# 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.)

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…

# Some of the implications of the “One Object” idea…

0. Review and Context: This post continues with the subject of one vs. many physical objects. 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… [^]

In this post, we cover the implications of the second description, i.e., of the “one object” idea.

1. The observed multiplicity of objects as corresponding to certain quantitative differences in the attributes possessed by the universe-object:

In the second description, there exists one and only one object, which is the entire universe itself. This singleton object carries a myriad of attributes—literally each and everything that you ever see/touch/etc. around you (including your physical body) exists as “just” an attribute of this singleton object. In the general case, such attributes exist with quantitatively different degrees in different parts of the singleton universe-object. Those contiguous regions of the singleton object where the quantitative degrees of the given attribute fall sufficiently closer in range are treated by our perceptual faculty as separate objects.

In the general philosophy, there is a certain observation: Everything is interconnected. However, following the second description, not only are all objects interconnected, but at a deeper level, they are literally one and the same object! It’s just that each perceptually separate object has been distinguished on the basis of some quantitative measures (or amounts) of some or the other attribute or property with which that distinguished region exists.

A few consequences are noteworthy.

2. Implications for what precisely the law of causality refers to:

In the second description, what physically exists is the single physical object (that is the physical universe) and nothing else but that physical object.

The physical actor, in the primary sense of the term, therefore always is the entire universe itself, acting as a whole. The “appearance” of multiple objects—and their separate actions—is only a consequence of the universe having varying properties in different parts of or logically within itself.

Just the way the attributes carried by the universe are inhomogeneous (i.e., they differ in quantitative measures over different parts), so are the actions. The quantitative measures of actions too are inhomogeneous. In the general case, for any of the actions taken by the universe, the same action in general occurs to different degrees in different parts.

In the deepest and the most fundamental sense, since there is only one physical actor viz. the entire physical universe, what the law of causality refers to it is nothing but this physical actor, i.e., to the entire universe taken as a whole.

However, since the very nature of the singleton object includes the fact that different parts of itself exist with different attributes of differing degrees and therefore can and do act differently, the law of causality can also be seen to apply, in a secondary or derivative sense, to these distinguishable parts taken in isolation. The differing natures of the inhomogeneous parts together constitute all the causes existing in the physical universe, and the nature of the actions that this singleton object takes, to differing measures in different parts of itself, constitute all the effects.

The fact that the universe-object exists with various physical attributes or properties, leads to different concepts with which the universe-object can be studied.

3. The idea of space as derived from the physical universe:

One most prominent, general and fundamental property which may be used for distinguishing different parts of the universe-object is the fact that the distinguishable parts, taken by themselves, are spatially extended, and the related fact that they carry the attribute of being located where they are.

Locations and extensions are given in the sensory perceptual evidence. Thus, extensions and locations are directly perceived. They in part form the perceptual basis for the concept of space.

Space is an abstract, mathematical concept. Using this higher level concept, we are able to ascribe places even to those combinations of spatial relations where there is no concrete object existing.

4. A (mathematical) space as an abstraction based on certain attributes of the (physical) universe:

The above discussion makes it clear that the universe does not exist in space. On the other hand, space may be said to exist “in” the universe. However, here, here, the word “in” is to be taken in an abstract logical sense, not in the sense of a concrete existence. Space does exist in the universe but not concretely.

Space is an abstraction based on certain fundamental, directly perceived, spatial attributes or properties possessed by the singular universe-object. The two most fundamental of such (spatial) attributes are extensions and locations; other spatial attributes such as connectivity/topology, of being enclosed or covered or placed inside/outside, etc. are merely higher-level ideas that isolate different ways in which groups of objects with various extensions and locations exist. The extensions and locations themselves pertain to certain quantitative but directly perceived differences over different parts of the universe-object. Thus, ultimately, all spatial properties are possessed by the perceptually distinguishable parts of the singleton universe-object.

Since the concept of space is mathematical and abstract, many different ideas or imaginations may be used in formulating the concept of a space. For instance, Euclidean vs. hyperbolic space, or continuous vs. discrete space, etc. Not only that, multiple instances of a given space also are easily possible. In contrast, the idea of instances, of quantities, does not apply to the universe-object; it remains the unique, singular, concept, one which, when taken as a whole, must remain beyond any quantitative characterization.

