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
“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 it’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 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 in the East direction, then move some distance in the North direction, and then climb some distance 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 , and readings are completely independent of each other. No matter how hard you slog along, say the -direction, it yields no fruit at all along the – or – 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 , and 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 does cover all the points in space, and if isn’t necessary to reach every point in space, and if falls short, then the inevitable conclusion is: indeed is the exact dimensionality of the physical space.
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 , 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.
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 -, -, 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 -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…”
[May be an editing pass tomorrow? Too tired today.]