Stay tuned to the NSF on the next evening…

Update on 2019.04.10 18:50 IST: 

Dimitrios Psaltis, University of Arizona in Tucson, EHT project scientist [^]:

The size and shape of the shadow matches the precise predictions of Einstein’s general theory of relativity, increasing our confidence in this century-old theory. Imaging a black hole is just the beginning of our effort to develop new tools that will enable us to interpret the massively complex data that nature gives us.”

Update over.


Stay tuned to the NSF on the next evening (on 10th April 2019 at 06:30 PM IST) for an announcement of astronomical proportions. Or so it is, I gather. See: “For Media” from NSF [^]. Another media advisory made by NSF roughly 9 days ago, i.e. on the Fool’s Day, here [^]. Their news “report”s [^].


No, I don’t understand the relativity theory. Not even the “special” one (when it’s taken outside of its context of the so-called “classical” electrodynamics)—let alone the “general” one. It’s not one of my fields of knowledge.

But if I had to bet my money then, based purely on my grasp of the sociological factors these days operative in science as practised in the Western world, then I would bet a good amount (even Indian Rs. 1,000/-) that the announcement would be just a further confirmation of Einstein’s theory of general relativity.

That’s how such things go, in the Western world, today.

In other words, I would be very, very, very surprised—I mean to say, about my grasp of the sociology of science in the Western world—if they found something (anything) going even apparently contrary to any one of the implications of any one of Einstein’s theories. Here, emphatically, his theory of the General Relativity.


That’s all for now, folks! Bye for now. Will update this post in a minor way when the facts are on the table.


TBD: The songs section. Will do that too, within the next 24 hours. That’s a promise. For sure. (Or, may be, right tonight, if a song nice enough to listen to, strikes me within the next half an hour or so… Bye, really, for now.)


A song I like:

(Hindi) “ek haseen shaam ko, dil meraa kho_ gayaa…”
Lyrics: Raajaa Mehdi Ali Khaan
Music: Madan Mohan
Singer: Mohammad Rafi [Some beautiful singing here…]

 

 

 

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The self-field, and the objectivity of the classical electrostatic potentials: my analysis

This blog post continues from my last post, and has become overdue by now. I had promised to give my answers to the questions raised last time. Without attempting to explain too much, let me jot down the answers.


1. The rule of omitting the self-field:

This rule arises in electrostatic interactions basically because the Coulombic field has a spherical symmetry. The same rule would also work out in any field that has a spherical symmetry—not just the inverse-separation fields, and not necessarily only the singular potentials, though Coulombic potentials do show both these latter properties too.

It is helpful here to think in terms of not potentials but of forces.

Draw any arbitrary curve. Then, hold one end of the curve fixed at the origin, and sweep the curve through all possible angles around it, to get a 3D field. This 3D field has a spherical symmetry, too. Hence, gradients at the same radial distance on opposite sides of the origin are always equal and opposite.

Now you know that the negative gradient of potential gives you a force. Since for any spherical potential the gradients are equal and opposite, they cancel out. So, the forces cancel out to.

Realize here that in calculating the force exerted by a potential field on a point-particle (say an electron), the force cannot be calculated in reference to just one point. The very definition of the gradient refers to two different points in space, even if they be only infinitesimally separated apart. So, the proper procedure is to start with a small sphere centered around the given electron, calculate the gradients of the potential field at all points on the surface of this sphere, calculate the sum of the forces exerted on the domain contained inside the spherical surface by these forces, and then take the sphere to the limiting of vanishing size. The sum of the forces thus exerted is the net force acting on that point-particle.

In case of the Coulombic potentials, the forces thus calculated on the surface of any sphere (centered on that particle) turn out to be zero. This fact holds true for spheres of all radii. It is true that gradients (and forces) progressively increase as the size of the sphere decreases—in fact they increase without all bounds for singular potentials. However, the aforementioned cancellation holds true at any stage in the limiting process. Hence, it holds true for the entirety of the self-field.

In calculating motions of a given electron, what matters is not whether its self-field exists or not, but whether it exerts a net force on the same electron or not. The self-field does exist (at least in the sense explained later below) and in that sense, yes, it does keep exerting forces at all times, also on the same electron. However, due to the spherical symmetry, the net force that the field exerts on the same electron turns out to be zero.

In short:

Even if you were to include the self-field in the calculations, if the field is spherically symmetric, then the final net force experienced by the same electron would still have no part coming from its own self-field. Hence, to economize calculations without sacrificing exactitude in any way, we discard it out of considerations.The rule of omitting the self-field is just a matter of economizing calculations; it is not a fundamental law characterizing what field may be objectively said to exist. If the potential field due to other charges exists, then, in the same sense, the self-field too exists. It’s just that for the motions of the self field-generating electron, it is as good as non-existent.

However, the question of whether a potential field physically exists or not, turns out to be more subtle than what might be thought.


2. Conditions for the objective existence of electrostatic potentials:

It once again helps to think of forces first, and only then of potentials.

Consider two electrons in an otherwise empty spatial region of an isolated system. Suppose the first electron (e_1), is at a position x_1, and a second electron e_2 is at a position x_2. What Coulomb’s law now says is that the two electrons mutually exert equal and opposite forces on each other. The magnitudes of these forces are proportional to the inverse-square of the distance which separates the two. For the like charges, the forces is repulsive, and for unlike charges, it is attractive. The amount of the electrostatic forces thus exerted do not depend on mass; they depend only the amounts of the respective charges.

The potential energy of the system for this particular configuration is given by (i) arbitrarily assigning a zero potential to infinite separation between the two charges, and (ii) imagining as if both the charges have been brought from infinity to their respective current positions.

It is important to realize that the potential energy for a particular configuration of two electrons does not form a field. It is merely a single number.

However, it is possible to imagine that one of the charges (say e_1) is held fixed at a point, say at \vec{r}_1, and the other charge is successively taken, in any order, at every other point \vec{r}_2 in the infinite domain. A single number is thus generated for each pair of (\vec{r}_1, \vec{r}_2). Thus, we can obtain a mapping from the set of positions for the two charges, to a set of the potential energy numbers. This second set can be regarded as forming a field—in the 3D space.

However, notice that thus defined, the potential energy field is only a device of calculations. It necessarily refers to a second charge—the one which is imagined to be at one point in the domain at a time, with the procedure covering the entire domain. The energy field cannot be regarded as a property of the first charge alone.

Now, if the potential energy field U thus obtained is normalized by dividing it with the electric charge of the second charge, then we get the potential energy for a unit test-charge. Another name for the potential energy obtained when a unit test-charge is used for the second charge is: the electrostatic potential (denoted as V).

But still, in classical mechanics, the potential field also is only a device of calculations; it does not exist as a property of the first charge, because the potential energy itself does not exist as a property of that fixed charge alone. What does exist is the physical effect that there are those potential energy numbers for those specific configurations of the fixed charge and the test charge.

This is the reason why the potential energy field, and therefore the electrostatic potential of a single charge in an otherwise empty space does not exist. Mathematically, it is regarded as zero (though it could have been assigned any other arbitrary, constant value.)

Potentials arise only out of interaction of two charges. In classical mechanics, the charges are point-particles. Point-particles exist only at definite locations and nowhere else. Therefore, their interaction also must be seen as happening only at the locations where they do exist, and nowhere else.

If that is so, then in what sense can we at all say that potential energy (or electrostaic potential) field does physically exist?

Consider a single electron in an isolated system, again. Assume that its position remains fixed.

Suppose there were something else in the isolated system—-something—some object—every part of which undergoes an electrostatic interaction with the fixed (first) electron. If this second object were to be spread all over the domain, and if every part of it were able to interact with the fixed charge, then we could say that the potential energy field exists objectively—as an attribute of this second object. Ditto, for the electric potential field.

Note three crucially important points, now.

2.1. The second object is not the usual classical object.

You cannot regard the second (spread-out) object as a mere classical charge distribution. The reason is this.

If the second object were to be actually a classical object, then any given part of it would have to electrostatically interact with every other part of itself too. You couldn’t possibly say that a volume element in this second object interacts only with the “external” electron. But if the second object were also to be self-interacting, then what would come to exist would not be the simple inverse-distance potential field energy, in reference to that single “external” electron. The space would be filled with a very weird field. Admitting motion to the property of the local charge in the second object, every locally present charge would soon redistribute itself back “to” infinity (if it is negative), or it all would collapse into the origin (if the charge on the second object were to be positive, because the fixed electron’s field is singular). But if we allow no charge redistributions, and the second field were to be classical (i.e. capable of self-interacting), then the field of the second object would have to have singularities everywhere. Very weird. That’s why:

If you want to regard the potential field as objectively existing, you have to also posit (i.e. postulate) that the second object itself is not classical in nature.

Classical electrostatics, if it has to regard a potential field as objectively (i.e. physically) existing, must therefore come to postulate a non-classical background object!

2.2. Assuming you do posit such a (non-classical) second object (one which becomes “just” a background object), then what happens when you introduce a second electron into the system?

You would run into another seeming contradiction. You would find that this second electron has no job left to do, as far as interacting with the first (fixed) electron is concerned.

If the potential field exists objectively, then the second electron would have to just passively register the pre-existing potential in its vicinity (because it is the second object which is doing all the electrostatic interactions—all the mutual forcings—with the first electron). So, the second electron would do nothing of consequence with respect to the first electron. It would just become a receptacle for registering the force being exchanged by the background object in its local neighborhood.

But the seeming contradiction here is that as far as the first electron is concerned, it does feel the potential set up by the second electron! It may be seen to do so once again via the mediation of the background object.

Therefore, both electrons have to be simultaneously regarded as being active and passive with respect to each other. They are active as agents that establish their own potential fields, together with an interaction with the background object. But they also become passive in the sense that they are mere point-masses that only feel the potential field in the background object and experience forces (accelerations) accordingly.

The paradox is thus resolved by having each electron set up a field as a result of an interaction with the background object—but have no interaction with the other electron at all.

2.3. Note carefully what agency is assigned to what object.

The potential field has a singularity at the position of that charge which produces it. But the potential field itself is created either by the second charge (by imagining it to be present at various places), or by a non-classical background object (which, in a way, is nothing but an objectification of the potential field-calculation procedure).

Thus, there arises a duality of a kind—a double-agent nature, so to speak. The potential energy is calculated for the second charge (the one that is passive), in the sense that the potential energy is relevant for calculating the motion of the second charge. That’s because the self-field cancels out for all motions of the first charge. However,

 The potential energy is calculated for the second charge. But the field so calculated has been set up by the first (fixed) charge. Charges do not interact with each other; they interact only with the background object.

2.4. If the charges do not interact with each other, and if they interact only with the background object, then it is worth considering this question:

Can’t the charges be seen as mere conditions—points of singularities—in the background object?

Indeed, this seems to be the most reasonable approach to take. In other words,

All effects due to point charges can be regarded as field conditions within the background object. Thus, paradoxically enough, a non-classical distributed field comes to represent the classical, massive and charged point-particles themselves. (The mass becomes just a parameter of the interactions of singularities within a 3D field.) The charges (like electrons) do not exist as classical massive particles, not even in the classical electrostatics.


3. A partly analogous situation: The stress-strain fields:

If the above situation seems too paradoxical, it might be helpful to think of the stress-strain fields in solids.

Consider a horizontally lying thin plate of steel with two rigid rods welded to it at two different points. Suppose horizontal forces of mutually opposite directions are applied through the rods (either compressive or tensile). As you know, as a consequence, stress-strain fields get set up in the plate.

From an external viewpoint, the two rods are regarded as interacting with each other (exchanging forces with each other) via the medium of the plate. However, in reality, they are interacting only with the object that is the plate. The direct interaction, thus, is only between a rod and the plate. A rod is forced, it interacts with the plate, the plate sets up stress-strain field everywhere, the local stress-field near the second rod interacts with it, and the second rod registers a force—which balances out the force applied at its end. Conversely, the force applied at the second rod also can be seen as getting transmitted to the first rod via the stress-strain field in the plate material.

There is no contradiction in this description, because we attribute the stress-strain field to the plate itself, and always treat this stress-strain field as if it came into existence due to both the rods acting simultaneously.

In particular, we do not try to isolate a single-rod attribute out of the stress-strain field, the way we try to ascribe a potential to the first charge alone.