Since there is nothing but the universe object to exist physically, the only spatially relevant statement we can make about the universe itself is this: if some part of the universe does indeed exist, then this part can be put in a quantitative relation with one of the instances of some or the other space.

The italicized part is based on the assumption that every part of the universe does carry spatial attributes. This itself is just an assumption; there is no way to directly validate it.

Note that the aforementioned statement does not imply that the physical universe can be said as being present everywhere. The universe does not exist everywhere.

To say that the physical universe is present everywhere is an epistemologically misconceived formulation. It is indicative of an intellectually sloppy, inconsistent way of connecting the two ideas: (i) physical universe (which is what actually exists, in the physical sense), and (ii) space (which is a mathematical and abstract concept).

“Everywhere” refers to a set of all possible places implied by a certain concept of space. Physical universe, on the other hand, refers what actually exists. It is possible that the procedure of constructing a concept of space includes places that have no correspondence to any part of the physical universe.

5. A space can be finite or infinite, but the physical universe is neither:

Space, being a mathematical concept, can be imagined as infinitely extended. However, the physical universe cannot be. And the reason that an infinitely extended physical universe is a nonsense idea is not because the physical universe is, or even can be known to be, finite.

The fact of the matter is, no quantitative statement can at all be made in respect of the physical universe taken as a whole.

Quantitative statements can only be made if some suitable mathematical procedure is available for making the requisite measurements. Now, any and all mathematical procedures are constructed only in reference to some or the other parts of the universe, not in reference to the entirety of the universe taken as a whole. The very nature of mathematics is like that. The epistemological procedures of differentiation and integration must first be performed before any mathematical procedure can at all be constructed or applied. (For instance, before inventing or applying even the simplest mathematical procedure of counting, you must have first performed integration of a group of similar concrete objects such as identical balls, and differentiated this group from the background of the rest of the she-bang.) But as soon as you say: “differentiate,” you already concede the idea that the entirety of the universe is not being considered in the further thought. To differentiate is to agree to selectively pick up only a part and thereby to agree to leave some other part(s). So, as soon as you perform differentiation, from that point on, you no longer are referring to all the parts at the same time. That’s why, no concrete mathematical procedure can at all be constructed which possibly can allow you to measure the universe as a whole. The very idea itself does not make sense. There can be a measure for this part of the universe or for that part. But there can be no measure for the universe taken as a whole. That’s why, its meaningless to talk of applying any quantitative attributes to the entirety of the physical universe taken as a whole—including the talk of the universe being even finite in extent.

No procedure can be said to have yielded even a finite amount as a measurement outcome, if the thing asserted as measured is taken to be the universe as a whole. As a result, no statement regarding even finitude can be made for the physical universe. (I here differ from the Objectivist position, e.g., Dr. Peikoff’s writings in OPAR; they believe that the universe is finite.)

It is true that every property shown by every actually observed part of the physical universe is finite. The inference from this statement to the conclusion that every part of the not-actually-observed but in-principle possibly existing part itself must also be finite, also is valid—within its context. However, the validity of this inference cannot be extended to the idea of a mathematical procedure that applies to all the parts of the universe at the same time. The objection is: we cannot speak of “all” parts itself unless we specify a procedure to include and exhaust every existing part—but no such procedure can ever be specified because differentiation and integration are at the base of the very conceptual level (i.e. at the base of every mathematical procedure).

The idea of an infinite physical universe [^] is flawed at a deep level. Infinity is a mathematical concept. Physical universe is what exists. The two cannot be related—there can be no mathematical procedure to relate the two.

Similarly, the idea of a finite physical universe also is flawed at a deep level.

Now, the idea that every part of the physical universe is finite, can be taken to be valid, simply because the procedure of measuring parts can at all be conducted, and such a procedure does in principle yield outcomes that are finite.

To speak of an infinite space, in contrast, also is OK. The idea here is to make a mental note to the effect that any  statements being made for some parts (possibly infinite number of parts) of this space need not have any correspondence with the spatial attributes of the actually existing physical universe-object—that the logical mapping from a part of a space to a physically existing spatial attribute would necessarily break down for every infinite part of an infinite space.

