# The One vs. the Many

This post continues from my last post. In that post, I had presented a series of diagrams depicting the states of the universe over time, and I had then asked you a simple question pertaining to the physics of it: what the series depicted, physically speaking.

I had also given an answer to that question, the one which most people would give. It would run something like this:

There are two blocks/objects/entities which are initially moving closer towards each other. Following their motions, they come closer to each other, touch each other, and then reverse the directions of their motions. Thus, there is a collision of sorts. (We deliberately didn’t go into the maths of it, e.g., such narrower, detailed or higher-level aspects such as whether the motions were uniform or whether they had accelerations/decelerations (implying forces) or not, etc.)

I had then told you that the preceding was not the only answer possible. At least one more answer that captures the physics of it, also is certainly possible. This other answer in fact leads to an entirely different kind of mathematics! I had asked you to think about such alternative(s).

In this post, let me present the alternative description.

The alternative answer is what school/early college-level text-books never present to students. Neither do the pop-sci. books. However, the alternative approach has been documented, in some or the other form, at least for centuries if not for millenia. The topic is routinely taught in the advanced UG and PG courses in physics. However, the university courses always focus on the maths of it, not the physics. The physical ideas are never explicitly discussed in them. The text-books, too, dive straight into the relevant mathematics. The refusal of physicists (and of mathematicians) to dwell on the physical bases of this alternative description is in part responsible for the endless confusion and debates surrounding such issues as quantum entanglement, action at a distance, etc.

There also is another interesting side to it. Some aspects of this kind of a thinking are also evident in the philosophical/spiritual/religious/theological thinking. I am sure that you would immediately notice the resonance to such broader ideas as we subsequently discuss the alternative approach. However, let me stress that, in this post, we focus only on the physics-related issues. Thus, if I at times just say “universe,” it is to be understood that the word pertains only to the physical universe (i.e. the sum total of the inanimate objects, and also the inanimate aspects of living beings), not to any broader, spiritual or philosophical issue.

OK. Now, on to the alternative description itself. It runs something like this:

There is only one physical object which physically exists, and it is the physical universe. The grey blocks that you see in the series of diagrams are not independent objects, really speaking. In this particular depiction, what look like two independent “objects” are, really speaking, only two spatially isolated parts of what actually is one and only one object. In fact, the “empty” or the “white” space you see in between the objects is not, really speaking, empty at all—it does not represent the literal void or the nought, so to speak. The region of space corresponding to the “empty” portions is actually occupied by a physical something. In fact, since there is only one physical object to all exist, it is that same—singleton—physical object which is present also in the apparently empty portions.

This is not to deny that the distinction between the grey and the white/“empty” parts is not real. The physically existing distinction between them—the supposed qualitative differences among them—arises only because of some quantitative differences in some property/properties of the universe-object. In other words, the universe does not exist uniformly across all its parts. There are non-uniformities within it, some quantitative differences existing over different parts of itself. Notice, up to this point, we are talking of parts and variations within the universe. Both these words: “parts” and “within” are to be taken in the broadest possible sense, as in  the sense of“logical parts” and “logically within”.

However, one set of physical attributes that the universe carries pertains to the spatial characteristics such as extension and location. A suitable concept of space can therefore be abstracted from these physically existing characteristics. With the concept of space at hand, the physical universe can then be put into an abstract correspondence with a suitable choice of a space.

Thus, what this approach naturally suggests is the idea that we could use a mathematical field-function—i.e. a function of the coordinates of a chosen space—in order to describe the quantitative variations in the properties of the physical universe. For instance, assuming a $1D$ universe, it could be a function that looks something like what the following diagram shows.

Here, the function shows that a certain property (like mass density) exists with a zero measure in the regions of the supposedly empty space, whereas it exists with a finite measure, say with density of $\rho_{g}$ in the grey regions. Notice that if the formalism of a field-function (or a function of a space) is followed, then the property that captures the variations is necessarily a density. Just the way the mass density is the density of mass, similarly, you can have a density of any suitable quantity that is spread over space.

Now, simply because the density function (shown in blue) goes to zero in certain regions, we cannot therefore claim that nothing exists in those regions. The reason is: we can always construct another function that has some non-zero values everywhere, and yet it shows sufficiently sharp differences between different regions.

