Leibniz, Newton, Gravitational Waves and the Second Triad

The Question Arises! The Second Triad

March 3, 2016


For some time, I have thought that it would be useful to go to the task of drawing out certain implications of Einstein’s theory of General Relativity with respect to the different views of the nature of space itself- especially those of Leibniz and Newton. With the recent news of the experimental evidence corresponding to the existence of “gravitational waves” and the dynamic of increased discussion on the nature of space which it has created around the world, I thought it would be an opportune moment to provide some of my thoughts which, I believe, provide perspectives on the issue which have been heretofore unconsidered. I believe that this provides a contribution to the pathway of investigation whereby certain fundamental concepts of physics can be, yet again, reexamined, changed, and overthrown in new and fruitful ways.

On the Concept of Non-Euclidean Space from the Standpoint of Leibniz and Newton

The question arises, how is it that we are to conceive of space if it be not Euclidean? Further, if we accept Einstein’s theory of relativity in which the curvature of space can be changed, how are we to conceive of what the space itself is? What is it which is being curved? Return to the debate between Leibniz and Newton and the question as to whether space is itself an entity which exists independently of the objects within it -as an existent feature of the universe on its own- or, whether space is nothing other than the relationships between what we consider to be extended physical bodies and physical processes whose existence we regard, generally, as given, or self evident? Let's examine the question of Einstein’s Relativity from this standpoint, especially the concept of gravitational waves. If one of the theorems of Relativity is that an accelerating mass will produce waves in space(time), what are the two ways of looking at this question as within the Leibniz/Newton debate? (I will also note, in passing, that there may be conceivable certain non-Euclidean spaces which in which the curvature is different in each part. Such spaces would escape nullify one of the arguments which Leibniz levied against absolute uniform space: that God could not have chosen in which place to put the universe.)

Relativity as a Modification of Newton’s Concept of Space

Let’s examine the question using a hypothetical physical system of the most simple kind- that of a single finite particle/body imbued with “mass”. If we say that we take Newton's notion of space as an absolute and self-existing entity, then we say that this space actually exists around our isolated particle in our system, and that the space around this isolated mass is curved in such a way as to “bend” the path of any other physical entity (such as other bodies or light) which may come to pass through it. But, remember, being Newtonians, we say that the existence of this space around our particle does not depend for its existence on anything which may or may not pass through it. Now, continuing with Newton’s view of space, Relativity says that if we were to accelerate our isolated mass then this would create a rippling effect, a wave, in the space itself. This wave would continue to propagate through the space away from the mass forever, decreasing in intensity as it moves outward. This wave, like the space which it exists in, does not require the presence of any other body or physical process to affirm its existence, for it exists in the space itself which needs no body to exist. Thus, we see that in the Newtonian view, the concept of gravitational waves seems conceptually distinct and intuitively plausible.

Relativity Considered form a Leibnizian Standpoint

Now, how would our belief in Relativity change if we take Leibniz’s view? As in our last system, we examine a simple physical system of one isolated particle/body existing on it’s own. We will immediately notice a conceptual difficulty in this task- in fact, not a difficulty, but an impossibility. For, from a Leibnizian point of view, conceiving of a completely isolated (that is to say- there is only one in the universe) finite extended body is itself a logical contradiction. Why? For the body must be finite relative to something else, and, if we admit of no other body against which we could compare our single body, then -being Leibnizians who reject the existence of any absolute space- there could be nothing outside -or other than- our finite particle; a logical contradiction- a conceptual impossibility. For there is nothing outside our particle, and there is no space beyond it, since space is not real, being only an artifact of the relationship between a plurality of extended bodies. Let’s say we could conceive of such a thing momentarily: Perhaps a little piece breaks off from our particle and is propelled outward for some reason- in this case a relationship between the particle and its outwardly projected part provides the concept of space between the two bodies, but once the part returns to the original body there is no space around the body and the body itself must cease to exist- at least as a finite body. Continuing to fancy ourselves capable of conceiving of such a physical system with one body and no absolute space, we go onto find that this lone body would be incapable of any acceleration since there is no space around it to accelerate in, or any other body existing which it could relatively accelerate towards. Now, if we were to say that this body were to pulsate like an elastic ball such that its parts accelerated relative to each other, then this would be an acceleration of the mass, and thus, believing Relativity, gravitational waves must be created- but here, the term “gravitational waves” means something different than it did previously. For now “gravitational waves” must mean a wave like propagation of a change in the metric relations of existing physical bodies/processes. But if there is no physical body in the universe other than our one body we started with, then the idea of “gravitational waves” loses all meaning, for there is no space which could possibly propagate them, and/or nothing else existing which could register their effect. But, all this is only intended to illustrate the way in which concepts are modified, even in simple cases as imagining one single particle, by the Leibnizian standpoint with respect to the Newtonian one- even though the distinction between them is considered conceptually nil by many. In any case, we can see that the concept of “space” is required to conceive of any isolated finite body.

