Zeno's Arrow: Langan, Bergson and mathematical physics

Christopher Langan - see Christopher M. Langan Interview

While reading Langan's "Quantum Metamechanics (QMM)" I stumbled upon these two paragraphs that have caused me to wonder (italics mine):

 Unfortunately,  there  are  problems  with  mathematical  quantization,  and  they  carry  over  into  physics.  This  is  easy  to  see  in  the  case  of  a  classical  manifold,  basically  a  space  consisting  of  zero-dimensional  (0D)  points  (real  numbers, elements of the real continuum R n ) and equipped with a metric that  is “locally Euclidean”, permitting a reasonable in-frame notion of distance and  locality. Immediately  we  detect  a  paradox: “of zero extent in a given space”  means “nonexistent in that space” – existence in a space means taking up space in it - and we cannot assert the existence of a space consisting of nonexistent points  that  take  up  no  space  at  all.  Even  if  we  could,  there  would  be  yet  another  problem associated with continuity, the adjacency paradox. An infinitesimal line  element or increment of linear motion must relocate a point-object from one  point to an adjacent point. But where points are zero-dimensional as continuity  demands,  adjacency  or  “being  in  mutual  contact”  effectively  identifies  them.  Adjacent points simply merge, and no relocation can occur. Linear motion is  out of the question.  

And then he goes on:

It follows that continuous motion requires finite termination or bounding of  the  interval  in  order  to  scale  and  sum  infinitesimal  increments,  produce  a  definite integral, and assign a length to the interval. But this still leaves nonzero  (albeit  sub-finite)  separations  between  each  pair  of  successive  points  along  a  path or physical trajectory, and an object must “jump out of” the manifold or  space in order to get from one point to the next.  Motion then becomes a series  of infinitesimal “quantum jumps” through hyperspace. This, of course, is not  what  mathematicians  and  physicists  typically  have  in  mind  when  they  talk  about  “the  continuous  (differentiable,  smooth)  motion  of  objects  or  waves  through  the  continuum”.  (We  ignore  for  now  the  various  workarounds  that  have  been  proposed  for  these  problems,  at  least  one  of  which  –  usually  the  Cauchy-Weierstrass  epsilon-delta  formalism  based  on  infinite  converging  sequences  –  is  generally  invoked  in  introductory  calculus  courses  in  order  to  deflect the “problem / paradox of infinitesimals”, which is never satisfactorily  resolved  with  respect  to  the  existence  or  nonexistence  of  0D  points  and  infinitesimal intervals between them.) 

I am extracting three main points from the above:

1) The question of how 0D points make a 1D line?

2) The problem of motion (related to Zeno's paradox)

3) The problem of "existence"

Source of illustration image: 

https://fr.wikipedia.org/wiki/Z%C3%A9non_d%27%C3%89l%C3%A9e#/media/Fichier:Zeno_of_Elea_Tibaldi_or_Carducci_Escorial.jpg

But essentially, if I am not mistaken, Christopher Langan seems to point out problems that are quite similar to those that have made Zeno of Elea unhappy, and which are still debated today (though mostly dismissed).

Before addressing these three point the way I see them as a mathematical physicist, let me quote the view of a renowned philosopher Henri Bergson on the subject. But first let me quote from Wikipedia this piece:

(...) In 1922, Bergson's book Durée et simultanéité, a propos de la theorie d'Einstein (Duration and Simultaneity: Bergson and the Einsteinian Universe) was published.^[33] Earlier that year, Albert Einstein had come to the French Society of Philosophy and briefly replied to a short speech made by Bergson.^[34] It has been alleged that Bergson's knowledge of physics was insufficient and that the book did not follow up contemporary developments on physics. On the contrary, in "Einstein and the Crisis of Reason", a leading French philosopher, Maurice Merleau-Ponty, accused Einstein of failing to grasp Bergson's argument. This argument, Merleau-Ponty says, which concerns not the physics of special relativity but its philosophical foundations, addresses paradoxes caused by popular interpretations and misconceptions about the theory, including Einstein's own.