As far as physics is concerned, infinity is only a useful device for simplifying—reifying out—the complications due to certain possible variations in the boundary conditions of physics problems. When the domain is finite, changes in boundary conditions make the problem so complex that is is impossible to yield a law in the form of a differential equation. The idea of an infinite domain allows us to do precisely that. I had covered this aspect in an earlier post, here [^].

6. Implications for the gaps between perceived objects, and the issue of whether empty space plays a causal role or not:

There is no such a thing as a really “empty” part in the physical universe; the idea is a contradiction in terms.

In contrast, on the basis of our above discussion, notice that there can be empty regions of space(s), in fact even infinitely large empty regions of space(s) where literally nothing may be said to exist.

However, the ideas of emptiness or filled-ness can refer only to space, not to the physical universe.

Since there is no empty part in the universe, the issue of what causal role such an empty part can or does play, does not arise. As to the empty regions of space, since there can be no mapping from such regions to the physical universe, once again, the issue of its causal role does not arise. An empty space (or an empty part of a space) does not physically exist, period. Hence, it has no causal role to play, period.

However, if by empty space you mean such things as the region between two grey “objects” (i.e. two grey parts of the physical universe), then: that region is not, really speaking, empty; a part of what actually is the physical universe does exist there; otherwise, during their motions, the grey parts could not have come to occupy this supposedly empty regions of the space. In other words, if literally nothing were to exist in the gap between two objects, then the attribute of grayness could never possibly travel over there. But no such restriction on the movement of distinguishable objects has ever been observed, reported, or rationally conceived of, directly or indirectly. Hence, in conclusion, the gap region is not really speaking empty.

7.  The issue of the local vs. the “non-local” actions:

In the second description, since only one causal agent exists, what-ever physical action happens, it is taken by this one and the only physical universe. As a particular implication of that fact, where-ever any physical action happens, it again is to be attributed to the same physical universe.

In taking a physical action, it is easily conceivable that wherever the physical universe is actually extended, it simultaneously takes action at all those locations—and therefore, in all those abstract places which correspond to these locations.

As a consequence, it is possible that the physical universe simultaneously takes the same action, but to differing degrees, in different places. Since the actor is a singleton, since it anyway is present wherever any action occurs at all, any and all mystification arising from ascribing a cause and its effect to two separate entities simply vaporizes away. So does any and all mystification arising from ascribing a cause and its effect to two spatially separated locations. The locations may be different, but the actor remains the same.

For the above reasons, in the second description, instantaneous action-at-a-distance no longer remains a spooky idea. The reason is: there indeed is no instantaneous action at a distance, really speaking. IAD is only a loose way of saying that there is simultaneous action of, by, in, etc., the same causal (and effectual) actor that is the singleton object of the physical universe.

In fact we can go ahead and even say that in the second description, every action always is necessarily a global action (albeit with zero magnitudes in some parts of the universe); that there is no such a thing as an in-principle local action.

However, the aforementioned statement does not mean that spatially separated causes and effects cannot be observed. All that it means is that such multiple-objects-like phenomena are not primary; they are only higher-level, abstract, consequences of the more fundamental processes that are necessarily global in nature.

In the second post of this series [^], we saw how the grey regions of our illustrative example can be distinguished from each other (and from the background object) by using some critical density value as the criterion of their distinction or separation.

Since the second description involves only a single object, it necessarily requires a procedure for separating this singleton universe-object into multiple objects. There are certain interesting ideas concerning such a separation, and we will have a closer look at this very idea of separation, in the next post.

Of all the posts in this series, it is this post where I remain the most unsatisfied as far as my expression is concerned. I think a lot of simplification is called for. But in the choice between a better but very late expression and a timely but poor, awkward, expression, I have chosen the latter.

May be I will come back later and try to improve the flow and the expression of this post.

Next time,  I will also try to write something on how the two objections to the aether idea (mentioned in the last post) can be overcome.

A Song I Like:

(Marathi) “maajhee na mee raahile”
Music: Bal Parte
Singer: Lata Mangeshkar
Lyrics: Shanta Shelke

[A very minor revision done on 4th May 2017, 15:19 IST. May be, I will effect some more revisions later on.]