Come to think of it, if we have only one rod and if we apply force to it, no stress-strain field would result (i.e. neglecting inertia effects of the steel plate). Instead, the plate would simply move in the rigid body mode. Now, in solid mechanics, we never try to visualize a stress-strain field associated with a single rod alone.

It is a fallacy of our thinking that when it comes to electrostatics, we try to ascribe the potential to the first charge, and altogether neglect the abstract procedure of placing the test charge at various locations, or the postulate of positing a non-classical background object which carries that potential.

In the interest of completeness, it must be noted that the stress-strain fields are tensor fields (they are based on the gradients of vector fields), whereas the electrostatic force-field is a vector field (it is based on the gradient of the scalar potential field). A more relevant analogy for the electrostatic field, therefore might the forces exchanged by two point-vortices existing in an ideal fluid.


4. But why bother with it all?

The reason I went into all this discussion is because all these issues become important in the context of quantum mechanics. Even in quantum mechanics, when you have two charges that are interacting with each other, you do run into these same issues, because the Schrodinger equation does have a potential energy term in it. Consider the following situation.

If an electrostatic potential is regarded as being set up by a single charge (as is done by the proton in the nucleus of the hydrogen atom), but if it is also to be regarded as an actually existing and spread out entity (as a 3D field, the way Schrodinger’s equation assumes it to be), then a question arises: What is the role of the second charge (e.g., that of the electron in an hydrogen atom)? What happens when the second charge (the electron) is represented quantum mechanically? In particular:

What happens to the potential field if it represents the potential energy of the second charge, but the second charge itself is now being represented only via the complex-valued wavefunction?

And worse: What happens when there are two electrons, and both interacting with each other via electrostatic repulsions, and both are required to be represented quantum mechanically—as in the case of the electrons in an helium atom?

Can a charge be regarded as having a potential field as well as a wavefunction field? If so, what happens to the point-specific repulsions as are mandated by the Coulomb law? How precisely is the V(\vec{r}_1, \vec{r}_2) term to be interpreted?

I was thinking about these things when these issues occurred to me: the issue of the self-field, and the question of the physical vs. merely mathematical existence of the potential fields of two or more quantum-mechanically interacting charges.

Guess I am inching towards my full answers. Guess I have reached my answers, but I need to have them verified with some physicists.


5. The help I want:

As a part of my answer-finding exercises (to be finished by this month-end), I might be contacting a second set of physicists soon enough. The issue I want to learn from them is the following:

How exactly do they do computational modeling of the helium atom using the finite difference method (FDM), within the context of the standard (mainstream) quantum mechanics?

That is the question. Once I understand this part, I would be done with the development of my new approach to understanding QM.

I do have some ideas regarding the highlighted question. It’s just that I want to have these ideas confirmed from some physicists before (or along-side) implementing the FDM code. So, I might be approaching someone—possibly you!

Please note my question once again. I don’t want to do perturbation theory. I would also like to avoid the variational method.

Yes, I am very comfortable with the finite element method, which is basically based on the variational calculus. So, given a good (detailed enough) account of the variational method for the He atom, it should be possible to translate it into the FEM terms.

However, ideally, what I would like to do is to implement it as an FDM code.

So there.

Please suggest good references and / or people working on this topic, if you know any. Thanks in advance.


A song I like:

[… Here I thought that there was no song that Salil Chowdhury had composed and I had not listened to. (Well, at least when it comes to his Hindi songs). That’s what I had come to believe, and here trots along this one—and that too, as a part of a collection by someone! … The time-delay between my first listening to this song, and my liking it, was zero. (Or, it was a negative time-delay, if you refer to the instant that the first listening got over). … Also, one of those rare occasions when one is able to say that any linear ordering of the credits could only be random.]

(Hindi) “mada bhari yeh hawaayen”
Music: Salil Chowdhury
Lyrics: Gulzaar
Singer: Lata Mangeshkar

 

The rule of omitting the self-field in calculations—and whether potentials have an objective existence or not

There was an issue concerning the strictly classical, non-relativistic electricity which I was (once again) confronted with, during my continuing preoccupation with quantum mechanics.

Actually, a small part of this issue had occurred to me earlier too, and I had worked through it back then.

However, the overall issue had never occurred to me with as much of scope, generality and force as it did last evening. And I could not immediately resolve it. So, for a while, especially last night, I unexpectedly found myself to have become very confused, even discouraged.

Then, this morning, after a good night’s rest, everything became clear right while sipping my morning cup of tea. Things came together literally within a span of just a few minutes. I want to share the issue and its resolution with you.

The question in question (!) is the following.


Consider 2 (or N) number of point-charges, say electrons. Each electron sets up an electrostatic (Coulombic) potential everywhere in space, for the other electrons to “feel”.

As you know, the potential set up by the i-th electron is:
V_i(\vec{r}_i, \vec{r}) = \dfrac{1}{4 \pi \epsilon_0} \dfrac{Q_i}{|\vec{r} - \vec{r}_i|}
where \vec{r}_i is the position vector of the i-th electron, \vec{r} is any arbitrary point in space, and Q_i is the charge of the i-th electron.

The potential energy associated with some other (j-th) electron being at the position \vec{r}_j (i.e. the energy that the system acquires in bringing the two electrons from \infty to their respective positions some finite distance apart), is then given as:
U_{ij}(\vec{r}_i, \vec{r}_j) = \dfrac{1}{4 \pi \epsilon_0} \dfrac{Q_i\,Q_j}{|\vec{r}_j - \vec{r}_i|}

The notation followed here is the following: In U_{ij}, the potential field is produced by the i-th electron, and the work is done by the j-th electron against the i-th electron.

Symmetrically, the potential energy for this configuration can also be expressed as:
U_{ji}(\vec{r}_j, \vec{r}_i) = \dfrac{1}{4 \pi \epsilon_0} \dfrac{Q_j\,Q_i}{|\vec{r}_i - \vec{r}_j|}

If a system has only two charges, then its total potential energy U can be expressed either as U_{ji} or as U_{ij}. Thus,
U = U_{ji} = U_{ij}

Similarly, for any pair of charges in an N-particle system, too. Therefore, the total energy of an N-particle system is given as:
U = \sum\limits_{i}^{N} \sum\limits_{j = i+1}^{N} U_{ij}

The issue now is this: Can we say that the total potential energy U has an objective existence in the physical world? Or is it just a device of calculations that we have invented, just a concept from maths that has no meaningful physical counterpart?

(A side remark: Energy may perhaps exist as an attribute or property of something else, and not necessarily as a separate physical object by itself. However, existence as an attribute still is an objective existence.)

The reason to raise this doubt is the following.


When calculating the motion of the i-th charge, we consider only the potentials V_j produced by the other charges, not the potential produced by the given charge V_i itself.

Now, if the potential produced by the given charge (V_i) also exists at every point in space, then why does it not enter the calculations? How does its physical efficacy get evaporated away? And, symmetrically: The motion of the j-th charge occurs as if V_j had physically evaporated away.

The issue generalizes in a straight-forward manner. If there are N number of charges, then for calculating the motion of a given i-th charge, the potential fields of all other charges are considered operative. But not its own field.

How can motion become sensitive to only a part of the total potential energy existing at a point even if the other part also exists at the same point? That is the question.


This circumstance seems to indicate as if there is subjectivity built deep into the very fabric of classical mechanics. It is as if the universe just knows what a subject is going to calculate, and accordingly, it just makes the corresponding field mystically go away. The universe—the physical universe—acts as if it were changing in response to what we choose to do in our mind. Mind you, the universe seems to change in response to not just our observations (as in QM), but even as we merely proceed to do calculations. How does that come to happen?… May be the whole physical universe exists only in our imagination?

Got the point?


No, my confusion was not as pathetic as that in the previous paragraph. But I still found myself being confused about how to account for the fact that an electron’s own field does not enter the calculations.

But it was not all. A non-clarity on this issue also meant that there was another confusing issue which also raised its head. This secondary issue arises out of the fact that the Coulombic potential set up by any point-charge is singular in nature (or at least approximately so).

If the electron is a point-particle and if its own potential “is” \infty at its position, then why does it at all get influenced by the finite potential of any other charge? That is the question.

Notice, the second issue is most acute when the potentials in question are singular in nature. But even if you arbitrarily remove the singularity by declaring (say by fiat) a finite size for the electron, thereby making its own field only finitely large (and not infinite), the above-mentioned issue still remains. So long as its own field is finite but much, much larger than the potential of any other charge, the effects due to the other charges should become comparatively less significant, perhaps even negligibly small. Why does this not happen? Why does the rule instead go exactly the other way around, and makes those much smaller effects due to other charges count, but not the self-field of the very electron in question?


While thinking about QM, there was a certain point where this entire gamut of issues became important—whether the potential has an objective existence or not, the rule of omitting the self-field while calculating motions of particles, the singular potential, etc.

The specific issue I was trying to think through was: two interacting particles (e.g. the two electrons in the helium atom). It was while thinking on this problem that this problem occurred to me. And then, it also led me to wonder: what if some intellectual goon in the guise of a physicist comes along, and says that my proposal isn’t valid because there is this element of subjectivity to it? This thought occurred to me with all its force only last night. (Or so I think.) And I could not recall seeing a ready-made answer in a text-book or so. Nor could I figure it out immediately, at night, after a whole day’s work. And as I failed to resolve the anticipated objection, I progressively got more and more confused last night, even discouraged.

However, this morning, it all got resolved in a jiffy.


Would you like to give it a try? Why is it that while calculating the motion of the i-th charge, you consider the potentials set up by all the rest of the charges, but not its own potential field? Why this rule? Get this part right, and all the philosophical humbug mentioned earlier just evaporates away too.

I would wait for a couple of days or so before coming back and providing you with the answer I found. May be I will write another post about it.


Update on 2019.03.16 20:14 IST: Corrected the statement concerning the total energy of a two-electron system. Also simplified the further discussion by couching it preferably in terms of potentials rather than energies (as in the first published version), because a Coulombic potential always remains anchored in the given charge—it doesn’t additionally depend on the other charges the way energy does. Modified the notation to reflect the emphasis on the potentials rather than energy.


A song I like:

[What else? [… see the songs section in the last post.]]
(Hindi) “woh dil kahaan se laaoon…”
Singer: Lata Mangeshkar
Music: Ravi
Lyrics: Rajinder Kishen


A bit of a conjecture as to why Ravi’s songs tend to be so hummable, of a certain simplicity, especially, almost always based on a very simple rhythm. My conjecture is that because Ravi grew up in an atmosphere of “bhajan”-singing.

Observe that it is in the very nature of music that it puts your mind into an abstract frame of mind. Observe any singer, especially the non-professional ones (or the ones who are not very highly experienced in controlling their body-language while singing, as happens to singers who participate in college events or talent shows).

When they sing, their eyes seem to roll in a very peculiar manner. It seems random but it isn’t. It’s as if the eyes involuntarily get set in the motions of searching for something definite to be found somewhere, as if the thing to be found would be in the concrete physical space outside, but within a split-second, the eyes again move as if the person has realized that nothing corresponding is to be found in the world out there. That’s why the eyes “roll away.” The same thing goes on repeating, as the singer passes over various words, points of pauses, nuances, or musical phrases.

The involuntary motions of the eyes of the singer provide a window into his experience of music. It’s as if his consciousness was again and again going on registering a sequence of two very fleeting experiences: (i) a search for something in the outside world corresponding to an inner experience felt in the present, and immediately later, (ii) a realization (and therefore the turning away of the eyes from an initially picked up tentative direction) that nothing in the outside world would match what was being searched for.

The experience of music necessarily makes you realize the abstractness of itself. It tends to make you realize that the root-referents of your musical experience lie not in a specific object or phenomenon in the physical world, but in the inner realm, that of your own emotions, judgments, self-reflections, etc.

This nature of music makes it ideally suited to let you turn your attention away from the outside world, and has the capacity or potential to induce a kind of a quiet self-reflection in you.

But the switch from the experience of frustrated searches into the outside world to a quiet self-reflection within oneself is not the only option available here. Music can also induce in you a transitioning from those unfulfilled searches to a frantic kind of an activity: screams, frantic shouting, random gyrations, and what not. In evidence, observe any piece of modern American / Western pop-music.

