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 ðŸ˜‰ .]

(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 anyÂ  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.Â  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,Â  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.]

Some of the implications of the “Many Objects” idea…

0. Context and Review:

This post continues from the last one. In the last post, we saw that the same perceptual evidence (involving two moving grey regions) can be conceptually captured using two entirely different, fundamental, physics ideas.

In the first description, the perceived grey regions are treated as physical objects in their own right.

In the second description, the perceived grey regions are treated not as physical objects in their own right, but merely as distinguishable (and therefore different) parts of the singleton object that is the universe (the latter being taken in its entirety).

We will now try to look at some of the implications that the two descriptions naturally lead to.

1. The “Many Objects” Viewpoint Always Implies an In-Principle Empty Background Object:

To repeat, in the first description, the grey regions are treated as objects in their own right. This is the “Many Objects” viewpoint. The universe is fundamentally presumed to contain many objects.

But what if there is one and only one grey block in the perceptual field? Wouldn’t such a universe then contain only that one grey object?

Not quite.

The fact of the matter is, even in this case, there implicitly are two objects in the universe: (i) the grey object and (ii) the background or the white object.

As an aside: Do see here Ayn Rand’s example (in ITOE, 2nd Edition) of how a uniform blue expanse of the sky by itself would not even be perceived as an object, but how, once you introduce a single speck of dust, the perceptual contrast that it introduces would allow perceptions of both the speck and the blue sky to proceed. Of course, this point is of only technical importance. Looking at the real world while following the first description, there are zillions of objects evidently present anyway.

Leaving aside the theoretically extreme case of a single grey region, and thus focusing on the main general ideas: the main trouble following this “Many Objects” description is twofold:

(i) It is hard to come to realize that something exists even in the regions that are “empty space.”

(ii) Methodologically, it is not clear as to precisely how one proceeds from the zillions of concrete objects to the singleton object that is the universe.Â  Observe that the concrete objects here are physical objects. Hence, one would look for a conceptual common denominator (CCD) that is narrower than just the fact that all these concrete objects do exist. One would look for something more physical by way of the CCD, but it is not clear what it could possibly be.

2. Implications of the “Many Objects” Viewpoint for Causality:

In the first description, there are two blocks and they collide. Let’s try to trace the consequences of such a description for causality:

With the supposition that there are two blocks, one is drawn into a temptation of thinking along the following lines:

the first block pushes on the second block—and the second block pushes on the first.

Following this line of thought, the first block can be taken as being responsible for altering the motion of the second block (and the second, of the first). Therefore, a certain conclusion seems inevitable:

the motion of the first block may be regarded as the cause, and the (change in) the motion of the second block may be regarded as the effect.

In other words, in this line of thought, one entity/object (the first block) supplies, produces or enacts the cause, and another entity/object (the second block) suffers the consequences, the effects. following the considerations of symmetry and thereby abstracting a more general truth (e.g. as captured in Newton’s third law), you could also argue that that it is the second object that is the real cause, and the first object shows only effects. Then, abstracting the truth following the consideration of symmetry, you could say that

the motion (or, broadly, the nature) of each of the two blocks is a cause, and the action it produces on the other block is an effect.

But regardless of the symmetry considerations or the abstractness of the argument that it leads to, note that this entire train of thought still manages to retain a certain basic idea as it is, viz.:

the entity/actions that is the cause is necessarily different from the entity/actions that is the effect.

Such an idea, of ascribing the cause and the effect parts of a single causal event (here, the collision event) to two different object not only can arise in the many objects description, it is the most common and natural way in which the very idea of causality has come to be understood.Â Examples abound: the swinging bat is a cause; the ball flying away is the effect; the entities to which we ascribe the cause and the effect are entirely different objects. The same paradigm runs throughout much of physics. Also in the humanities. Consider this: “he makes me feel good.”

Every time such a separation of cause and effect occurs, logically speaking, it must first be supposed that many objects exist in the universe.

It is only on the basis of a many objects viewpoint that the objects that act as causes can be metaphysically separated, at least in an event-by-event concrete description, from the objects that suffer the corresponding effects.

3. Implications of the “Many Objects” Viewpoint, and the Idea of the “Empty” Space:

Notice that in the “many objects” description, no causal role is at all played by those parts of the universe that are “empty space.” Consider the description again:

The grey blocks move, come closer together, collide, and fly away in the opposite directions after the collision.

Notice how this entire description is anchored only to the grey blocks. Whatever action happens in this universe, it is taken by the grey blocks. The empty white space gets no metaphysical role whatsoever to play.

It is as if any metaphysical locus standi that the empty space region should otherwise have, somehow got completely sucked out of itself, and this locus standi then got transferred, in a way overly concentrated, into the grey regions.

Once this distortion is allowed to be introduced into the overall theoretical scheme, then, logically speaking, it would be simple to propagate the error throughout the theory and its implication. Just apply an epistemologically minor principle like Occam’s Razor, and the metaphysical suggestion that this entire exercise leads to is tantamount to this idea:

why not simply drop the empty space out of any physical consideration? out of all physics theory?

A Side Remark on Occam’s Principle: The first thing to say about Occam’s Principle is that it is not a very fundamental principle. The second thing to say is that it is impossible to state it using any rigorous terms. People have tried doing that for centuries, and yet, not a single soul of them feels very proud when it comes to showing results for his efforts. Just because today’s leading theoretical physics love it, vouch by it, and vigorously promote it, it still does not make Occam’s principle play a greater epistemological role than it actually does. Qua an epistemological principle, it is a very minor principle. The best contribution that it can at all aspire to is: serving as a vague, merely suggestive, guideline. Going by its actual usage in classical physics, it did not even make for a frequently used epistemological norm let alone acted as a principle that would necessarily have to be invoked for achieving logical consistency. And, as a mere guideline, it is also very easily susceptible to misuse. Compare this principle to, e.g., the requirement that the process of concept formation must always show both the essentials: differentiation and integration. Or compare it to the idea that concept-formation involves measurement-omission. Physicists promote Occam’s Principle to the high pedestal, simply because they want to slip in their own bad ideas into physics theory. No, Occam’s Razor does not directly help them. What it actually lets them do, through misapplication, is to push a wedge to dislodge some valid theoretical block from the well-integrated wall that is physics. For instance, if the empty space has no role to play in the physical description of the universe [preparation of misapplication], then, by Occam’s Principle [the wedge], why not take the idea of aether [a valid block] out ofÂ  physics theory? [which helps make physics crumble into pieces].

It is in this way that the first description—viz. the “many objects” description—indirectly but inevitably leads to a denial of any physical meaning to the idea of space.

If a fundamental physical concept like space itself is denied any physical roots, then what possibly could one still say about this concept—about its fundamental character or nature? The only plausible answers would be the following:

That space must be (a) a mathematical concept (based on the idea that fundamental ideas had better be physical, mathematical or both), and (b) an arbitrary concept (based on the idea that if there is no hard basis of the physical reality underlying this concept, then it can always be made as soft as desired, i.e. infinitely soft, i.e., arbitrary).

If the second idea (viz., the idea that space is an arbitrary human invention) is accorded the legitimacy of a rigorously established truth, then, in logic, anyone would be free to bend space any which way he liked. So, there would have to be, in logic, a proliferation in spaces. The history of the 19th and 20th centuries is nothing but a practically evident proof of precisely this logic.

Notice further that in following this approach (of the “many objects”), metaphysically speaking, the first casualty is that golden principle taught by Aristotle, viz. the idea that a literal void cannot exist, that the nothing cannot be a part of the existence. (It is known that Aristotle did teach this principle. However, it is not known if he had predecessors, esp. in the more synthetic, Indic, traditions. In any case, the principle itself is truly golden—it saves one from so many epistemological errors.)

Physics is an extraordinarily well-integrated a science. However, this does not mean that it is (or ever has been) perfectly integrated. There are (and always have been) inconsistencies in it.

The way physics got formulated—the classical physics in particular—there always was a streak of thought in it which had always carried the supposition that there existed a literal void in the region of the “gap” between objects. Thus, as far as the working physicist was concerned, a literal void could not exist, it actually did. “Just look at the emptiness of that valley out there,” (said while standing at a mountain top). Or, “look at the bleakness, at the dark emptiness out there between those two shining bright stars!” That was their “evidence.” For many physicists—and philosophers—such could be enough of an evidence to accept the premise of a physically existing emptiness, the literal naught of the philosophers.

Of course, people didn’t always think in such terms—in terms of a literal naught existing as a part of existence.

Until the end of the 19th century, at least some people also thought in terms of “aether.”

The aether was supposed to be a massless object. It was supposed that “aether” existed everywhere, including in the regions of space where there were no massive objects. Thus, the presence of aether ensured that there was no void left anywhere in the universe.

However, as soon as you think of an idea like “aether,” two questions immediately arise: (i) how can aether exist even in those places where a massive object is already present? and (ii) as to the places where there is no massive object, if all that aether does is to sit pretty and do nothing, then how is it really different from those imaginary angels pushing on the planets in the solar system?

Hard questions, these two. None could have satisfactorily answered these two questions. … In fact, as far as I know, none in the history of physics has ever even raised the first question! And therefore, the issue of whether, in the history of thought, there has been any satisfactory answer provided to it or not, cannot even arise in the first place.

It is the absence of satisfactory answers to these two questions that has really allowed Occam’s Razor to be misapplied.

By the time Einstein arrived, the scene was already ripe to throw the baby out with the water, and thus he could happily declare that thinking in terms of the aether concept was entirely uncalled for, that it was best to toss it into in the junkyard of bad ideas discarded in the march of human progress.

The “empty” space, in effect, progressively got “emptier” and “emptier” still. First, it got replaced by the classical electromagnetic “field,” and then, as space got progressively more mathematical and arbitrary, the fields themselves got replaced by just an abstract mathematical function—whether the spacetime of the relativity theory or the $\Psi$ function of QM.

4. Implications of the “Many Objects” Viewpoint and the Supposed Mysteriousness of the Quantum Entanglement:

In the “many objects” viewpoint, the actual causal objects are many. Further, this viewpoint very naturally suggests the idea of some one object being a cause and some other object being the effect. There is one very serious implication of this separation of cause and effect into many, metaphysically separate, objects.

With that supposition, now, if two distant objects (and metaphysically separate objects always are distant) happen to show some synchronized sort of a behavior, then, a question arises: how do we connect the cause with the effect, if the effect is observed not to lag in time from the cause.

Historically, there had been some discussion on the question of “[instantaneous] action at a distance,” or IAD for short. However, it was subdued. It was only in the context of QM riddles that IAD acquired the status of a deeply troubling/unsettling issue.

5. Miscellaneous:

5.1

Let me take a bit of a digression into philosophy proper here, by introducing Ayn Rand’s ideas of causality at this point [^]. In OPAR, Dr. Peikoff has clarified the issue amply well: The identity or nature of an entity is the cause, and its actions is the effect.

Following Ayn Rand, if two grey blocks (as given in our example perceptual field) reverse their directions of motions after collision, each of the two blocks is a cause, and the reversals in the directions of the same block is the effect.

Make sure to understand the difference in what is meant by causality. In the common-sense thinking, as spelt out in section 2. of this post, if the block A’ is the cause, then the block B’ is the effect (and vice versa). However, according to Ayn Rand, if the block A’ is the cause, then the actions of this same block A’ are the effect. It is an important difference, and make sure you know it.

Thus, notice, for the time being, that in Ayn Rand’s sense of the terms, the principle of causality actually does not need a multiplicity of objects.

However, notice that the causal role of the “empty” space continues to remain curiously unanswered even after you bring Ayn Rand’s above-mentioned insights to bear on the issue.

5.2:

The only causal role that can at all be ascribed to the “empty” space, it would seem, is for it to continuously go on “monitoring” if a truly causal body—a massive object—was impinging on itself or not, and if such a body actually did that, to allow it to do so.

In other words, the causal identity of the empty space becomes entirely other-located: it summarily depends on the identity of the massive objects. But the identity of a given object pertains to what that object itself is—not to what other objects are like. Clearly, something is wrong here.

In the next post, we shall try to trace the implications that the second description (i.e. The One Object) leads to.

A Song I Like:

(Hindi) “man mera tujh ko maange, door door too bhaage…”
Singer: Suman Kalyanpur
Music: Kalyanji Anandji
Lyrics: Indivar

[PS: May be an editing pass is due…. Let me see if I can find the time to come back and do it…. Considerable revision done on 28 April 2017 12:20 PM IST though no new ideas were added; I will leave the remaining grammatical errors/awkward construction as they are. The next post should get posted within a few days’ time.]

Why is the physical space 3-dimensional?

Why I write on this topic?

Well, it so happened that recently (about a month ago) I realized that I didn’t quite understand matrices. I mean, at least not as well as I should. … I was getting interested in the Data Science, browsing through a few books and Web sites on the topic, and soon enough realized that before going further, first, it would be better if I could systematically write down a short summary of the relevant mathematics, starting with the topic of matrices (and probability theory and regression analysis and the lot).

So, immediately, I fired TeXMaker, and started writing an “article” on matrices. But as is my habit, once I began actually typing, slowly, I also began to go meandering—pursuing just this one aside, and then just that one aside, and then just this one footnote, and then just that one end-note… The end product quickly became… unusable. Which means, it was useless. To any one. Including me.

So, after typing in a goodly amount, may be some 4–5 pages, I deleted that document, and began afresh.

