What am I reading?

This joblessness, for catching up on my reading, I have been having a more detailed look at Bohm’s theory.

In the past, I had written a longish post on it [^]. However, I thought I could perhaps have a re-look at this theory, and try to write something more concisely. Here are my current thoughts (though not very concisely).

BTW, in my last post, though it was a bit too free-wheeling and longish, I had not noted anything about Bohm’s personal life or character. So let me note down something about it, and thereby get it out of the way, before we come to his physics.

I haven’t read any biographical book on Bohm (nor am I interested in reading one), but from what I gather by browsing brief articles on the ‘net, I think that you can’t hold that McCarthy affair against him, even if as a young man, he sincerely believed in Marxism. [Yes, I myself continue to believe in Capitalism, but read on anyway.] I also don’t hold his association with Jiddu Krishnamurti against him. [Yes, JK was a real funny British creation, even if based on an Indian version of mysticism.] If I must comment on Bohm’s personal life, the first thing I would say, i.e., apart from noting his bewildering naiveté, is that he obviously deserved a PhD advisor/boss better than Oppenheimer, a country better than the USA (or the way it treated him anyway), and an intellectual Guru better than JK. He turned (partly) lucky on only one of the three counts. Unfortunate.

He also deserved an audience better than the 20th century physicists. And, his physics, I now believe, deserves a bit better estimate than what I think I accorded it the last time.

Bohm’s theory, that way, is not much different from the standard mainstream QM. His theory, I think, essentially is:

(a) deterministic
(b) non-local
(c) with an ontological separation of the quantum into the wave and the particle as two distinct kinds of entities,
(d) and, truly remarkably, having particles inhabiting only a 3D space.

It’s obvious that modern physicists would hate him for (a), and they do.

It would be expected that they should love him for (b), but they don’t. Their passion on the count of (a) has been so strong that they can’t even notice (b).

They wouldn’t a care a hoot about (c) simply because it’s “all philosophical” to them. On this count, they do deliver completely as expected.

And, they to this day haven’t allowed themselves to know that they also hate him because of (d). Since they don’t know it, they just silently chew their lips as they hurriedly skip over this feature of Bohm’s theory.

In contrast, my biggest problems with Bohm’s theory have been (b) and (c).

I was on my guard regarding (a) on two counts: (i) so many attempts at giving a deterministic theory have been so negligent of so many QM features or so much observational data, or have been so outright foolish, that even I couldn’t keep too much enthusiasm for a deterministic theory—one tends to think that in view of the success of probability in classical statistical mechanics, the probability in QM must be a simple interpretation issue. (ii) In philosophization, the determinism-oriented people slip so easily into a denial of free will.

Still, I now realize that we should applaud Bohm for (a), i.e., determinism. We could even be thankful to him for upholding it despite a bitter opposition.

And, if you ask me, we should be even more grateful to him for (d), i.e., for keeping his particles only in a 3D space. (I have to finish my series of posts on space, and when I return to it, I will make it a point to address this issue.)

Now, let’s look at the points (b) and (c), i.e., the non-locality and the ontological separation, in more detail.

Regarding the non-locality, it’s only recently—as recently as this month—that I seem to have finally come to agree that I don’t have a good argument to necessarily deny instantaneous action at a distance (IAD) in every physical theory. (When David Harriman had noted in the mid-naughties on some forum that IAD was not an issue of philosophy, that it is not a task of philosophers to ponder whether one end of the see-saw goes up literally at the same exact time that the other end is pushed down, I had thought that it should be possible to figure this issue out on the philosophic grounds alone, more particularly, on the epistemological grounds. Now I no longer seem to think so.)

But that does not mean that I have jumped over on to the IAD side in general? No! Not at all.

All that I have realized here is that you can’t deny IAD on the basis of the principle of identity, or on epistemological grounds. In other words, the idea is not arbitrary, i.e., it is not devoid of any fundamental cognitive merit. No matter how ridiculous it may sound, a proper theory of physics could still, perhaps, have IAD built into it. Despite Einstein’s relativity.

In my own theorization, of course, I would continue to have locality. My insistence on having locality in a physical theory (or the reason to deny IAD) never was based on the relativistic objection. It was based on a simple consideration: I always thought that when I tossed a ball, or a typed a key, I was not directly and instantaneously affecting the path of a pebble rolling somewhere at the bottom of the Grand Canyon. That, if A, B, and C are three objects situated in space next to each other in the given sequence, then a disturbance from A must first travel to B before it gets to C. This has been just a “native” conviction for me, that’s all. In XI standard, while reading Newtonian mechanics, my mind couldn’t stay focused on calculating acceleration of a ball once it is hit by a bat. The reason wasn’t a lack of a mathematical reasoning ability. The reason was, knowing that a ball was not a particle, I would wonder how the hit must be propagating inside the finite ball, and what it would take to understand this issue really well (the stress waves, I learnt later, but couldn’t explain the issue well right back in XI standard to friends as to why the then text-book explanation based on impulse and all falls short—I only insisted that it does). Wanting to explain the stationary via the transient—or at least wanting to relate the two—has been native to me, to my natural thought processes. (That’s how the sub-title of this blog.) … So, I would continue building my theorization via the local and propagation-al processes.

For the same reason, I also have had this resistance to accept the viability of IAD in a theory of physics. But, finally, I seem to have built some argument to show that IAD could be a reasonable view to take.

IAD would be a relatively easier to accept in a fully deterministic i.e. materialistic world, one that is devoid of any willed (or even just animate) physical action. In the literally clockwork universe, IAD would be easier to believe. How?