However, when done right, music can also induce a state of self-reflection, and by evoking certain kind of emotions, it can even lead to a sense of orderliness, peace, serenity. To make this part effective, such a music has to be simple enough, and orderly enough. That’s why devotional music in the refined cultural traditions is, as a rule, of a certain kind of simplicity.

The experience of music isn’t the highest possible spiritual experience. But if done right, it can make your transition from the ordinary experience to a deep, profound spiritual experience easy. And doing it right involves certain orderliness, simplicity in all respects: tune, tone, singing style, rhythm, instrumental sections, transitions between phrases, etc.

If you grow up listening to this kind of a music, your own music in your adult years tends to reflect the same qualities. The simplicity of rhythm. The alluringly simple tunes. The “hummability quotient.” (You don’t want to focus on intricate patterns of melody in devotional music; you want it to be so simple that minimal mental exertion is involved in rendering it, so that your mental energy can quietly transition towards your spiritual quest and experiences.) Etc.

I am not saying that the reason Ravi’s music is so great is because he listened his father sing “bhajan”s. If this were true, there would be tens of thousands of music composers having talents comparable to Ravi’s. But the fact is that Ravi was a genius—a self-taught genius, in fact. (He never received any formal training in music ever.) But what I am saying is that if you do have the musical ability, having this kind of a family environment would leave its mark. Definitely.

Of course, this all was just a conjecture. Check it out and see if it holds or not.

… May be I should convert this “note” in a separate post by itself. Would be easier to keep track of it. … Some other time. … I have to work on QM; after all, exactly only half the month remains now. … Bye for now. …


Learnability of machine learning is provably an undecidable?—part 3: closure

Update on 23 January 2019, 17:55 IST:

In this series of posts, which was just a step further from the initial, brain-storming kind of a stage, I had come to the conclusion that based on certain epistemological (and metaphysical) considerations, Ben-David et al.’s conclusion (that learnability can be an undecidable) is logically untenable.

However, now, as explained here [^], I find that this particular conclusion which I drew, was erroneous. I now stand corrected, i.e., I now consider Ben-David et al.’s result to be plausible. Obviously, it merits a further, deeper, study.

However, even as acknowledging the above-mentioned mistake, let me also hasten to clarify that I still stick to my other positions, especially the central theme in this series of posts. The central theme here was that there are certain core features of the set theory which make implications such as Godel’s incompleteness theorems possible. These features (of the set theory) demonstrably carry a glaring epistemological flaw such that applying Godel’s theorem outside of its narrow technical scope in mathematics or computer science is not permissible. In particular, Godel’s incompleteness theorem does not apply to knowledge or its validation in the more general sense of these terms. This theme, I believe, continues to hold as is.

Update over.


Gosh! I gotta get this series out of my hand—and also head! ASAP, really!! … So, I am going to scrap the bits and pieces I had written for it earlier; they would have turned this series into a 4- or 5-part one. Instead, I am going to start entirely afresh, and I am going to approach this topic from an entirely different angle—a somewhat indirect but a faster route, sort of like a short-cut. Let’s get going.


Statements:

Open any article, research paper, book or a post, and what do you find? Basically, all these consist of sentences after sentences. That is, a series of statements, in a way. That’s all. So, let’s get going at the level of statements, from a “logical” (i.e. logic-thoretical) point of view.

Statements are made to propose or to identify (or at least to assert) some (or the other) fact(s) of reality. That’s what their purpose is.


The conceptual-level consciousness as being prone to making errors:

Coming to the consciousness of man, there are broadly two levels of cognition at which it operates: the sensory-perceptual, and the conceptual.

Examples of the sensory-perceptual level consciousness would consist of reaching a mental grasp of such facts of reality as: “This object exists, here and now;” “this object has this property, to this much degree, in reality,” etc. Notice that what we have done here is to take items of perception, and put them into the form of propositions.

Propositions can be true or false. However, at the perceptual level, a consciousness has no choice in regard to the truth-status. If the item is perceived, that’s it! It’s “true” anyway. Rather, perceptions are not subject to a test of truth- or false-hoods; they are at the very base standards of deciding truth- or false-hoods.

A consciousness—better still, an organism—does have some choice, even at the perceptual level. The choice which it has exists in regard to such things as: what aspect of reality to focus on, with what degree of focus, with what end (or purpose), etc. But we are not talking about such things here. What matters to us here is just the truth-status, that’s all. Thus, keeping only the truth-status in mind, we can say that this very idea itself (of a truth-status) is inapplicable at the purely perceptual level. However, it is very much relevant at the conceptual level. The reason is that at the conceptual level, the consciousness is prone to err.

The conceptual level of consciousness may be said to involve two different abilities:

  • First, the ability to conceive of (i.e. create) the mental units that are the concepts.
  • Second, the ability to connect together the various existing concepts to create propositions which express different aspects of the truths pertaining to them.

It is possible for a consciousness to go wrong in either of the two respects. However, mistakes are much more easier to make when it comes to the second respect.

Homework 1: Supply an example of going wrong in the first way, i.e., right at the stage of forming concepts. (Hint: Take a concept that is at least somewhat higher-level so that mistakes are easier in forming it; consider its valid definition; then modify its definition by dropping one of its defining characteristics and substituting a non-essential in it.)

Homework 2: Supply a few examples of going wrong in the second way, i.e., in forming propositions. (Hint: I guess almost any logical fallacy can be taken as a starting point for generating examples here.)


Truth-hood operator for statements:

As seen above, statements (i.e. complete sentences that formally can be treated as propositions) made at the conceptual level can, and do, go wrong.

We therefore define a truth-hood operator which, when it operates on a statement, yields the result as to whether the given statement is true or non-true. (Aside: Without getting into further epistemological complexities, let me note here that I reject the idea of the arbitrary, and thus regard non-true as nothing but a sub-category of the false. Thus, in my view, a proposition is either true or it is false. There is no middle (as Aristotle said), or even an “outside” (like the arbitrary) to its truth-status.)

Here are a few examples of applying the truth-status (or truth-hood) operator to a statement:

  • Truth-hood[ California is not a state in the USA ] = false
  • Truth-hood[ Texas is a state in the USA ] = true
  • Truth-hood[ All reasonable people are leftists ] = false
  • Truth-hood[ All reasonable people are rightists ] = false
  • Truth-hood[ Indians have significantly contributed to mankind’s culture ] = true
  • etc.

For ease in writing and manipulation, we propose to give names to statements. Thus, first declaring

A: California is not a state in the USA

and then applying the Truth-hood operator to “A”, is fully equivalent to applying this operator to the entire sentence appearing after the colon (:) symbol. Thus,

Truth-hood[ A ] <==> Truth-hood[ California is not a state in the USA ] = false


Just a bit of the computer languages theory: terminals and non-terminals:

To take a short-cut through this entire theory, we would like to approach the idea of statements from a little abstract perspective. Accordingly, borrowing some terminology from the area of computer languages, we define and use two types of symbols: terminals and non-terminals. The overall idea is this. We regard any program (i.e. a “write-up”) written in any computer-language as consisting of a sequence of statements. A statement, in turn, consists of certain well-defined arrangement of words or symbols. Now, we observe that symbols (or words) can be  either terminals or non-terminals.

You can think of a non-terminal symbol in different ways: as higher-level or more abstract words, as “potent” symbols. The non-terminal symbols have a “definition”—i.e., an expansion rule. (In CS, it is customary to call an expansion rule a “production” rule.) Here is a simple example of a non-terminal and its expansion:

  • P => S1 S2

where the symbol “=>” is taken to mean things like: “is the same as” or “is fully equivalent to” or “expands to.” What we have here is an example of an abstract statement. We interpret this statement as the following. Wherever you see the symbol “P,” you may substitute it using the train of the two symbols, S1 and S2, written in that order (and without anything else coming in between them).

Now consider the following non-terminals, and their expansion rules:

  • P1 => P2 P S1
  • P2 => S3

The question is: Given the expansion rules for P, P1, and P2, what exactly does P1 mean? what precisely does it stand for?

Answer:

  • P1 => (P2) P S1 => S3 (P) S1 => S3 S1 S2 S1

In the above, we first take the expansion rule for P1. Then, we expand the P2 symbol in it. Finally, we expand the P symbol. When no non-terminal symbol is left to expand, we arrive at our answer that “P1” means the same as “S3 S1 S2 S1.” We could have said the same fact using the colon symbol, because the colon (:) and the “expands to” symbol “=>” mean one and the same thing. Thus, we can say:

  • P1: S3 S1 S2 S1

The left hand-side and the right hand-side are fully equivalent ways of saying the same thing. If you want, you may regard the expression on the right hand-side as a “meaning” of the symbol on the left hand-side.

It is at this point that we are able to understand the terms: terminals and non-terminals.

The symbols which do not have any further expansion for them are called, for obvious reasons, the terminal symbols. In contrast, non-terminal symbols are those which can be expanded in terms of an ordered sequence of non-terminals and/or terminals.

We can now connect our present discussion (which is in terms of computer languages) to our prior discussion of statements (which is in terms of symbolic logic), and arrive at the following correspondence:

The name of every named statement is a non-terminal; and the statement body itself is an expansion rule.

This correspondence works also in the reverse direction.

You can always think of a non-terminal (from a computer language) as the name of a named proposition or statement, and you can think of an expansion rule as the body of the statement.

Easy enough, right? … I think that we are now all set to consider the next topic, which is: liar’s paradox.


Liar’s paradox:

The liar paradox is a topic from the theory of logic [^]. It has been resolved by many people in different ways. We would like to treat it from the viewpoint of the elementary computer languages theory (as covered above).

The simplest example of the liar paradox is , using the terminology of the computer languages theory, the following named statement or expansion rule:

  • A: A is false.

Notice, it wouldn’t be a paradox if the same non-terminal symbol, viz. “A” were not to appear on both sides of the expansion rule.

To understand why the above expansion rule (or “definition”) involves a paradox, let’s get into the game.

Our task will be to evaluate the truth-status of the named statement that is “A”. This is the “A” which comes on the left hand-side, i.e., before the colon.

In symbolic logic, a statement is nothing but its expansion; the two are exactly and fully identical, i.e., they are one and the same. Accordingly, to evaluate the truth-status of “A” (the one which comes before the colon), we consider its expansion (which comes after the colon), and get the following:

  • Truth-hood[ A ] = Truth-hood[ A is false ] = false           (equation 1)

Alright. From this point onward, I will drop explicitly writing down the Truth-hood operator. It is still there; it’s just that to simplify typing out the ensuing discussion, I am not going to note it explicitly every time.

Anyway, coming back to the game, what we have got thus far is the truth-hood status of the given statement in this form:

  • A: “A is false”

Now, realizing that the “A” appearing on the right hand-side itself also is a non-terminal, we can substitute for its expansion within the aforementioned expansion. We thus get to the following:

  • A: “(A is false) is false”

We can apply the Truth-hood operator to this expansion, and thereby get the following: The statement which appears within the parentheses, viz., the “A is false” part, itself is false. Accordingly, the Truth-hood operator must now evaluate thus:

  • Truth-hood[ A ] = Truth-hood[ A is false] = Truth-hood[ (A is false) is false ] = Truth-hood[ A is true ] = true            (equation 2)

Fun, isn’t it? Initially, via equation 1, we got the result that A is false. Now, via equation 2, we get the result that A is true. That is the paradox.

But the fun doesn’t stop there. It can continue. In fact, it can continue indefinitely. Let’s see how.

If only we were not to halt the expansions, i.e., if only we continue a bit further with the game, we could have just as well made one more expansion, and got to the following:

  • A: ((A is false) is false) is false.

The Truth-hood status of the immediately preceding expansion now is: false. Convince yourself that it is so. Hint: Always expand the inner-most parentheses first.

Homework 3: Convince yourself that what we get here is an indefinitely long alternating sequence of the Truth-hood statuses that: A is false, A is true, A is false, A is true

What can we say by way of a conclusion?

Conclusion: The truth-status of “A” is not uniquely decidable.

The emphasis is on the word “uniquely.”

We have used all the seemingly simple rules of logic, and yet have stumbled on to the result that, apparently, logic does not allow us to decide something uniquely or meaningfully.