This time round, I wrote only the abstract for a “future” document, and that too only in a point-by-point manner—you know, the way they specify those course syllabi? This strategy did help. In doing that, I realized that I still had quite a few issues to get straightened out well. For instance, the concept of the dual space [^][^].

After pursuing this activity very enthusiastically for something like a couple of days or so, my attention, naturally, once again got diverted to something else. And then, to something else. And then, to something else again… And soon enough, I came to even completely forget the original topic—I mean matrices. … Until in my random walk, I hit it once again, which was this week.

Once the orientation of my inspiration thus got once again aligned to “matrices” last week (I came back via eigen-values of differential operators), I now decided to first check out Prof. Zhigang Suo’s notes on Linear Algebra [^].

Yes! Zhigang’s notes are excellent! Very highly recommended! I like the way he builds topics: very carefully, and yet, very informally, with tons of common-sense examples to illustrate conceptual points. And in a very neat order. A lot of the initially stuff is accessible to even high-school students.

Now, what I wanted here was a single and concise document. So, I decided to take notes from his notes, and thereby make a shorter document that emphasized my own personal needs. Immediately thereafter, I found myself being engaged into that activity. I have already finished the first two chapters of his notes.

Then, the inevitable happened. Yes, you guessed it right: my attention (once again) got diverted.

What happened was that I ran into Prof. Scott Aaronson’s latest blog post [^], which is actually a transcript of an informal talk he gave recently. The topic of this post doesn’t really interest me, but there is an offhand (in fact a parenthetical) remark Scott makes which caught my eye and got me thinking. Let me quote here the culprit passage:

“The more general idea that I was groping toward or reinventing here is called a hidden-variable theory, of which the most famous example is Bohmian mechanics. Again, though, Bohmian mechanics has the defect that itâ€™s only formulated for some exotic state space that the physicists care about for some reason—a space involving pointlike objects called “particles” that move around in 3 Euclidean dimensions (why 3? why not 17?).”

Hmmm, indeed… Why 3? Why not 17?

Knowing Scott, it was clear (to me) that he meant this remark not quite in the sense of a simple and straight-forward question (to be taken up for answering in detail), but more or less fully in the sense of challenging the common-sense assumption that the physical space is 3-dimensional.

One major reason why modern physicists don’t like Bohm’s theory is precisely because its physics occurs in the common-sense 3 dimensions, even though, I think, they don’t know that they hate him also because of this reason. (See my 2013 post here [^].)

But should you challenge an assumption just for the sake of challenging one? …

It’s true that modern physicists routinely do that—challenging assumptions just for the sake of challenging them.

Well, that way, this attitude is not bad by itself; it can potentially open doorways to new modes of thinking, even discoveries. But where they—the physicists and mathematicians—go wrong is: in not understanding the nature of their challenges themselves, well enough. In other words, questioning is good, but modern physicists fail to get what the question itself is, or even means (even if they themselves have posed the question out of a desire to challenge every thing and even everything). And yet—even if they don’t get even their own questions right—they do begin to blabber, all the same. Not just on arXiv but also in journal papers. The result is the epistemological jungle that is in the plain sight. The layman gets (or more accurately, is deliberately kept) confused.

Last year, I had written a post about what physicists mean by “higher-dimensional reality.” In fact, in 2013, I had also written a series of posts on the topic of space—which was more from a philosophical view, but unfortunately not yet completed. Check out my writings on space by hitting the tag “space” on my blog [^].

My last year’s post on the multi-dimensional reality [^] did address the issue of the $n > 3$ dimensions, but the writing in a way was geared more towards understanding what the term “dimension” itself means (to physicists).

In contrast, the aspect which now caught my attention was slightly different; it was this question:

Just how would you know if the physical space that you see around you is indeed was 3-, 4-, or 17-dimensional? What method would you use to positively assert the exact dimensionality of space? using what kind of an experiment? (Here, the experiment is to be taken in the sense of a thought experiment.)

I found an answer this question, too. Let me give you here some indication of it.

First, why, in our day-to-day life (and in most of engineering), do we take the physical space to be 3-dimensional?

The question is understood better if it is put more accurately:

What precisely do we mean when we say that the physical space is 3-dimensional? How do we validate that statement?

The answer is “simple” enough.

Mark a fixed point on the ground. Then, starting from that fixed point, walk down some distance $x$ in the East direction, then move some distance $y$ in the North direction, and then climb some distance $z$ vertically straight up. Now, from that point, travel further by respectively the same distances along the three axes, but in the exactly opposite directions. (You can change the order in which you travel along the three axes, but the distance along a given axis for both the to- and the fro-travels must remain the same—it’s just that the directions have to be opposite.)

What happens if you actually do something like this in the physical reality?

You don’t have to leave your favorite arm-chair; just trace your finger along the edges of your laptop—making sure that the laptop’s screen remains at exactly 90 degrees to the plane of the keyboard.

If you actually undertake this strenuous an activity in the physical reality, you will find that, in physical reality, a “magic” happens: You come back exactly to the same point from where you had begun your journey.

That’s an important point. A very obvious point, but also, in a way, an important one. There are other crucially important points too. For instance, this observation. (Note, it is a physical observation, and not an arbitrary mathematical assumption):

No matter where you stop during the process of going in, say the East direction, you will find that you have not traveled even an inch in the North direction. Ditto, for the vertical axis. (It is to ensure this part that we keep the laptop screen at exactly 90 degrees to the keyboard.)

Thus, your $x$, $y$ and $z$ readings are completely independent of each other. No matter how hard you slog along, say the $x$-direction, it yields no fruit at all along the $y$– or $z$– directions.

It’s something like this: Suppose there is a girl that you really, really like. After a lot of hard-work, suppose you somehow manage to impress her. But then, at the end of it, you come to realize that all that hard work has done you no good as far as impressing her father is concerned. And then, even if you somehow manage to win her father on your side, there still remains her mother!

To say that the physical space is 3-dimensional is a positive statement, a statement of an experimentally measured fact (and not an arbitrary “geometrical” assertion which you accept only because Euclid said so). It consists of two parts:

The first part is this:

Using the travels along only 3 mutually independent directions (the position and the orientation of the coordinate frame being arbitrary), you can in principle reach any other point in the space.

If some region of space were to remain unreachable this way, if there were to be any “gaps” left in the space which you could not reach using this procedure, then it would imply either (i) that the procedure itself isn’t appropriate to establish the dimensionality of the space, or (ii) that it is, but the space itself may have more than 3 dimensions.

Assuming that the procedure itself is good enough, for a space to have more than 3 dimensions, the “unreachable region” doesn’t have to be a volume. The “gaps” in question may be limited to just isolated points here and there. In fact, logically speaking, there needs to be just one single (isolated) point which remains in principle unreachable by the procedure. Find just one such a point—and the dimensionality of the space would come in question. (Think: The Aunt! (The assumption here is that aunts aren’t gentlemen [^].))

Now what we do find in practice is that any point in the actual physical space indeed is in principle reachable via the above-mentioned procedure (of altering $x$, $y$ and $z$ values). It is in part for this reason that we say that the actual physical space is 3-D.

The second part is this:

We have to also prove, via observations, that fewer than 3 dimensions do fall short. (I told you: there was the mother!) Staircases and lifts (Americans call them elevators) are necessary in real life.

Putting it all together:

If $n =3$ does cover all the points in space, and if $n > 3$ isn’t necessary to reach every point in space, and if $n < 3$ falls short, then the inevitable conclusion is: $n = 3$ indeed is the exact dimensionality of the physical space.

QED?

Well, both yes and no.

Yes, because that’s what we have always observed.

No, because all physics knowledge has a certain definite scope and a definite context—it is “bounded” by the inductive context of the physical observations.

For fundamental physics theories, we often don’t exactly know the bounds. That’s OK. The most typical way in which the bounds get discovered is by “lying” to ourselves that no such bounds exist, and then experimentally discovering a new phenomena or a new range in which the current theory fails, and a new theory—which merely extends and subsumes the current theory—is validated.

Applied to our current problem, we can say that we know that the physical space is exactly three-dimensional—within the context of our present knowledge. However, it also is true that we don’t know what exactly the conceptual or “logical” boundaries of this physical conclusion are. One way to find them is to lie to ourselves that there are no such bounds, and continue investigating nature, and hope to find a phenomenon or something that helps find these bounds.

If tomorrow we discover a principle which implies that a certain region of space (or even just one single isolated point in it) remains in principle unreachable using just three dimensions, then we would have to abandon the idea that $n = 3$, that the physical space is 3-dimensional.

Thus far, not a single soul has been able to do that—Einstein, Minkowski or Poincare included.

No one has spelt out a single physically established principle using which a spatial gap (a region unreachable by the linear combination procedure) may become possible, even if only in principle.

So, it is 3, not 17.

QED.

All the same, it is not ridiculous to think whether there can be 4 or more number of dimensions—I mean for the physical space alone, not counting time. I could explain how. However, I have got too tired typing this post, and so, I am going to just jot down some indicative essentials.

Essentially, the argument rests on the idea that a physical “travel” (rigorously: a physical displacement of a physical object) isn’t the only physical process that may be used in establishing the dimensionality of the physical space.

Any other physical process, if it is sufficiently fundamental and sufficiently “capable,” could in principle be used. The requirements, I think, would be: (i) that the process must be able to generate certain physical effects which involve some changes in their spatial measurements, (ii) that it must be capable of producing any amount of a spatial change, and (iii) that it must allow fixing of an origin.

There would be the other usual requirements such as reproducibility etc., though the homogeneity wouldn’t be a requirement. Also observe Ayn Rand’s “some-but-any” principle [^] at work here.

So long as such requirements are met (I thought of it on the fly, but I think I got it fairly well), the physically occurring process (and not some mathematically dreamt up procedure) is a valid candidate to establish the physically existing dimensionality of the space “out there.”

Here is a hypothetical example.

Suppose that there are three knobs, each with a pointer and a scale. Keeping the three knobs at three positions results in a certain point (and only that point) getting mysteriously lit up. Changing the knob positions then means changing which exact point is lit-up—this one or that one. In a way, it means: “moving” the lit-up point from here to there. Then, if to each point in space there exists a unique “permutation” of the three knob readings (and here, by “permutation,” we mean that the order of the readings at the three knobs is important), then the process of turning the knobs qualifies for establishing the dimensionality of the space.

Notice, this hypothetical process does produce a physical effect that involves changes in the spatial measurements, but it does not involve a physical displacement of a physical object. (It’s something like sending two laser beams in the night sky, and being able to focus the point of intersection of the two “rays” at any point in the physical space.)

No one has been able to find any such process which even if only in principle (or in just thought experiments) could go towards establishing a $4$-, $2$-, or any other number for the dimensionality of the physical space.

I don’t know if my above answer was already known to physicists or not. I think the situation is going to be like this:

If I say that this answer is new, then I am sure that at some “opportune” moment in future, some American is simply going to pop up from nowhere at a forum or so, and write something which implies (or more likely, merely hints) that “everybody knew” it.

But if I say that the answer is old and well-known, and then if some layman comes to me and asks me how come the physicists keep talking as if it can’t be proved whether the space we inhabit is 3-dimensional or not, I would be at a loss to explain it to him—I don’t know a good explanation or a reference that spells out the “well known” solution that “everybody knew already.”

In my (very) limited reading, I haven’t found the point made above; so it could be a new insight. Assuming it is new, what could be the reason that despite its simplicity, physicists didn’t get it so far?

Answer to that question, in essential terms (I’ve really got too tired today) is this:

They define the very idea of space itself via spanning; they don’t first define the concept of space independently of any operation such as spanning, and only then see whether the space is closed under a given spanning operation or not.

In other words, effectively, what they do is to assign the concept of dimensionality to the spanning operation, and not to the space itself.

It is for this reason that discussions on the dimensionality of space remain confused and confusing.

Food for thought:

What does a $2.5$-dimensional space mean? Hint: Lookup any book on fractals.

Why didn’t we consider such a procedure here? (We in fact don’t admit it as a proper procedure) Hint: We required that it must be possible to conduct the process in the physical reality—which means: the process must come to a completion—which means: it can’t be an infinite (indefinitely long or interminable) process—which means, it can’t be merely mathematical.

[Now you know why I hate mathematicians. They are the “gap” in our ability to convince someone else. You can convince laymen, engineers and programmers. (You can even convince the girl, the father and the mother.) But mathematicians? Oh God!…]

A Song I Like:

(English) “When she was just seventeen, you know what I mean…”
Band: Beatles

[May be an editing pass tomorrow? Too tired today.]

[E&OE]

What do physicists mean by “multidimensional” physical reality?

Update on 2015.09.07, 07 AM: I have effected a few corrections. In particular, I have made it explicit that the third quantity isn’t the strength of an independently existing third property, but merely a quantity that is registered when the two independent quantities are both being varied. Sorry about that. If the need be, I will simplify this discussion further and write another blog post clarifying such points, some time later.

The last time, I said that I am falling short on time these days. This shortfall, generally speaking, continues. However, it just so happens that I’ve essentially finished a unit each for both the UG courses by today. Therefore, I do have a bit of a breather for this week-end (only); I don’t have to dig into texts for lecture preparations this evening. (Also, it turns out that despite the accreditation-related overtime work, we aren’t working on Sundays, though that’s what I had mentioned the last time round). All in all, I can slip in a small note, and the title question seems right.