To finish with our Leibnizian analysis of space and “gravitational waves” we examine a hypothetical system which is not much more complex than our previous one: two finite extended bodies. In Leibniz’s framework, this system provides us with no contradiction as the last one did. (But only, it seems, if we only imagine the system from the point of view of one of the bodies.) If one body (with mass) undergoes acceleration, then, by Relativity, a “gravitational wave” will occur. Now what does that mean? Well, we know, being Leibnizians, that the gravitational wave doesn’t propagate through space, since space does not exist to propagate anything in itself. We must mean, then, that there is a change in the metric relations of the second body as a result of the acceleration of the first which will be registered to occur as if a wave disturbance propagated through an extended medium which encompassed both bodies- but, really, there is no such medium, and the word “wave” in the phrase “gravitational wave” only refers to the pattern which a change in the metric relations of bodies in relative position to each other will undergo in time.

The Same Question Applied to Our Current Concept of the World

Many people today, and probably throughout history, think that the debate between Leibniz and Newton has no real significance for scientific practice, conceptualization, or application, because both viewpoints seem to lead to the same exact results in the final analysis of any given phenomena. But, if we examine these contrary viewpoints a little further in their bearing on what we (think we) know of processes in the universe, we may find difference we seek. Indeed, it is upon the recognition of that difference that a new discovery hinges.

Newton: In the “real world”, if we are Newtonians who have modified our notion of absolute space with Einstein’s Relativity, the actually existing space of the universe is curved by the presence of masses. Remember, we believe that the space in between the masses actually exists- it’s existence does not depend on any body or light ray or whatever which might pass between the masses in question. The curvature of this space necessitates a particular relationship between bodies/physical processes, like light rays, and when one such physical process does, at some point, pass through this already existing space, its metric relations will change. Continuing our treatment of relativity from the Newtonian standpoint, we say, further, that if a mass accelerates in space, then there will be created in that space a curvature change which will propagate outwards as a wave- a gravitational wave. This gravitational wave, as said before, moves through space in between bodies and exists even if it does not encounter a physical body or process to reveal its existence. Again, this is a somewhat intuitively plausible and accessible conception.

Leibniz: Now, let's look at Relativity theory applied to our world from Leibniz’s standpoint. If we have a body/mass existing in a relative relationship to other masses, and the relative position of our mass accelerates with respect to another, then this will create a “gravitational wave”. This “gravitational wave” is not a thing in itself, for, if the “space” between the masses/bodies is devoid of anything else, then the gravitational wave has no existence between the bodies. This is a “wave” in the metric relations of physical processes/bodies, and only if we assume a continuous distribution of physical material amongst all things can we say that this “wave” is really a continuous wave at all. Otherwise, we must be content in saying that there is only a “wave-like” change in the metrical properties of physical processes as if there were a wave through a medium like Newtonian-space which pervaded and determined the metric relations of everything.

Is There Empty Space?

Now, what is our situation in the universe with which we seem to know something about? Which viewpoint, Newton or Leibniz are we to take? If we admit the existence of a continuous distribution of physical processes, or matter, then we would have a hard time deciding, although we might lean towards the Newtonian view as being more intuitively pleasing. If we were to admit the existence of areas of the universe in which there existed no physical process, then we would be even more inclined to Newton’s view for the same reason. Certainly, we have come to know that certain portions of the “space” in our galaxy are only populated with about one atom per 10 cubic centimeters (what to speak of the space between galaxies?!). It seems that there really is nothing else between the atoms themselves. What then?- are we to agree with Leibniz who would say that the “gravitational wave” only seems to pass though something in between those atoms!? A wave passing through nothing!? How quaintly irrational dear Leibniz!! Such may be the questions- and they would not be unjustified. What are we to do in such a situation?

Some years ago, we heard a particularly wise man remind us that “There is no empty space!”. We also recall that Leibniz himself said: “For everything is a plenum, so that all matter is bound together, and every motion in this plenum has some effect upon distant bodies..” But, what could these two wise men mean given the atomic density of space just cited? Ah, there must be something between those atoms: light. Light, or radiation, is that which is, perhaps, distributed completely over the cosmic firmament. Light could be that which carries the changes in metrical relations as in a continuous wave when no matter is present.

Why the Wave?