Liar’s paradox and the set theory:

The importance of the liar paradox to our present concerns is this:

Godel himself believed, correctly, that the liar paradox was a semantic analogue to his Incompleteness Theorem [^].

Go read the Wiki article (or anything else on the topic) to understand why. For our purposes here, I will simply point out what the connection of the liar paradox is to the set theory, and then (more or less) call it a day. The key observation I want to make is the following:

You can think of every named statement as an instance of an ordered set.

What the above key observation does is to tie the symbolic logic of proposition with the set theory. We thus have three equivalent ways of describing the same idea: symbolic logic (name of a statement and its body), computer languages theory (non-terminals and their expansions to terminals), and set theory (the label of an ordered set and its enumeration).

As an aside, the set in question may have further properties, or further mathematical or logical structures and attributes embedded in itself. But at its minimal, we can say that the name of a named statement can be seen as a non-terminal, and the “body” of the statement (or the expansion rule) can be seen as an ordered set of some symbols—an arbitrarily specified sequence of some (zero or more) terminals and (zero or more) non-terminals.

Two clarifications:

  • Yes, in case there is no sequence in a production at all, it can be called the empty set.
  • When you have the same non-terminal on both sides of an expansion rule, it is said to form a recursion relation.

An aside: It might be fun to convince yourself that the liar paradox cannot be posed or discussed in terms of Venn’s diagram. The property of the “sheet” on which Venn’ diagram is drawn is, by some simple intuitive notions we all bring to bear on Venn’s diagram, cannot have a “recursion” relation.

Yes, the set theory itself was always “powerful” enough to allow for recursions. People like Godel merely made this feature explicit, and took full “advantage” of it.


Recursion, the continuum, and epistemological (and metaphysical) validity:

In our discussion above, I had merely asserted, without giving even a hint of a proof, that the three ways (viz., the symbolic logic of statements or  propositions, the computer languages theory, and the set theory) were all equivalent ways of expressing the same basic idea (i.e. the one which we are concerned about, here).

I will now once again make a few more observations, but without explaining them in detail or supplying even an indication of their proofs. The factoids I must point out are the following:

  • You can start with the natural numbers, and by using simple operations such as addition and its inverse, and multiplication and its inverse, you can reach the real number system. The generalization goes as: Natural to Whole to Integers to Rationals to Reals. Another name for the real number system is: the continuum.
  • You can use the computer languages theory to generate a machine representation for the natural numbers. You can also mechanize the addition etc. operations. Thus, you can “in principle” (i.e. with infinite time and infinite memory) represent the continuum in the CS terms.
  • Generating a machine representation for natural numbers requires the use of recursion.

Finally, a few words about epistemological (and metaphysical) validity.

  • The concepts of numbers (whether natural or real) have a logical precedence, i.e., they come first. The entire arithmetic and the calculus must come before does the computer-representation of some of their concepts.
  • A machine-representation (or, equivalently, a set-theoretic representation) is merely a representation. That is to say, it captures only some aspects or attributes of the actual concepts from maths (whether arithmetic or the continuum hypothesis). This issue is exactly like what we saw in the first and second posts in this series: a set is a concrete collection, unlike a concept which involves a consciously cast unit perspective.
  • If you try to translate the idea of recursion into the usual cognitive terms, you get absurdities such as: You can be your child, literally speaking. Not in the sense that using scientific advances in biology, you can create a clone of yourself and regard that clone to be both yourself and your child. No, not that way. Actually, such a clone is always your twin, not child, but still, the idea here is even worse. The idea here is you can literally father your own self.
  • Aristotle got it right. Look up the distinction between completed processes and the uncompleted ones. Metaphysically, only those objects or attributes can exist which correspond to completed mathematical processes. (Yes, as an extension, you can throw in the finite limiting values, too, provided they otherwise do mean something.)
  • Recursion by very definition involves not just absence of completion but the essence of the very inability to do so.

Closure on the “learnability issue”:

Homework 4: Go through the last two posts in this series as well as this one, and figure out that the only reason that the set theory allows a “recursive” relation is because a set is, by the design of the set theory, a concrete object whose definition does not have to involve an epistemologically valid process—a unit perspective as in a properly formed concept—and so, its name does not have to stand for an abstract mentally held unit. Call this happenstance “The Glaring Epistemological Flaw of the Set Theory” (or TGEFST for short).

Homework 5: Convince yourself that any lemma or theorem that makes use of Godel’s Incompleteness Theorem is necessarily based on TGEFST, and for the same reason, its truth-status is: it is not true. (In other words, any lemma or theorem based on Godel’s theorem is an invalid or untenable idea, i.e., essentially, a falsehood.)

Homework 6: Realize that the learnability issue, as discussed in Prof. Lev Reyzin’s news article (discussed in the first part of this series [^]), must be one that makes use of Godel’s Incompleteness Theorem. Then convince yourself that for precisely the same reason, it too must be untenable.

[Yes, Betteridge’s law [^] holds.]


Other remarks:

Remark 1:

As “asymptotical” pointed out at the relevant Reddit thread [^], the authors themselves say, in another paper posted at arXiv [^] that

While this case may not arise in practical ML applications, it does serve to show that the fundamental definitions of PAC learnability (in this case, their generalization to the EMX setting) is vulnerable in the sense of not being robust to changing the underlying set theoretical model.

What I now remark here is stronger. I am saying that it can be shown, on rigorously theoretical (epistemological) grounds, that the “learnability as undecidable” thesis by itself is, logically speaking, entirely and in principle untenable.

Remark 2:

Another point. My preceding conclusion does not mean that the work reported in the paper itself is, in all its aspects, completely worthless. For instance, it might perhaps come in handy while characterizing some tricky issues related to learnability. I certainly do admit of this possibility. (To give a vague analogy, this issue is something like running into a mathematically somewhat novel way into a known type of mathematical singularity, or so.) Of course, I am not competent enough to judge how valuable the work of the paper(s) might turn out to be, in the narrow technical contexts like that.

However, what I can, and will say is this: the result does not—and cannot—bring the very learnability of ANNs itself into doubt.


Phew! First, Panpsychiasm, and immediately then, Learnability and Godel. … I’ve had to deal with two untenable claims back to back here on this blog!

… My head aches….

… Code! I have to write some code! Or write some neat notes on ML in LaTeX. Only then will, I guess, my head stop aching so much…

Honestly, I just downloaded TensorFlow yesterday, and configured an environment for it in Anaconda. I am excited, and look forward to trying out some tutorials on it…

BTW, I also honestly hope that I don’t run into anything untenable, at least for a few weeks or so…

…BTW, I also feel like taking a break… May be I should go visit IIT Bombay or some place in konkan. … But there are money constraints… Anyway, bye, really, for now…


A song I like:

(Marathi) “hirvyaa hirvyaa rangaachi jhaaDee ghanadaaTa”
Music: Sooraj (the pen-name of “Shankar” from the Shankar-Jaikishan pair)
Lyrics: Ramesh Anavakar
Singers: Jaywant Kulkarni, Sharada


[Any editing would be minimal; guess I will not even note it down separately.] Did an extensive revision by 2019.01.21 23:13 IST. Now I will leave this post in the shape in which it is. Bye for now.

Learnability of machine learning is provably an undecidable?—part 2

Update on 23 January 2019, 17:55 IST:

In this series of posts, which was just a step further from the initial, brain-storming kind of a stage, I had come to the conclusion that based on certain epistemological (and metaphysical) considerations, Ben-David et al.’s conclusion (that learnability can be an undecidable) is logically untenable.

However, now, as explained here [^], I find that this particular conclusion which I drew, was erroneous. I now stand corrected, i.e., I now consider Ben-David et al.’s result to be plausible. Obviously, it merits a further, deeper, study.

However, even as acknowledging the above-mentioned mistake, let me also hasten to clarify that I still stick to my other positions, especially the central theme in this series of posts. The central theme here was that there are certain core features of the set theory which make implications such as Godel’s incompleteness theorems possible. These features (of the set theory) demonstrably carry a glaring epistemological flaw such that applying Godel’s theorem outside of its narrow technical scope in mathematics or computer science is not permissible. In particular, Godel’s incompleteness theorem does not apply to knowledge or its validation in the more general sense of these terms. This theme, I believe, continues to hold as is.

Update over.


In this post, we look into the differences of the idea of sets from that of concepts. The discussion here is exploratory, and hence, not very well isolated. There are overlaps of points between sections. Indeed, there are going to be overlaps of points from post to post too! The idea behind this series of posts is not to present a long thought out and matured point of view; it is much in the nature of jotting down salient points and trying to bring some initial structure to them. Thus the writing in this series is just a step further from the stage of brain-storming, really speaking.

There is no direct discussion in this post regarding the learnability issue at all. However, the points we note here are crucial to understanding Godel’s incompleteness theorem, and in that sense, the contents of this post are crucially important in framing the learnability issue right.

Anyway, let’s get going over the differences of sets and concepts.


A concept as an abstract unit of mental integration:

Concepts are mental abstractions. It is true that concepts, once formed, can themselves be regarded as mental units, and qua units, they can further be integrated together into even higher-level concepts, or possibly sub-divided into narrower concepts. However, regardless of the level of abstraction at which a given concept exists, the concretes being subsumed under it are necessarily required to be less abstract than the single mental unit that is the concept itself.

Using the terms of computer science, the “graph” of a concept and its associated concrete units is not only acyclic and directional (from the concretes to the higher-level mental abstraction that is the concept), its connections too can be drawn if and only if the concretes satisfy the rules of conceptual commensurability.

A concept is necessarily a mental abstraction, and as a unit of mental integration, it always exists at a higher level of abstraction as compared to the units it subsumes.


A set as a mathematical object that is just a concrete collection:

Sets, on the other hand, necessarily are just concrete objects in themselves, even if they do represent collections of other concrete objects. Sets take birth as concrete objects—i.e., as objects that don’t have to represent any act of mental isolation and integration—and they remain that way till the end of their life.

For the same reason, set theory carries absolutely no rules whereby constraints can be placed on combining sets. No meaning is supposed to be assigned to the very act of placing braces around the rule which defines admissibility of objects as members into a set (or that of enumeration of their member objects).

The act of creating the collection that is a set is formally allowed to proceed even in the absence of any preceding act of mental differentiations and integrations.

This distinction between these two ideas, the idea of a concept, and that of a set, is important to grasp.


An instance of a mental abstraction vs. a membership into a concrete collection:

In the last post in this series, I had used the terminology in a particular way: I had said that there is a concept “table,” and that there is a set of “tables.” The plural form for the idea of the set was not a typo; it was a deliberate device to highlight this same significant point, viz., the essential concreteness of any set.

The mathematical theory of sets didn’t have to be designed this way, but given the way it anyway has actually been designed, one of the inevitable implications of its conception—its very design—has been this difference which exists between the ideas of concepts and sets. Since this difference is extremely important, it may be worth our while to look at it from yet another viewpoint.

When we look at a table and, having already had reached the concept of “table” we affirm that the given concrete table in front of us is indeed a table, this seemingly simple and almost instantaneously completed act of recognition itself implicitly involves a complex mental process. The process includes invoking a previously generated mental integration—an integration which was, sometime in the past, performed in reference to those attributes which actually exist in reality and which make a concrete object a table. The process begins with the availability of this context as a pre-requisite, and now involves an application of the concept. It involves actively bringing forth the pre-existing mental integration, actively “see” that yet another concrete instance of a table does indeed in reality carry the attributes which make an object a table, and thereby concluding that it is a table.

In other words, if you put the concept table symbolically as:

table = { this table X, that table Y, now yet another table Z, … etc. }

then it is understood that what the symbol on the left hand side stands for is a mental integration, and that each of the concrete entities X, Y, Z, etc. appearing in the list on the right hand-side is, by itself, an instance corresponding to that unit of mental integration.

But if you interpret the same “equation” as one standing for the set “tables”, then strictly speaking, according to the actual formalism of the set theory itself (i.e., without bringing into the context any additional perspective which we by habit do, but sticking strictly only to the formalism), each of the X, Y, Z etc. objects remains just a concrete member of a merely concrete collection or aggregate that is the set. The mental integration which regards X, Y, Z as equally similar instances of the idea of “table” is missing altogether.