We often hear that the physical reality, according to physicists, is not the $3$-dimensional reality that we perceive. Instead, it is supposed to be some $n$-dimensional entity. For instance, we are told that space and time are not independent; that they form a $4$-dimensional continuum. (One idea which then gets suggested is that space and time are physically inter-convertible—like iron and gold, for instance. (You mean to say you had never thought of it, before?)) But that’s only for the starters. There are string theorists who say that physical universe is $10$-, $n$-, or $\infty$-dimensional.

What do physicists mean when they say that reality is $n$-dimensional where $n >3$? Let’s try to understand their viewpoint with a simple example. … This being a brief post, we will not pursue all the relevant threads, even if important. … All that I want to touch upon here is just one simple—but often missed—point, via just one, simple, illustration.

Take a straight line, say of infinite length. Take a point on this line. Suppose that you can associate a physical object with this point. The object itself may have a finite extent. For example, the object may be extended over a small segment of this line. In such a case, we will associate, say the mid-point of the segment with this object.

Suppose this straight line, together with the $1$-dimensionally spread-out object, defines a universe. That is a supposition; just accept that.

The $1$-dimensional object, being physical, carries some physical properties (or attributes), denoted as $p_1, p_2, p_3, \cdots$. For example, for the usual $3$-dimensional universe, each object may have some extent (which we have already seen above), as well as some mass (and therefore density), color, transmissivity, velocity, spinning rate, etc. Also, position from a chosen origin.

Since we live in a $3$-dimensional universe, we have to apply some appropriate limiting processes to make sense of this $1$-dimensional universe. This task is actually demanding, but for the sake of the mathematical simplicity of the resulting model, we will continue with a $1$-dimensional universe.

So, coming back to the object and its properties, each property it possesses exists in a certain finite amount.

Suppose that the strength of each property depends on the position of the object in the universe. Thus, when the object is at the origin (any arbitrary point on the line chosen as the reference point), the property $p_1$ exists with the strength $s_1(0)$, the property $p_2$ exists with the strength $s_2(0)$, etc. In short the $i$th property $p_i$ exists with a strength $s_i(x)$ where $x$ is the position of the object in the universe (as measured from the arbitrarily selected origin.) Suppose the physicist knows (or chooses to consider) $n$ number of such properties.

For each of these $n$ number of properties, you could plot a graph of its strength at various positions in the universe.

To the physicist, what is important and interesting is not the fact that the object itself is only $1$-dimensionally spread; it is: how the quantitative measures $s_i(x)$s of these properties $p_i$s vary with the position $x$. In other words, whether or not there is any co-variation that a given $i$th property has with another $k$th property, or not, and if yes, what is the nature of this co-variation.

If the variation in the $i$th property has no relation (or functional dependence) to the $k$th property, then the physicist declares these two properties to be independent of each other. (If they are dependent on each other, the physicist simply retains only one of these two properties in his basic or fundamental model of the universe; he declares the other as the derived quantity.)

Assuming that a set of some $n$ chosen properties such that they are independent of each other, his next quest is to find the nature of their functional dependence on position $x$.

To this end, he considers two arbitrarily selected points, $x_1$ and $x_2$. Suppose that his initial model has only three properties: $p_1$, $p_2$ and $p_3$. Suppose he experimentally measures their strengths at various positions $x_1, x_2, x_3, x_4, \cdots$.

While doing this experimentation, suppose he has the freedom to vary only one property at a time, keeping all others constant. Or, vary two properties simultaneously, while keeping all others constant. Etc. In short, he can vary combinations of properties.

By way of an analogy, you can think of a small box carrying a few on-off buttons and some readout boxes on it. Suppose that this box is mounted on a horizontal beam. You can freely move it in between two fixed points $x = x_1$ and $x = x_2$. The on-off’ buttons can be switched on or off independent of each other.

Suppose you put the first button $b_1$ in the on’ position and keep the the rest of the buttons in the off’ position. Then, suppose you move the box from the point $x_1$ to the point $x_2$. The box is designed such that, if you do this particular trial, you will get a readout of how the property $p_1$ varied between the two points; its strength at various positions $s_1(x)$ will be shown in a readout box $b_1$. (During this particular trial, the other buttons are kept switched off, and so, the other readout boxes register zero).

Similarly, you can put another button $b_2$ into the on’ position and the rest in the off’ position, and you get another readout in the readout box $b_2$.

Suppose you systematize your observations with the following notation: (i) when only the button $b_1$ is switched on (and all the other buttons are switched off), the property $p_1$ is seen to exist with $s_1(x_1)$ units at the position $x = x_1$ and $s_1(x_2)$ units at $x = x_2$; this readout is available in the box $b_1$. (ii) When only the button $b_2$ is switched on (and all the other buttons are switched off), the property $p_2$ exists with $s_2(x_1)$ units at $x = x_1$ and $s_2(x_2)$ units at $x = x_2$; this readout is available in the box $b_2$. So on and so forth.

Next, consider what happens when more than one switch is put in the on’ position.

Suppose that the box carries only two switches, and both are put in the on’ position. The reading for this combination is given in a third box: $b_{(1+2)}$; it refers to the variation that the box registers while moving on the horizontal beam. Let’s call the strengths registered in the third box, at $x_1$ and $x_2$ positions, as $s_{(1+2)}(x_1)$ and $s_{(1+2)}(x_2)$, respectively; these refer to the $(1+2)$ combination (i.e. both the switches $1$ and $2$ put in the on’ position simultaneously).

Next, suppose that after his experimentation, the physicist discovers that the following relation holds:

$[s_{(1+2)}(x_2) - s_{(1+2)}(x_1)]^2 = [s_1(x_2) - s_1(x_1)]^2 + [s_2(x_2) - s_2(x_1)]^2$

(Remember the Pythogorean theorem? It’s useful here!) Suppose he finds the above equation holds no matter what the specific values of $x_1$ and $x_2$ may be (i.e. whatever be the distances of the two arbitrarily selected points from the same origin).

In this case, the physicist declares that this universe is a $2$-dimensional vector space, with respect to these $p_1$ and $p_2$ properties taken as the bases.

Why? Why does he call it a $2$-dimensional universe? Why doesn’t he continue calling it a $1$-dimensional universe?

Because, he can take a $2$-dimensional graph paper by way of an abstract representation of how the quantities of the properties (or attributes) vary, plot these quantities $s_1$ and $s_2$ along the two Cartesian axes, and then use them to determine the third quantity $s_{(1+2)}$ from them. (In fact, he can use any two of these strengths to find out the third one.)

In particular, he happily and blithely ignores the fact that the object of which $p_i$ are mere properties (or attributes), actually is spread (or extended) over only a single dimension, viz., the $x$-axis.

He still insists on calling this universe a $2$-dimensional universe.

That’s all there is to this $n$-dimensional nonsense. Really.

But what about the $n$-dimensional space, you ask?

Well, the physicist just regards the extension and the position themselves to form the set of the physical properties $p_i$ under discussion! The physicist regards distance as a property, even if he is going to measure the strengths or magnitudes of the properties (i.e. distances, really speaking) only in reference to $x$ (i.e. positions)!!

But doesn’t that involve at least one kind of a circularity, you ask?

The answer is embedded right in the question.

Understand this part, and the entire mystification of physics based on the “multi-dimensional” whatever vaporizes away.

But don’t rely on the popular science paperbacks to tell you this simple truth, though!

Hopefully, the description above is not too dumbed down, and further, hopefully, it doesn’t have too significant an error. (It would be easy for me (or for that matter any one else) to commit an error—even a conceptual error—on this topic. So, if you spot something, please do point it out to me, and I will correct the description accordingly. On my part, I will come back sometime next week, and read this post afresh, and then decide whether what I wrote makes sense or not.)

A Song I Like:

For this time round, I am going to list a song even if I don’t actually evaluate it to be a very great song.

In fact, in violation of the time-honored traditions of this blog, what I am going to do is to list the video of a song. It’s the video of a 25+ years old song that I found I liked, when I checked it out recently. As to the song, well, it has only a nostalgia value to me. In fact, even the video, for the most part, has only a nostalgia value to me. The song is this:

(Hindi) “may se naa minaa se na saaki se…”
Music: Rajesh Roshan
Lyrics: Farooq Qaisar

Well, those were the technical details (regarding this song). To really quickly locate the song (and the video), forget the lyrics mentioned above. Instead, just google “aap ke aa jaane se,” and hit the first video link that the search throws up. (Yes, it’s the same song.)

As I said, I like this video mainly for its nostalgic value (to me). It instantaneously takes me back to the 1987–88 times. The other reasons are: the utter natural ease with which both the actors perform the dance here (esp. Neelam!). They both in fact look like they are authentically enjoying their dancing. Watch Neelam’s steps, in particular. She was reputed to be a good dancer, and you might think that this song must have been a cake-walk for her. Well, check out her thin (canvas-like) shoes, and the kind of rough ground in the mountains and in the fields over which she seems so effortlessly to take those steps. Govinda, in comparison, must have had it a bit easier (with his thicker, leather shoes), but in any case, in actuality, it must have been some pretty good hard work for both of them—it’s just that the hard work doesn’t show in the song. … Further, I also like the relative simplicity of the picturization. And, the catchy rhythm. Also, the absence, here, of those gaudy gestures which by now are so routine in Hindi film songs (and in fact were there even in the times of this song, and in fact also for about a decade or more earlier). I mean: those pelvic thrusts, that passing off of a thousand of people doing their PT exercises on a new, sprawling suburban street in Mumbai/Gurgaon/Lutyens’ Delhi as an instance of dance, etc.

… I don’t know if you end up liking this song or not. To me, however, it unmistakably takes me to the times when I was a freshly minted MTech from IIT Madras, was doing some good (also satisfying) work in NDT, had just recently bought a bike (the Yamaha RX 100), and was looking forward to life in general with far more enthusiasm (and in retrospect, even naivete) than I can manage to even fake these days. So, there.

[As I said, drop a line if there are mistakes in the main post. Main mistake (or omission) corrected. As I said, drop a line if there are further mistakes in the main post. And, excuse me for some time, esp. the next week-end, esp. the next Saturday late night (IST). I may not find any time the next Sunday, because I would once again be in the middle of teaching a couple of new units over the next 2–3 weeks.]

[E&OE]

Putting context for space in place

[Update on 24th October, 2013: The post has now undergone a bit of streamlining and a few minor additions/clarifications here and there.]

I took something of a vacation this week (i.e. during the week ending today), and so, posting this post got postponed. (No, can’t call it a real vacation—I don’t have a job—but, it was, let’s say, a tour. I will post a few photographs sometime later. Anyway, let’s get back to our series on space, first.)

1. Place:

Before we could measure volumes of extensions, it would be necessary to introduce another concept, viz., that of “place.”

The concept of place, following the terminology I use, is a slightly higher-level abstraction. But it pertains to location, not extension. Its meaning can be best approached by first considering its usage. Here, first of all, recall the meaning of location.

To speak of the location of the sea is to simply point out at that object itself and say: “there,” thereby meaning the characteristic of location that it has. We say that the sea exists as a definite object with its perceptually given location, and similarly, for the sand and the distant hill, etc.

But to say something about the place of the sea is (even if it sounds tautological): placing it at its location. To place the sea is to make a reference to the location of the sea, as in relation to the locations of other objects.

Let’s look at the detail of what it means to ascribe a place to an object, e.g., to the sea.

You first have to make a reference to the (perceptually given) location of the sea, and at the same time, also to the (perceptually given) location of the sand. In fact, you have to make a reference also to the locations of all the other relevant objects in your perceptual field, e.g., the locations of the woods, of the hill, of the distant hill, etc.

You then have to conceive of an abstract relationship connecting all of these locations, by regarding each concrete location as a unit in an abstract unit-perspective. [^] In effect, what you do is to first observe a certain similarity to the locations, viz. the fact they are all locations, and then tell yourself that in view of that existing similarity, each of the perceptually given location is not so unique after all; instead, mentally, you can regard it as just another instance, of a certain kind, in a series of them—an instance of place. A particular concrete belonging to a group of many similar concretes is seen as just a bird of an abstract feather, so to speak.

You treat each perceptually evident location, thus, as a mentally noted instance falling within a certain mentally generated abstract perspective that is created by your mind; you thus begin to hold the perceived location in your mind in reference to that abstraction. Ayn Rand called it the unit-perspective. You first generate the abstract unit-perspective in reference to the concretely existing similarity, and then, the particular location of a particular object can be regarded as just a concrete instance of that abstract kind. The next step consists of finalizing this mental grasp of this abstract unit-perspective, by giving it a name: “place.”

The concept of place is formed following a process such as the above [^][^], and thereafter, you are ready to say: the perceptually evident location of the sea is not unique all by itself, but it is just one of those… say, places. The location of the sea is just one place. And, the location of the sand is another thing of the same kind: another place. And, the location of the distant hill is just another thing of the same kind—it’s just another place.

You thus realize that locations can be treated as places, by considering many locations at the same time in your mind, observing the fact that they are similar in that they all are locations, and differing only in the particular measures (the particular locations) where they are found. In other words, you relate the different objects in reference to a common characteristic—location—that each object exists possessing with its own, different, measure.

During the process by which you thus realize that locations are places, in reality, the locations themselves have remained exactly the same; all that has objectively changed pertains to what has happened in your mind. You have added a mental note, and further: you have created a new mental object, in a way—to regard each location as one of an abstract (or “only mental”) kind of a thing.

