this strategy of analysis is illustrated by reference to the elastic collision.

Laws and theories are not logical necessities but empirical associations derived from observations in the distributions of matter. At the limits of observational technology, scientific and philosophical endeavors are extended by extrapolation that is mediated and regulated by mathematical principles.

An examination of the elastic collision illustrates an example where Newton’s third law [Whenever one body exerts a force on another, the second always exerts on the first a force which is equal in absolute magnitude, opposite in direction (or sign), and has the same line of action.] appears to be validated at the macro level, but where the extrapolation process reveals a potential conceptual flaw. The conservation-of-momentum principle of a system is the only mechanical principle applicable to the problem.

At the macro level the mechanistic explanation routinely presented is that during contact an impulse is generated between bodies that is equally divided between participating bodies creating a separation mode. By stipulating that no kinetic energy may be lost and that momentum is conserved, mathematically the elastic collision is executed, characterized by a rate of departure equal to the rate of closure. The after effects of an elastic collision between two spherical bodies of equal matter would be equivalent to an exchange of positions of the bodies.

An observation whose theoretical consequence is often ignored is that collisions between more rigid bodies tend toward more elastic collisions. If there exist particles with no parts they should be absolutely rigid. The name criton is assigned to such postulated particles with no substructure. (Such particles would occupy a position analogous to Leibniz’s dynamic units, except that Leibniz’s units are not composed of matter.) By extrapolation critons should execute perfectly elastic collisions.

Two critons composed only of single units of matter would express no relative motion. However, particles are observed to exhibit relative motion. To account for motion a vectorial entity (velose) is proposed to be associated with the critons. Thus the criton-velose association constitutes an aggregate. The relationships among the velocity factor (v), mass or matter component of an individual criton (m), and its associated velose (p) is expressed as follows: v = p/m. Therefore, p = mv, which is the same expression as that for momentum. Anytime the ratio p/m between two critons is not equal, relative motion will be expressed.

Collisions between critons, because they are rigid, cannot create an impulse for redistribution of motion to the respective bodies; i.e. Impulse = change in momentum/time and no compression of particles is possible to create a force. It is proposed that regardless of matter content, colliding bodies approach each other with equal but oppositely directed compliments of velose (momentum). At contact, rigid bodies exchange equal quantities of velose (momentum) relative to the contact point. If this occurs an elastic collision is executed without the necessity to specify that kinetic energy is conserved. This type of exchange is consistent with the postulate that all motion is relative.

Returning to the classical representation of the elastic collision, Newton’s third law is a manifestation of equal but opposite exchanges of momentum during an interval. The impulse between bodies may be attributed to the time dependent process of velose (momentum) exchanges occurring between populations of rigid bodies that constitute the particles that may be observed. Inelasticity is a function of interval length of momentum exchanges and suggests a process whereby each exchange initiates a momentum equilibrium process within the respective colliding bodies.

I am not sure I fully follow your argument. However, a 'particle' with no parts cannot be rigid because rigidity is defined as the constancy of relation of parts of something under all conditions of interaction - I can't think if any other way one could define it. To be rigid is for your left hand part to stay the same distance from your right hand part. Without parts rigidity vanishes. As I understand it Leibniz came to his conclusions, which are more or less identical to the precepts of modern fundamental physics, precisely because he thought through these issues. His 'prior position' on things was merely a recognition of a contradiction in the naive concept of particle. There could never be such a thing as a particle - it is a self-contradictory pseudoconcept - and physics has now recognised that.

Dear Jonathan,

A rigid body may be defined as a system of particles held together by forces obeying Newton’s third law in such a way that the distance between any two particles remains unchanged. This conceptualization has not been documented at the observational level. Component bodies deform to some extent under the influence of applied forces such as collisions. The question is: Can a body be broken into units that have no substructure and would such entities be non-deformable?

If the ultimate units of “reality” are of such size that with their properties they cannot be individually detected at the experimental level, it allows the interpretation that immaterial or non-physical phenomena operate at the most fundamental level. Such a condition presents the situation that, starting with the observable matter components of the laboratory, at least three extrapolative paths exist: (1) Matter extrapolates to ultimate matter components. (2) Matter is infinitively divisible. (3) Matter with its associated properties extrapolates to Leibniz’s monads or some analogous non-physical entities.

