Continuity and Discontinuity

WHAT IS CONTINUOUS AND DISCONTINUOUS ?
Can we move continuously, that is to say, progressively through all the intermediate values, without leaps or breaks, from inactivity to activity, for a neuron transmitting a saccade, for a particle emitting a light photon, for a terrestrial plate entering into movement, for a society changing its social bases, for a brain evoking an image, for a living being in the process of growing, for a species in the process of changing qualitatively, for example by "inventing" birth in the egg or the spinal column, or for a radioactive nucleus that destabilizes itself by emitting radiation ? The answer is no. This is what we are saying when we choose to affirm that the dynamics that drive the world are discontinuous. This does not mean that there would never be cases where these discontinuities would be sufficiently regular to allow us to write an apparently continuous law. This apparent continuity is not then a description of reality, but only an average value of phenomena that we pretend to add up. When fluid dynamics speaks of the continuity hypothesis, it is not at all a question of physicists who would hypothesize that fluids would not be made up of discrete units, molecules, of course, but only the hypothesis that such and such a phenomenon does not manifest too large gaps at the level where the phenomenon is observed. Fluid physicists who speak of the "continuity hypothesis" mean that one can suppose that "any material point that is infinitely close to another material point will remain so in the future." However, anyone who has observed grains in a ray of light knows that this continuum hypothesis is false even if it works well at the mathematical level. As for continuity in analysis, a function "jumps," this means that points very close to the starting space end up on very distant points at the arrival. However, for a continuous application, these jumps are impossible, because if we consider a starting point and its image at the arrival, we know that a whole neighborhood of this starting point must arrive at the neighborhood of the arrival point ! In this sense, with regard to the fundamental meaning of reality, matter as energy, continuity does not exist. This is what quantum physics stipulates, which announces, in addition, that any measurement is itself tainted by a fundamental discontinuity (problem of measurement) !!!

Discontinuity is often defined, in my opinion wrongly, as the absence or interruption of continuity. Worse still, some authors believe that it is a product of continuity, which they believe is fundamental to nature. This is a mistake. Discontinuity is the interruption (or the opposite) of immobility or fixity. To move forward, one must take a step. This is already a discontinuity. Continuity is the average appearance of a roughly regular series of discontinuities : steps forward.

Discontinuity is therefore the characteristic property of a dynamic that proceeds by jumps. It is fundamental to dynamic processes. Indeed, change and movement do not proceed by infinitesimal transformations but by jumps, by quanta.

In physics, if we assume a continuous evolution in time, it is because we presuppose that time itself evolves continuously, which is far from obvious. There is at least one insurmountable barrier : the Planck time. It is very small but we cannot go below it.

Continuity is merely the illusion produced by a series of sufficiently small regular jumps observed from a sufficiently distant distance. Any apparent continuity, examined with sufficient magnification, shows multiple discontinuities.

Matter, inert as well as living, particles, living species, man, his relationships, his perceptions, his thoughts and his society, only emerge from their fixity by leaps, leaps which establish movement as well as change.

Every living or inert species has appeared, and this birth is a discontinuity in history. Every individual is born one day and must also die. These are two more discontinuities. All change is based on modifications that are not only quantitative but also qualitative, and therefore discontinuous.

Discontinuities of matter

The structures of matter manifest a fundamental discontinuity : there is no intermediary between the levels of the star and the galaxy, of the atom and the particle, of the living cell with nucleus and without nucleus, of the unicellular and multicellular being, etc.

The dynamic itself manifests an equally fundamental discontinuity : Zeno discovered this long ago.

The interactions are just as discontinuous since they are quantum, whether it is light or electromagnetism or other interaction forces.

The apparent continuity of certain parameters comes from the fact that they emerge from collective activity and are obtained on average. This is the case, for example, of electric current in which the agitation of free electrons causes average values ​​such as current or power to appear. Or it comes from the level of resolution of the interaction. Thus, the viewer at the cinema has the illusion of continuity of images because the photos have passed quickly enough for him not to distinguish them one by one. The brain establishes the junction that suggests continuity, in the same way that it establishes it for points or segments that are sufficiently close. The same is true for an interaction whose characteristic time is too long to distinguish small differences. Thus, on a large scale, the Planck quanta is much too small to be distinguished one by one. Hence the macroscopic reality as we know it. At the tap, we also believe we see a continuous juice, but the molecules arrive one by one.

Erwin Schrödinger reports in "Quantum Physics and World Representation" :

"If we consider the development of physics during the last half-century, we get the impression that the discontinuous view of nature has been imposed on us largely against our will. We seem to be entirely satisfied with the continuous. Max Planck was seriously frightened by the idea of ​​a discontinuous exchange of energy which he had introduced (1900) to explain the distribution of energy in black-body radiation. He made great efforts to weaken his hypothesis and to eliminate it as far as possible, but it was in vain. Twenty-five years later, the inventors of wave mechanics entertained for some time with the greatest ardor the hope of having prepared the way for a return to the classical continuous description, but again this hope was disappointed. Nature itself seemed to reject a continuous description, and this refusal seems to have no relation to the aporias of mathematicians concerning the continuous."

