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 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.

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 , 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 -, -, 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…”

Band: Beatles

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

[E&OE]

I think that a fundamental mistake is to believe that mathematics is the “language of the universe” (ie. of physical reality), whereas, in fact, I believe it is merely the language of our minds. There has never been a perfect mathematical model of any physical phenomena, because it is just that – a model – so math is just a highly compact modeling language that provides a means of generating neural patterns that have some useful but finite (limited) coherence with physical reality, helping to structure the neural patterns into useful mechanisms for the detection of other related patterns, growing the perception of reality… and somebody will always come along and show that the current model is an approximation, give us a new model, and that, too, will ultimately be an approximation (Newton -> Einstein -> etc). If there were a perfect mathematical model of reality, on any scale for any thing, then my suspicion is that it would be infinitely complex and infinite in scope/extent, despite the finiteness of what it models, thus placing it well beyond the reach of the human mind… and so, is that really mathematics, then? Is infinity a number or just a direction? NaN, methinks!

Bengineer,

Thanks for your thoughtful comments. Your remarks also are thought-provocative, but since my replies would become too big, let me stick to writing just a few notings.

1. Yes, I couldn’t agree more that maths is

not“the language of the universe.” That is too ambitious a statement, and it completely over-blows the case for maths. (Whenever I hear it, I always instinctively find myself comparing the role of maths to the roles of each of: physical units, the law of the dimensional homogeneity, and diagrams (including engineering graphics or “blue-prints,” complete with bill of materials.)2. But still, the fact is, maths

isavitalpart of the theory-building in physics (and in every field or topic where using methods of physics is appropriate, such as in parts of chemistry, and in all of engineering, data science, etc.)Thinking on-the-fly and off-hand, there are two different roles that (proper) maths plays in physics and engineering:

(2.1) Imparting a quantitative precision to the statements of physical laws. E.g., saying that gravity obeys the inverse square law is not only more concise but also more precise and therefore more informative; it tells us not only that gravity drops as you go away from its source, but also that it drops not as slow as 1/r or as fast as 1/r^3. Which allows us to predict where we may position our satellites. See David Harriman’s book on Induction in Physics for a stronger (IMO a bit too strong) a defence of this position.

(2.2) Due to its

abstractnature, statements in mathematical terms also tend to become concise, and hence allow us to pick out analogies (or conceptual classifications) very fast. E.g., all second-order differential equations are similar; the king here is the second-order term, not the first-, because it’s the former that shapes the nature of the solution, e.g. difference between potential, diffusion and wave phenomena. So, insights from electric potential can be applied to gravitational, etc.(2.3) The

downsideon both the above two points—(2.1) and (2.2) is this: use of maths in physics places an increasingly high premium on your ability to retain the appropriate context—both prior mathematical and physical context. In modern physics and maths, it in fact over-burdens and over-taxes this ability. All confusions (including those pertaining to space, infinity) arise from this characteristics of maths—and its abuse in physics. So, I hate the lionizers of maths because they not only never explicitly acknowledge the role of the right context, but they even have a tendency to hurriedly shove it under the carpet. … Modern physicists and mathematicians don’t even care to do that; they in fact casually throw the very idea of the necessity of a physical context itself out of the window, and then even proceed to blabber about in a languid manner. … In short, they actively encourage the bad of context-dropping.3. You bring up a fine point concerning the idea of perfection in knowledge, and the infinitely complex, or the infinite in scope/content.

[To other readers: Here, to formulate and express his thought, Bengineer is draws on the technique of expansion into a suitable basis (say polynomial, though the Fourier basis too would imply exactly the same set of things).]

Ummm… Once again, I am helped by Ayn Rand’s epistemology seminar (Intro. to Objectivist Epistemology, 2nd ed.). The thing is, she says that “perfection” is primarily a normative (or moral) concept, not epistemological (or cognitive) one. OTOH, physics is primarily a descriptive science, not normative—morality arises only by implication, not primarily, when we talk of physics. So, the applicable concept in physics is “(level of) precision,” and not “perfection.”