…It isn’t just a matter of mentally grouping the locations together, but something more fundamental than that. When you speak of a group, you do refer to many objects; yet, the focus is more on the plurality of it. There is no reference to the commonality connecting them. Instead, what you do in concept formation is creating a certain bit of knowledge that refers to each individual thing within the group by itself, even if it refers only in an abstract light of relationship which is shared by them. It isn’t just: many things; it is: many things of a kind. Taking two locations together as a group isn’t the same as seeing them as two places. …

… Your play-mates could see the physical objects, see that they are located where they are, and even could mentally take many of them at the same time, but they wouldn’t know, unless you tell them, that you have now begun to see these locations also as different places. …Whether they also had begun to see it the same way or not, whether you were the first to generate this abstract grasp of the location characteristics or whether it was one of them, whether it is your responsibility to make them see what you see, etc—none of such considerations is relevant here. The point is only this much: what you did with your mind, and whether an object you hold in your mind also separately exists in the concrete reality out there or not. In the case of concepts, it doesn’t; it remains only in your mind. If, on the other hand, the mental integration to regard locations as places were to concretely exist in the reality out there, then no volitional process of observing the existing similarities and differences, reaching an abstract unit-perspective, and thereby beginning regarding the existing concretes as units, would at all be necessary. The essence is epistemological, not metaphysical—the chief difference between Ayn Rand and Aristotle.

Since objects possess locations, you can, now, in reference to the concept of place, begin to say that objects possess places, too. Indeed, right as a part of forming the concept, you would have observed that each object can be regarded to have its own place. And, the nature of your abstract grasp would be such that, once you had it, not only every location you actually see, but any location whatsoever that you may ever see in future (or any one that you ever have seen in the past) would still be regarded only as a place.

To say something about the place of something is to make a statement about it in the light of the abstract concept of place. In contrast, to say something about the location of something is to stop making any reference to any abstract mental object(s), and to directly point out to it.

Even though even to get location you would have to have at least two objects, qua perception, your consciousness doesn’t have to do anything deliberately, volitionally, for you to get it. Just looking at the objects is enough—your perceptual faculty automatically performs the necessary differentiation and integration and lets you perceive the two objects as existing with two locations.

Perception is severely limited. It’s impossible to determine the locations of any objects not directly perceived. Conception (i.e. making and using concepts) is powerful. In the process of forming a concept, it is necessary to abstractly relate what potentially are an unspecified number of objects.

According to the terminology I follow, the location is a spatial characteristic that an object has, all by itself. It is a part of its own physical existence, its own identity, and hence directly available in perception.

However, the place of an object essentially also involves the location(s) of some other object(s), and further, it’s an abstraction: it is not directly available in perception; you have to exercise volition and conceive of it.

[And, please remember, for convenience, throughout this discussion, we now no longer are making a direct reference to the background objects. Thus, all the objects here are the foreground objects. Actually, it’s not a major conceptual issue; conceptually, it’s a rather marginal issue, but it’s best tackled at a more advanced stage of our discussion.]

Whereas the concept of location is defined in reference to concrete objects (their perceived characteristics), the concept of place is defined in relation to locations.

2. The power of the concept of place:

Since abstract relationships are formed and held only as mental objects, they do not thereby concretely exist in reality. Locations do exist in concrete reality—they concretely exist as characteristics of the physical objects. But places do not exist all by themselves in the concrete reality; they are just our way of organizing the concretely existing objects within an abstract perspective. Places do not exist except as the concrete instances organized together via a uniting perpective with which a certain conceptual consciousness sees them. If all people die, locations would continue to exist (because the concrete objects would be where they would be), but places would cease to exist.

This nature of the concept of place is not its limitation; on the other hand, it is what gives it its real power. How come?

You can think of some one place—some concrete instance of the concept of place—even if there is no concrete object actually to be found at that place at all, ever!

The very word “at” implies the abstract concept of place. If a place is said to exist, what it thereby implies is that a definite location is being abstractly referred to, even if there is no concrete object existing there. If a place is said to exist, what that instance of place means is the location a physical object would concretely have if it were to be perceptually found there. If, as seen in a “place-wise” relationship abstractly uniting a group of existing physical objects it is possible to regard a certain measure of their connecting place-ness characteristic as valid, then that measurement need not wait for a concrete object’s actual location to coincide with it, before it can be validly regarded as a place.

None of this true with the concept of location. If out of a group of objects, some one particular object were suddenly to be completely annihilated, that location would cease to exist—you could not have the basis for perceiving the locatedness of that object. However, so long as other objects exist, given a thinking mind that has reached the concept of place, the place of that object would still continue to exist. Some locations may come and some locations may go, but all their places have always been there and will always go on—so long as a mind to hold the concept of place, exists.

In going from the concept of location to that of place, the perceptual field itself remains the same; the only change is: the addition of a conceptual viewpoint with which to “see” it.

Thus, what dogs and cats can see are locations; only human beings are capable of seeing places—but they can do so only after seeing locations (and their pre-existing similarities and differences). Lost dogs and cats are sometimes able to trace their way back to their owner’s home. Evidently, they are able to form and retain in their memory not just groups of concrete locations but also connected chains of such locations. But, as far as it can be determined, they still are unable to regard the locations as places—something that a human baby can do with such ease. The chain connecting locations is itself retained only at a concrete level; it is just a part of a passive, perceptual level memory; the unit-perspective of regarding the concrete locations as instances of abstract places is entirely absent.

One final clarification. In reaching the concept of place, we didn’t make any reference to the concept of “point.” The point is merely a concept that imparts an abstract precision to the concept of place. But it is not essential to have it in order to form the concept of place, in the first place! Locations of concrete objects are enough.

3. Reference System:

The fact that places are abstract measures of multiple objects itself means that the instances of places, taken together, form a system, even if only an implicit one.

A reference system is nothing but a way of deriving a system of places, given certain definite locations. Its purpose is to allow us go in a “reverse” direction, i.e. in going from abstracts to concretes. Its purpose is to let us go to a chosen destination location i.e. a place, from our current location i.e. another place.

In the terminology I use, it is not necessary to have the more abstract concepts of points, lines, planes, etc., and to organize these in a systematic manner, in order to have a reference system. The latter are necessary only for defining a reference frame. In contrast, a reference system can do with just the concrete instances of locations, none of which actually has a zero extension in any sense. Thus, the concept of reference system is more primitive; it could be grasped even by a primitive man.

If a primitive man (or a trekker, for that matter!) is planning his journey from one village to another, he may decide to, say, prepare and pack his lunch right in the morning, walk down a known walk-way through a jungle, halt at a mango tree in a clearing near a stream of water for his lunch, rest for a while, and then walk further till he reaches a huge banyan tree by the time the sun has gone some half-way down in the sky, take a right turn from there, and follow for some way a small path through some fields, before he hits his destination. The pathway, the mango tree/stream, the banyan tree, and the path through the fields, together, forms a reference system, albeit of a primitive sort.

Why do we call it a system? Why not, say, a reference “concrete”? Because, you see, presumed here is the fact that these are already being seen as places. And, presumed here also is the fact that there are other path-ways criss-crossing this territory too. A certain group of certain landmarks, when taken together, help one in defining not just one path, but many of them. This group of landmarks is the fewest number of locations which together allow one to place the greatest number of (or all of) destination points. They bring about the greatest order to the greatest number of (or all of) the possible path-ways through the territory, and thereby helps reduce the mental burden. Carrying a knowledge of the territory is easier in reference to these landmarks as compared to noting every arbitrary concrete location that befalls the eye on every possible journey. This characteristic is what makes them a reference system. It’s a system to measure the places of locations, really speaking.

It’s only when the demands of precision of measurements increases, that we have to go from a reference system to a reference frame.

Notice the two, related, aspects of a reference system. (i) It is a tool for deriving abstract places from concrete locations, and, (ii) symmetrically, once the system is available, it also is a tool for deriving concrete locations from the abstract places.

Another point: Each reference system also involves a method of undertaking the above-mentioned two processes. I call it the reference method.

4. Moving objects as without a location:

A further discussion of spatial concepts cannot be undertaken without referring, at least implicitly, to a world in which there can be some motion, at least for some time.

What is motion? The motion of an object is not a change of its location; it is the process which is necessary to bring about that change. This distinction is important.

An object in motion is perceivably in a different existential state than the one in which it is when it is stationary. The concept of location can be defined only in a motionless world. For an object in motion, the concept of a definite location does not at all apply. [I came to know this point by reading up catalog description of Dr. Binswanger’s lecture cassettes, not by reading Aristotle.]

Inasmuch as motion of an object does not destroy its identity, but instead, actually is a part of its identity—a part of what it does—it is obvious that the objectively existing characteristic of its location should also remain with it, in some way. Yet, while an object is in the process of motion, we are unable to perceive its specific location. [And, the issue extends to many other characteristics as well.]

In fact, as far as perceptions go, what we perceive for a moving object is not a definite object at all, but just a vague sense of like “it’s there but also not there.” Call it a smearing kind of a sense, if you wish. Whether the perception occurs via a visual component (as illustrated by the long-exposure photographs of moving cars) or via a tactile component (e.g. the rolling of a football on your back, or of a cricket ball on your fore-arm when you are idly rolling it down from the wrist and flicking it up with your elbow), the sense of there being a vague streak of something, as in contrast to the sense of there being a definite location, remains.

So, using the most exact terms, we cannot perceive a location for a moving object at all. To attempt ascribing to a moving object the attribute of a particular measure of location is to take the term outside of its defining scope (viz. that it is directly perceivable), and therefore, outside of the realm of its applicability.

It is obvious that motion does not destroy the identity of an object, but quite on the contrary is a part of its identity—moving is simply a part of what the object does. But we cannot therefore say that all the concrete measures of all the characteristics of that object must therefore always remain constant in the process of moving. The latter is a contradiction in terms. If all the characteristics were to retain all their specific measures, then no object could at all undergo any change; i.e. no change at all would be possible in the universe. Motion does not destroy the fact that the moving object has some location; what it does is only to make it impossible to perceive the particular measures with which it exists, so long as the object remains in that state (of motion).

By way of an indirect and limiting argument—say by dropping a ball from a height and catching it after progressively longer durations of time, i.e., by successively stopping the same kind of a motion over greater measures, we could possibly say that the location does have some definite value at every moment during that process (of motion), but since the measure is undergoing a change beyond the finite capacity of a consciousness to be aware of it while the process actually lasts, the location having a definite value is only an inference—it cannot be directly perceived. The only sense in which we can say something about the location of an moving object is that qua an identity, the object would have some definite measures for all its characteristics, including location—not that it has this specific quantity against that one. A definitive statement that an object has a location can only be made in that context in which this characteristic can at all be isolated, i.e. is perceivable, viz., when it is motionless.

This circumstance is far from being whimsical, miraculous, or invalidating knowledge. No matter how small the change in location effected by the motion may be, before that change at all begins, and once that change is complete, the location is self-evident. And, as the result of the motion, we can see that it has undergone a change. Yet, the two states—the state of being in motion, and the state of being at one of the endpoints—are in two different classes altogether. If it is felt that miraculous is the transformation of a vague streak into a definite thing, then equally miraculous it should sound to have a definite thing transform itself into a vague streak, and still come back to being its usual definite kind of a “self” after a while. The fact of the matter is, none of these aspects are miraculous; they all are lawful. The only difficulty is the one introduced by trying to apply a term beyond the scope of its derivation and therefore of its applicability.

5. Moving objects as with (changing) places:

How about place? Do the moving objects at least have [definite] places, even if they do not have [definite] locations?

The answer is, of course, yes, but only in the applicable context.

A moving object may be seen as possessing definite places during its motion, even if it does not have any location. How come? Because, places are abstract, that’s why. [And, BTW, this is another instance where you can appreciate the power of abstraction.]

For a place to exist, it’s not at all even necessary that an object must exist with the location implied by that place. So, the bigger issue of existence (of every object) itself has no bearing on the issue at hand. If so, how can the issue of a mere change being undergone by an object have any? And, indeed, it does not. For a place to exist, even if only some objects are motionless (or, more fundamentally, if they are at all perceivable), then that is good enough. You can then derive places from locations. And, once you have a reference system of places, then, it does not matter whether other objects even continue to exist or not, let alone whether they move or not. You can always use the concept of place to describe their motion.

However, the context does matter.

Here, an important point is the precision with which placing an object is at all possible, given the reference system of deriving places from locations (and of determining locations from places). You cannot determine even the place of an object any more precisely than what the reference system, and the reference method, together allow. And, any limitations that creep in while building a method and a system, do apply also on the application side.

For instance, it is very obvious to every one that, in a game of fortune-hunting, if you use a crude method of placing things such as: locate that fortune buried somewhere in the ground, at 100 steps due West and 500 steps due North, then, there is a very good probability that you wouldn’t get it in the first try: the place where you end up wouldn’t necessarily be the location of the fortune. Your method of measurements will be too crude to determine your place to the required level of accuracy.