For instance, we could say that the graph has $\rho_{0} \neq 0$ value in the “empty” region, whereas it has a $\rho_{g}$ value in the interior of the grey regions.

Notice that in the above paragraph, we have subtly introduced two new ideas: (i) some non-zero value, say $\rho_{0}$, as being assigned even to the “empty” region—thereby assigning a “something”, a matter of positive existence, to the “empty”-ness; and (ii) the interface between the grey and the white regions is now asserted to be only “sufficiently” sharp—which means, the function does not take a totally sharp jump from $\rho_{0}$ to $\rho_{g}$ at a single point $x_i$ which identifies the location of the interface. Notice that if the function were to have such a totally sharp jump at a single point, it would not in fact even be a proper function, because there would be an infinity of density values between and including $\rho_{0}$ and $\rho_{g}$ existing at the same point $x_i$. Since the density would not have a unique value at $x_i$, it won’t be a function.

However, we can always replace the infinitely sharp interface of zero thickness by a sufficiently sharp (and not infinitely sharp) interface of a sufficiently small but finite thickness.

Essentially, what this trick does is to introduce three types of spatial regions, instead of two: (i) the region of the “empty” space, (ii) the region of the interface (iii) the interior, grey, region.

Of course, what we want are only two regions, not three. After all, we need to make a distinction only between the grey and the white regions. Not an issue. We can always club the interface region with either of the remaining two. Here is the mathematical procedure to do it.

Introduce yet another quantitative measure, viz., $\rho_{c}$, called the critical density. Using it, we can in fact divide the interface dispense region into further two parts: one which has $\rho < \rho_c$ and another one which has $\rho \geq \rho_c$. This procedure does give us a point-thick locus for the distinction between the grey and the white regions, and yet, the actual changes in the density always remain fully smooth (i.e. density can remain an infinitely differentiable function).

All in all, the property-variation at the interface looks like this:

Indeed, our previous solution of clubbing the interface region into the grey region is nothing but having $\rho_c = \rho_0$, whereas clubbing the interface in the “empty” space region is tantamount to having $\rho_c = \rho_g$.

In any case, we do have a sharp demarcation of regions, and yet, the density remains a continuous function.

We can now claim that such is what the physical reality is actually like; that the depiction presented in the original series of diagrams, consisting of infinitely sharp interfaces, cannot be taken as the reference standard because that depiction itself was just that: a mere depiction, which means: an idealized description. The actual reality never was like that. Our ultimate standard ought to be reality itself. There is no reason why reality should not actually be like what our latter description shows.

This argument does hold. Mankind has never been able to think of a single solid argument against having the latter kind of a description.

Even Euclid had no argument for the infinitely sharp interfaces his geometry implies. Euclid accepted the point, the line and the plane as the already given entities, as axioms. He did not bother himself with locating their meaning in some more fundamental geometrical or mathematical objects or methods.

What can be granted to Euclid can be granted to us. He had some axioms. We don’t believe them. So we will have our own axioms. As part of our axioms, interfaces are only finitely sharp.

Notice that the perceptual evidence remains the same. The difference between the two descriptions pertains to the question of what is it that we regard as object(s), primarily. The considerations of the sharpness or the thickness of the interface is only a detail, in the overall scheme.

In the first description, the grey regions are treated as objects in their own right. And there are many such objects.

In the second description, the grey regions are treated not as objects in their own right, but merely as distinguishable (and therefore different) parts of a single object that is the universe. Thus, there is only one object.

So, we now have two alternative descriptions. Which one is correct? And what precisely should we regard as an object anyway? … That, indeed, is a big question! 🙂

More on that question, and the consequences of the answers, in the next post in this series…. In it, I will touch upon the implications of the two descriptions for such things as (a) causality, (b) the issue of the aether—whether it exists and if yes, what its meaning is, (c) and the issue of the local vs. non-local descriptions (and implications therefore, in turn, for such issues as quantum entanglement), etc. Stay tuned.

A Song I Like:

(Hindi) “kitni akeli kitni tanha see lagi…”
Singer: Lata Mangeshkar
Music: Sachin Dev Burman
Lyrics: Majrooh Sultanpuri

[May be one editing pass, later? May be. …]