Thus, no idea of similarity (or of differences) among the members at all gets involved, because there is no mental abstraction: “table” in the first place. There are only concrete tables, and there is a well-specified but concrete object, a collective, which is only formally defined to be stand for this concrete collection (of those specified tables).

Grasp this difference, and the incompleteness paradox brought forth by Godel begins to dissolve away.


The idea of an infinite set cuts out the preceding theoretical context:

Since the aforementioned point is complex but important, there is no risk in repeating (though there could be boredom!):

There is no place-holder in the set theory which would be equivalent to saying: “being able to regard concretes as the units of an abstract, singular, mental perspective—a perspective reached in recognition of certain facts of reality.”

The way set theory progresses in this regard is indeed extreme. Here is one way to look at it.

The idea of an infinite set is altogether inconceivable before you first have grasped the concept of infinity. On the other hand, grasping the concept of infinity can be accomplished without any involvement of the set theory anyway—formally or informally. However, since every set you actually observe in the concrete reality can only be finite, and since sets themselves are concrete objects, there is no way to conceive of the very idea of an infinite sets, unless you already know what infinity means (at least in some working, implicit, sense). Thus, to generate the concrete members contained in the given infinite set, you of course need the conceptual knowledge of infinite sequences and series.

However, even if the set theory must use this theoretical apparatus of analysis, the actual mathematical object it ends up having still captures only the “concrete-collection” aspect of it—none other. In other words, the set theory drops from its very considerations some of the crucially important aspects of knowledge with which infinite sets can at all be conceived of. For instance, it drops the idea that the infinite set-generating rule is in itself an abstraction. The set theory asks you to supply and use that rule. The theory itself is merely content in being supplied some well-defined entities as the members of a set.

It is at places like this that the infamous incompleteness creeps into the theory—I mean, the theory of sets, not the theory that is the analysis as was historically formulated and practiced.


The name of a set vs. the word that stands for a concept:

The name given to a set (the symbol or label appearing on the left hand-side of the equation) is just an arbitrary and concrete a label; it is not a theoretical place-holder for the corresponding mental concept—not so long as you remain strictly within the formalism, and therefore, the scope of application of, the set theory.

When they introduce you to the set theory in your high-school, they take care to choose each of the examples only such a way that there always is an easy-to-invoke and well-defined concept; this per-existing concept can then be put into a 1:1 correspondence with the definition of that particular set.

But if you therefore begin thinking that there is a well-defined concept for each possible instance of a set, then such a characterization is only a figment of your own imagination. An idea like this is certainly not to be found in the actual formalism of the set theory.

Show me the place in the axioms, or their combinations, or theorems, or even just lemmas or definitions in the set theory where they say that the label for a set, or the rule for formation of a set, must always stand for a conceptually coherent mental integration. Such an idea is simply absent from the mathematical theory.

The designers of the set theory, to put it directly, simply didn’t have the wits to include such ideas in their theory.


Implications for the allowed operations:

The reason why the set theory allows for any arbitrary operands (including those which don’t make any sense in the real world) is, thus, not an accident. It is a direct consequence of the fact that sets are, by design, concrete aggregates, not mental integrations based on certain rules of cognition (which in turn must make a reference to the actual characteristics and attributes possessed by the actually existing objects).

Since sets are mere aggregations, not integrations, as a consequence, we no longer remain concerned with the fact that there have to be two or more common characteristics to the concrete objects being put together, or with the problem of having to pick up the most fundamental one among them.

When it comes to sets, there are no such constraints on the further manipulations. Thus arises the possibility of being apply any operator any which way you feel like on any given set.


Godel’s incompleteness theorem as merely a consequence:

Given such a nature of the set theory—its glaring epistemological flaws—something like Kurt Godel’s incompleteness theorem had to arrive in the scene, sooner or later. The theorem succeeds only because the set theory (on which it is based) does give it what it needs—viz., a loss of a connection between a word (a set label) and how it is meant to be used (the contexts in which it can be further used, and how).


In the next part, we will reiterate some of these points by looking at the issue of (i) systems of axioms based on the set theory on the one hand, and (ii) the actual conceptual body of knowledge that is arithmetic, on the other hand. We will recast the discussion so far in terms of the “is a” vs. the “has a” types of relationships. The “is a” relationship may be described as the “is an instance of a mental integration or concept of” relationship. The “has a” relationship may be described as “is (somehow) defined (in whatever way) to carry the given concrete” type of a relationship. If you are curious, here is the preview: concepts allow for both types of relationships to exist; however, for defining a concept, the “is an instance or unit of” relationship is crucially important. In contrast, the set theory requires and has the formal place for only the “has a” type of relationships. A necessary outcome is that each set itself must remain only a concrete collection.

 

Absolutely Random Notings on QM—Part 1: Bohr. And, a bad philosophy making its way into physics with his work, and his academic influence

TL;DR: Go—and keep—away.


I am still firming up my opinions. However, there is never a harm in launching yet another series of posts on a personal blog, is there? So here we go…


Quantum Mechanics began with Planck. But there was no theory of quanta in what Planck had offered.

What Planck had done was to postulate only the existence of the quanta of the energy, in the cavity radiation.

Einstein used this idea to predict the heat capacities of solids—a remarkable work, one that remains underappreciated in both text-books as well as popular science books on QM.

The first pretense at a quantum theory proper came from Bohr.


Bohr was thinking not about the cavity radiations, but about the spectra of the radiations emitted or absorbed by gases.

Matter, esp. gases, following Dalton, …, Einstein, and Perin, were made of distinct atoms. The properties of gases—especially the reason why they emitted or absorbed radiation only at certain distinct frequencies, but not at any other frequencies (including those continuous patches of frequencies in between the experimentally evident sharp peaks)—had to be explained in reference to what the atoms themselves were like. There was no other way out—not yet, not given the sound epistemology in physics of those days.

Thinking up a new universe still was not allowed back then in science let alone in physics. One still had to clearly think about explaining what was given in observations, what was in evidence. Effects still had be related back to causes; outward actions still had to be related back to the character/nature of the entities that thus acted.

The actor, unquestionably by now, was the atom. The effects were the discrete spectra. Not much else was known.

Those were the days were when the best hotels and restaurants in Berlin, London, and New York would have horse-driven buggies ushering in the socially important guests. Buggies still was the latest technology back then. Not many people thus ushered in are remembered today. But Bohr is.


If the atom was the actor, and the effects under study were the discrete spectra, then what was needed to be said, in theory, was something regarding the structure of the atom.

If an imagined entity sheer by its material/chemical type doesn’t do it, then it’s the structure—its shape and size—which must do it.

Back then, this still was regarded as one of the cardinal principles of science, unlike the mindless opposition to the science of Homeopathy today, esp. in the UK. But back then, it was known that one important reason that Calvin gets harassed by the school bully was that not just the sheer size of the latter’s matter but also that the structure of the latter was different. In other words: If you consumed alcohol, you simply didn’t take in so many atoms of carbon as in proportion to so many atoms of hydrogen, etc. You took in a structure, a configuration with which these atoms came in.


However, the trouble back then was, none had have the means to see the atoms.

If by structure you mean the geometrical shape and size, or some patterns of density, then clearly, there was no experimental observations pertaining to the same. The only relevant observation available to people back then was what had already been encapsulated in Rutherford’s model, viz., the incontestable idea that the atomic nucleus had to be massive and dense, occupying a very small space as compared to an atom taken as a whole; the electrons had to carry very little mass in comparison. (The contrast of Rutherford’s model of c. 1911 was to the earlier plum cake model by Thomson.)

Bohr would, therefore, have to start with Rutherford’s model of atoms, and invent some new ideas concerning it, and see if his model was consistent with the known results given by spectroscopic observations.

What Bohr offered was a model for the electrons contained in a nuclear atom.


However, even while differing from the Rutherford’s plum-cake model, Bohr’s model emphatically lacked a theory for the nature of the electrons themselves. This part has been kept underappreciated by the textbook authors and science teachers.

In particular, Bohr’s theory had absolutely no clue as to the process according to which the electrons could, and must, jump in between their stable orbits.


The meat of the matter was worse, far worse: Bohr had explicitly prohibited from pursuing any mechanism or explanation concerning the quantum jumps—an idea which he was the first to propose. [I don’t know of any one else originally but independently proposing the same idea.]

Bohr achieved this objective not through any deployment of the best possible levels of scientific reason but out of his philosophic convictions—the convictions of the more irrational kind. The quantum jumps were obviously not observable, according to him, only their effects were. So, strictly speaking, the quantum jumps couldn’t possibly be a part of his theory—plain and simple!

But then, Bohr in his philosophic enthusiasm didn’t stop just there. He went even further—much further. He fully deployed the powers of his explicit reasoning as well as the weight of his seniority in prohibiting the young physicists from even thinking of—let alone ideating or offering—any mechanism for such quantum jumps.

In other words, Bohr took special efforts to keep the young quantum enthusiasts absolutely and in principle clueless, as far as his quantum jumps were concerned.


Bohr’s theory, in a sense, was in line with the strictest demands of the philosophy of empiricism. Here is how Bohr’s application of this philosophy went:

  1. This electron—it can be measured!—at this energy level, now!
  2. [May be] The same electron, but this energy level, now!
  3. This energy difference, this frequency. Measured! [Thank you experimental spectroscopists; hats off to you, for, you leave Bohr alone!!]
  4. OK. Now, put the above three into a cohesive “theory.” And, BTW, don’t you ever even try to think about anything else!!

Continuing just a bit on the same lines, Bohr sure would have said (quoting Peikoff’s explanation of the philosophy of empiricism):

  1. [Looking at a tomato] We can only say this much in theory: “This, now, tomato!”
  2. Making a leeway for the most ambitious ones of the ilk: “This *red* tomato!!”

Going by his explicit philosophic convictions, it must have been a height of “speculation” for Bohr to mumble something—anything—about a thing like “orbit.” After all, even by just mentioning a word like “orbit,” Bohr was being absolutely philosophically inconsistent here. Dear reader, observe that the orbit itself never at all was an observable!

Bohr must have in his conscience convulsed at this fact; his own philosophy couldn’t possibly have, strictly speaking, permitted him to accommodate into his theory a non-measurable feature of a non-measurable entity—such as his orbits of his electrons. Only the allure of outwardly producing predictions that matched with the experiment might have quietened his conscience—and that too, temporarily. At least until he got a new stone-building housing an Institute for himself and/or a Physics Nobel, that is.

Possible. With Herr Herr Herr Doktor Doktor Doktor Professor Professors, anything is possible.


It is often remarked that the one curious feature of the Bohr theory was the fact that the stability of the electronic orbits was postulated in it, not explained.

That is, not explained in reference to any known physical principle. The analogy to the solar system indeed was just that: an analogy. It was not a reference to an established physical principle.

However, the basically marvelous feature of the Bohr theory was not that the orbits were stable (in violation of the known laws of electrodynamics). It was: there at all were any orbits in it, even if no experiment had ever given any evidence for the continuously or discontinuously subsequent positions electrons within an atom or of their motions.

So much for originator of the cult of sticking only to the “observables.”


What Sommerfeld did was to add footnotes to Bohr’s work.

Sommerfeld did this work admirably well.

However, what this instance in the history of physics clearly demonstrates is yet another principle from the epistemology of physics: how a man of otherwise enormous mathematical abilities and training (and an academically influential position, I might add), but having evidently no remarkable capacity for a very novel, breakthrough kind of conceptual thinking, just cannot but fall short of making any lasting contributions to physics.

“Math” by itself simply isn’t enough for physics.

What came to be known as the old quantum theory, thus, faced an impasse.

Under Bohr’s (and philosophers’) loving tutorship, the situation continued for a long time—for more than a decade!


A Song I Like:

(Marathi) “sakhi ga murali mohan mohi manaa…”
Music: Hridaynath Mangeshkar
Singer: Asha Bhosale
Lyrics: P. Savalaram


PS: Only typos and animals of the similar ilk remain to be corrected.

 

My small contribution towards the controversies surrounding the important question of “1, 2, 3, …”

As you know, I have been engaged in writing about scalars, vectors, tensors, and CFD.

However, at the same time, while writing my notes, I also happened to think of the “1, 2, 3, …” controversy. Here is my small, personal, contribution to the same.


