Abstract: The Dichotomy is an aporia that Zeno of Elea illustrated with the paradoxes of Achilles, the Arrow and the Stadium. Twenty-five centuries later, the dichotomy seems to be resolved, and in several different ways. This is precisely a problem. These solutions are contradictory to each other, keeping the paradox all its relevance. Zeno wanted to demonstrate that movement is an illusion. In a block universe, is he not fundamentally right? We need an additional dimension, complexity, to start the movement, as well as time and consciousness.
Presentation
1) Zeno’s Dichotomy: I explain Zeno’s motivation for presenting his famous paradox. Then I list the different attempts, mathematical and philosophical, to erase it, since its conclusions seem aberrant. But these attempts contradict each other because of the debate on the continuous or discontinuous nature of reality. The paradox remains all its relevance.
2) Emergentism: Getting out of the rut makes us revisit the subject of emergence, cleared by the British emergentists and then buried. Finding consensus, in science, generates political problems that are briefly analyzed.
3) Getting started: I present a renovated emergentist theory of reality that, by solving the mystery of time, manages to explain the paradox of movement.
The conclusion explains why uncertainty about the continuous or discontinuous nature of reality is no longer an obstacle.
1 – Zeno’s Dichotomy
Zeno of Elea, in the 5th century BC, in The Dichotomy, did not design his paradox to mock an Achilles incapable of catching the tortoise. He wanted to demonstrate, following his master Parmenides, that movement is an illusion. « Before reaching the end of the road, one must have reached the middle. Then the middle of what remains. And so on to infinity. Since infinity cannot be completed, movement is impossible. » This is the so-called progressive formulation of the paradox, its regressive formulation being: Before starting the second half of the journey, one must have traveled the first half. Before that, the first quarter. And so on to infinity. With this regression, movement has not started.
Impossible movement? And yet Achilles easily catches up with the tortoise. Zeno’s reasoning therefore seems an aberration. Where is the error? In fact, Zeno did not assert that movement is non-existent but that it is an illusion. Fundamentally, nothing has moved. Which brings him closer to a theory developed twenty-four centuries later by Einstein and still solid: the block universe. In Einsteinian space-time, nothing moves… because time itself is an immobile dimension, and we can no longer say where the impression that something is moving in it comes from. The movement still exists in the form of equations but is no longer animated. Its agitation is illusory.
Too Long an Execution
In the meantime, over the course of these twenty-four centuries, it has been necessary to denounce the stupidity of Zeno, whose fallacious reasoning contradicts the simplest of our daily observations. Why would we bother going to work if we had no chance of reaching it? Zeno with his aporias threatened to transform us into cicadas waiting peacefully in their homes for someone to come and feed them… Firing squad! Shoulder weapon! Shoot this dangerous protester!
And there were many shots at the paradox. Too many? By multiplying, they have become contradictory, which leaves one perplexed. Did they all shoot at the same thing? And if they each shot at a non-vital part, is the paradox shot not still alive?
The atomicist response
According to atomism, infinite division is impossible. We end up stumbling upon the smallest unit of reality. Democritus called it the atom. It was discovered, then dismembered again. Today, quantum theory has brought back into the spotlight the ultimate discontinuity of matter with Planck’s unit. If a distance is not indefinitely divisible, it can therefore be traveled. Zeno is wrong.
What do we discover here? The dispute sparked by Zeno is also that between the continuous and the discontinuous. The Einsteinian block universe, which supports the impossibility of movement desired by Zeno, is based on the continuous. The atomist denigration is based on the discontinuous. In the absence of certainty about the ultimate nature of reality, it is no longer so easy to decide.
The potentialist response
Aristotle wanted to preserve the two most natural ideas, that of the continuous world and the reality of movement. His response to the paradox is called potentialist. He recognizes the possibility of an intrinsically infinite division of length but also of its finite limit as a “whole”. Infinity is only potential. If a division operates it necessarily comes from the outside. Zeno can thus divide length into as many steps as he wishes but this act is necessarily finite, as well as the number of steps. The distance can be covered.
What do we have here again? It is the dispute between the upward look, that of the points inscribed on the distance to be covered, and the downward look, that of the runner who is going to swallow them up. The points on the line say: “We are infinite in number! It is impossible for you to pass by each of us, you will never have finished”. The runner answers them: “With each of my steps I leave an infinity of you behind me, a single step should be enough”.
Touching the fundamental
Thus Zeno’s paradox refers us to two fundamental debates on the nature of reality. The first, the quarrel between continuous and discontinuous, is identified but not resolved. The second, the quarrel between the upward and downward looks, is not rigorously codified (except on this blog where it is the common thread) and is not resolveable, each of the looks having an independent existence, coming from the start and the result.