If matter is infinitively divisible, why isn’t the observed array of physical entities much greater than observed since the potential different combinations would be infinite? Similarly, if each monad is distinctly different, how can they be combined to produce the apparent uniform, repeating units that are observed in the distributions of matter? One property that the ultimate entities must possess is an impetus to form associations.

Could you direct me to the source of the argument that physics has now recognized that a particle is a self-contradictory, pseudo concept?

Dear William,

I feel you asking me to explain a truism, or tautology and that is quite a big ask. If a unit has no substructure how would one define its deformability? If it has no right and left hand parts to get nearer or further apart how could deformability be applied to it?

I detect some problems in the way you lay out the arguments. 'Detection at the experimental level' conceals the fact that all our observations rely on inferences. Those inferences are often subconscious - as when we 'see a ball'. But there are always inferences and so 'detection' of an electron is an inference from a blip on a screen or a crackle of a Geiger counter but 'detection' of an elephant is similarly an inference from the arrival of various photons at the retina. The detection of Higgs particles at the experimental level required inferences relating to complex statistical estimations but they were detected. So if we can infer any properties of ultimate components we can 'detect' these.

You ask if matter is based on ultimate matter components but what are you meaning by matter? Matter is a set of dispositions - to reflect light, or fall to the floor or deform or whatever. The constituents of matter must somehow explain these dispositions but it is easy to see that they will give rise to the aggregate dispositions through the combinations of rather different sorts of dispositions. A flat shiny surface will reflect light with angles of incidence and reflection being the same but any element much smaller than the wavelength of light will not have this sort of disposition. Leibniz's monads have different dispositions from their aggregates but I am not sure in what sense that makes them 'non-physical'. The way he describes them looks very much like the modes of excitation of fields that are the basic dynamic units in modern field theory. Leibniz has no distinction between physical and non-physical, except the distinction between aggregate descriptions and descriptions of individual units.

You ask why if matter is infinitely divisible the array of entities is not greater. But the array of entities is as rich as there are entities. As Leibniz says, and Feynman confirms in his lectures any two modes that are indistinguishable are the same thing. I think the confusion is that people still think that the units are things like 'electrons'. In field theory the units are modes like 's orbitals'. There is no fact of the matter whether an orbital is 'occupied' by electron A or electron B. It is just that orbital. And 's orbital' is just a convenient approximation because the reality is that this is a mode of progression that is almost entirely determined by relation to a nucleus, but NOT entirely so, to the extent that it is trivially determined by the presence of every other mode of excitation in the universe. And since the universe is not symmetrical every s orbital at every point in the universe is different. You might say that this is still not what you would get with infinite divisibility but if you then accept the concept of quantum foam, or at least the indeterminacy of everything then you do get infinite possibilities. In the end it becomes clear that the idea of infinite divisibility is itself not clearly enough defined to know whether to say it applies. All these intuitive concepts based on the idea of matter as 'stuff' evaporate if you press them hard enough.

And there is no problem in different monads combining to form order. If their difference lies in asymmetry of the universe then there is no obstacle to them lining up to form 'ordered' arrays that are themselves breakages of symmetry. The difference between monads, qua s orbitals, in a row of 100 is that the first has 99 nuclear potentials just to the left, the second has 98, etc. And as the structure grows the orbitals change to relate more distinctly to the array as in a Kronig Penney model. What in Leibniz's terms is identical in all s orbitals is an 'internal principle of change' given by, in modern terms, spin a half, change -1, energy level 1 (for s) mass ... etc. and if you pack units with that same principle together you get something pretty regular, but each unit is distinct.

I am not sure where you will find someone saying that physics has now accepted that a particle is a pseudo-concept. It has just been quietly dropped in favour of modes of excitation or maybe strings, although those latter are in trouble at the minute. My understanding is that the argument is the basis for Descartes's discussion of matter, and that in Leibniz, and goes back to Plato and the dissatisfaction with Democritus. Even JJ Thompson and Rutherford when they proposed models for atoms did so in a way that took on board the contradiction in Democritus, maybe especially JJT. Over the years my experience of talking to physicists with serious reputations is that the concept of non-composite 'particle' as something that might have a 'size' or 'rigidity' was something people needed to get past early on in doing physics.