Before we come to that point, let us pause to consider another characteristic of Bohm’s theory—the place where my quantum approach (or call it attempt at to build one) differs from his. Recall my past posts on the nature of space, on what I call the foreground objects (say the physical things you see such as apples, trees, buildings, planets, etc.) and the background object (or the aether).

The point concerns what dynamical attributes are carried by which—the usual material (or massive) objects and the aether/field/”empty space”. Since a physical theory must have both of them, I now realize, it should be possible to think of a whole spectrum of theories based on how they partition these two aspects.

In Newton’s particle and finite-body mechanics, it’s the material objects that carry all the dynamically important attributes; the empty “absolute” space simply sits idle. In contrast, in Maxwell’s classical electromagnetism, both the material objects and the fields carry the necessary physical (dynamical) attributes, and an interaction between the two is necessary for a complete physical description. In Bohm’s mechanics, this trend reaches its logical extreme: it’s the Bohmian field that is the true dynamical causal agent; the particles are completely passive.

There, of course, is a position that is even more “extreme” continuing in the same direction, but it falls outside of the spectrum because it is so thoroughly illogical: the mainstream QM. Here, like in Bohm’s mechanics, it’s the “other thing” (say Schrodinger’s wavefunction) that does everything dynamical, but the difference is this: you can’t even say that particles are completely passive because, the mainstream QM insists, the particles can’t even be said to exist unless when observed, and the wavefunction can’t be seen as a 3D phenomenon in the general case of many particles. So, logically speaking, it’s only Bohm’s theory that represents the extreme end of the possible spectrum.

So, there. Newton–Maxwell–Bohm. All the other proper theories fall in between. For instance, molecular dynamics falls in between Newton’s and Maxwell’s, and Higg’s theory, I suppose, could be taken to lie between Maxwell’s and Bohm’s. Bohm’s theory indeed is at the logical extreme (leaving aside the mainstream QM that randomly falls off the table).

Now, if the “empty” thing/field is the real physical agent, IAD becomes more easily believable. Why? Because, quantitatively, there exist only one causal agent, all by itself. When this entity acts, it must act as a whole. And, now, the key point: This action of the whole doesn’t have to be divisible across the parts. The action indeed may be quantitatively different for the different parts (e.g. the force being generated in one part may be more than that being generated in some other part), but inasmuch as it’s the only  object in the entire universe, whatever it does is only a single action. Such an action may be taken as carrying a kind of IAD.

Strictly speaking, it’s not exactly IAD in the usual sense of the term. It’s not some action that one object exerts over some other object lying at some distance and somehow instantaneously. It’s an instantaneous action at every point of the same object. It’s a bit like morphing an image: say, a circle expanding to a bigger circle, or a ring carrying some waves transverse to its central fiber. Here, all points are taken to move simultaneously, and so, you could arguably describe it by saying that the motion of one point has an instantaneous effect at another point.

That’s the best possible argument I could come up with, in support of the IAD.

I still have a feel that it all is a nonsense, but let’s be clear about distinguishing a mere feel from a reasoned argument.

Now, if you can ascribe all the essential dynamics to that single object i.e. the Bohmian field, then the possibility of IAD within that field is, how to put it, without a soundly opposing argument.

Then, once you sprinkle some particles in it, the rest of the Bohmian mechanics follows.

But, do note very carefully what is being conceded here. All that I have so far conceded is that the presence of a sound argument necessitating a denial of this kind of a theoretical IAD—the one occurring in a “field” where the field is the exclusive actor in the entire universe. It’s only the universe that ever acts; the parts have no such freedom in such a world, but they may be abstractly seen to have instantaneous influences on each other. All that I am saying is that I have no argument against this kind of an IAD.

But I thereby do not concede that this kind of a theorization (the one involving IAD) is necessary to explain the quantum phenomena. IMO, a good, logical QM theory can also be local in nature—nay, it must be.

Now, even if you grant IAD to the Bohmian field, there is another issue that Bohmian mechanics runs into, viz., the ontological separation between the particles and the field.

If the field is the exclusive actor—as required by the IAD—then it leaves no ontological place in the theory for any particles at all. If so, why are they there?

(Or, if you like: if the American society is a single object that can do all the productive work necessary for itself, then why sprinkle immigrants into it?)

Thus, if the particles are ontological (i.e. if they at all exist in this world as objects), then the field cannot be the exclusive actor in the universe, and so, IAD is ruled out. On the other hand, if the IAD is to be retained in the theory, then to make it the exclusive actor, the particles have to be taken out of that theory; they cannot be more that mere visualization aids.

In the first case, the particles are like the tracer particles in an actual flow of a real fluid—they do affect the flow locally, they are not dynamically passive entities, and so IAD for them is as spooky as lifting your arm and thereby causing a dust particle in the next room or a mountain on the Mars instantaneously move up, too.

In the second case, the particles are like the arrows drawn on a photograph of a real flow—they cannot affect the flow but neither are they actually moving in the actual flow in the reality out there.

You see, IAD is a tough thing to accommodate in a physical theory—whether in the diffusion of carbon in steel, or mainstream QM, or Bohmian QM.

If the Bohmian mechanics is that bad, then why am I reading it? especially since I do seem to know better? Good question.

Answer: Because, even if it is that bad, it is no more worse than that. It certainly is not as bad as its critics make it out to be. In fact, this theory actually becomes the better exactly for the reasons its mainstream critics hate it: determinism and 3D space. And the introduction of these two features make it a far more easily understandable wrong theory. As compared to others. … You see, a theory based on particles moving in only a 3D space does not have to bother about bringing results from 4, 5, 10 or 1000 dimensions back to a space of three dimensions—the symmetries or otherwise of the collapse of dimensions. And, precisely because it’s deterministic with definite trajectories, with particles always moving forward in time, it is easy to grasp, believe even if only temporarily, visualize even if the variables are only hidden, and, possibly also easier to calculate, at least in many situations. Classical determinism, with the feature of a 3D space reduces the cognitive load enormously.