But my point here is: even if you do use the infinitesimal calculus and thus, the limiting cases of those geometrical points, lines, etc., and use them to create a reference system, even then, a certain limitation still comes up. You are not a point. Neither are the marking lines on the foot-rule. You could easily miss the fortune if it were kept in a small tube buried vertically, especially if you use a relatively cruder method as repeated use of a single footrule. The basic point is this. If you knew your current place to some level of accuracy, then you could deduce the abstract destination place (within the implied level of accuracy). And, further, having thus deduced the abstract destination place, thereafter “going” to the finite location from the infinitesimal point of that destination place, would also introduce yet another source by which you could get off-the-mark. The abstract system of determining places may be “infinitely accurate;” but neither the method of going to the abstract from the concrete, nor from the abstract to the concrete, is at all possible except within certain finite ranges (say “tolerances”) of accuracy. Remember, all concretes are definite. You cannot derive a standard of accuracy by assuming a context of only abstract manipulations of only abstract entities, tear that standard out of that context so as to drop the latter, and attempt to apply that standard to a process of concretely inter-relating the concrete entities. It is called Context-Dropping [^].

(BTW, to stay epistemologically consistent, you also cannot apply a more precise standard for the reference method+system while going from the concrete to the abstract, and a less precise standard in going from the abstract to the concrete, or vice versa. You cannot mix precision of the two aspects of reference methods or systems.

And, for that matter, you cannot ever take the infinitely accurate as the standard while involving anything concretely real even just in part. It cannot in fact apply either in going to the abstract from the concrete, or in going to the concrete from the abstract. The only place where it can at all be “thought” to apply is while being completely in the abstract realm—by relegating the considerations of going in either direction of the concrete-to-abstract relationships, entirely into context.)

Anyway, so, if you have an abstract reference system that makes use of abstract geometrical objects like points, lines, surfaces, etc., then, using it (and also introducing some other assumptions such as that of a continuous change and what it, in turn means), a moving object may indeed be considered to have even an infinitely accurate place at every instant during its motion. But, since the abstract-concrete connections cannot be “infinitely accurate,” that hardly helps the simple-minded quest of a high-school student wanting to invent an infinitely accurate measurement system. … Not a bother. Enough of them learn, even if only in some implicit terms, why it’s a futile quest. It’s the theoretical physicists, mathematicians, theoretical computer scientists, and philosophers who alone persist in that quest even in their adulthood. And build careers of the kind they do! (Sorry, can’t call their outputs “work.”)

6. Displacement as a result of motion:

Let us leave aside a further study of the nature of the process i.e. motion, and instead, let us focus on just the start and end of a finitely lasting process of motion. This way, we are, at least at the two end-points of it, in the more comfortable zone of the motionless world: both the starting and the ending states are without motion. As such, they have definite locations. And, their places are far more easily measured, too.

One point we have neglected thus far is this: Moving an object involves two physical events of displacings/occupations. One is: the de-occupation of its initial place by the moving object once the motion begins, and the occupation of the final place by the moved object once the motion ends. In both cases, there also are other (de)occupations: The occupation of the initial place by some other object (or at least by the background object—call it the “empty space” or aether), and the de-occupation of the final place by some other object.

These additional consequences do take place even if our focus is only on a single moving foreground object. Motion involves changes not only in the object that moves but also in the other objects (and if in none of them—as in the water being displaced by a boat, then at least in the background object). However, since the “empty space” does not seem to offer any significant resistance/encouragement to such displacings/occupations, we tend to ignore this part of those physical happenings. But they are there. And, here, once again, if used correctly, the power of abstraction can truly begin to shine.

So long as you know how to correctly derive places from locations (i.e. so long as you know the right method of deriving places), you can afford to ignore the physical displacings/”replacements”. Since places are abstract quantities, the physical displacings attendant with a motion do not affect them—I mean, the places.

What the preceding observation suggests is a very simple paradigm of doing physics: Keep the system of determining places as simple as possible, and keep any physical effects due to the physical displacings, to their own characteristics, i.e. to the characteristics other than the spatial characteristics of the physical objects.

This view of doing physics was successful until the relativity theorists—Poincare, Einstein, or Minkowsky or someone else, I don’t care who all—ruined it, and replaced it by an ugly way of doing physics: overload the idea of the system of determining places with as much other, non-spatial physical considerations as possible. Spatialize every physical change, i.e., treat every change as if it affected/resulted from some change in a system to define spatial relationship. Even if such changes be time, or even force! (The latter two involve the fallacy of concept-stealing in progressively uglier forms. Today, in the context of dark matter, physicists ascribe even matter to space, thereby bringing the inversion to the logical extremum to which it is at all possible to take. … Cheer up, because, precisely for that reason, they can’t go any further, and the field is all left for us to reverse all those ugly changes.)

But, of course, hierarchically, such topics are way, way too advanced. So, please treat this all only as an aside, and leave it at that.

Actually, that way, we never do in fact fill an empty space. We always only displace objects—i.e. move each from one place to another place—i.e., change their locations.

As far as measuring volumes is concerned, we never have to fill space. It’s enough to displace objects—change their places, as in reference to other objects. Suppose that we wish to measure the volume of a brick. We can place other objects next to the brick, then remove the brick so as to leave a cavity, and then fill the cavity with a number of pebbles, or water. The amount of pebbles (or water) necessary to fill the cavity would make for a simple way of measuring that volume. In this entire process, every time we moved any foreground object to some other place, what it actually did was to displace the background object. But it never does fill empty space. There is no empty space to begin with.

If the background object—let’s call it the aether—does not apparently resist being displaced, it does not mean that this background object does not exist. Remember, the aether is a part of perception. All that it means is that we haven’t yet become aware of the other results that removing a portion of it produces—results other than the changes given right in our perceptual field, viz., that it gets displaced.

7. A bit about geometrical objects like points:

Are we at least now ready to take out our rulers and compasses, you ask?

Not quite. There are just a few more observations that I would like to indicate (though not discuss—simply because, people should be clear enough on most of these), before you could take out your rulers and compasses.

Extension and location are the two most fundamental spatial characteristics; both come simultaneously, i.e., from the same perceptual field. Therefore, to focus on any one for a detailed study, e.g. to measure any one, the best epistemological policy is to drop the other from active consideration and to relegate it into context (in a proper way). (Similarly, for the higher-level characteristics.)

Thus, to measure the volume aspect of extension, we need to keep the shape constant, or better still, via a limits-involving argument, to completely remove it from the active consideration and relegate it back into mere context. To measure locations, we need to keep extension out of active consideration, or, in a limiting process, remove it from any active consideration, and relegate it back into mere context.

It’s thus that the ideas of points, lines and surfaces become necessary. They are the limiting cases. A point is the limiting case of ever smaller spherical extensions. A line is the limiting case of a sequence of ever thinner and longer ropes. A surface is the limiting case of ever thinner and wider peel or coating. In each case, we have first specified a shape, then varied certain of its extension measurements in a systematic manner, and then in focusing on the characteristics of the trends displayed by these sequences, relegated the extension completely out of the active consideration and into the context.

Since extension is thus, say, “reified” out (i.e., in the epistemologically proper view of the process, abstractified out), you get the “reifing” in (or abstractification) of the other characteristics, viz., location. As you make the extension of a sphere “vanish” out, you “get” its location more precisely “in”. For most precise determination of places, then, you should be referring to the place of a sphere of zero extension, i.e. point, which can be accomplished by projecting the infinitesimal to the zero in the limiting case. Similarly, for the curve, and the surface. And, thereby, you now have a collection of the most precise reference objects—they all are abstract, and not just that: they all arise as the limiting cases. They are limitingly abstract aspects of certain extensional characteristics of the concretely existing.

The Euclidean axioms belong to this world in this way.

Once you have them, you can use them to specify, and then to study, the properties (i.e. the higher-level spatial relationships) of shapes—i.e. geometry.

Going by the catalog descriptions of the Objectivist events, someone had proposed treating geometry as a part of physics, some years ago. I immediately knew that he was being wrong. Inasmuch as the geometric entities like points, curves (or straight lines), surfaces (or planes) are the limiting cases, they are just concepts of methods (i.e., the mental objects standing in for the final result of a systematically executed method that relates some abstract or even concrete objects). Physics studies what exists, not how to supply the abstract or mental methods of measurements for measuring that which (physically) exists; the latter is the province of mathematics. Extension and location are physical concepts. Place is a concept that mediates between physics and mathematics—it is at once both a physical and a mathematical concept; reference system is another one of a similar nature. Points, curves, surfaces is where mathematics begins. Distance may be taken as another mediating concept; in contrast, length is decidedly a mathematical concept. Geometry is a part of mathematics.

… If you wish, it may now be time to think what a straight line is, what a curved line is, what direction is, and what angle is (it’s a property, not of a line segment, but of two of them taken together)…. And once you are done doing that, then to think of taking out your rulers and compasses.

8. An exercise for you:

Now that I have typed so much (even if not in the best possible order, and not using best possible expression), there is some interesting exercise I may expect you to do.

Based on these three posts, derive as many definitions of space as you can. No, really! Write them down. Keep them aside.

Then, go watch this video: “The Fabric of the Cosmos: What Is Space?” [^].

I strongly recommend this video because it carries a certain unique combination of interesting extremes: (i) it presents what essentially are ideas that are extremely bad (or at least, merely bad), (ii) but it presents them in an extremely brilliant (i.e. well-essentialized) manner, (iii) with very amazing visualizations to go with them (i.e., the video has very high “production values”), and (iv) as if just to counter any sense of doom/despair/gloom/hopelessness that the simultaneous existence of these three elements in such a combination might generate (e.g., the sense of gloom as expressed in: “if Americans do spend so many dollars on such pathetic ideas, and if no good ideas are found similarly presented on PBS or elsewhere, then what fighting chance could a good idea possibly have”?) it also has an extremely nice presenter, Brian Green of Columbia, as its narrator. (No, Lisa Randall, of Harvard, couldn’t have pulled this thing off so well. She would be too sharp in her presentation and yet, I guess, in an attempt to make the pathetic ideas sound reasonable, she would somewhat compromise presenting their very pathetic essence in a very direct manner. And, it’s for this reason that hers wouldn’t be so memorable a video. Sean Carroll, of CalTech, could have had a very good chance at it, but he would sometimes come across, I guess, a bit too “modern.” Only Green, among the three, seems to naturally have that personality to smoothly pull the trick of presenting as clueless a stuff as this set of wrong fundamental ideas, and yet get away without inducing an enduring sense of gloom in you…)

So, anyway, go, watch that video, and once done watching it (or may be while you are watching it), jot down any additional definitions of space that you can think of having. [Yes, you are allowed to cheat. You may refer to the transcript of the video available at the same link.]

I will come to presenting a few definitions of space in the next post (I mean, the next post of this series, that is—some other stuff may come up, in the meanwhile). And, oh, also, think of that problem of measuring volumes, and how the concepts of place and reference systems (i.e. something connected with locations) allow measurements of volumes (i.e. something connected with extension). … Think about it, though I would probably not come to discuss it in detail myself, and so, regard it as an exercise left for you. (Actually, I now realize, I myself have supplied all the crucial points for this latter exercise right in the main text of this post.)

So, the next time in this series, we will pick up the definitions of space… Or perhaps, the issue of dimensionality and infinity of space… The questions like: Is this world basically three-dimensional? Can a fourth dimension exist? Is space infinite? Is the universe infinite? … Questions like that…. Let me think about it—whether I should discuss these topics before discussing definitions of space, or not. …. What do you think?

* * * * *Â Â  * * * * *Â Â  * * * * *

A Song I Like:

(Hindi) “jhir jhir barase saavanee ankhiyaan, saanwariyaa ghar aaa…”
Music: Vasant Desai
Singer: Lata Mangeshkar
Lyrics: Harindranath Chattopadhyaya (??) / Gulzaar (??)

[TBD: I will streamline this post a bit tomorrow—e.g., convert the _emphases_ to the italicized emphases, make a statement here or there more easily intelligible, etc. Update: Done on 24th October, 2013.]

[E&OE]

Shaping up “space”

I continue with the meaning of the concept of space in this post.

0.0 What we saw last time:

In the last post in this series, we saw that all physical objects have the characteristics of extension and location. We began with the perceptually evident reality; then considered only the motionless aspect of the world; and then saw how extension and location are ostensively defined in reference to the objects in that motionless world. We thus did make a lot of progress.

And, even though I am not fully satisfied with my formulations in that post (and for that matter, even would like to rethink even some of my asserted positions), I would like to leave these matters aside for some time. But let me jot down just one issue about which I am not fully satisfied, by way of an example.

0.1. Boundary as an advanced concept:

While isolating the concept of extension, while writing my post, I seemed to have ended up over-emphasizing the use of “beginning” and “end.” The way I wrote it, it seems as if, hierarchically speaking, the concept of “boundary” could at least easily vie for the same hierarchical level as that of extension, if not being even more basic to it. The way I wrote it, the concept of extension seems to require a reference to be made to an object’s boundaries.

But, this is not the position I actually had in mind. It was pretty clear to me that while extensions and locations do exist in the concretely real physical reality, boundaries don’t.

Boundaries are mental objects (“imaginary” ones), and their hierarchical position is therefore at some higher, more abstract, level. A boundary does not exist in the physical world; it is not grasped perceptually. You do not rest at the boundary of a wall (or the backside of your chair); you rest against the wall (or the backside). It, therefore, should be possible to write about this issue in a way that de-emphasizes the references to “beginnings” and “ends.”

… Thus, I seem to have ended up writing what in fact is not my actual position. … While writing on intricate and deep matters completely independently, such things do happen.

Another thing. I am not sure if the English word “location” is the right one to use for the characteristic I have explained in my last post. … I have been confused about the choice of a suitable word to use here, for a long time by now.

0.2. A bit about the terminology I use, and something about the personal background in which it grew:

When I thought about these things seriously for the first time, which was some time in the early 1990s while I was a student at UAB, some of my thinking was actually done in Marathi.