The physical world evidently consists of a myriad variety of things. Attributes are the metaphysically inseparable aspects that together constitute the identity of a thing. To exist is to exist with all the attributes. But getting to know the identity of a thing does not mean having a knowledge of all of its attributes. The identity of a thing is grasped, or the thing is recognized, on the basis of just a few attributes/characteristics—those which are the defining attributes (including properties, characteristics, actions, etc.), within a given context.

Similarities and differences are perceptually evident. When two or more concretely real things possess the same attribute, they are directly perceived as being similar. Two mangoes are similar, and so are two bananas. The differences between two or more things of the same kind are the differences in the sizes of those attribute(s) which are in common to them. All mangoes share a great deal of attributes between them, and the differences in the two mangoes are not just the basic fact that they are two separate mangoes, but also that they differ in their respective colors, shapes, sizes, etc.

Sizes or magnitudes (lit.: “bigness”) refer to sizes of things; sizes do not metaphysically exist independent of the things of which they are sizes.

Numbers are the concepts that can be used to measure the sizes of things (and also of their attributes, characteristics, actions, etc.).


It is true that sizes can be grasped and specified without using numbers.

For instance, we can say that this mango is bigger than that. The preceding statement did not involve any number. However, it did involve a comparative statement that ordered two different things in accordance with the sizes of some common attribute possessed by each, e.g., the weight of, or the volume occupied by, each of the two mangoes. In the case of concrete objects such as two mangoes differing in size, the comparative differences in their sizes are grasped via direct perception; one mango is directly seen/felt as being bigger than the other; the mental process involved at this level is direct and automatic.

A certain issue arises when we try to extend the logic to three or more mangoes. To say that the mango A is bigger than the mango B, and that the mango B is bigger than the mango C, is perfectly fine.

However, it is clear from common experience that the size-wise difference between A and B may not exactly be the same as the size-wise difference between B and C. The simple measure: “is bigger than”, thus, is crude.

The idea of numbers is the means through which we try to make the quantitative comparative statements more refined, more precise, more accurately capturing of the metaphysically given sizes.

An important point to note here is that even if you use numbers, a statement involving sizes still remains only a comparative one. Whenever you say that something is bigger or smaller, you are always implicitly adding: as in comparison to something else, i.e., some other thing. Contrary to what a lot of thinkers have presumed, numbers do not provide any more absolute a standard than what is already contained in the comparisons on which a concept of numbers is based.


Fundamentally, an attribute can metaphysically exist only with some definite size (and only as part of the identity of the object which possesses that attribute). Thus, the idea of a size-less attribute is a metaphysical impossibility.

Sizes are a given in the metaphysical reality. Each concretely real object by itself carries all the sizes of all its attributes. An existent or an object, i.e., when an object taken singly, separately, still does possess all its attributes, with all the sizes with which it exists.

However, the idea of measuring a size cannot arise in reference to just a single concrete object. Measurements cannot be conducted on single objects taken out of context, i.e., in complete isolation of everything else that exists.

You need to take at least two objects that differ in sizes (in the same attribute), and it is only then that any quantitative comparison (based on that attribute) becomes possible. And it is only when some comparison is possible that a process for measurements of sizes can at all be conceived of. A process of measurement is a process of comparison.

A number is an end-product of a certain mathematical method that puts a given thing in a size-wise quantitative relationship (or comparison) with other things (of the same kind).


Sizes or magnitudes exist in the raw nature. But numbers do not exist in the raw nature. They are an end-product of certain mathematical processes. A number-producing mathematical process pins down (or defines) some specific sense of what the size of an attribute can at all be taken to mean, in the first place.

Numbers do not exist in the raw nature because the mathematical methods which produce them themselves do not exist in the raw nature.

A method for measuring sizes has to be conceived of (or created or invented) by a mind. The method settles the question of how the metaphysically existing sizes of objects/attributes are to be processed via some kind of a comparison. As such, sure, the method does require a prior grasp of the metaphysical existents, i.e., of the physical reality.

However, the meaning of the method proper itself is not to be located in the metaphysically differing sizes themselves; it is to be located in how those differences in sizes are grasped, processed, and what kind of an end-product is produced by that process.

Thus, a mathematical method is an invention of using the mind in a certain way; it is not a discovery of some metaphysical facts existing independent of the mind grasping (and holding, using, etc.) it.

However, once invented by someone, the mathematical method can be taught to others, and can be used by all those who do know it, but only in within the delimited scope of the method itself, i.e., only in those applications where that particular method can at all be applied.


The simplest kind of numbers are the natural numbers: 1, 2, 3, \dots. As an aside, to remind you, natural numbers do not include the zero; the set of whole numbers does that.

Reaching the idea of the natural numbers involves three steps:

(i) treating a group of some concrete objects of the same kind (e.g. five mangoes) as not only a collection of so many separately existing things, but also as if it were a single, imaginary, composite object, when the constituent objects are seen as a group,

(ii) treating a single concrete object (of the same aforementioned kind, e.g. one mango) not only as a separately existing concrete object, but also as an instance of a group of the aforementioned kind—i.e. a group of the one,

and

(iii) treating the first group (consisting of multiple objects) as if it were obtained by exactly/identically repeating the second group (consisting of a single object).

The interplay between the concrete perception on the one hand and a more abstract, conceptual-level grasp of that perception on the other hand, occurs in each of the first two steps mentioned above. (Ayn Rand: “The ability to regard entities as mental units \dots” [^].)

In contrast, the synthesis of a new mental process that is suitable for making quantitative measurements, which means the issue in the third step, occurs only at an abstract level. There is nothing corresponding to the process of repetition (or for that matter, to any method of quantitative measurements) in the concrete, metaphysically given, reality.

In the third step, the many objects comprising the first group are regarded as if they were exact replicas of the concrete object from the second (singular) group.

This point is important. Primitive humans would use some uniform-looking symbols like dots (.) or circles (\bullet) or sticks (`|‘), to stand for the concrete objects that go in making up either of the aforementioned two groups—the group of the many mangoes vs. the group of the one mango. Using the same symbol for each occurrence of a concrete object underscores the idea that all other facts pertaining to those concrete objects (here, mangoes) are to be summarily disregarded, and that the only important point worth retaining is that a next instance of an exact replica (an instance of an abstract mango, so to speak) has become available.

At this point, we begin representing the group of five mangoes as G_1 = \lbrace\, \bullet\,\bullet\,\bullet\,\bullet\,\bullet\, \rbrace, and the single concretely existing mango as a second abstract group: G_2 = \lbrace\,\bullet\,\rbrace.


Next comes a more clear grasp of the process of repetition. It is seen that the process of repetition can be stopped at discrete stages. For instance:

  1. The process P_1 produces \lbrace\,\bullet\,\rbrace (i.e. the repetition process is stopped after taking \bullet once).
  2. The process P_2 produces \lbrace\,\bullet\,\bullet\,\rbrace (i.e. the repetition process is stopped after taking \bullet twice)
  3. The process P_3 produces \lbrace\,\bullet\,\bullet\,\bullet\,\rbrace (i.e. the repetition process is stopped after taking \bullet thrice)
    etc.

At this point, it is recognized that each output or end-product that a terminated repetition-process produces, is precisely identical to certain abstract group of objects of the first kind.

Thus, each of the P_1 \equiv \lbrace\,\bullet\,\rbrace, or P_2 \equiv \lbrace\,\bullet\,\bullet\,\rbrace, or  P_3 \equiv \lbrace\,\bullet\,\bullet\,\bullet\,\rbrace, $\dots$ is now regarded as if it were a single (composite) object.

Notice how we began by saying that P_1, P_2, P_3 etc. were processes, and then ended up saying that we now see single objects in them.

Thus, the size of each abstract group of many objects (the groups of one, of two, of three, of n objects) gets tied to a particular length of a terminated process, here, of repetitions. As the length of the process varies, so does the size of its output i.e. the abstract composite object.

It is in this way that a process (here, of repetition) becomes capable of measuring the size of the abstract composite object. And it does so in reference to the stage (or the length of repetitions) at which the process was terminated.

It is thus that the repetition process becomes a process of measuring sizes. In other words, it becomes a method of measurement. Qua a method of measurement, the process has been given a name: it is called “counting.”

The end-products of the terminated repetition process, i.e., of the counting process, are the mathematical objects called the natural numbers.


More generally, what we said for the natural numbers also holds true for any other kind of a number. Any kind of a number stands for an end-product that is obtained when a well-defined process of measurement is conducted to completion.

An uncompleted process is just that: a process that is still continuing. The notion of an end-product applies only to a process that has come to an end. Numbers are the end-products of size-measuring processes.

Since an infinite process is not a completed process, infinity is not a number; it is merely a short-hand to denote some aspect of the measurement process other than the use of the process in measuring a size.

The only valid use of infinity is in the context of establishing the limiting values of sequences, i.e., in capturing the essence of the trend in the numbers produced by the nature (or identity) of a given sequence-producing process.

Thus, infinity is a concept that helps pin down the nature of the trend in the numbers belonging to a sequence. On the other hand, a number is a product of a process when it is terminated after a certain, definite, length.

With the concept of infinity, the idea that the process never terminates is not crucial; the crucial thing is that you reach an independence  from the length of a sequence. Let me give you an example.

Consider the sequence for which the n-th term is given by the formula:

S_n = \dfrac{1}{n}.

Thus, the sequence is: 1, \dfrac{1}{2}, \dfrac{1}{3}, \dfrac{1}{4}, \dots.

If we take first two terms, we can see that the value has decreased, from 1 to 0.5. If we go from the second to the third term, we can see that the value has decreased even further, to 0.3333. The difference in the decrement has, however, dropped; it has gone from 1 - \dfrac{1}{2} = 0.5 to \dfrac{1}{2} - \dfrac{1}{3} =  0.1666666\dots. Go from the third to the fourth term, and we can see that while the value goes still down, and the decrement itself also has decreased, it has now become 0.08333 . Thus, two trends are unmistakable: (i) the value keeps dropping, but (ii) the decrement also becomes sluggish.  If the values were to drop uniformly, i.e. if the decrement were to stay the same, we would have immediately hit 0, and then gone on to the negative numbers. But the second factor, viz., that the decrement itself is progressively decreasing, seems to play a trick. It seems intent on keeping you afloat, above the 0 value. We can verify this fact. No matter how big n might get, it still is a finite number, and so, its reciprocal is always going to be a finite number, not zero. At the same time, we now have observed that the differences between the subsequent reciprocals has been decreasing. How can we capture this intuition? What we want to say is this: As you go further and further down in the sequence, the value must become smaller and ever smaller. It would never actually become 0. But it will approach 0 (and no number other than 0) better and still better. Take any small but definite positive number, and we can say that our sequence would eventually drop down below the level of that number, in a finite number of steps. We can say this thing for any given definite positive number, no matter how small. So long as it is a definite number, we are going to hit its level in a finite number of steps. But we also know that since n is positive, our sequence is never going to go so far down as to reach into the regime of the negative numbers. In fact, as we just said, let alone the range of the negative numbers, our sequence is not going to hit even 0, in finite number of steps.

To capture all these facts, viz.: (i) We will always go below the level any positive real number R, no matter how small R may be, in a finite number of steps, (ii) the number of steps n required to go below a specified R level would always go on increasing as R becomes smaller, and (iii) we will never reach 0 in any finite number of steps no matter how large n may get, but will always experience decrement with increasing n, we say that:

the limit of the sequence S_n as n approaches infinity is 0.

The word “infinity” in the above description crucially refers to the facts (i) and (ii), which together clearly establish the trend in the values of the sequence S_n. [The fact (iii) is incidental to the idea of “infinity” itself, though it brings out a neat property of limits, viz., the fact that the limit need not always belong to the set of numbers that is the sequence itself. ]


With the development of mathematical knowledge, the idea of numbers does undergo changes. The concept number gets more and more complex/sophisticated, as the process of measurement becomes more and more complex/sophisticated.

We can form the process of addition starting from the process of counting.

The simplest addition is that of adding a unit (or the number 1) to a given number. We can apply the process of addition by 1, to the number 1, and see that the number we thus arrive at is 2. Then we can apply the process of addition by 1, to the number 2, and see that the number we thus arrive at is 3. We can continue to apply the logic further, and thereby see that it is possible to generate any desired natural number.