So, as a bottom line, Bohm’s theory is wrong, but “good”!

It sure does not resolve the QM riddles, e.g., the wave-particle duality, but it does essentialize these riddles very well, even, brilliantly. In any case, it does so better than any of the existing QM interpretations. That’s why, it is a good idea to study it.

For most people, this theory should be a good step to get out of the totally mystical abyss of the mainstream QM, even though it wouldn’t get you completely out of it—it might get you out, perhaps, say, some half-way through. But, yes, the air will be fresher, and you will see a greater expanse of the sky.

Little wonder that Ayn Rand-admiring physicists like Dr. Travis Norsen or Dr. Eric Dennis took as much enthusiastically to it as they did. … If there were no Ayn Rand, and no ancient Indian wisdom, one can still be certain, one would have been an Aristotlean. No comparison of the scale even suggested in any sense, but merely as a matter of stating a fact, if I were not to have my approach—or at least some early success with it—I would have ended up being a Bohmian.

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If you want some good pointers to the literature on the Bohmian mechanics, go to “Bohmian-Mechanics.net” [^]. Make sure to see the frequently asked questions [^], PDF document [^].

Just one more point before closing. There are many other, more detailed or more technical objections against Bohm’s theory. For instance, people object that Bohmian mechanics is inapplicable for photons, for relativistic situations, etc. Some of these don’t hold any water; other objections should go away in future (may be within 10–20 years). I mean, generally, I think, you can expect the scope of Bohmian mechanics to be the same as that of the mainstream QM. If there is a mainstream QM theory to explain a certain phenomenon, then, in principle, it must be possible to extend the existing Bohmian approach (even if not the exact mechanics currently existing) to include those same features, too. That’s what I anticipate. With the QM, unless it is made a local theory, all workable interpretations are in a way equivalent, and selection of any one is just a matter of suitability to attack a given problem, or even of personal choice! Bohm’s theory is more than an interpretation (who else has only a 3D space? determinism? forward time?), even if its development as of today may not be as complete as compared to the other interpretations.

[E&OE]

So, you think physicists got it wrong?

So, you think physicists got it wrong?

If so, why not tell them—or, even if they wouldn’t listen, at least to the world—what precisely it is?

The obvious reference is to the latest FQXi essay contest. The topic they have selected for this edition of the essay contest is:

“Which of Our Basic Physical Assumptions Are Wrong?”

For more details, see here [^]. Note that the last date of submitting your essays is August 31, 2012. There also is a chance that “proper” physicists may end up reading your essay; see the “Who is FQXi” page here [^].

However, in case you didn’t know about FQXi, also note that this is not the first time that they are conducting such an essay contest. Check out the winning essays from the earlier contests: 2008 (on “The Nature of Time”) [^], 2009 (on “What Is Ultimately Possible in Physics?”) [^], and 2010-11 (“Is Reality Digital or Analog?”) [^].

To read the essays already submitted for the current (still open) contest—and the ongoing public discussions on them—follow this URL [^].

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In case you are curious to know my opinion of this essay contest, the reason why I didn’t participate in it so far, whether I would participate at least now, etc.:

Why I didn’t participate thus far. Well, there are different reasons for it, not a single one.

As to the very first contest, I would have liked to participate in it. However, I simply wasn’t even aware of the FQXi itself at that time. In fact, despite my fairly extensive browsing of physics-related sites, I didn’t come to know of the first edition of the essay contest any time before it was already over.

What would I have written for the first contest (“The Nature of Time”)?

I would have written about the nature of space before coming to that of time. If further curious, my position would have been in many ways quite similar to Ron Pisaturo’s [^]. A few asides: (i) I didn’t know about Pisaturo or his position at that time. (ii) Pisaturo, in his articles, addresses more points than I would have. In fact, some of these points existed only faintly on my radar; it was he, who, in addressing them, highlighted their existence/importance to me. (iii) Regarding the nature of space itself (not to mention other issues like the finitude or otherwise of the physical universe), my position was (and remains) independently arrived at. In fact, I found out via an exchange of a few emails with him that there could be some differences in our positions, may be even some essential ones, esp. at the level of details. In particular, it’s concerning whether space is a concept of mathematics, physics, or both. Now, as far as my own position is concerned, I had been jotting down my points in small pocket notebooks (the paper version!) that I usually carry around. I hope to find the time, and more importantly, the right frame of mind, to convert these into an essay. I would certainly like to do that, but only after I am more than halfway through writing my QM book. Which means: after about a year or more. Ok. Enough about the first contest.

For the immediate next contest (“What Is Ultimately Possible in Physics”), while the topic selection here was rather smart, I personally didn’t think that it was well focused enough. And so, in all probability, I wouldn’t have participated in it. However, it didn’t matter one way or the other because I happened to miss the deadline once again. (FQXi contests are not held periodically i.e. regularly.)

As to the last contest (“Is Reality Digital or Analog”), I thought that the topic was, at least at the very first sight, a bit frivolous. However, as it happened, what I thought of it on the second thoughts also didn’t matter anyway because I once again missed the deadline, this time round by just a few days or so.