Gilles Cohen-Tannoudji explains in "Time and its Arrow" (collective work directed by Etienne Klein and Michel Spiro) :

"Heisenberg’s inequality marks the irruption of the discontinuous where it was not expected, in interactions. While the discontinuous was accepted in matter, since it is essentially the foundation of the atomic hypothesis, it was thought that interactions were completely continuous. It is indeed the thought of the continuous which constitutes the foundation of Newton’s theory of universal gravitation, and Maxwell’s theory of electromagnetism is a wave theory, and what is more continuous than a wave or a field ? Neither special relativity nor general relativity changes anything : in classical physics, interactions are entirely continuous. Now the quantum of action is fundamentally a quantum of interaction : there is no interaction unless an action at least equal to the quantum of action is brought into play. We must therefore admit the idea that, just as there are elementary particles of matter, the fermions, there must be elementary particles of interaction. And, In fact, it is proven that the fundamental interactions are indeed carried, conveyed, transmitted, by authentic elementary particles, the bosons. The photon is the boson of the electromagnetic interaction, the W+, W- and Z° bosons are the bosons of the weak interaction and the gluons are the bosons of quantum chromodynamics (the strong interaction at the quark level)."

Henri Poincaré
in “The Future of Mathematics” :

"In most problems of mathematical physics, the equations to be integrated are linear ; they serve to determine unknown functions of several variables and these functions are continuous. Why ? Because we have written the equations by viewing matter as continuous. But matter is not continuous : it is made up of atoms, and, if we had wanted to write the equations as an observer with eyesight sharp enough to see atoms would have done, we would not have had a small number of differential equations serving to determine certain unknown functions, we would have had a large number of algebraic equations serving to determine a large number of unknown constants."

Quantum physicist Erwin Schrödinger in "What is Life ?" :

“Darwin is mistaken in regarding the small, continuous, and accidental variations that cannot fail to occur, even among the most homogeneous populations, as the matter on which selection operates. For it has been proven that they are not hereditary. (..) The Dutchman de Vries discovered that (evolution originated) from a very small number of individuals (..) appearing with slight but abrupt changes, the term ’abrupt’ meaning, not that the change is very considerable, but that it represents a discontinuity, in the sense that there are no intermediate forms between the unchanged individuals and the few that have changed. De Vries calls this phenomenon a mutation. This reminds a physicist of quantum theory – no intermediate energies between two neighboring energy levels. He would be tempted to call, figuratively, De Vries’ theory of mutations, the quantum theory of biology. (..) Mutations are in fact due to quantum jumps of the gene molecule. (..) The great revolution of quantum theory was that discontinuity characteristics were discovered in the Book of Nature, in a context where anything other than continuity appeared absurd according to the views accepted until that time.

There are also mathematical discontinuities, when there are one or more abrupt changes in numerical values.

Mathematics and science have long relied on the idea of ​​the continuity of so-called real numbers, then defined the continuity of "real functions of real variables," based on the idea that one can get as close as one wants to a value that has been reached. This notion of continuity is based on that of the infinitely small, a notion that it assumes to be valid. All these concepts must be subject to criticism.

As for human consciousness, which precisely produces the illusion of continuity, it is itself discrete. "The flow of our consciousness is therefore not to be considered as a permanent continuous change but rather as a succession of stable states. (…) Numerous data from experimental psychology and human neurophysiology support this "quantum" aspect of our conscious states : Donchin and Coles, 1998 ; Raymond et al, 1992 ; VanRullen and Koch, 2003 and 2005," writes Lionel Naccache in "The New Unconscious."

Physical phenomena are fundamentally and generally discontinuous. Time, space, matter, energy, and interactions are all discontinuous phenomena. It is only the mathematical relationships between average values ​​(which have no direct physical existence) examined on a scale far larger than the phenomenon that can give an appearance of continuity, regularity, or linearity.

For example, in quantum physics, we can cite as fundamental phenomena that are discontinuous, brutal, qualitative and not only quantitative and not gradual : collisions between particles giving two photons, quantum jumps of the electron in the atom, transformations between proton and neutron, changes in the shape of the electron cloud of the atom, emissions and absorptions of photons by particles, quantum jumps that move the mass particle within the virtual cloud, transformations of a photon into a particle and antiparticle pair and vice versa, transformations of a gluon, a meson, a pion into quark and antiquark, absorptions/emissions of gluons or quark/antiquark pairs by the proton, appearances/disappearances of matter and antimatter pairs within the quantum vacuum, exchanges of virtual photons, virtual gluons between matter particles, the change of color of quarks, transformations of neutrino states, etc.

The theoretical notion of continuity presupposes others :

Compactness :

If we form a sequence of points from this set, its elements cannot move far from each other and concentrate on certain values. The compactness property also allows us to transfer certain properties from the local to the global. That is, a property that is true in the neighborhood of each point becomes valid uniformly throughout the compact set.

Convexity :

A geometric object is said to be convex when, each time we take two points A and B, the segment [A,B] which joins them is entirely contained therein.