The so-called natural numbers thus state the sizes of groups of identical objects, as measured via the process of counting. Since natural numbers encapsulate the sizes of such groups, they obviously can be ordered by the sizes they encapsulate. One way to see how the order 1, then 2, then 3, \dots, arises is to observe that in successively applying the process of addition starting from the number 1, it is the number 2 which comes immediately after the number $1$, but before the number 3, etc.

The process of subtraction is formed by inverting the process of addition, i.e., by seeing the logic of addition in a certain, reverse, way.

The process of addition by 1, when repeatedly applied to a given natural number, is capable of generating all the natural numbers greater than the given number. The process of subtraction by 1, when repeatedly applied to a given natural number, is capable of generating all the natural numbers smaller than the given number.

When the process of subtraction by 1 is applied right to the number 1 itself, we reach the idea of the zero. [Dear Indian, now you know that the idea of the number zero was not all that breath-taking, was it?]

In a further development, the idea of the negative numbers is established.


Thus, the concept of numbers develops from the natural numbers (1, 2, 3, \dots) to whole numbers (0, 1, 2, \dots) to integers (\dots, -2, -1, 0, 1, 2, \dots).

At each such a stage, the idea of what a number means—its definition—undergoes a definite change; at any such a stage, there is a well-defined mathematical process, of increasing conceptual complexity, of measuring sizes, whose end-products that idea of numbers represents.


The idea of multiplication follows from that of repeated additions; the idea of division follows from that of the repeated subtractions; the two process are then recognized as the multiplicative inverses of each other. It’s only then that the idea of fractions follows. The distinction between the rational and irrational fractions is then recognized, and then, the concept of numbers gets extended to include the idea of the irrational as well as rational numbers.

A crucial lesson learnt from this entire expansion of knowledge of what it means to be a number, is the recognition of the fact that for any well-defined and completed process of measurement, there must follow a certain number (and only that unique number, obviously!).


Then, in a further, distinct, development, we come to recognize that while some process must exist to produce a number, any well-defined process producing a number would do just as well.

With this realization, we then come to a stage whereby, we can think of conceptually omitting specifying any specific process of measurement.

We thus come to retain only the fact while some process must be specified, any valid process can be, and then, the end-product still would be just a number.

It is with this realization that we come to reach the idea of the real numbers.


The purpose of forming the idea of real numbers is that they allow us to form statements that would hold true for any number qua a number.


The crux of the distinction of the real numbers from any of the preceding notion of numbers (natural, whole, integers) is the following statement, which can be applied to real numbers, and only to real numbers—not to integers.

The statement is this: there is an infinity of real numbers existing between any two distinct real numbers R_1 and R_2, no matter how close they might be to each other.

There is a wealth of information contained in that statement, but if some aspects are to be highlighted and appreciated more than the others, they would be these:

(i) Each of the two numbers R_1 and R_2 are recognized as being an end-product of some or the other well-defined process.

The responsibility of specifying what precise size is meant when you say R_1 or R_2 is left entirely up to you; the definition of real numbers does not take that burden. It only specifies that some well-defined process must exist to produce R_1 as well as R_2, so that what they denote indeed are numbers.

A mathematical process may produce a result that corresponds to a so-called “irrational” number, and yet, it can be a definite process. For instance, you may specify the size-measurement process thus: hold in a compass the distance equal to the diagonal of a right-angled isoscales triangle having the equal sides of 1, and mark this distance out from the origin on the real number-line. This measurement process is well-specified even if \sqrt{2} can be proved to be an irrational number.

(ii) You don’t have to specify any particular measurement process which might produce a number strictly in between R_1 and R_2, to assert that it’s a number. This part is crucial to understand the concept of real numbers.

The real numbers get all their power precisely because their idea brings into the jurisdiction of the concept of numbers not only all those specific definitions of numbers that have been invented thus far, but also all those definitions which ever possibly would be. That’s the crucial part to understand.

The crucial part is not the fact that there are an infinity of numbers lying between any two R_1 and R_2. In fact, the existence of an infinity of numbers is damn easy to prove: just take the average of R_1 and R_2 and show that it must fall strictly in between them—in fact, it divides the line-segment from R_1 to R_2 into two equal halves. Then, take each half separately, and take the average of its end-points to hit the middle point of that half. In the first step, you go from one line-segment to two (i.e., you produce one new number that is the average). In the next step, you go from the two segments to the four (i.e. in all, three new numbers). Now, go easy; wash-rinse-repeat! … The number of the numbers lying strictly between R_1 and R_2 increases without bound—i.e., it blows “up to” infinity. [Why not “down to” infinity? Simple: God is up in his heavens, and so, we naturally consider the natural numbers rather than the negative integers, first!]

Since the proof is this simple, obviously, it just cannot be the real meat, it just cannot be the real reason why the idea of real numbers is at all required.

The crucial thing to realize here now is this part: Even if you don’t specify any specific process like hitting the mid-point of the line-segment by taking average, there still would be an infinity of numbers between the end-points.


Another closely related and crucial thing to realize is this part: No matter what measurement (i.e. number-producing) process you conceive of, if it is capable of producing a new number that lies strictly between the two bounds, then the set of real numbers has already included it.

Got it? No? Go read that line again. It’s important.

This idea that

“all possible numbers have already been subsumed in the real numbers set”

has not been proven, nor can it be—not on the basis of any of the previous notions of what it means to be a number. In fact, it cannot be proven on the basis of any well-defined (i.e. specified) notion of what it means to be a number. So long as a number-producing process is specified, it is known, by the very definition of real numbers, that that process would not exhaust all real numbers. Why?

Simple. Because, someone can always spin out yet another specific process that generates a different set of numbers, which all would still belong only to the real number system, and your prior process didn’t cover those numbers.

So, the statement cannot be proven on the basis of any specified system of producing numbers.

Formally, this is precisely what [I think] is the issue at the core of the “continuum hypothesis.”

The continuum hypothesis is just a way of formalizing the mathematician’s confidence that a set of numbers such as real numbers can at all be defined, that a concept that includes all possible numbers does have its uses in theory of measurements.

You can’t use the ideas like some already defined notions of numbers in order to prove the continuum hypothesis, because the hypothesis itself is at the base of what it at all means to be a number, when the term is taken in its broadest possible sense.


But why would mathematicians think of such a notion in the first place?

Primarily, so that those numbers which are defined only as the limits (known or unknown, whether translatable using the already known operations of mathematics or otherwise) of some infinite processes can also be treated as proper numbers.

And hence, dramatically, infinite processes also can be used for measuring sizes of actual, metaphysically definite and mathematically finite, objects.

Huh? Where’s the catch?

The catch is that these infinite processes must have limits (i.e., they must have finite numbers as their output); that’s all! (LOL!).


It is often said that the idea of real numbers is a bridge between algebra and geometry, that it’s the counterpart in algebra of what the geometer means by his continuous curve.

True, but not quite hitting the bull’s eye. Continuity is a notion that geometer himself cannot grasp or state well unless when aided by the ideas of the calculus.

Therefore, a somewhat better statement is this: the idea of the real numbers is a bridge between algebra and calculus.

OK, an improvement, but still, it, too, misses the mark.

The real statement is this:

The idea of real numbers provides the grounds in algebra (and in turn, in the arithmetics) so that the (more abstract) methods such as those of the calculus (or of any future method that can ever get invented for measuring sizes) already become completely well-defined qua producers of numbers.

The function of the real number system is, in a way, to just go nuts, just fill the gaps that are (or even would ever be) left by any possible number system.


In the preceding discussion, we had freely made use of the 1:1 correspondence between the real numbers and the beloved continuous curve of our school-time geometry.

This correspondence was not always as obvious as it is today; in fact, it was a towering achievement of, I guess, Descartes. I mean to say, the algebra-ization of geometry.

In the simplest (1D) case, points on a line can be put in 1:1 correspondence with real numbers, and vice-versa. Thus, for every real number there is one and only one point on the real-number line, and for any point actually (i.e. well-) specified on the real number-line, there is one and only one real number corresponding to it.

But the crucial advancement represented by the idea of real numbers is not that there is this correspondence between numbers (an algebraic concept) and geometry.

The crux is this: you can (or, rather, you are left free to) think of any possible process that ends up cutting a given line segment into two (not necessarily equal) halves, and regardless of the particular nature of that process, indeed, without even having to know anything about its particular nature, we can still make a blanket statement:

if the process terminates and ends up cutting the line segment at a certain geometrical point, then the number which corresponds to that geometrical point is already included in the infinite set of real numbers.


Since the set of real numbers exhausts all possible end-products of all possible infinite limiting processes too, it is fully capable of representing any kind of a continuous change.


We in engineering often model the physical reality using the notion of the continuum.

Inasmuch as it’s a fact that to any arbitrary but finite part of a continuum there does correspond a number, when we have the real number system at hand, we already know that this size is already included in the set of real numbers.

Real numbers are indispensable to us the engineers—theoretically speaking. It gives us the freedom to invent any new mathematical methods for quantitatively dealing with continua, by giving us the confidence that all that they would produce, if valid, is already included in the numbers-set we already use; that our numbers-set will never ever let us down, that it will never ever fall short, that we will never ever fall in between the two stools, so to speak. Yes, we could use even the infinite processes, such as those of the calculus, with confidence, so long as they are limiting.

That’s the [theoretical] confidence which the real number system brings us [the engineers].


A Song I Don’t Like:

[Here is a song I don’t like, didn’t ever like, and what’s more, I am confident, I would never ever like either. No, neither this part of it nor that. I don’t like any part of it, whether the partition is made “integer”-ly, or “real”ly.

Hence my confidence. I just don’t like it.

But a lot of Indian [some would say “retards”] do, I do acknowledge this part. To wit [^].

But to repeat: no, I didn’t, don’t, and wouldn’t ever like it. Neither in its 1st avataar, nor in the 2nd, nor even in an hypothetically \pi-th avataar. Teaser: Can we use a transcendental irrational number to denote the stage of iteration? Are fractional derivatives possible?

OK, coming back to the song itself. Go ahead, listen to it, and you will immediately come to know why I wouldn’t like it.]

(Hindi) “1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 \n …” [OK, yes, read the finite sequence before the newline character, using Hindi.]
Credits: [You go hunt for them. I really don’t like it.]


PS: As usual, I may come back and make this post even better. BTW, in the meanwhile, I am thinking of relying on my more junior colleagues to keep me on the track towards delivering on the promised CFD FDP. Bye for now, and take care…

 

In maths, the boundary is…

In maths, the boundary is a verb, not a noun.

It’s an active something, that, through certain agencies (whose influence, in the usual maths, is wholly captured via differential equations) actually goes on to act [directly or indirectly] over the entirety of a [spatial] region.

Mathematicians have come to forget about this simple physical fact, but by the basic rules of knowledge, that’s how it is.

They love to portray the BV (boundary-value) problems in terms of some dead thing sitting at the boundary, esp. for the Dirichlet variety of problems (esp. for the case when the field variable is zero out there) but that’s not what the basic nature of the abstraction is actually like. You couldn’t possibly build the very abstraction of a boundary unless if first pre-supposed that what it in maths represented was an active [read: physically active] something!

Keep that in mind; keep on reminding yourself at least 10^n times every day, where n is an integer \ge 1.

 


A Song I Like:

[Unlike most other songs, this was an “average” one  in my [self-]esteemed teenage opinion, formed after listening to it on a poor-reception-area radio in an odd town at some odd times. … It changed for forever to a “surprisingly wonderful one” the moment I saw the movie in my SE (second year engineering) while at COEP. … And, haven’t yet gotten out of that impression yet… .]

(Hindi) “main chali main chali, peechhe peeche jahaan…”
Singers: Lata Mangeshkar, Mohammad Rafi
Music: Shankar-Jaikishan
Lyrics: Shailendra


[May be an editing pass would be due tomorrow or so?]

 

Off the blog. [“Matter” cannot act “where” it is not.]

I am going to go off the blogging activity in general, and this blog in most particular, for some time. [And, this time round, I will keep my promise.]


The reason is, I’ve just received the shipment of a book which I had ordered about a month ago. Though only about 300 pages in length, it’s going to take me weeks to complete. And, the book is gripping enough, and the issue important enough, that I am not going to let a mere blog or two—or the entire Internet—come in the way.