I think there was some discussion on HBL on a related topic roughly around that time, and I, in fact had a subscription to the HBL at that time. That topic, I think, began with the discussion on whether 0.999999… equals 1.0 or not; then went a bit on to series and infinitesimals in a geometrical context; and then, another related thread made appearance: whether, as we go “all the way down,” does the physical stuff at that level have sharp boundaries or not. BTW, this is a far, far better way of formulating what the FQXi contest topic had merely hinted at. … My answers, without providing full justifications here: the stuff “all the way down” cannot have (infinitely) sharp boundaries, because infinity does not exist in the physical reality (HB had this same position); 0.9… does equal to 1.0, but only in the limiting sense—the former does not “go to” the latter. Here, surprisingly, HB differed from me, in the sense, he didn’t at least immediately agree with me; he kept quiet—perhaps was thinking about it. (In case you missed the reason why I might have found it surprising: the infinitesimal is nothing but the infinitely small. Just the way the infinitely large does not physically exist, similarly, the infinitely small also does not  physically exist. Both are mathematical concepts—concepts of methods.)

… Anyway, coming back to the FQXi contest, notice the difference that the FQXi topic had from that discussed at the HBL: both digital and analog are, primarily, mathematical concepts, not physical. The fact that they can be successfully applied to physical reality does not, by itself, make them physical. That’s the reason why I said that the terms in which the issue got discussed at HBL were better—the formulation there captured the essential issue more directly, in fact, quite explicitly. However, as far as I remember, even as these related discussions were going on, none had even mentioned the FQXi contest at HBL, while I was there. So, I missed that edition too.

So, this is the first time that I have run into a FQXi contest while there still is some time left for it.

Would I participate now?

As of today, frankly, I don’t know. … As you can see by now, as far as I am concerned, it’s the topic that matters more than anything else, actually.

Come to think of it, I am not afraid of putting even the inchoate among my thoughts, in an essay contest like this. And, that’s to a large extent because, I am most certainly not at all afraid of participating in it and also not winning anything—not even a fourth prize. One doesn’t enter an essay contest in order to win a prize (just the way one does not take an examination to score the highest marks/ranks). In case you are sufficiently idiotic to not get it, notice that what I said in the last line is not an argument against having prizes in contests (or taking them home if you win them). It is merely a way of highlighting the fact that prizes do not deterministically elicit better responses (just the way top examinations ranks do not necessarily always go to the best guy). (If you are not convinced, substitute “fatalistically” in place of “deterministically.”)

The main function of prizes is to attract publicity, and thereby, possibly increase one’s chances of finding, or reaching out to, the right people, the right minds. Prizes serve to attract a better audience rather than a better set of participants. That is, statistically speaking, of course.

You don’t necessarily have to win prizes in order to reach out to a better audience. (And, what’s a better audience, you ask? Obvious. It’s an audience that is itself capable, employs you, pays you, respects you, etc.—overall, values you on a rational basis.) That is the reason why participation matters more than winning.

(The Olympics participants usually get it right—and most of the humanities folks, never do. For example, consider: If it were to be just a matter of exceeding one’s own past performances, or to see the limits of one’s abilities, why not go to a secluded place, exceed your abilities to your heart’s content, and then, never let anyone else get even the wind of it? Ditto, even if your motivation is less exalted, and consists solely of exceeding others’ abilities—beating others. Here, suppose that there are just the two of you, you and your opponent (or the ten of you, or ten teams), and suppose that you (or your team) win (wins) over (all) your opponent(s)—but strictly under the condition that no one else ever gets to know of it. None. You continue to know that you exceeded your past records, or that you beat others, but there is no audience for it, no better consequences to follow in your own life, out of it… And, now, also consider a contrasting scenario: What if you do get to connect to the right kind of an audience even if you don’t win a contest in which you participate. What would it be like? Here, I am tempted to speculate: It would be just like any of our (India’s) sports teams, especially, our cricket team. … So, either way, it is the participation that matters more than the winning. QED, nah?)

So, the idea of participating in a contest like FQXi is quite OK by me. So, coming back to the topic for the current edition of this contest:

As soon as I read about the contest (which was something like the last week or so), I got the sense that the topic selection was, once again, rather smart—but also that the topic was a bit too open-ended, though probably not too broad. Reading through the vast variety of the essays that people have submitted so far only confirmed, in a way, this apprehension of mine.

If a well-informed physicist friend were to ask me in an informal but serious chat the question  of the topic (“Which of our basic physical assumptions are wrong?”), the first set of things to strike me would have been rather philosophical in nature. But then, this is not an essay contest in philosophy as such, though, I guess, certain parts of philosophy clearly are not out of place here—in fact, dealing, as the contest does, with questioning the foundations, philosophy of physics, and also relevant principles from general philosophy, are clearly welcome. However, the main part that philosophy can play here would be limited to identifying the broad context; the essay cannot be concerned with expounding philosophy itself, as such.  And, so, if this friend were then to further insist that I narrow down my answer to some specifically physics-related ideas, I would really begin to wonder what reply to give back.

That, in particular, was the position in which I found myself for the past few days.

I found that, if I have to think of some issues or ideas specifically from physics to answer that question, I could easily think of not just one or two but at least five-six issues, if not ten or more of them. And, I found that I could not really pick out one over the other without also substantially involving general philosophy as well—comparing and contrasting these issues in the light of philosophy, i.e. using philosophy to put every one of them in a common broad context embracing them all—in which case, it would become (at least a small) book and cease to be just an essay i.e. an article.

So, honestly speaking, this essay contest has, in a way, foxed me.