A consequence of this absence of codification of the double look, the paradox is considered either absurd or real, depending on whether only one of the two looks is used, or both jointly. Note that Aristotle correctly begins his investigation with the double look. He approaches infinity by its intrinsic face (the infinity of points on the line) and by its extrinsic face (the runner who gathers infinity in a single step to cover). Unfortunately, he then decides to favor the extrinsic face, seen by the downward look, so that the runner does not experience any paradox: he can therefore finish covering his line, which is nevertheless infinitely dotted. But Aristotle, by excluding the upward look, explained nothing of the paradox itself.
Fashionable views
Later, the appearance of infinitesimal calculus will on the contrary favor the upward look. Mathematical reality becomes infinitely divisible. But above all, the success of mathematics in the precise description of physical reality leads to a switch in minds: mathematics is no longer considered as a language but as the fundamental nature of reality. Today, many physicists are convinced that the universe is made of information, that it is a pure mathematical structure. It is possible. But this idea hides a little cheating: it makes people believe that all mathematics is real by the simple fact of being intrinsically coherent. Excessive enthusiasm. Different postulates found different mathematics and some are not found in reality. This is how mathematics remains a language, some parts inscribed in the fundamental structure of reality, while others remain virtual, that is to say, exist only in the form of neural configurations in the brains of mathematicians.
This little cheating is important. It is often possible, by choosing the appropriate postulates, to find a mathematical formalism that supports an interpretation that is nevertheless disconnected from reality. Mathematical solutions have thus been proposed to Zeno’s paradox, but they do not provide the slightest answer to the fundamental debates that we have just mentioned. They are only solutions within their own arbitrary formalism. Pure mathematical rhetoric, we could say with acidity.
The most common solution dates back to the 19th century and uses geometric statements: The sum of a geometric sequence with general term 1/2n, n ranging from 1 to infinity, tends towards 1 (finite) and not infinity. There is no longer a paradox… within geometric language. This language uses logic but is not logic in itself. It develops it from its own postulates. It is in fact a particular mathematical formulation of the general paradox proposed by Zeno, and not its resolution.
The paradox of immobility?
The paradox continues to generate so much perplexity that some contemporary authors have reversed it into the “paradox of immobility”, that is to say that it would be immobility and not movement that would be illusory. It is in the air of time, everything that seems immobile being in reality formed of constantly moving particles. The advantage is that the paradox collapses by itself. It is the idea that the runner is immobile that is absurd. And that the goal to be reached is immobile does not make sense, it also moves!
Have you noticed where this solution comes from? This time it comes from a pure downward look. It is the observer of the movement who reasons in this way. By refining the resolution of his vision down to the particles, everything is moving. Extraordinary! The conflict of the double look becomes a caricature! The downward look claims that everything moves, while the upward look starting from a block universe affirms on the contrary that nothing moves!
In summary, we have various solutions to Zeno’s paradox, but each belongs to a particular look or language, and they contradict each other! Sometimes movement is an illusion, sometimes it is immobility. And by the way, what is an illusion? Are our minds deities observing reality from an “outside” to delude themselves in this way?
Convergence with the problem of time
Let us try to bring our minds back to reality. A major point of gathering is that the mystery of movement converges with that of time. Movement is an immobile point set in motion by time. Zeno’s Dichotomy is both the paradox of movement and the paradox of time. Both illusory? When brought together, they cancel each other out, we see. Indeed, dividing the remaining distance to infinity consists of equally dividing the time needed to travel each step to infinity. These two infinities fit together perfectly to form a distance… completed at the end of the allotted time.
This is the simplest solution on a global scale. But it does not answer the intrinsic problems of time and movement. It simply opposes the problematic notion of infinity to itself, without saying whether infinity/continuity really exist, or whether movement is a reality. We will have to do something less simple.
2 – Emergentism
Here is how to really overcome this paradox. It requires us to make a simple but radical conceptual revision of reality. The theory I am going to propose allows us to initiate movement from an immobile universe and gradually set it in motion when it becomes more complex. This solution is based on ‘Temporium’, the book in which I explain the start of time from a block universe, where time does not “pass”, to subjective time, which passes at a speed specific to each brain.
What radical revision are we talking about? It was already attempted at the end of the 19th century by the so-called British emergentist school. It was about giving volume to complexity by making it a dimension in its own right of reality. Alexander and then Broad defined independent “levels of existence” stacked by evolution according to a pyramid of complexity. They thus oppose the reductionists who “flatten” the Universe and reduce it to only fundamental micromechanisms, complexity being a simple emanation of the equations that govern them.