Contiguity :

It assumes a more or less exact adhesion, that is to say that each element is in contact with the next.

Density :

It assumes that near every point there is another, as close as you want.

Connectivity :

This property indicates the absence of breaks

Continuity presupposes all of these properties at once.

The notion of quanta put forward by Planck and Einstein gave a completely different meaning to the discontinuity of matter. It is not mass (or energy) that contains an integer number of elementary quantities. It is a product of energy and time that is in an integer number of grains. This quantity is called action. That action is quantified means that it is not only the mass of matter that is discontinuous but also the interactions. At that time, it was not far from being considered that light (the grains called photons) but also space, time and the void (virtual photons) were also quantified, a new qualitative leap that quantum physics would have to make, completely abandoning the old ideal of continuity...
"Contrary to what we often hear, the discontinuity that Planck discovers here (with quanta) affects not matter, but interactions. (...) Planck’s calculations show that the exchanges of electromagnetic energy are carried by grains, whereas they were believed to be continuous. (...) What Planck discovered was that in every interaction there is an exchange and, moreover, that there is a minimum exchange below which there is no more interaction. (...) It is to Planck that the credit goes for having carried out the first "one-two" against continuity. In 1905, Einstein concluded Planck’s "one-two" with a decisive uppercut : he attributes to radiation itself, and not only to energy exchanges, a corpuscular structure. Radiation, essentially discontinuous, is, according to him, formed of a set of corpuscles each carrying a quantum of energy. (...) Radiation is not emitted in a continuous manner." writes Etienne Klein in "Regards sur la matière"

Etienne Klein and Bernard D’Espagnat added in "Regards sur la matière" : "The quantum, as we will see, has a minuscule value, but the idea of ​​the quantum has become as unavoidable as a mastodon. It is indeed proof that one can be both ghostly and essential. Truth of paradoxes, Zeno of Elea already argued."

Gilles Cohen-Tannoudji explains in "Time and its Arrow" (collective work directed by Etienne Klein and Michel Spiro) :

"Heisenberg’s inequality marks the irruption of the discontinuous where it was not expected, in interactions. While the discontinuous was accepted in matter, since it is essentially the foundation of the atomic hypothesis, it was thought that interactions were completely continuous. It is indeed the thought of the continuous which constitutes the foundation of Newton’s theory of universal gravitation, and Maxwell’s theory of electromagnetism is a wave theory, and what is more continuous than a wave or a field ? Neither special relativity nor general relativity changes anything : in classical physics, interactions are entirely continuous. Now the quantum of action is fundamentally a quantum of interaction : there is no interaction unless an action at least equal to the quantum of action is brought into play. We must therefore admit the idea that, just as there are elementary particles of matter, the fermions, there must be elementary particles of interaction. And, In fact, it is proven that the fundamental interactions are indeed carried, conveyed, transmitted, by authentic elementary particles, the bosons. The photon is the boson of the electromagnetic interaction, the W+, W- and Z° bosons are the bosons of the weak interaction and the gluons are the bosons of quantum chromodynamics (the strong interaction at the quark level)."

Henri Poincaré in “Last Thoughts” :

"We are no longer only asking ourselves whether the differential equations
of Dynamics should be modified, but whether the laws of motion can still be expressed by differential equations. And this would be the most profound revolution that Natural Philosophy has undergone since
Newton. The clear genius of Newton had clearly seen (or thought he saw, we are beginning to wonder) that the state of a moving system, or more generally that of the universe, could only depend on its immediately preceding state, that all variations in nature must occur in a continuous manner.
Certainly, it was not he who had invented this idea : it was found in the thought of the ancients and the scholastics, who proclaimed the adage : Natura non facit saltus ; but it was stifled there by a host of weeds which prevented it from developing and which the great philosophers of the 17th century ended up pruning.
Well, it is this fundamental idea which is in question today ; we wonder if we should not introduce into natural laws discontinuities, not apparent, but essential (…)"

We see what the state of the question is ; the old theories, which
until now seemed to account for all known phenomena, have come up against an
unexpected obstacle. It seemed that a modification was necessary. A hypothesis
first presented itself to Mr. Planck’s mind, but so strange that one was
tempted to seek all means of freeing oneself from it ; these means, one has
sought them in vain until now. And this does not prevent the new theory
from raising a host of difficulties, many of which are real and are not
simple illusions due to the laziness of our mind which is reluctant to change its
habits.
It is impossible for the moment to predict what the final outcome will be ;
will another, entirely different explanation be found ? Or, on the
contrary, will the partisans of the new theory succeed in removing the obstacles preventing its unreserved adoption ? Will discontinuity reign over the physical universe and is its triumph definitive ?

Louis de Broglie , in “New Physics and Quanta” :

"Without quanta, there would be neither light nor matter, and, if we may paraphrase a Gospel text, we can say that nothing that has been made has been made in them."

Physicist Leon Lederman :

"If the electron is a point, where is the mass, where is the charge ? How do we know the electron is a point ? Can I get my money back ?"