In a way, it had become a challenge for me to see if I could find just one or two issues out of all those numerous issues. Without there being adequate space to put all of those issues in context, treating just one or two of them would come to mean, I thought, that I consider the selected issues to be at least more pertinent if not more foundational than the others left out of the essay. And, there, I realized, my home-work is not yet well done. I don’t have a very clear idea as to why I should pick out this issue over that one. That’s why, at least as far as I am concerned, the essay topic had, in fact, become a challenge to me.

I then decided to see if I could challenge the challenge (!). Namely, what if I pick up a few issues almost at random, and write something about them, without thereby necessarily implying that these selected issues must be taken as the hierarchically more foundational/at the core/important than the others? Would it then be possible for me to write something?

BTW, there would have been another point against participating, which no longer matters: Sometimes, the discussion at FQXi seemed to digressed too much into inconsequential matters. Submitting an essay is to commit to having discussions. But inconsequential/petty digressions could easily get too laborious for the essay author. Here, however, I have noticed that as the contest and its management matures, the degree of such largely pointless digressions seems to be going down. I think you now can more easily ignore the issues/folks you don’t want to tackle/answer, especially so if you really don’t care much about winning the prize. So, that’s another point in favor of participating.

So, the churning in my mind regarding the topic, regarding whether to participate in it or not, is still going on, even as I write this blog post. However, I think I am getting increasingly inclined towards the idea of writing something anyway and dumping it there. … Let me see if I can do something along that line. And, the only way to see whether that is doable or not is to actually sit down and start writing something. I will do that. … If something “sensible” comes out it, you will see me submitting an entry by August 31. If not, here is a promise: I will at least share a bit from whatever that I wrote (and decided not to submit for the contest), here at my personal blog, and possibly also other blogs/public fora.

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No “A Song I Like” section, once again. I still go jobless. Keep that in mind.

[Minor corrections and updates were made on July 23, 2012.]
[E&OE]

Can an Infinitesimal Have Parts?

Context and Motivation:

The title question of this post has been lingering in my mind for quite some time—actually, years (nay, decades). Some two decades ago or so, I thought I had reached some good understanding of it. But then, some of the discussion at a recent iMechanica thread “A point and a particle” [^] seemed to suggest otherwise. The issue again got raised, in a somewhat indirect manner, in relation to this comment [^] on yet another iMechanica thread today. In between, there also were a few message exchanges that I had at HBL last year, not all of which made it to the published HBL exchange. There, too, my own position was at odds with that of Dr. Harry Binswanger, an Objectivist professor of philosophy (and the way he sometimes describes himself, an amateur scientist).

The essential difference is this: People seem to think, for example, that:

(i) you can take a small but finite line-segment, subject it to an infinitely long limiting process, and what you get in the end is a point; or,

(ii) as the chord of a circle is systematically made ever smaller by bringing its two end-points closer, even as always keeping them on the circle, eventually, the circle, in comparison with the straight-chord, seems to get flattened out so much that eventually, in an infinite process, it becomes indistinguishable from a straight-line, and so, the circular arc becomes the chord (which is the same as saying that the chord becomes the arc); or,

(iii) a particle’s geometry is fully described by a point; etc.

All of these examples, in some way, touch on the title question. For instance, since a point does not have any parts, and if in an infinite process a line goes to a point, then, obviously, an infinitesimal cannot have parts. And so on…

Now, I seem to disagree with the views expressed by people, as above. I also think that some of the basic confusions arising in quantum mechanics (e.g. those concerning the quantum spin) in part arise out of this issue.

[Therefore, an immediate declaration: If someone gets a better idea of what QM really is like, after reading this post, thank me, and also, regardless of that and more importantly: give me appropriate and explicit intellectual credit. To my knowledge, the topic has not been treated so directly and in the following way anywhere else before.]

Background:

Consider an arbitrary but “nice” enough a function: y = f(x). Consider two points P(x_1,y_1) and Q(x_2,y_2) lying on the curve but a finite distance apart. The slope of the line-segment PQ is given by: m_f \equiv \dfrac{y_2 - y1}{x_2-x_1} \equiv \dfrac{\Delta y}{\Delta x}, where the subscript f put on m indicates that this is a finite-distance case.  As you know, there is an infinity of points in between the end-points of any finite line-segment.

To determine the slope of the curve at the point P, we take the limit of the ratio m_f as the distance between x_2 and x_1 approaches zero. In symbolic terms: m_P = \lim_{\Delta x \to 0} \dfrac{\Delta y}{\Delta x}, where m_P is supposed to be the slope of the curve at the point P.

Clarifications—The Idea of Slope:

The italicized parts in the above statement are important. Firstly, it is implicitly (and somewhat blithely) assumed that a curve can have a slope, which can be approximated by that of a line-segment such as PQ. Secondly, it is even more implicitly (and even more blithely) assumed that there exists something such as a slope at a point. Let’s examine both a bit more closely.

What does the notion of slope mean? The extreme case of 0^0 and the pathological case of 90^0 apart, what the notion basically means is that you are going to either gain or lose your current height as you travel (in some direction).

Notice that immediately implicit here is the idea of there being two different locations whose heights are being compared! You cannot define slope without there being two distinct reference points. Hence, you also should not use the term in those contexts where only one reference point is given. If so, how can we speak of a slope at a point?

Realize that the above objection applies as much to the points lying on a straight line-segment as those on a curved line-segment. Even single points on straight lines cannot, strictly speaking, can be said to have a slope—only the straight line-segment, as a whole (or any finite parts of it) may be said to have one. If so, what does it mean when we speak of a slope at a point?

My answer: Primarily, it means nothing! It’s just a loose way of putting things. What it really means is the entire limiting process, and the result of it (if there is any valid result coming off that limiting process).