But now we only return to the same point. Both Newton and Leibniz seem to be on equal footing again, and there seems to be no difference whatsoever in choosing one over the other in how to think about the universe- particularly “gravitational waves”. In fact, again, we might seem to come to a point at which Newton’s view grant's us greater solace. For the question still arises: how is this change in metrical relationship propagated through the matter which is “all bound together”, which fills space? How would one particle or light beam “transfer curvature” to another contiguous particle or light beam as if it were a wave? This could not happen from sort of special collision, since we conceive these curvature characteristics to change during whatever kind of collision/intersection process the altered matter/light may be undergoing upon reception of the gravitational wave-pulse. It seems that we are at a loss to provide a reason as to why this would happen. On the other hand, Newton’s concept of space solves all difficulties- for we can readily explain the wave like propagation of changing curvature/metric-relations amongst bodies/light as a result of an actual wave of changing curvature propagating through actually existing -independently existing- space, within which all those things exist. Thus, even from the standpoint of Leibniz’s axiom that “all thing occurs for a necessary and sufficient reason”, is not Newton’s (modified) concept of space better in explaining gravitational waves? It seems so.

It is just here that we come to that difference between the two conceptions of space which we hoped to uncover: It is a difference with respect to scientific practice/conceptualization. For, while Newton’s modified concept of space provides us with a certain reason, albeit a descriptive one, as to why the propagation of metric curvature takes place in physical processes, Leibniz’s notion of space does not. (Fn. 1) But, just because Leibniz’s notion of space does not provide such an explanation, we are not obliged to believe that no explanation could be found as long as we adhere to Leibniz's view of space. In fact, if we were to find such an explanation while adhering to this view, we may, in fact, find that that explanation is scientifically superior to the Newtonian conception-that that reason may be an as of yet undiscovered principle which, once uncovered, will increase mankind's power over the universe.

The Equivalence Principle and the Second Triad

Whether we be Newtonians or Leibnizians in our thinking as to the nature of space, we must admit that, ultimately, our theories as to the curvature of space must correspond to observations we make of what we consider to be “straight” physical processes. While any physical entity may be used to make this determination, the physical process which is most susceptible for this purpose is the one which we most readily associate with the notion of straightness in space: light. Thus, when we ask the question: “How is space curved?” we are -essentially- asking the question: “Why does does light propagate in the way that it does?”, or, alternatively, “Why do the effects which we associate with light register in our observational apparatus in the way that they do?”. But wait!...wait. We will notice that this is the same question which is asked by anyone who has observed the phenomena associated with the double slit experiment. “Why do the effects which we associate with light register in our observational apparatus in the way they do?”. Let me be so bold as to put forward a hypothesis which invokes the Equivalence Principle: The answers to those two identical questions will be identical.

An objection may arise to this hypothesis: The question of the shape of space is different than the question of the nature of that which moves through it. The question as to the shape of space, deals with what will determine the potential relations of physical processes, but the question as to the paradoxical effects of light when considered in view of the quantum addresses the nature of light itself. Therefore, whatever the answer to the question respecting the nature of light we might find, it is not related to the question of how space is curved.

This objection, however, relies on an assumption: the Newtonian conception of the nature of space. Since the Newtonian idea is that space has an existence which is in no way dependent upon the existence of any physical process which may occur in it, the question as to the metrical relations of space is separate from the question of the nature of the light which “moves through” space. Thus, the hypothesis put forward above presupposes the rejection of the Newtonian view for that of Leibniz’s.

We see thus, initially, and as if through a glass darkly, but nonetheless with faith, that we come upon a relevant consideration respecting the as of yet uncompleted Second Triad identified by LaRouche. The resolution of the paradoxes of the quantum with the question of the metrical relations of physical processes- the resolution of the revolutionary concept of Planck in the small with that of Einstein in the large.

Quantum Considerations

Implications of current concepts of quantum processes which bear on the scientific investigation of the nature of the metrical relations of physical processes should be addressed. I will elaborate a few which come to my mind.

  • The Uncertainty Principle. If our hypotheses as to the metrical relations of space must ultimately find justification in their correspondence to observation, then the limitations inherent in the observation process have a bearing on the significance of those hypotheses. The uncertainty principle places a limitation on the potential accuracy of all physical observations, including those observations which we make to determine the metrical characteristics of space. Thus, we find that we can never, in principle, come to determine the shape of space with indefinite precision. In my view, this is an indication that there is actually no definite shape of space in reality.

