[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.)
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?
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A Song I Like:
(Hindi) “jhir jhir barase saavanee ankhiyaan, saanwariyaa ghar aaa…”
Music: Vasant Desai
Singer: Lata Mangeshkar
Lyrics: Harindranath Chattopadhyaya (??) / Gulzaar (??)
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.]