The slope of a line at a point (whether that line is straight or curved, it does not matter) is just the definite “tendency” shown in the trends of the actual slopes of all the small but finitely long straight line-segments in the close neighbourhood of the given point. You cannot speak of a slope at a point in any other terms. Not even for straight-lines. Straight lines just happen to be a special case wherein all the slope values are the same, and so, determining the trend is a trivial matter. Yet, the principle of having to make a reference to an actual trend of certain property displayed by a definitely ordered sequence of finitely long segments, in an appropriate limiting process, does remain there. It is only in this sense that lines can have slopes at various points. Ditto, for the curved lines.

Clarifications—A Line “Going” “to” a Point:

Now, there is something even funnier. At least in applied science and engineering, we often speak of the above kind of a limiting process, in terms like the following:

Take Q close to P, as close as possible. In the limit, as the length of PQ “goes” “to” “zero,” the slope of the segment PQ “goes” “to” “the slope of the curve” “at” “P“.

All the words put in the scare-quotes (“”) are important.

What does it mean for a length of a straight line-segment PQ to go to zero? It means: P and Q are coincident—i.e. they are one and the same point. (There is no such a thing as two different points occupying the same point—it’s either two names for the same mathematical object, or a contradiction in terms.)

So, can a slope have a curve? The very idea is meaningless outside the context of a limiting process. Yes, you may gain or lose height as you traverse the curve, sure. But does it mean that the curve has a slope? Nope. Not unless your context has the right limiting process in it.

Clarifications—Points, Lines, and the Nature of Limiting Processes:

Now, a bit about the nature of limiting process.

Realize that there is a fundamental difference between a point and a line. (For our purposes, both may be taken as given axiomatically, as abstractions of the locations and the edges of the actually existing objects. That there also is suggested an infinite process in reaching such abstractions is a subtle point that we choose to ignore for the time being.)

The units of a point and a line are different. You cannot compare a point and a line in any commensurate manner whatsoever, full-stop. (Incommensurability is quite frequent in mathematics, more often than what most people realize.)

A line segment may be put in one:one correspondence with an (orderly) infinite set of points, and in this way, it may abstractly be seen to consist of points. However, realize that infinity does not exist. The one:one correspondence process, should you wish to conduct one in actuality, will never terminate, and hence, you will never get a line starting from points, or vice versa: a point, starting from a line. Incidentally, that’s just another way of realizing that a line is incommensurate with a point. Then how is it that we can talk meaningfully of such a process?

What we mean when we talk of a line as being made of an infinity of points is this:

Take a finite line-segment, say from the point P to Q. Take a point P lying on it. Find the finite lengths of M from each of its end-points.  (Aside: It is here that the defining processes of a point, a line, etc. that we have chose to ignore in this post, create some tricky issues. We will deal with them later, in another post.)

Now, take a sub-segment from any of the two end-points to the middle point (whose location, in the general case, is arbitrary; it need not exactly divide the segment into two equal halves.) Suppose we take the sub-segment PM. Now, conduct a limiting process by reducing the size of PM, while holding M fixed. (BTW, observe that every limiting process involves holding something the same even as varying something else.) Making the sub-segment monotonically smaller in size means that the end-point of the segment in the reduced size corresponding to P, say, P' monotonically increasingly gets closer to M. But, it never quite reaches M.

The only case in which P' could reach M is if it is coincident with—i.e. is the same point as—M. However, in this case, there cannot be two distinct end-points left to serve as the end-points of the diminishing sub-segment, and hence, no sub-segment left to speak of.

Hence, we have to say that the point P' never quite reaches M—not even in this infinite limiting process. The most crucial point of the logic is already thus given. The rest is a bit of house-keeping so that even if we revise the entire description here by expressing a point via a limiting process, the essential logic as spelt about remains unaffected.

Now, repeat the process for another, distinct, point N \neq M, lying on the same original line-segment. Since M and N are not one and the same point, and since the “getting closer” process for any arbitrary sub-part of the line-segment cannot terminate for either of them, and further, since both lie on the same original finite segment and thereby enjoy an ordering relation between them (e.g. that M < N etc.), we must conclude that there must be an infinity of N points corresponding to any arbitrarily given point M. Just make M coincident with (or the same as) Q, and the inevitable conclusion follows, namely, that there must be an infinity of such processes for them to span all the distinct points lying over the entire original line-segment.

The existence of this infinity of such “getting closer” processes is what we actually mean when we say a line is “made of” an infinity of points.

Emphatically, it does not mean that a point and a line are commensurate. It only means that the endpoints of a line can be made as close to a given point lying on that line as you wish. That’s all.

Clarifications—An Infinitesimal of a Finite:

Now, we are ready to tackle the idea of infinitesimal.

An infinitesimal of a line-segment is an imaginary projection of the result that would be had if a line-segment were to be made ever smaller in a limiting infinite (i.e. definite but unterminating) process.

Notice that we didn’t jump directly to what the term “infinitesimal” means in a general sense. We simply made a statement in respect of the infinitesimal of a line-segment. This distinction is important. The reason is that there is no such thing as a general infinitesimal!

You can have infinitesimals of (finite) lines, surfaces, volumes, etc. Or, of quantities that, essentially, are some kind of densities of some other quantities which have been defined in a “wholesale” manner over finite lines (or surfaces, volumes, etc.). But you cannot have infinitesimals “in general,” as such.

Infinitesimals not only acquire their meaning only in some definite kind of an infinite limiting process, but they also do so only in reference to the certain finite thing (and its associated properties) which is being subjected to that process. A process without an input or an output is a contradiction in terms. An infinitesimal can only result when you begin in the first place with a finite.