We see from the above that Bergson was probably deeply interested in the foundations of physics. In "Creative Evolution" (Routledge 2023) Bergson considers the problem of motion. He first notices that:

(...)  In order to  move forward with moving reality, we would have to place ourselves  back within that reality. Place yourself within that which is changing  and you will immediately grasp both change itself and the successive  states in which it could, at any moment, be immobilized. But you will  never  reconstitute  movement  from  these  successive  states,  seen  from  the outside as real immobile parts (and no longer as virtual immobile  parts). Depending on the situation, you might call them “qualities,”  “forms,” “positions,” or “intentions.”  You can multiply them as much  as you like, and thereby continuously bring two consecutive states closer  together. But when it comes to the intermediary movement between  the two states, you will always experience the same disappointment as  the child who tries to crush some smoke by pressing together his two  open hands.  The movement will slip away into the interval because  every  attempt  to  reconstitute  change  from  states  involves  the  absurd  proposition that movement is made up of immobile parts.

And then continues: 

Philosophy  noticed  this  absurdity  the  very  moment  it  opened  its  eyes. 86  Zeno of Elea’s arguments, though formulated with a very different  intention, in fact say nothing else. Shall we consider the flying arrow? According to Zeno, the arrow is  immobile at each instant, since it would only have the time to move—  that is, to occupy at least two successive positions— if it is granted at  least two instants. At any given moment, the arrow is thus at rest at a  given point. Immobile at each point along its path, the arrow is thus  immobile during the entire time that it moves. Yes, provided we assume the arrow can ever be at one point of its  path. Yes,  provided  the  arrow,  which  belongs  to  moving  reality  [le  mouvant], ever coincided with a position, which belongs to immobile  reality [l’immobilité]. But the arrow never is at any point of its path. The  most we can say is that the arrow could be at a certain point, in the sense  that it passes through that point and would be free to stop there. True,  if it had stopped there, it would have remained there, and at that point  we would no longer be dealing with a movement. The truth is that if  the arrow leaves point A in order to land at point B, then the movement  AB is, as a movement, just as simple and just as indecomposable as is  the tension in the bow that launches it. Just as the shrapnel, exploding  prior to reaching the ground, extends an indivisible danger across the zone of the explosion, so too the arrow that goes from A to B deploys,  in a single stroke, its indivisible mobility, although this time across a  certain extension of durée. Imagine an elastic band that you stretch from  A to B; can you divide up its extension? The flight of the arrow is this  extension itself: it is just as simple and, like it, indivisible. It is a single  and unique leap. You focus on a point C in the interval that is traversed,  and you say that at a certain moment the arrow was at C. But if it had  been there, this means it would have stopped there, and now you would  no longer have the flight from A to B, but rather two flights, one from  A to C,  the other from C to B, with an interval of rest in between. By definition, a single movement is an entire movement between two stopping  points: if there are intermediary stopping points, then it is no longer a  single movement. In essence, the illusion comes from the fact that the  movement, once completed, has deposited along its path an immobile trajectory along which we can count as many immobile parts as we like. From this we conclude that at each instant the movement, while being  accomplished, deposited beneath itself a position with which it coincided.  But we thereby fail to see that the trajectory is created in a single stroke,  even though it takes a certain amount of time, and that even though  we can divide at will the trajectory once it has been created, we could  not divide its creation, which is not a thing but an act in progress. To  assume that the moving object is at a given point of its path amounts  to making a cut with scissors at precisely that point, thereby cutting the  path in two and substituting two trajectories for the single trajectory  that we were first considering. It is to distinguish two successive acts  where, by definition, there is only one. In short, it is to transport into the  very flight of the arrow everything that might be said about the interval  that the arrow flies through; it is to accept, a priori, the absurdity that  movement coincides with the immobile.

Let me start with 1) The question of how 0D points make a 1D line?