I had read it once, almost cover-to-cover, some 25 years ago, while I was a student in UAB.

Reading a book cover-to-cover—I mean: in-sequence, and by that I mean: starting from the front-cover and going through the pages in the same sequence as the one in which the book has been written, all the way to the back-cover—was quite odd a thing to have happened with me, at that time. It was quite unlike my usual habits whereby I am more or less always randomly jumping around in a book, even while reading one for the very first time.

But this book was different; it was extraordinarily engaging.

In fact, as I vividly remember, I had just idly picked up this book off a shelf from the Hill library of UAB, for a casual examination, had browsed it a bit, and then had began sampling some passage from nowhere in the middle of the book while standing in an library aisle. Then, some little time later, I was engrossed in reading it—with a folded elbow resting on the shelf, head turned down and resting against a shelf rack (due to a general weakness due to a physical hunger which I was ignoring [and I would have have to go home and cook something for myself; there was none to do that for me; and so, it was easy enough to ignore the hunger]). I don’t honestly remember how the pages turned. But I do remember that I must have already finished some 15-20 pages (all “in-the-order”!) before I even realized that I had been reading this book while still awkwardly resting against that shelf-rack. …

… I checked out the book, and once home [student dormitory], began reading it starting from the very first page. … I took time, days, perhaps weeks. But whatever the length of time that I did take, with this book, I didn’t have to jump around the pages.


The issue that the book dealt with was:

[Instantaneous] Action at a Distance.

The book in question was:

Hesse, Mary B. (1961) “Forces and Fields: The concept of Action at a Distance in the history of physics,” Philosophical Library, Edinburgh and New York.


It was the very first book I had found, I even today distinctly remember, in which someone—someone, anyone, other than me—had cared to think about the issues like the IAD, the concepts like fields and point particles—and had tried to trace their physical roots, to understand the physical origins behind these (and such) mathematical concepts. (And, had chosen to say “concepts” while meaning ones, rather than trying to hide behind poor substitute words like “ideas”, “experiences”, “issues”, “models”, etc.)

Twenty-five years later, I still remain hooked on to the topic. Despite having published a paper on IAD and diffusion [and yes, what the hell, I will say it: despite claiming a first in 200+ years in reference to this topic], I even today do find new things to think about, about this “kutty” [Original: IITM lingo; English translation: “small”] topic. And so, I keep returning to it and thinking about it. I still am able to gain new insights once in an odd while. … Indeed, my recent ‘net search on IAD (the one which led to Hesse and my buying the book) precisely was to see if someone had reported the conceptual [and of course, mathematical] observation which I have recently made, or not. [If too curious about it, the answer: looks like, none has.]


But now coming to Hesse’s writing style, let me quote a passage from one of her research papers. I ran into this paper only recently, last month (in July 2017), and it was while going through it that I happened [once again] to remember her book. Since I did have some money in hand, I did immediately decide to order my copy of this book.

Anyway, the paper I have in mind is this:

Hesse, Mary B. (1955) “Action at a Distance in Classical Physics,” Isis, Vol. 46, No. 4 (Dec., 1955), pp. 337–353, University of Chicago Press/The History of Science Society.

The paper (it has no abstract) begins thus:

The scholastic axiom that “matter cannot act where it is not” is one of the very general metaphysical principles found in science before the seventeenth century which retain their relevance for scientific theory even when the metaphysics itself has been discarded. Other such principles have been fruitful in the development of physics: for example, the “conservation of motion” stated by Descartes and Leibniz, which was generalized and given precision in the nineteenth century as the doctrine of the conservation of energy; …

Here is another passage, once again, from the same paper:

Now Faraday uses a terminology in speaking about the lines of force which is derived from the idea of a bundle of elastic strings stretched under tension from point to point of the field. Thus he speaks of “tension” and “the number of lines” cut by a body moving in the field. Remembering his discussion about contiguous particles of a dielectric medium, one must think of the strings as stretching from one particle of the medium to the next in a straight line, the distance between particles being so small that the line appears as a smooth curve. How seriously does he take this model? Certainly the bundle of elastic strings is nothing like those one can buy at the store. The “number of lines” does not refer to a definite number of discrete material entities, but to the amount of force exerted over a given area in the field. It would not make sense to assign points through which a line passes and points which are free from a line. The field of force is continuous.

See the flow of the writing? the authentic respect for the intellectual history, and yet, the overriding concern for having to reach a conclusion, a meaning? the appreciation for the subtle drama? the clarity of thought, of expression?

Well, these passages were from the paper, but the book itself, too, is similarly written.


Obviously, while I remain engaged in [re-]reading the book [after a gap of 25 years], don’t expect me to blog.

After all, even I cannot act “where” I am not.


A Song I Like:

[I thought a bit between this song and another song, one by R.D. Burman, Gulzar and Lata. In the end, it was this song which won out. As usual, in making my decision, the reference was exclusively made to the respective audio tracks. In fact, in the making of this decision, I happened to have also ignored even the excellent guitar pieces in this song, and the orchestration in general in both. The words and the tune were too well “fused” together in this song; that’s why. I do promise you to run the RD song once I return. In the meanwhile, I don’t at all mind keeping you guessing. Happy guessing!]

(Hindi) “bheegi bheegi…” [“bheege bheege lamhon kee bheegee bheegee yaadein…”]
Music and Lyrics: Kaushal S. Inamdar
Singer: Hamsika Iyer

[Minor additions/editing may follow tomorrow or so.]

 

Relating the One with the Many

0. Review and Context: This post is the last one in this mini-series on the subject of the one vs. many (as understood in the context of physics). The earlier posts in this series have been, in the chronological and logical order, these:

  1. Introducing a very foundational issue of physics (and of maths) [^]
  2. The One vs. the Many [^]
  3. Some of the implications of the “Many Objects” idea… [^]
  4. Some of the implications of the “One Object” idea… [^]

In the second post in this series, we had seen how a single object can be split up into many objects (or the many objects seen as parts of a single object). Now, in this post, we note some more observations about relating the One with the Many.

The description below begins with a discussion of how the One Object may be separated into Many Objects. However, note that the maths involved here is perfectly symmetrical, and therefore, the ensuing discussion for the separation of the one object into many objects also just as well applies for putting many objects together into one object, i.e., integration.


In the second and third posts, we handled the perceived multiplicity of objects via a spatial separation according to the varying measures of the same property. A few remarks on the process of separation (or, symmetrically, on the process of integration) are now in order.

1. The extents of spatial separation depends on what property you choose on the basis of which to effect the separation:

To begin with, note that the exact extents of any spatial separations would vary depending on what property you choose for measuring them.

To take a very “layman-like” example, suppose you take a cotton-seed, i.e. the one with a soft ball of fine cotton fibres emanating from a hard center, as shown here [^]. Suppose if you use the property of reflectivity (or, the ability to be seen in a bright light against a darker background), then for the cotton-seed, the width of the overall seed might come out to be, say, 5 cm. That is to say, the spatial extent ascribable to this object would be 5 cm. However, if you choose some other physical property, then the same object may end up registering quite a different size. For instance, if you use the property: “ability to be lifted using prongs” as the true measure for the width for the seed, then its size may very well come out as just about 1–2 cm, because the soft ball of the fibres would have got crushed to a smaller volume in the act of lifting.

In short: Different properties can easily imply different extensions for the same distinguished (or separated)“object,” i.e., for the same distinguished part of the physical universe.

2. The One Object may be separated into Many Objects on a basis other than that of the spatial separation:

Spatial attributes are fundamental, but they don’t always provide the best principle to organize a theory of physics.

The separation of the single universe-object into many of its parts need not proceed on the basis of only the “physical” space.

It would be possible to separate the universe on the basis of certain basis-functions which are defined over every spatial part of the universe. For instance, the Fourier analysis gives rise to a separation of a property-function into many complex-valued frequencies (viz. pairs of spatial undulations).

If the separation is done on the basis of such abstract functions, and not on the basis of the spatial extents, then the problem of the empty regions vaporizes away immediately. There always is some or the other “frequency”, with some or the other amplitude and phase, present at literally every point in the physical universe—including in the regions of the so-called “empty” space.

However, do note that the Fourier separation is a mathematical principle. Its correspondence to the physical universe must pass through the usual, required, epistemological hoops. … Here is one instance:

Question: If infinity cannot metaphysically exist (simply because it is a mathematical concept and no mathematical concept physically exists), then how is it that an infinite series may potentially be required for splitting up the given function (viz. the one which specifies the variations the given property of the physical universe)?

Answer: An infinite Fourier series cannot indeed be used by way of a direct physical description; however, a truncated (finite) Fourier series may be.

Here, we are basically relying on the same trick as we saw earlier in this mini-series of posts: We can claim that what the truncated Fourier series represents is the actual reality, and that that function which requires an infinite series is merely a depiction, an idealization, an abstraction.

3. When to use which description—the One Object or the Many Objects:

Despite the enormous advantages of the second approach (of the One Object idea) in the fundamental theoretical physics, in classical physics as well as in our “day-to-day” life, we often speak of the physical reality using the cruder first approach (the one involving the Many Objects idea). This we do—and it’s perfectly OK to do so—mainly because of the involved context.

The Many Objects description of physics is closer to the perceptual level. Hence, its more direct, even simpler, in a way. Now, note a very important consideration:

The precision to used in a description (or a theory) is determined by its purpose.

The purpose for a description may be lofty, such as achieving fullest possible consistency of conceptual interrelations. Or it may be mundane, referring to what needs to be understood in order to get the practical things done in the day-to-day life. The range of integrations to be performed for the day-to-day usage is limited, very limited in fact. A cruder description could do for this purpose. The Many Objects idea is conceptually more economical to use here. [As a polemical remark on the side, observe that while Ayn Rand highlighted the value of purpose, neither Occam nor the later philosophers/physicists following him ever even thought of that idea: purpose.]

However, as the scope of the physical knowledge increases, the requirements of the long-range consistency mandate that it is the second approach (the one involving the One Object idea) which we must adopt as being a better representative of the actual reality, as being more fundamental.

Where does the switch-over occur?

I think that it occurs at a level of those physics problems in which the energetics program (initiated by Leibnitz), i.e., the Lagrangian approach, makes it easier to solve them, compared to the earlier, Newtonian approach. This answer basically says that any time you use the ideas such as fields, and energy, you must make the switch-over, because in the very act of using such ideas, implicitly, you are using the One Object idea anyway. Which means, EM theory, and yes, also thermodynamics.

And of course, by the time you begin tackling QM, the second approach becomes simply indispensable.

A personal side remark: I should have known better. I should have adopted the second approach earlier in my life. It would have spared me a lot of agonizing over the riddles of quantum physics, a lot of running in loops over the same territory (like a dog chasing his own tail). … But it’s OK. I am glad that at least by now, I know better. (And, engineers anyway don’t get taught the Lagrangian mechanics to the extent physicists do.)

A few days ago, Roger Schlafly had written a nice and brief post at his blog saying that there is a place for non-locality in physics. He had touched on that issue more from a common-sense and “practical” viewpoint of covering these two physics approaches [^].

Now, given the above write-up, you know that a stronger statement, in fact, can be made:

As soon as you enter the realm of the EM fields and the further development, the non-local (or the global or the One Object) theories are the only way to go.


A Song I Like:

[When I was a school-boy, I used to very much like this song. I would hum [no, can’t call it singing] with my friends. I don’t know why. OK. At least, don’t ask me why. Not any more, anyway 😉 .]

(Hindi) “thokar main hai meri saaraa zamaanaa”
Singer: Kishore Kumar
Music: R. D. Burman
Lyrics: Rajinder Krishan


OK. I am glad I have brought to a completion a series of posts that I initiated. Happened for the first time!

I have not been able to find time to actually write anything on my promised position paper on QM. … Have been thinking about how to present certain ideas better, but not making much progress… If you must ask: these involve entangled vs. product states—and why both must be possible, etc.

So, I don’t think I am going to be able to hold the mid-2017 deadline that I myself had set for me. It will take longer.

For the same reasons, may be I will be blogging less… Or, who knows, may be I will write very short general notings here and there…

Bye for now and take care…