Rise and fall
British emergentism wanted to be scientific. It tried to introduce the concept of “laws of composition” governing the different levels of existence independently. But the success of reductionism in establishing bridges between levels buried these claims. Quantum mechanics explained chemical bonds; molecular biology took charge of clarifying the physiology of living things. Rather than looking for laws specific to the levels, we looked more closely at what happens below. Science has become “flat-earth”, losing in the process any chance of explaining the phenomena specific to each level of existence. All this has gone into the great dustbin of Illusions. The phenomena simply no longer interest scientists, who find no contribution to their models in them.
Complexity is thus seen sometimes as indispensable to account for the diversity and sophistication of phenomena, sometimes as a nuisance for those who hope to reduce the Universe to a fundamental equation. A return of the quarrel between animists and supporters of the one God? The British emergentists were judged as rearguard theologians at a time when Quantum Mechanics revolutionized knowledge and became the major deity of physicists. It even serves as the basis for a quantum theory of consciousness launched by renowned physicists but which falls more into the mystical than scientific field. Quantum is today as imbued with the sacred as the old white-bearded Creator celebrated by Christianity.
From zealotry to the grotesque
Is it one religion chasing another? Quantum physicists would be offended to see their Standard Model presented in this way. They have very expensive instruments to validate it through experiment. They put God into equations, undeniable progress compared to the previous one, human too human… Nevertheless the flaw of this model is its lack of universality and intrinsic explanation, not to mention the multiple flaws it still contains. There is no trace of time, consciousness, or resolution of the major problems raised by Zeno’s paradox. In short, it is a photograph of fundamental reality and nothing else. Many details are missing and above all… who took the photo of the outside of this perfectly flat universe?? What experiences the illusions that emanate from it?
The Quantum Standard Model, due to its remarkable successes, triggers excessive zealotry in those who practice it, blinding them to elementary philosophical problems, for example the unexplained presence of the minds that conceived it. God has progenitors, which is also different from the previous version. My irony has no other purpose than to awaken you to the possibility of upsetting this beautiful order, and it is difficult to do so within the community of physicists. Only the big names venture there but quickly risk solitude. Lee Smolin, for example, proposed to reverse Einsteinian theory by making space relative and no longer time. A conceptual upheaval too severe to gain support, despite the concrete answers provided to the mystery of the passage of time.
My proposal is even more grotesque and I do not have the celebrity of a Smolin, so you will be few to read it and even fewer to understand it. It is about following in the footsteps of the emergentists to validate complexity as a fundamental dimension of reality, this time avoiding the pitfalls that sent their work to the oblivion of the History of Science.
3 – Getting started
How to set in motion a motionless universe background? Because this is the question raised in the end by Zeno. With his paradox he wanted to convince us that movement is an illusion, but since we want to believe in this illusion we must set in motion a universe that is perhaps motionless at bottom. The real way to extinguish the paradox is not to decide between mobility and immobility, nor between discontinuous and continuous, but to show how one can give birth to the other.
Setting in motion a motionless background? We need to make it scroll through time. Einstein made our task much more complicated by immobilizing the temporal dimension alongside the spatial ones. Now with our impression of movement, we are in fact ejected from the Einsteinian framework. Things should only move for an eye outside of reality, which would see a timeline scrolling in front of it, animating events as in a flipbook. Being fairly certain that our impression exists, since we are in direct contact with it, here we are outside the Einsteinian framework. This framework is therefore not all of reality. And it is a question of understanding how we are connected to it.
The complex mille-feuille
The solution is hidden in complexity. By neglecting the work of emergentists, we have lost an essential dimension of reality, and have never completely found it again. An emergent level is not only endowed with different properties but with a specific time, slower than the previous one. Indeed, it is a stable Whole encompassing the parts. These can go through a multitude of different configurations without the Whole changing in nature. The temporal beats of the parts and of the Whole, measuring the rhythm of their transformations, are clearly offset.
Thus complex reality appears as a mille-feuille of layers endowed with their own time. Nevertheless, this is not enough to make these times scroll by. Our consciousness should be stuck in its level of reality, without any impression of passage. To start time and make it move in a single direction, entropy must be added. The explanation is a bit difficult and I refer you to ‘Temporium’ to understand the details. The emergentists lacked a way to concretize the globality of a system. Ironically, it is quantum mechanics, which contributed to burying British emergentism, which provides this means.