Since an infinitesimal must always refer to its input finite thing (be it a length, a surface, etc. or a density variable defined with respect to these), therefore, it must always carry some units—which are the same as that of the finite thing.

The “infinitesimal-izing” process (to coin a new word!) does not touch the units of the finite thing, and hence, neither does the end-result of that process—even if the result be only via an imaginary projection. Thus, the infinitesimal of a line always retains the units of, say, m, and that of a surface, m^2, etc.

The above precisely is the reason why we can “cancel out” dx dy with da where the first expression is a product of lengths, and the second one is an area—and wherein all the quantities are infinitesimals. Infinitesimals have units; equations formulated in infinitesimal terms must follow the law of dimensional homogeneity.

Clarifications—Can Infinitesimals Have Parts?

Now, having examined the nature of infinitesimals to (hopefully) sufficient extent, we are finally ready to answer the title question: “Can an infinitesimal have parts?”

I will not directly answer the question in yes or no terms. My answer should be obvious to you by now. (If not, kick yourself a couple of times, and proceed to read further or, equally well, abandon this blog forever.)

First, observe that it is only a finite line-segment which, when subject to an infinitesimal-izing process, becomes an infinitesimal.

Apart from its two end-points, you can always take a third point lying on that finite segment such that it divides the segment into two (not necessarily equal) parts. Say, L = L_1 + L_2. Now, observe that as you take L to an infinitesimally small quantity, you also thereby subject L_1 and L_2  to the same infinitesimal-izing process such that the equation dL = dL_1 + dL_2 holds as a result. (The reason we can directly put this relation in this way is that the rates with which each becomes small is identical. In contrast, the area gets smaller at a rate faster than that of the length—another way of seeing that an infinitesimal always has dimensions i.e. units.)

Now, returning back to today’s discussion. At iMechanica, I have raised a couple of points:

(i) Do we define stress in relation to a plane? Or do we do so in relation to a thin plate made infinitesimally small? The difference, now you can see, is this: a plane has no thickness. But a plate does. Its thickness has the units of length, which can’t be made zero. Hence the question.

(ii) Is the elemental cube (used for defining variations in stress, say to the first order) have a finite length? Or is it (or can it be) infinitesimal?

Once again, I will not provide a direct answer to these questions. However, I will leave you with a very very obvious clue (apart from what all I have mentioned above)—but one, which, nevertheless, raises further curious issues. These are essentially nothing but the same as the issues we have chosen to ignore today—what are points? lines? surfaces? do they exist? Anyway, the clue, presently, is the following.

Take a brick. You can always make its size ever smaller in a limiting process so as to get an infinitesimal Cartesian volume element. Agreed? OK.

Now, take a pack of playing cards. Subject it to a similar limiting process. And, ask yourself the above two questions.  The answer(s) should be obvious!! (As to the tricky part: Ask yourself: Can you assume zero thickness in between two adjacent playing cards in the same pack? Your answer to the question of whether stress is defined in relation to a plane or an infinitesimally thin plate, will in part differ depending on how you answer this question!)

[PS: I think I might edit this post a bit. If I do so, I will also note down any major change (e.g. that of the logic or of hierarchical precedence, etc.) that I make. For instance, I am not at all happy with the way I have explained the idea of “an infinity of points in a line, even though a line never goes to a point.” That part hasn’t at all come out well. I expect to make changes there—or, may be, perhaps, write another post to once again give a try to that part. … Hey, after all, this is not a paper on mathematics—just a blog post, OK? 🙂 ]

[A side note: I know that the limit notation as rendered here on the Web page does not look nice, but that’s an issue primarily with the WordPress support of LaTeX. I am not going to hack around with \dfrac etc. just to get the \lim look nice here!]

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A Song I Like:
(Hindi) “dil beqaraar saa hai…”
Singer: Lata Mangeshkar (I like her version better than Rafi’s)
Music: Kalyanji-Anandji
Lyrics: Majrooh Sultanpuri

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A List of My Enemies

1. Dr. Nitin B. Kulkarni. MBBS (BJMC), PhD (Michigan), MD (Psychiatry). Founder, Ayn Rand Thinkers’ Club, Pune (ARTC). Formerly (in 1991), a recipient of a minor travel etc. grant from the Ayn Rand Institute. Currently, Psychiatrist and Assistant Medical Director, Patton State Hospital, CA, USA.

2. Mrs. Nina Nitin Kulkarni (Miss Sadhana V. Nagarkar). B.Com., MA (Psychology), etc. Currently (since 1994 (?)), Nitin Kulkarni’s current wife. Also, my ex-wife, of a marriage that lasted essentially for less than a month, and which should not have taken place in the first place.

3. Mrs. Pradnya Martz (Miss Pradnya Prabhakar Walhekar). Convent educated. B. Arch. (JJ School), Masters in Architecture (Massachusetts, Amherst). Currently, employed with the Oberlin College, OH, USA. The Kulkarnis’ “friend.”

4. Dr. Amar Ghorpade. MBBS (Miraj), Grad. Studies in Psychiatry (Lousiana State, Baton Rouge). Currently, Associate Chief of Psychiatry, New York Methodist Hospital.

And, all their Objectivist and other supporters and/or users, including those in FBI, CIA, etc. This is quite a bunch, and they are necessarily to be included any time I mention the above four.