… The way those times turned out for me, I was being in a sustained background state of completely unexpected personal losses (heaped on me by others), was in a foreign country, and overall, also was quite isolated (there was no Internet back then; air-mail would take one week, one way), and so, those unexpected losses were beyond my capacity to bear.

One day, in one of my several (but only partly successful) attempts to snap out of it, by engaging my mind into something new, I had thought of trying my hand at translating some essential philosophical terms into Marathi. It was this attempt which first had, unwittingly, jolted me for a while into an even deeper sense of isolation (and an attendant sharpening of the sense of sorrow), but, immediately later on, the touch with the mother-tongue, even if in this way, also had this magical effect of bringing clarity to the deepest of my thinking on many issues.

It was magical because, you see, in those times, in my attempts to try and keep up with the world, I would always try to think seamlessly only in English, so as to improve on my command over it—a process which I had consciously tried to imbibe for quite some time by then. (In a way, I was trying to keep up with my English medium-educated friends—as well as foes. I was the only Indian student at UAB who was neither Brahmin nor English medium-educated nor someone who had visited a foreign country before—and, of course, I had no relatives or family friends anywhere abroad, let alone in the USA.) But such attempts of mine, I then realized, were in some ways doing more harm to me than good. Thinking in Marathi brought a certain clarity to my thoughts much more quickly and far more easily.

In particular, about certain deeper concepts, esp. the first-level, ostensively defined concepts, it brought about an incomparable kind of clarity. And it’s thus that my thoughts on these matters were first done using certain Marathi terms (and possibly their Sanskrit bases) like “ThikaaN,” “jaagaa,” “sthaL,” “sthir,” and why, even the simple “ithe” vs. “tithe” distinction, etc.

In my informal (though computer-typed) notes in English on this topic (some of which I still keep, though I never consult them in my later writings), I had used the word “spot,” for what I now have called “location.”

[BTW, another personal aside. Speaking of those times, I didn’t even know touch-typing back then, but following the tip of a Chinese graduate assistant working in the Computer Center at UAB—foreign students typically had no computers at home back then—I kept up the practise of keeping both my wrists on the computer table all the time, and that’s how, slowly, I had also mostly picked up touch-typing—I mean, without using any typing tutor software or so (though I did use one later on, while being a student at C-DAC).]

Anyway, coming back to this word “location,” well, some other word like “place” perhaps could be more suitable for it. Perhaps. However, I have resisted using it here, because “place” appears in Aristotle’s writings (which I have not yet read even by now), and I am not sure if people would not get confused about the meaning I have in mind vs. the one that Aristotle had indicated. It seems to me that using “location” here would minimize any possible confusions.

But, anyway, let us completely ignore the merely terminological issues for the time being, and instead focus on the actual, live, thought-process which precedes those terms. And, thus, let’s first go further, first finish this series on space, and then, surely, I will come back and revise this whole series again, ironing out the wrinkles in the expressions, as well as probably also revising my intellectual positions. Indeed, I am thinking of dumping a “scientific” version of this series even at arXiv! … So, there.

1. The minimum perceptual field involves two objects:

A certain issue which I had relegated to the miscellaneous notes section in my last post, I now realize, assumes a much greater importance. This issue concerns the “what-if” scenario put forth by Ayn Rand. As she indicated, there is no perception at all so long as there is nothing in the universe except for only a vast expanse of a uniform pale blue. But introduce just a single speck of dust (or a sizeable circular blob of a uniform dark blue) against that background, and suddenly, you would begin seeing—i.e. perceiving.

The point which I now want to emphasize is this. What you then begin to perceive isn’t just the dark blue blob but also the pale blue background. In particular, strictly speaking, what you actually perceive isn’t a single object; it’s two objects, simultaneously—even though, habitually, when asked to conceptually identify, most of us would not identify the pale blue background as an object in its own right. Yet, our perceptual field does actually contain both of them as two separate objects.

How come?

A part of your perception (of that what-if scenario) is the fact that you can see that the dark blue is what the pale blue isn’t, and also, exactly symmetrically, the pale blue is what the dark blue isn’t. For instance, the circle is where the background isn’t, and the background is where the circle isn’t. (The idea of “where” refers to both location and extension. More on that, later.)

Now, take the last sentence, drop the particular measurements (e.g. those pertaining to the locations and extensions) so as to isolate the more basic, existence, issue: The dark blue is—and, the pale blue is. Both are. In particular, the pale blue, too, exists.

2. Why people have wrong intuitions about the minimum perceptual field, about aether, etc.

Realize that Ayn Rand’s minimum perceptual field is only an artificial scenario. We consider this highly isolated scenario only in order to help our thought processes.

In actual reality, we never see only two objects of uniform colours (i.e., objects so uniform that various parts within an object, too, are perceptually completely indistinguishable). We always come across real objects, in fact very many of them, all at the same time. They are of various shapes, shades, brightnesses, smells, hot-/cool-ness, textures, tastes, heavy-/light-ness, etc. And, they also differ in their responses to our attempts at handling or manipulating them. Or, plain, in what they do to us. And, therefore, we also tend to focus more on their anticipated potential to do something to us. The teleological end of cognition is to live your life.

When it comes to dealing with all the varied objects of the real world, we make many, many other observations as well, some of them quite crucial to survival (or at least in avoiding pain and enjoying pleasure). We bring to bear the sum-totality of all such observations as a background context in our adult perceptions. Our experience leads us to (correctly) believe that it is usually the foreground object which is the most interesting one because it does so many more important things to us—things which require our attention with so much greater urgency—as compared to any background objects. A hidden thorn (or a needle) gives us an unexpected kind of a pain precisely because it was not in the foreground. When we are hungry and mother brings us some food, the sweets are, say, interesting; the plate isn’t. The sky—the background of all backgrounds—never ever seems to do anything to us (or to any thing, for that matter); the lightening in the clouds and the rains sure do.

… Now, what happens is that we bring these cognitive habits to bear even onto that “what-if” scenario. And it is these habits which might make us believe, but wrongly so, that there would be perception of just one object—that of the dark blue. Yet, the fact is, the pale blue also begins to get perceived at the same time as the dark blue does. One essence of perception—if it may be described this way—is that one (perceived) object is what the other (perceived) object isn’t, and vice versa. As such, the background object also is actually perceived as a separate object in its own right.

If the background object is the “empty space,” we typically don’t identify it as an object, mainly because it seemingly does not offer any resistance to its own displacement—you can always pick a ball and keep it at a place where there is nothing but “empty space,” and thereby displace the “empty space” which had existed earlier there. We don’t notice any such resistance, not in our day-to-day experience anyway; it doesn’t do us something dramatic the way a wall, a thorn, a body of water, or even plain wind does. But therefore to ascribe pure nothingness to that “empty space” is wrong. This is a subtle point to which we will return later.

As a somewhat interesting aside, I will now merely assert, without explaining, the fact that even if you have a complete darkness in place of that pale blue background, what you would perceive still would be two objects. The bright (“visible”) object, and the absolutely dark (“invisible”) object.

…Incidentally, this is another interesting issue that I am now leaving as an exercise for you. The first one was (see my last post): even if you don’t see the outer boundary of the pale blue, how come it is still perceived at all; why the “indefinite” extension (or location) of the background does not stop us from perceiving it; why this nature of the background object does not stump us from visualizing it (unlike either an object of no extension or one of infinite extension).

In a way, the fact that there are two objects even in the simplest possible perception, is something that we have already used, while discussing the characteristics of extension and location. Epistemologically, we had noted, a grasp of any characteristic requires at least two objects possessing the same characteristics but in different measures.

… Now, in the what-if scenario, the characteristic perhaps easiest to grasp, in fact, could very well be the colour! It’s just that since we here are concerned with the spatial matters that we have highlighted the characteristics of extension and location.

As far as the spatial characteristics go, I assert (without separate proof), that extension and location are the simplest possible spatial characteristics. In the simplest scenario (the dark blue over the pale blue), they correspond with certain differences in which you use your mental focus. Extension corresponds to what you grasp when your mental focus moves away from the inner parts of the object in question to its outer parts and continues on to the next object (extension). Location corresponds to the characteristics that you grasp in the reverse—when your mental focus moves from the next object, and towards the inner parts of the object in question. I consider these to be the simplest possible, perceptual-level, activities, and so, further assert that there are no other spatial characteristics at their hierarchical level.

The concept of space is deep. But it is not as deep as Einstein (and the relativity theorists) asserted it to be. Contraries to the concept of space are easily possible (e.g. colour is a contrary to extension and location, and therefore, to every possible conception of space).

And, BTW, while at it, let’s note just in the passing that the spatial characteristics are perceived right in the motionless world. In contrast, what the relativity theorists did, was to posit time at par with space. That is plain impossible. You actually define space only in the motionless world.

3. Towards measuring extensions:

While discussing the concept of extension, I had asked you to ignore the questions of the how, and the how much, of the extensions of objects, but instead, to focus on the existence-related fact that each object has [some, finite] extension. Though we did not discuss this idea in detail for the concept of location, the same consideration, of course, holds also for the concept of location: each object has [some, finite] location. The specific measures of those finite extensions/locations are dropped while forming their concepts; concepts are formed via a process of measurement-omission.

I am not sure, but anyway guess, that now might be a good time to begin focusing on the issue of measurements of extensions and locations. But let’s not therefore dive directly into an inventory of the specific methods of measurements. Instead, let’s continue keeping our focus primarily on the referents in the concrete world, at least initially.

First, take a moment to fully recall the kind of perceptual field we had considered in the last post, viz., that of the motionless world at the beach, or a similar one at the mountain top. (To remind: by the term “motionless world,” we mean: the motionless aspects of the real, motion-having world which is perceptually evident.)

Now, though it is time to begin examining the issue of measuring extensions and locations of the concretely real physical objects, it still is not time to bring out your foot-rules and compasses. Not yet. … Many aspects of measuring extensions and locations precede these!

Indeed, the next simplest concept, I think, is that of shape, which primarily arises in the context of measuring extensions, and not locations.

[Here, I am skipping over other possible simple spatial concepts such as the specifically relational ones, e.g., “is next to/is on the upside of/is on the downside of” or “is a neighbour of,” etc.]

Just one more important note, before we go to “shape.”

For convenience, our further discussion in this post will assume that measuring extensions or locations requires at least two foreground kind of objects present in a single perceptual field. (I mean to say: as apart from that ever-present background object.)

One important reason I kept postponing posting of this post right on the 5th of October (by which time its first draft was anyway ready) was to see if I could present this issue of the background object in simple enough and clear enough a form. But I could not. I then decided to cut it out, and first to make a simpler presentation by assuming that both of the two minimum necessary objects are only of the foreground kind.

Actually, most of the following discussion does apply also to the contexts wherein only one foreground and one background object exists. However, I believe that it is an advanced issue, not a primary one. For a primary, simpler, discussion, it should be enough to refer to only the foreground objects.

4. Shape as a measure of extension:

One of the simplest possible way to measure the extension of an object is not its “size,” but something simpler to it: the shape with which that object remains extended.

The concept of shape is formed by perceiving (at least) two foreground objects of differing shapes.

Here, suppose you do not agree to the above statement. Suppose, you say that you can take just one (foreground) object, and deform it so as to change its shape. This way, you say, you could make do with just one foreground object, and still would be able to perceive different shapes, and therefore, would be able to get to the concept of shape.

However, realize that here you still are implicitly having two referent objects also in this case: the one currently in front of you (with the current shape), and another one in your memory (with the previous shape). A deformation is a kind of a change, and it involves some motion. And so, forming the concept of shape with just one object requires both motion and memory.

On the other hand, if we can in principle form some concept also in a changeless world, then that’s what we should be aiming for, because, this way, it would be defined in a simpler and more fundamental a manner.

Introducing the idea of “motion” right at this place in the hierarchy means wondering about many more relatively much more complex issues, e.g.: (a) either the agent (i.e. another object) that brings about the change (i.e. the motion), or at least the relatively higher-level consideration of the cause of the change—what part of the identity of that one object makes it act in such a way as to effect the observed change; (b) the nature of the change, e.g., whether it is sudden or continuous, whether it is a completed process or an ongoing one, etc.; (c) the metaphysical status of memory and its validation, etc. All these are more complex, more advanced considerations. And, in any case, you still end up referring, at least implicitly (via memory), to two referents. So, instead, let’s stick with the bare minimum referents, and say that the concept of shape can be grasped by looking at two objects of differing shape, right in a changeless, motionless world.

An obvious point here is that shape, too, is grasped perceptually. The way I currently think about it, it’s a part of the concept of extension. Objects exist as extended with their characteristic shapes.

Actually, shape and size are two characteristics with which an object may be taken as extended. However, before we can know how to measure the size of an object, we must know how to measure its shape.

The shapes may be irregular (which would be the case for most natural objects) or regular (which would be the case for most man-made objects). Thus, objects may be seen to exist as extended with more regular shapes like rectangles (windows), spheres (balls), circles (plates), or more irregular ones like ovals (mother’s face), sticks, strings, this animal vs. that animal, or toys replicating various shapes, etc.

The process of classifying objects according to their shapes, too, is a process of measurement, even though it does not involve our usual numbers. It’s a kind of measurement that does not even permit a linear kind of ordering within the different shapes. For instance, you could say that some kind of a linear order exists between: ball, egg, and apple. Also, between: ball, egg, and, say, dumb-bells. But can you place apple and dumb-bells in a linear order? (If you say apple is more primitive, let me ask you: why is a a sharp and inwardly curving surface necessarily more primitive than a smooth and outwardly curving one?)