The arrow advances because it is entropic
I defined the global stage of a system, its “Whole”, as the stable configuration of its multiple probable states. The discontinuous melted into the continuous. This stable configuration, however, does not appear instantly when the system is formed. There is an entropic delay between the two, with a precise direction. The equations are not reversible at the beginning. A temporal arrow appears. The Whole integrates in a way a childbirth, an adult life in equilibrium and a death when the details of its constitution put an end to the equilibrium. It is not yet this arrow that makes time scroll since the Whole has not extracted itself from it to experience it. The passage comes from the sliding of the layers of reality within the same complex entity. They have different temporal beats and variable entropic arrows while constituting a single entity for the Whole located at the top of the complex edifice.
In the brain this summit is the conscious workspace. Consciousness is the phenomenon experienced in this space. It comes from the sliding of the innumerable layers of complexity formed by neural graphs over their layers of biological complexity over their molecular and particulate complexity, up to this hypothetical foundation of reality that we do not yet perceive.
Here is time set in motion. Movement then becomes a simple formality. Each thing in motion is an animated Whole in its own time. Each of our minds evolves in its own time, sees movements whose speed is evaluated only by it. Fortunately, the proper times of our consciousnesses are close to each other, which facilitates communication. But we need common instruments, clocks, to agree on the speed of objects, because our impressions are always slightly dissimilar.
Moving floors
Things move while their most fundamental constituents probably do not move, as Zeno and the most recent science assert. How is this possible? Because the immobile constituents, through their relationships, create a mobile Whole, which moves on their “surface”, this surface being an emerging level in complexity, surmounted in turn by the next level. Each level moves according to its own time, and the sliding of times as well as movements between the layers of complexity creates the impression of something “moving” and not just affected by a displacement. It is in this slight shift constituted by an emerging level that this outline of exteriorization to reality appears, which allows us to “watch” what is underlying in complexity move.
The impression of movement is not reserved for consciousness. All layers of reality have this impression on their scale. The conscious impression is the surimposition of all the layers of our physical and neural constitution, which is what makes it rich. It does not emerge from nowhere at this location. No need to resort to a mystical explanation now. Time, movement, consciousness, and all the sparkling phenomena attached to them, are born from a background of an immobile universe.
Conclusion
Solving Zeno’s paradox is not demonstrating its absurdity, as has been done up to now. Because saying it is absurd requires making arbitrary postulates about reality, its continuity, and time. If we must guard against this, it becomes impossible to simply declare the paradox aberrant. On the contrary, a fair solution must explain why it seems aberrant, without being so. The one I have proposed allows it. The runner finishes the distance, despite the paradox that prevents him from doing so, but he does not do so on the same level of reality.
In the level where the paradox descends, the infinitely divisible distance, the runner no longer exists. He himself has become an infinity of points and, at the supposed root of reality, stops permanently. Only the relationships between the points change to signal his movement. But at this level nothing really moves. We are on the backdrop of reality. Movement does not exist there. The mind that nevertheless experiences this movement then says to itself that it is an illusion, having decomposed it so much that it has annihilated it. And this is indeed the case. But it is the decomposition that is responsible and not the movement that does not exist.
For these immobile relationships are organized into levels of complexity, which sets in motion both time and movement in a privileged direction, from the past to the future. The speeds differ in each level, and the integration (surimposition) of these in the final experience provides the sensation of the passage of time and of the runner who moves. These impressions are not illusory but very real, as much as the fundamental immobile level, which can also be an emanation of something that is inaccessible to us.
So, is this reality continuous or discontinuous?
The runner reaches the intended goal in the allotted time. Does this demonstrate that the fundamental level of reality from which the interactions at the origin of this movement start is not infinitely divisible? Indeed, if the interactions had to travel an infinite number of points to extend to the goal, they would never reach it and the runner would also fail, in his higher level of reality. But this reasoning has a flaw: it assumes that the interactions need time to extend. However, time does not yet exist at their level. It is therefore impossible to conclude on the fundamental continuity or discontinuity of reality.
Note that time has already shrunk incredibly at the quantum level to allow for the equally incredible sum of interactions represented by the movement of the runner to his goal. And in truth, the complex solution that I have presented allows us to resolve the continuity/discontinuity problem differently: the levels of reality are both discontinuous and continuous, discontinuous by their constitution, continuous as globalities. This is how models betting on continuity or discontinuity can all be effective, depending on the face of things they address. But they will never show the entirety of these things. To see reality in its entirety, a double perspective is needed, carrying a double model. A paradoxical perspective. Like that of Zeno of Elea.
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