A couple of characters who were only borderline cases back then (17–20 years ago) but who by now have faded out. I mention them just in order to sharpen the difference of the above four from the rest of them:

1. Dr. Sandip Tulshiram Patil. MBBS (BJMC, Pune). PhD. (?).
2. Mr. Shirish Kulkarni. Chartered Accountant, Pune.

The only members/associates of the ARTC who did not turn out to be my enemies, then, or later on:
1. Dr. Kiran Kharat. MBBS (BJMC, Pune). MS Ortho (completed?) (BJMC, Pune). MS Ortho (England). Currently, based in the UK, with visiting associations in hospitals in Pune.
2. Mrs. Netra Shirish Kulkarni. B. Sc. Shirish’s wife.

I have made many friends in life, and then, comparatively very rarely, also some enemies. The 80/20 rule, recently discussed in an article in the DNA newspaper, applies also to friendships turning into enmities. What people usually do is to forget, even forgive, and carry on with their lives. Which, all in all, works out well for relationships, for maintaining the fabric of a society. That’s usually everyone’s experience, and mine is no different—it is difficult to become or have enemies, by that 80/20 rule. It does take something out of the ordinary to become an enemy. But, yes, if there is an enmity, I am not going to evade it either.

So, I have made enemies too, even if they are far rarer as compared to friends. By way of numbers, that is. And, by way of values? Nowhere even near. (BTW, might as well mention it here. I consider also Swami Vivekananda a friend. In fact, that’s what the thought to cross my mind within the first 30 seconds was, when I read about his recent 150th anniversary, at Dr. Dey’s blog a couple of days ago. A friend. … Oh! I know what you mean, but never mind! The Swami did keep a huge company of friends, and it’s OK to call him that—a friend—provided you yourself are pure in extending your friendship to him, anyway. (BTW, but for the bad atmosphere created by BJP-RSS-Internet Hindus (and their Christian/Muslim etc. counterparts elsewhere), it would not have been really necessary to state it separately that it is OK to criticize intellectual positions of friends, too—provided it is done right.))

But even if I have made many enemies (e.g. those “Java” etc. bustards from the San Francisco Bay Area who first treated me as enemies and hence I had no choice but to fight back), the first four characters mentioned here in this post are rather different. They did require a separate mention. And, note that before going ahead and thus stating their names publicly, I have waited for years, suffering career setbacks, financial setbacks, even psychic attacks and all. Indeed, I think many of you do know that I do believe that many of these psychic attacks originate from within the USA, that at least some of them follow the US government’s policies and decisions (and those of the governmental agencies such as the FBI, CIA, etc.). It is this part—psychic attacks—which has made it necessary for me to put these names in this manner here. I don’t mean to say that the psychic attacks originate with these actively evil or at least extraordinarily dishonest and ingrate a set of characters. My point is that if tomorrow I come to know that the game originated with them, or that they added fuel to the attackers’ fire, I wouldn’t be surprised in the least. (My mother may get surprised, perhaps also my father—but I wouldn’t. And, I hope, at least some of my friends wouldn’t, whether they knew these characters first-hand or not.)

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A Song I Like:

(Marathi): “ek ek virate taaraa, aasmant ye aakaaraa,
ubhaa devaraayaa kshitijaavarati,
uThi shreedharaa re, sarali raati…”
Lyrics: (Shanta Shelke ??)
Music: ??
Singer: (Suman Kalyanpur ??)

[Note: This again is a song whose credits I neither can remember nor can I access via an Internet search. It certainly was a poem prescribed in the school-time texts for us. I have only a vague of recollection of having heard it as a recorded song. I remember the tune very distinctly, of course, and I can even upload or send you an audio file of my (bad) hummings if it will help you locate the song. I feel the lyricist most probably was Shanta Shelke (but then, I sometimes end up attributing to her many songs which actually were not hers). The singer could be Suman Kalyanpur.

… For non-Marathi readers: By way of formal genre, this song sure is a “bhupaali,” but a very different one. It is a very light song. It is a song that welcomes the beginning of a new day. It does so neither with the noises of the road traffic nor with the stern finality of a military siren. Nor even with the discernibly regimented voice of a classical number. Instead, this song seems to welcome the day with the natural sounds surrounding an idyllic village (one which is not a holiday resort). I will post a rough translation of the poem itself later on. … In terms of the tune, it is a very peace- and calm-inducing song. It is that sort of calm which is suited not to retire to sleep but to gently break it. Not to make a new beginning but to help you prepare your soul to do so. The tune is an ideal accompaniment to that sense of awareness which precedes any awareness of any mental activity for any sort of planning for the actual day. The song goes with just the barest grasp of a fresh, nice, welcoming day, that’s all. … If you are familiar with Marathi literature (and hence with the metaphors, idioms, etc., used in it), this song carries that same touch of wholesomeness as is found in such expressions as this one: “The place was new, but for some odd reason, it seemed familiar, as familiar as the smell of mother’s used saree.”

….Some songs you remember in your troubled times. Others are the songs, you realize only later on, you should have remembered in those troubled times, but somehow, didn’t. And there are still other songs… They are so simple and light, they simply descend on you regardless of the times you are in. This song is one of those. To the psychic attackers: I didn’t choose this song because there is this word “taaraa” in it, to remind Tara Malkani. Nope. I didn’t. I “chose” it because it is one of those songs that finds me out when I need them, so to speak.]

[Written in a hurry, I will edit this once more, and then leave them.

… Oh, BTW, this blog is not meant to be kid-sister-friendly, and so, I could have used words appropriate to describe these characters without a sense of hesitation. But frankly, I don’t use such words unless there is that emotional intensity present in the very moment in which they are to be used. I mean, one doesn’t insult even the “bad” words, does one? Right now, I feel nothing, and so, I didn’t use those words. But yes, I have used such words in the past, and God knows I will do so again, too! I may delete this last comment during the pending editing—or then, again, I may not!!]

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