  • The Double Slit. The objection may arise to the just stated interpretation (as it would apply to another physical process limited in this way- the notion of the definite path through space of a photon, electron or etc.) as follows: Just because there is not, in principle, a method whereby we can measure the path of the electron with indefinite precision, this does not mean that a definite path does not exist. Initially it seems irrelevant as to whether we choose to believe one way or another, but there is something which impels s to decide. The refutation of this objection lies in the phenomena of wave-like patterns of registration of the electron position in the double-slit experiment. For, if the electron had a definite path through space, then no sufficient reason could be given as to why such a registration pattern would be made manifest. Thus, there is no definite path of the electron (or photon of light) through “space”- or, rather, in Leibnizian terms, there is no definite mathematical relationship which characterizes the relationship of physical processes relative to each other. Once this is established, the search for causal principles which are conceived to express themselves in a mathematically definite way will be abandoned, except for limited technical purposes. This is coherent with Cusa’s insistence that, in principle, there exists nothing in the universe which cannot be more precisely known than what we might come to know of it. (Just as the issue of the curvature of space concerns itself with the metrical relations of all matter/physical processes, not just light, so too does the quantum paradox concern itself with all processes, and not just light per se. For example, the electron and other particles -even molecules- have exhibited the double slit wave phenomenon.)

  • Cosmophysical Factors. The work of Schnull and others (if there are any) which demonstrates the correlated characteristics of cosmic factors in the very large with quantum processes should be considered. The scale of cosmic factors which he identifies as correlated to the changes on the quantum level are of the scales associated with changes in the metrical relation of space (significant gravitational effects). This corresponds to the direction of my hypothesis, and I would put a premium on work similar to this for that reason. Another point of connection is that the quantum process of radioactive decay is associated with relativistic changes in mass.

  • A couple other thoughts:

  • That a photon is conceivable as a spatially extended wave or particle is contradicted by the very notion of the quantum of energy or action. This gives the question as to why the “speed” of light should exist at all a new element of consideration- even besides the mysterious fact of its constancy for all observers.

  • A photon does not perceive any elapsed time between its emission and absorption. From its point of view, the only “time”, or change, which exists is the succession of its quantized action- whatever that may be.

In my view, a different approach needs to be taken to understanding the metrical relations of physical processes where, after a certain point, we no longer attempt to seek out a definite mathematical characterization of these relations through observational refinement and, instead, search out those general conceptions which will enable us to understand -even if not mathematically- why we seem to observe changes in the metrical relations of things.

The Third Pillar of the Second Triad

And here we seem to have returned, by a novel route, to LaRouche’s identification of Vernadsky’s method as providing the pathway towards the consummation of the work of Planck and Einstein -towards the completion of the second triad. As a part of his work, Vernadsky raised the issue of the way in which life seems to change what we think of as “space”. Life, being a higher order principle than anything conceived of in physics today, may very well be of just that order that we require to present us with a reason as to how the metric relations of physical processes are changed.

As I mentioned in my paper “The Principle of Least Action”, coming to understand the way in which the metrical relations of physical processes are changed through a mastery of higher principles of life and cognition is not only a nice sounding phrase for an idea, but, it actually seems to be a requirement in the scientific progression (that is, in the progression of power over the universe) of mankind given the limitations implicit in our current concept of “physical power” associated with the idea of “energy transfers” from “fuels”.


As I have mentioned elsewhere, the causal concepts which we invoke to provide us with the reason as to why things are the way they are will be shaped to a great extent by what we consider those things to be in the first place. If we really are to abandon a mathematically constrained approach towards understanding the world -or even a conception of anything susceptible, in principle, to precise mathematical characterization- how then are we to conceive of the things in the world at all? What are we left with as conceptual anchors upon which we understand our experience? In response to these queries, I quote Ben: “...If we want to know more, what do we have? We have what the human mind does. That's what enables us to know things about the Universe. That's what tells us about the Universe, and that's what really tells us about how the Universe is fundamentally organized.”

It seems that poetry really must replace mathematics in physics after all.


1.) Indeed, the modified Newtonian conception will remain descriptive so long as the concept of space remains a geometrical one. There is no attempt, in this approach, to explain why the space must be “warped” by mass in the way it is. This would require a new conception of space itself- a more physical conception. Einstein seems to have learned in this direction at one time: “It would have been more correct if I had limited myself, in my earlier publications, to emphasizing only the nonexistence of an ether velocity, instead of arguing the total nonexistence of the ether, for I can see that with the word ether we say nothing else than that space has to be viewed as a carrier of physical qualities.” [http://einsteinpapers.press.princeton.edu/vol9-trans/161?ajax] However, once such a conception is posited, it seems we can always revert to the idea of a Euclidean absolute space within which this physical space is imbedded- especially if this physical space is conceived of as finite.

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