The use of 0D suggests that the author talks about "dimensions". Usually we define the dimension within the category of topological spaces ( see for instance Lebesgue covering dimension). Points of any topological space have automatically dimension zero, whatever the dimension of the whole space is. In metric spaces we also have the concept of a fractal dimension (for instance Hausdorff dimension). There points have also dimension 0. So I do not see any problem here. But the author also mentions R^n. For instance, for n=1, the space of all real numbers. Real numbers are well defined using the standard axioms of set theories.(see e.g. here). With the usual topology the space of real numbers is 1-dimensional and its elements, by definition, are points - the real numbers. Perhaps I am missing something, but I do not understand the problem. Perhaps the problem is in undefined concept of "making". Do real numbers "make" the set of all real numbers? In a well defined sense they do "make": elements of any set "make" this set. But, perhaps, Langan has a different definition of "making"? That is not clear to me.

Let us move to 2) The problem of motion (related to Zeno's paradox)

Here we need to distinguish between a "mathematical point" and a "material point". Mathematical point is just a point, an element of some set, and the concept of "motion" does not apply, unless it is somehow defined in a specific framework. Material point is, on the other hand, a concept discussed within classical mechanics. Material points do move, and their motion is usually described using solutions of differential equations, once the forces are known. At each given instant of time a material point is described by its "state", and position of this point is just a one property of its state. The other property may be, for instance, its momentum (in Hamiltonian dynamics) or velocity (in Lagrangian dynamics). Material point may also have attributes that help us to determine external forces. For instance it may have mass m and charge e (though usually the ratio e/m enters the equation. If the only external force is gravity, and if we accept the Equivalence Principle  (as in Einstein's General Relativity), the mass is irrelevant, all material points move on geodesics, and the only data needed to determine the state are position and velocity at a given time. 

The concept of "state" enters quantum mechanics. In the standard quantum mechanics the sate is characterized by the "wave function", and in de Broglie-Bohm pilot-wave mechanics the state is a pair (the quantum wave function and the position of the classical particle).

The concept of state is a tricky one as it usually involves parameters that are not "directly observable" (what is "directly observable" IS A GOOD QUESTION). For instance an instant photo of a bullet may tell us all we need to know about its position at the instant of time, but its velocity may be only guessed or totally unknown (depending on the exposer time). This should not present any philosophical difficulties - I think (cf. Plato cave allegory).  

Once we make the distinction between mathematical (or philosophical) point and a material point of mechanics - the paradox, as it seems to me, disappear completely.

Finally 3) The problem of "existence"

In mathematics we prove existence of solutions of certain equations (sometimes even uniqueness). There seems to be no problems with that. Existence of Borel non-measurable sets is more tricky - the proof of existence depends on whether we accept the axiom of choice or not. Mostly we do. But the "existence" property discussed by Langan seems to belong to philosophy, not just mathematics. There it depends on the accepted philosophical system. "Healthy" systems usually agree that tables and chairs exist. When it comes to momenta, masses or wave functions - here various options are being considered. There are, as it seems, different levels of existence. Philosophers, and even physicists, argue. John Archibald Wheeler, for instance, often stressed that "no phenomenon is a phenomenon unless it is an observed phenomenon". Though, unfortunately, he did not explain clearly what the term "observed" stands for. So, as the result the debate has started "Is the Moon There When Nobody Looks?"

As for me, if I would be forced to subscribe to some particular school of philosophy concerning the problem of existence, I would subscribe to this one

From experience.  - The  irrationality  of a  thing is  no argument  against its existence,  rather a condition of it.  

Nietzsche: Human, All Too Human: A Book for Free Spirits (ed. Cambridge University Press, 1996) - ISBN: 9780521567046, p. 182

P.S. Found by chance this interview with Jordan Peterson. He is a fantastic example of how to talk about difficult subjects without using jargon. An excellent example for me to take. Will read his 12 Rules of Life - to learn more.