So, there are measurements here—the same characteristic of extension exists with different measures when it comes to different objects—even if the measures are not even necessarily rank-able, leave alone expressible as multiples of each other or so. You can only say in a broad sense that, as far as shapes go, the measures of the shape of a sphere and an egg are, in some way, closer to each other than they are with the measures of the shapes of the four-legged animals. But beyond that, it all is mostly direct enumeration of your narrower classifications of shapes.

An important way to measure shapes also is via reference to their topology; however, it’s not a topic of sufficient fundamental importance in physics, and so, we will leave it at that. [The non-importance of topology to fundamental physics precisely was the reason I didn’t go over the relations like “is next to,” “is upwards of” or “is connected to,” in detail. The CS folks have been going abuzz with these concepts, but that does not alter their factual nature and make them of any greater fundamental importance in physics—regardless of what the modern physicists, e.g., the “connected worm-hole” types, think. Come to think of it, networks and topology are of considerable interest in engineering (e.g., networks of fluid pipes, electrical power networks, telephone networks, work-flow networks, the networks of meshes in computational science and engineering, etc.). But they are not of any consequence in the fundamental physics theories—at least those (among the valid ones) built till date.]

Measurements of extensions of objects is easy, and sometimes even at all possible, when their shapes are similar—i.e. when the measurements of their shapes fall close enough to each other that thinking of them as being essentially constant, is possible.

In short-and-sweet terms: you can measure the size of an object when the shape is held constant.

You can easily say that a football is bigger than a tennis-ball, because both of them have the same, spherical, shape. But, you know, you can always confuse a child by asking him which one is bigger: the football or grandpa’s walking stick. If he says the stick, you show him how the stick is really thinner, and therefore, smaller than the football. If he says the ball, you show him how the stick is indeed longer, and therefore, really bigger than the ball. (A smart kid—and one who is sufficiently honest—will argue back. (… If he were merely an “introvert,” he wouldn’t have got as far as talking this much with you, in the first place!))

Volume:

You might think that to measure the size of an object, you wouldn’t have to hold its shape as constant. One could just compare the volume of water displaced by a solid object when the it is completely immersed in the water, you might say. But, actually, measuring volume ofÂ  one thing via displacement of something else, is an advanced consideration. Displacements are possible only in a motion-carrying world.

And, even in a motion-carrying world, before you can say that the volume of a golden crown is equal to the volume of the body of water it displaces, you still first have to define what you mean by “volume of the displaced water,” of course.

To define the meaning of the latter term, you have no other option except going back to the perceptual concretes. You have to realize that “volume,” in the sense of a certain basic “3D-ness” or a basic “solid-ness” means a certain measure of its extension. And, since extension is measured by both shape and size, you have to keep the shape the same, and change the sizes.

That, incidentally, is precisely what you do when you define the volume measures. You have a cylinder of uniform cross-section, and thus, not just the shape but also the cross-section is held constant. It then is a simple matter to be able to measure volume of a given body of water itself, in reference to the remaining aspect of the extension it has within its container, viz., the height it assumes in the measuring cylinder. Yet, notice two things about this scenario: (i) we are measuring volume of water itself, not of another object like a golden crown, and (ii) liquids are not really physical objects in the primary sense of the term, only solids are. Liquids always require containers to hold them, which makes their extension a bit more complicated (than simplified!): the extension now depends on the shape of other objects. Epistemologically, this is a complicating consideration. No matter how you try to circumscribe it, it involves, directly or indirectly, some kind of a motion—and we are trying to avoid that for as long as possible.

BTW, by volume, I also do not mean the triple integral—which, incidentally, is only a technique, not for defining volume, but simply for going from an infinitesimally small volume to a finite volume, according to a highly precise summing-up scheme. But the whole procedure first assumes that the idea of “volume” of an object itself is already known.

And notice, I said: the volume of an object, and even, the volume of a (liquid) object displaced by another (solid) object being measured.

But I didn’t say: volumes occupied by objects in space.

… We still haven’t reached the far more sophisticated concept of space that would allow us a usage like that. We still don’t have a concept of space defined in that sense. Not yet.

All that we have, thus far, are only the concretely perceived objects in a motionless world, including their spatial characteristics, and whatever sense of the term “space” that might be had with just these basic considerations.

But we still are not ready to fill space with objects, or measure volumes via such a procedure. All of that is way high up, hierarchically speaking. We will get there, in due course of time.

For the time being, let’s just note that volume is a tricky concept to deal with, in a motionless world. It involves displacement, and the only way to measure displacement is to make a reference to a cavity-like feature of the measuring device. So, to measure the volume of a solid object X, you have to make reference to the size of the water body Y displaced by it, and the size of Y when placed in a solid container body Z having a peculiar shape—that of a container, i.e., carrying a “solid” or “volumetric” cavity. Two additional objects (Y and Z) are being required, simply for the measurement of a characteristic of a body X! Rather tricky!! And, it involves displacements, too, i.e. motion.

So, let’s leave the discussion of volume in that form.

Can there be any other spatial characteristics or attributes that can be measured in a motionless world? If not, can we think of some other concepts—the simple or basic or primitive concepts—which arise in reference to space, and which can be meaningful also in a motionless world? Can you?

We will come to a few of such concepts in the next post, and it’s only after examining them that we will be finally ready to go into a motion-full world!

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We are, in a way, shaping up the concept of space, though this fact may not be very obvious at this stage. For basic (or foundational) concepts, the time spent in just “clearing up the earth” is as good as, if not better than, directly “putting up the tent,” so to speak. Many issues of physics attributed to space really don’t belong “in” it, and so, even though the present discussion may seem a bit too lackadaisical, it, eventually, would only help. [Rather like a century coming off the bat of… Rahul Dravid.]

* * * * *Â Â  * * * * *Â Â  * * * * *

A Song I Like:

(Hindi) “kahin ek maasum, naazuk see laDki…”
Lyrics: Kamal Amarohi (??)/Jan Nisaan Akhtar (??)/Someone else (??)
Music: Khayyam

[E&OE]

Getting going about “space”

Before we begin:

I wanted to directly go on to my recent series, the one on the theme of what all different broad views of the physical world can be taken to be implicit in the states of the knowledge of physics, as it existed during the various eras of its development. Continuing linearly in that series, in this post, we should have been pursuing how Fourier’s theory marks the beginning of yet another “world.” Despite the fact that it came as early as the beginning of the 19th century—a time when, speaking off-hand, Laplace and Poisson were peaking, Gauss had just about got going, and not even Lagrange had yet made any mark on the scene let alone Hamilton—Fourier’s theory still is, markedly, not a “Newtonian” theory in its basic spirit.

Fourier’s theory doesn’t have the instantaneous action-at-a-distance (IAD) simply as a secondary or accidental feature the way gravity does in Newton’s gravitational theory. The fact is, Fourier’s theorization couldn’t have progressed even an iota, nay, it couldn’t even possibly have got off the mark, without first assuming a certain form of IAD as being a very essential feature of the physical world.

Anyway, the fact of the matter is:Â  understanding IAD requires sufficient clarity on the issue of nature of space (and also on the differential equation paradigm of the physics-model building, though people should be clear enough on this one). And so, at least some discussion of the nature of space is absolutely necessary.

… And, that way, it would have been necessary even if the relativity theory were never to be formulated in the form that it was, by Poincare and Einstein, simply because both Newtonian gravity and Fourier’s heat conduction had already involved IAD, but to a different degree of emphasis. I mean to say: If Newton was merely agnostic about IAD, then Fourier was a zealot for IAD—albeit a refined one, of course. And, subtle. … Very subtle, if you ask me. … And, IAD is something which, even on the basis of common sense alone, and right on the face of it, seems incoherent with any physics theory. And, therefore, demands a deeper examination of the concept of space, first.

In this post, I will try to address all the essential points about space that I wish to cover. However, the topic is both intricate and very “deep,” and so, I may not be able to finish it one go. (Hey, even I have limitations when it comes to typing, OK?) If so, we will pick up some portion of it in the next post.

Preliminary:

What does the word “space” mean? … Here, we can’t look up to the physicists and mathematicians. They don’t know the answer, but they do have the brains to confuse us—and also themselves—quite a lot. Just look at them! (LOL!). So, instead, we have to think independently, afresh, beginning at the beginning.

Actually, the word “space” is a bit overloaded a term. There are many shades to its meaning; it stands for many aspects. But I guess, jotting them down all right at the beginning is neither practical nor necessary. So, what we will do is to pick up the different nuances as we encounter them. So, first, let’s get going, starting from the very beginning, by going “all the way down,” i.e., to the base of all knowledge, i.e., to the perceptually evident reality.

A relevant, and directly available—i.e. perecptual—field:

Suppose you are seated at some remote and lonely beach on some evening, contemplating the meaning of the word “space.” … Imagine the vast expanse of the deep blue of the sea lying in front of you, seemingly coming closer and still closer towards you along with those waves, and ending with them somewhere at that thin beige crescent which is the patch of the sand surrounding you; with some thick green woods standing mostly quiet behind the sand; and then, a paler but even wider expanse of the blue sky “standing” completely aloof over it all, ignoring everything below it—even some idle white streak of a cloud hanging up there mostly cluelessly. Down below, that darkish green patch of the woods continues on for some distance, getting hazier and hazier, and merges into the greyish black outline of a distant hill, which then goes cutting across some way into that flat line where the two blues meet. … You may also imagine a few homeward-bound birds in the sky, a happily aimless stray dog roaming around on the beach, a seemingly motionless ship or two in the distant motionless sea, and the dark orange of the setting sun, so to complete your picture. … The night then sneaks in slowly, and brings with it a cool breeze, and some time later, it’s all mostly dark and silent, with the twinkling stars mischievously asking you whether you succeeded in getting the answer to your question or not. … The question of space. …

… Or, imagine that you are seated at some high point in the hills on some evening, with some greenish-blue mountains remaining majestically self-absorbed only in themselves for eternity—a range after another range progressively fading away into distance, and an even thinner and paler sky that still refuses to go away even if it now seems ever more expansive and even more aloof. … And, as the night falls, the stars, once again, wouldn’t forget to pull their mischief—of raising that question to you. …

Once back in the city, as you walk on a late monsoon evening on a bridge, past those bright orange sodium-vapor lamps erected on those dullish silver lamp-posts, the stars somehow become completely invisible, even if the skies are clear. Instead, what you see are some whizzing clouds of those pathetically thin flies randomly whizzing above the heads of the passers-by. There is no easily discernible pattern to the motion of those flies. In fact, the motion of any one fly is even plain visually impossible to follow. The flies make for a buzzing kind of a cloud, but far more indefinite for it all to be at all called a cloud. The “cloud” here seems to be engaged in a never-ending flux of sorts.

If the metaphysical nature of reality were such as to be constantly in a chaotic flux—a literal chaos, one far worse than the cloud of those flies buzzing overhead—then no knowledge would at all be possible in that universe.

Knowledge—all knowledge—begins with definite objects that are easily distinguishable in the perceptual field of awareness. The perceptual field is the basis of all knowledge. A sense of an enduring, definite, reality is where we can at all begin. And all our knowledge, if valid, must be reducible to such perceptual concretes. Even if there be some motion to some object, it must be perceptible. It must orderly—not constituting a random metaphysical “flux.”

At the beach, it’s the unmoving patches like that of the sand and of the distant hill that provide an implicit sort of an anchor to our visual field. That way, there is a lot of motion going on also at the beach. The dance of the foam upon the breaking of the waves is evanescent, but it can still be visually grasped because it is, in a way, orderly. Neither does the sea ever forget its limits nor do the bubbles get formed at unpredictable places in it. The motion of the flapping of the wings of the birds occurs relatively slower, and so, it is even more easily graspable. The outline of the idly loitering dog is slower still. And the sun does get noticeably longer to completely disappear from the view. At night, the stars move, but it would take some alertness and some patience to notice that fact. And, the distant hill does not move at all, neither does that patch of the sand.

You can make out the rapid flapping of the wings against a relatively slower motion of an approaching wave. But to distinguish every possible motion—including that of the sun and the stars—you first require that stand-still outline of that distant hill, or the constancy, as it were, of the horizon where the sea “ends.”

In the most general sense, a state of motion can be perceptually grasped only against the backdrop of a state of its absence. We require solid, definite objects that also are enduringly stand-still in a perceptual field, before we can at all become aware of the motion of some other objects.

So, let us first consider the stand-still objects—the sand at the beach and the hill nearby, or the unyielding patch of the ground on which you were seated on the mountain-top, and the stand-still ranges of those distant mountains. What all things about them can we make out right from such, simple, perceptual fields? What features of interest to physics do they hold?

The two characteristics of a motionless physical world:

What interesting aspects are held by stand-still things—that is, even if there isn’t any motion at all? [And, please remember, the discussion from this point onwards until explicitly noted, refers only to a motionless world—i.e. the perceptually enduring aspect of the as-perceived world.]

For one thing, the things in the motionless physical world still are distinguishable—i.e. perceptible.

And, among the perceptually grasped characteristics of the kind that allow any distinction to be at all perceived (i.e., those characteristics which permit our perception to at all have a sense of there being this object vs. that object), a couple of characteristics stand out. These are: the locations and the extensions of the individual objects.

In my (at least current) opinion, it does not matter which one we tackle first. Metaphysically, just like with any other characteristic, they both—extension and location—exist simultaneously, and so, a metaphysical consideration cannot settle the issue of deciding the hierarchical priority between the two. Now, epistemologically, it seems to me that they constitute a certain pair of characteristics such that both of them become at all graspable only simultaneously—i.e. in reference to the exactly same perceptual field (though their respective grasps would differ in time). And, as far as my current thinking goes, it therefore does not matter which one we begin with, for the purposes of our discussion. … BTW, this epistemological point is rather deep, and may be, I will come to it later on in this post. But, first, let’s deal with them, individually. And, remember the context: We are dealing with a motionless world, here.

Extension:

The patch of the sand has, broadly speaking, a definite beginning perceived at the shore-line, and a definite end perceived at the dark green woods. And, similarly, for the outline of a narrow path through the woods that you might see, while seated at a mountain-top. And, similarly, for the distant hills.

The very existence of these (and all such) objects is such that each of them is seen (i.e. perceived) to have a certain beginning and a certain end to them. Even in a world completely devoid of motion, for each object constituting it, there would still exist these kinds of beginnings and ends. Even if your eyes were taken as not grossly moving, these would exist—they are an objectively existing feature of the perceived objects. They are what allows us to make out the outlines of those objects in the first place; they are a part of perceptually distinguishing one object from the other.

When we thus speak of these directly perceived beginnings and ends, the one characteristic which we are trying to isolate here is not so much the boundary of the object, but (what I currently think is) something simpler: the extent of the object. Rather, a fact somewhat even more basic to it: the metaphysically given fact of the “extended-ness” of an object.

All [concretely real] physical objects are perceptually grasped as being extended. “Extended-ness” is a characteristic that is directly available right in the perceptual field.

And, it is there with every physical object that we perceive. To repeat (because this point is important): All objects are extended; extension is a characteristic of every object.

Using terms from the Objectivist epistemology, extension is a concept whose scope is co-extensive with that of the physically existing objects themselves.

Note, the primary issue here is not that pertaining to how [much] extended a given object is. The size of the extension [of an object] is, hierarchically, a later issue; it is the one which can arise only by first assuming that there is some definite extension to every object. For the time being, let’s leave aside the issue of quantities (or as people familiar with the Objectivist epistemology would know: the issue of the omitted measurements). It should be enough to just note here that the fact that physical objects exist with different sizes of their “extended-ness”es, is also something that is given right in the perceptual field. For instance, the dog is perceptually seen to be smaller than the beach, or the trees shorter than the hill. But, hierarchically, this whole issue of size (or of comparative measurements) is an advanced one.

More crucial for the time being is the fact that the concretely real objects—all of them—are only definitely extended.

In particular, contraries to this truth never are a part of the perceptual field. Given the nature of the perceived entities, the contraries cannot be—that is what the inductive generalization that objects are definitely extended, means. Since we are dealing with a first-level generalization here, only ostensive definitions are possible. All that we can do is to indicate its truth in an indirect manner. Say, by pointing out the approach to its meaning, as we have done above, and then, by pointing out the contradictions its contraries raise.

It is precisely because every object is definitely extended that we are completely stumped if we try to “visualize” an object with no extension. It’s the case wherein an end of an object is asserted to be the same as a beginning of the same object, and thus, extension cannot any longer be regarded as a characteristic of that “object.” And, it is this reason why “an object without extension” stumps us.

It is also precisely because every object is definitely extended that we also are completely stumped if we try to “visualize” an object with an infinite extension. This, now, is the case wherein either a beginning, or an end (or both) is (are) missing, not just in an isolated instance of perceptual fields, but in principle, i.e., as if even with a missing beginning or end, you could still have a concretely real object. Now, if either a beginning or an end is absent, then what this peculiarity actually does is to once again make it impossible to regard extension as a characteristic for that alleged “object.” In other words, the procedure wipes out the fact that extension is a necessary characteristic of any object. We, once again, are stumped only because no concretely real physical object can at all be without an extension.

The “some–any” principle of the Objectivist epistemology [^] requires a characteristic to exist in some (definite) quantity, even if it may exist with any (definite) quantity. Qua characteristics of physically existing objects, extension can only be finite quantity. If the term “extension” of a concretely real physical object has any meaning at all, it can only exist in some definite quantities.

Please remember this part of the discussion when we come to the issue of infinity or otherwise of space and universe. … However, let me also hasten to advise you to refrain from directly jumping to conclusions. Remember, there are many senses in which the word “space” may be taken. Here, we have been considering the most fundamental sense with which the word begins to derive its meaning. And, in fact, all that we have thus far seen is only one characteristic at the basis of the concept, viz., that of the extension—not space itself. The word “space” does require the characteristic of extension for its context, but it is not in itself sufficient.

Location:

There is another characteristic, given right in the same perceptual field, i.e., the one referring to the objects of the motionless world. So, let us take a moment to refresh the motionless aspects of our starting perceptual field, as, e.g., the one at the sea-side or on the mountain top, etc. What else, apart from extension, do we see by way of, say, “properties” of those objects?

We see, for example, the patch of the sand, and of the sea. And realize another simple truth, viz., that each of them is [found] there, where it is [found]. The sand and the sea are situated in a certain kind of a perceptually grasped (spatial) relationship with each other. The sand is next to the sea—or, if you wish to put it the symmetrically other way, the sea is next to the sand. Similarly, the sea is next to the hill—or the hill is next to the sea. And so on, for any pair of the perceptually distinguishable objects. … The sand begins where the sea ends; and the sea begins where the sand ends. The sand is there, where it is; and the sea is there, where it is. Each perceptually grasped object is there, where it is perceputally seen to be; it is simply a part of the perceptually given distinction that comes with the very act of perceiving that physical object.

Thus, there is a certain characteristic of “there-ness” that is possessed by each physically existing object. Let’s name this characteristic as: “location.”

Apart from extension, location is the other physically existing characteristic displayed by every physical object [in the motionless world]. Just the way each physical object is necessarily extended, similarly, each physical object also necessarily has a location.

The words like “here” and “there” denote a certain basic characteristic which is a part of the very identity of any perceived physical object.

Thus, the scope of the concept of location, too, is “co-extensive” with that of extension.

“Location,” too, has a certain definiteness of its measurements, as a part of its context.

And, though I wouldn’t go into a detailed explanation of it, once again, two main classes of errors are possible: we are stumped when we are asked to imagine (i) an object that is no-where, or (ii) an object that is any-where. The reason we cannot imagine either of them as belonging in the concretely real physical world is because: each is alleged to exist without possessing the essential characteric of a definite location.

[BTW, if you have got the above point right, you may want to think of the reason why, when it came to the error of supposing an infinity of “there-ness,” I have used the word “any-where,” and not “every-where.” Try to figure out the reason for that. Have fun! :)]

A couple of miscellaneous [though perhaps a bit advanced] epistemological points:

There are few points which involve some subtleties of epistemology. However, they aren’t entirely of a technical epistemological interest alone. In fact, a discussion of them could very well have the effect of anticipating a few important points of physics. So, even if you aren’t very comfortable with epistemology as such, do try to read on—it’s easily possible that you might very easily “get it” anyway! (The points may be subtle, but they are not too difficult.)

(E1) In case you missed a certain tricky point above, let me highlight it here: Location, I have asserted, is a characteristic of an individual object. The tricky point? If you noticed it, I was thereby also asserting that location does not primarily refer to a relation between two or more objects.

… In other words, I seem to be putting forth a view of space that is not, say, so much “relational” in nature—am I not? … BTW, here, you may want to see Dr. Peikoff’s writing on the concept of space [^]. And, while you read the ensuing discussion, it might be fun for you to decide whether I agree with Dr. Peikoff’s position or not! … But, coming back to the issue at the hand here, yes, I am putting forth a viewpoint of space that is not “so much” relational in nature. Let me explain.

In fact, I will go even further and state my wider epistemological conviction. There can be no characteristics existing as apart from the objects with which we might relate them! Not even when it comes to the relational characteristics. There can be no “purely” relational characteristics that exist separately from (or independent of) the objects of which they are the characteristics.

If two or more objects can at all be related to each other, it is only via some characteristic(s) (or attribute(s), etc.) which each of them must first separately possesses as a part of its own, separate, identity. Only if this condition is fulfilled that we can at all relate them.

And, thus, even the so-called relational characteristics do not exist all by themselves; they do not exist in a void of sorts, floating somehow disconnected from the objects, and somehow connecting with the objects only when it is the time for us to relate them. … You cannot join two isolated blobs (of objects) by a line (of a relationship) if the attribute you note for that line wasn’t already a part of the identity (i.e. a characteristic or an attribute, etc.) of each of those two blobs taken by itself.

Relations can be isolated as separate concepts, sure. But their status is not, therefore, at par with the objects which they help relate. The objects come first; the relations come at a higher level of abstraction. And, a particular relation can at all “come” (or be formed by a mind) if and only if each of those objects, taken separately, already possesses that same characteristics—on the basis of which they can later be related.

(E2) Another point, in a way, going seemingly against the above point:

Actually, even though I did not explicitly note this point while discussing the concept of extension, even “extension” is a characteristic which can only be grasped only if there are at least two objects separate physical objects (each of which would separately possess a definite measure of its extension).

If there were to be only one object in the world, you couldn’t grasp either its extended-ness or its located-ness (or “there”-ness). Indeed, you couldn’t even grasp that object. This is a very subtle point. It refers to a “what if” scenario which Ayn Rand had once used, in that seminar (in the 2nd edition of ITOE). Let me briefly touch upon it.

Suppose all you had for the universe was only a vast expanse of some uniform (and never-changing) pale blue, and nothing else, then you couldn’t grasp any thing, Ayn Rand said. Not even just the fact that there is some expanse of the blue. You couldn’t grasp even that—not even purely on the perceptual level. But, add just one speck of dust, and perception at all becomes possible, she had pointed out.

[As an aside: A reminder which might perhaps help those who are already familiar with the Objectivist epistemology: Both differentiation and integration are necessary elements of any process of awareness—even for the awareness only on the perceptual level. And, both of them come at the same time. If consciousness is a difference-detector, it also is a commonality-integrator, or vice versa. Aside over.]

… To resume back the main discussion: no perception is at all possible with just a uniform expanse of blue, but it becomes possible as soon as you add a speck of dust.

Now, for convenience, you may want to imagine a sizeable circle of a uniform shade of dark blue, instead of just a speck of dust. … Once this circle is introduced against that pale blue backdrop, you would begin perceiving not only that circle. And, the same perceptual field would also contain the most bare essentials for you to perceive the fact that this circular object was extended, and further, that it was located where it was.

In other words, what you could say, in effect is not just: “Something!”, but also—because it’s a physical object—“Some thing, there!”

In the very act of saying “there,” (i.e. grasping the location) you would also be implicitly referring to the fact that the dark blue circle was extended—that it was extended over a certain region where the pale blue was not, and the pale blue was where the dark blue circle was not, and vice versa.

An alternative way of putting the same fact is also possible: In the act of seeing that the dark blue circle had an extension, you would also be referring to the fact that it was extended from “here,” meaning one part of its boundary, to “there,” meaning some other part of its boundary.

In particular, the same perceptual field would be enough for you to grasp both the characteristics of location and extension. Just the way both differentiation and integration are necessary to any cognition—perception or conceptualization—similarly, both location and extension become possible at the same time.

But, “wait a minute,” you might say, here. “Location, in our above discussion, referred to finding out the where-ness of one object by referring this object to some other object next to it. The sand was next to the sea, and the sea was next to the sand, and both were definitely extended. But, here, the pale blue is not definitely extended. There is no visible “outside” boundary to the pale blue. Only the dark blue circle has a boundary. Since there is no boundary to the pale blue, by your own definition, it cannot be taken as a physical object. So, there is nothing sitting next to the dark blue circle. So, the circle may have extension, but it has no location. If so, how come both these concepts become possible at the same time?”

In answer, I would say: “If you are smart enough to think of that objection, then you must also be smart enough to figure out the answer as well!” No, really!! I mean it!!! So, try to crack it on your own. And, if you can’t, I will let you know what I think of it, some time later on in this series.

… For the time being, notice that, like with any proper, relevant, and deep issue of epistemology, some paradoxes which seem to raise their heads first in the physics theory obviously have some kind of a source in some deep epistemological and metaphysical theories. If I say that philosophical issues like these are relevant to the physics issues such as the existence or otherwise of the aether, the infinity or otherwise of the physical space and/or the universe, then my assertion should at least be plausible even if not outright obvious.

But of course, before we can begin talking about the infinity of space and all, first, we have to understand what “space” means. And, thus far, what we have done is only to isolate two characteristics of the concretely existing physical objects: viz., their extensions and locations.

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But, anyway, enough is enough, for today. … It’s not that a lot of territory is still left…. Just two more steps (or so), and we should be on our way to “completely” understanding what the word “space” basically means. I will come back to write about those remaining points soon enough. We will then be in a position to worry about the issue of IAD, of Fourier’s theory (i.e., whether it necessitates a further splitting of our physical worlds, or not), of Newton’s theory of gravitation, of aether, and the rest of the “shebang.”

So, hang in there. At your [own] locations, and with your [own] extensions (I mean to say: without worrying about exercise, diet, etc.!)

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A Song I Like:

(Hindi) “deewaanaa mujh saa nahin, is ambar ke neeche…”
Music: R. D. Burman