Within its context, Newton’s theory

was, is, remains, will always remainvalid; what changes from the Newtonian to the Relativistic/QMechanical progression is: a context that includes an increased scope (different for each of R and QM), and therefore, an increased level of sophistication of physical thought, which in turn allows an increased level of (mathematical) precision. (Parenthetically, I do think that David Harriman, despite his enormously great insights, missed the middle, connective part: sophistication of physical thought.) But the important take-away here is this: if taken within its own context and region of applicability, each theory of physics is perfect.Now, about the remarkably insightful remark which you made, viz. “despite the finiteness of what it models.” … I grinned involuntarily right on the first reading.

It’s this point which any modern physicist/mathematician would have very easily missed! And, it’s a very, very, valid point. E.g., in heat conduction, even to model something as simple as the “triangular” temperature profile initially, even though both the rod and the temperatures are finite, you still need an infinity of terms! Examples abound in every branch of physics. And yet, in my reading, no mathematician, physicist, or historian of maths has ever stopped to ponder about it! (But, yes, I had noticed that too!!)

4. Shameless self-promotion: You may want to check out my posts: https://ajitjadhav.wordpress.com/2015/08/12/is-the-physical-universe-infinite/ and https://ajitjadhav.wordpress.com/2015/08/16/why-do-physicists-use-infinity/

Thanks, again, and best!

–Ajit

[PS: And, once again, I ended up writing at length! Sorry about that.]

[E&OE]

No need for apologies…. enjoyed the read.

All the best.

All what you are saying does it not also depend upon what kind of cognitive faculties you yourself have? What sort of capabilities and limitations your own mind has? In other words what the result of any observation is does it not depend at least partly upon the cognitive faculties of the observer, so making any observation subjective? If other humans get the same observation as you do, does that make your observation objective or only inter-subjective to humans?

ontologicalrealist,

There are two ways to answer your questions. Let me pursue them both.

1. The straight-forward way, answering each point as it comes up:

Yes, what I say does depend on the kind of cognitive faculty I have. My mind, despite its limitations, allows me to live well, so, the so-called “limitations” don’t matter. (For man, his mind is the tool of survival qua man.) Yes, the result of my observation does depend on my cognitive faculties. (I can’t see anything using microwaves or ultraviolet light, or hear that sound which bats emit in moving around.) No, that does not make my observations subjective. (I do see this laptop screen and the keyboard keys, and that’s a part of the objective reality which exists independent of whether I look at my laptop or not.) If other humans (e.g. you) too can see laptops or computer screens, I like it, because it means, e.g., that they can read my blog posts and answers [and at least this way come to appreciate how great a guy I am]. But no, even if I may sometimes depend on their observations, I still have to cross-check and validate what they are telling me, before I can use them in reaching my conclusions. (I thought the silencer of my car was gone beyond having it repaired; one welder thought along the same lines too, but then, I showed it to another welder, and he fixed it. The silencer was objectively fixable.) In any case, I have to ensure the objectivity of my observations using my focus (free will), the right epistemology, and with the reality as the ultimate referent. In any case, I don’t use an agreement with others as my primary criterion. [An easy example here is this: even if you never come to agree that I am a great guy, I would always go on believing so… Always and always. … Forever.] As to humans in general, IMO, the same thing: right epistemology, hard-work (focus, free-will), and the reality as both the ultimate standard and reference.

[…Turns out that the straight-forward way wasn’t all that straight-forward; was it?]

2. Another way, speaking in overall terms:

Let me ask you something. Are you impressed by Kant? Do you agree with him? Or, I mean, even if you aren’t following Kant himself, was there a book which you read or an argument you heard or so, which can help explain how it is that you’ve come to adopt this line of questioning?… Would like to know.

And, thanks for dropping by and writing in, anyway!

Best,

–Ajit

[E&OE]

Hi Ajit,

You are right, I am much impressed by Kant. But I must add that I am not a follower of Kant or any other philosopher, religion or ideology etc. at all. I am my own man. I also want to add that I am much impressed by Ayn Rand from whom I have learned much. I also know that she was very opposed to Kantian epistemology.

As you are an admirer of Ayn Rand like myself, perhaps I can have at least a little hope that you may understand why I think what I think (if you are an independent thinker like myself).

Let me ask you a question. I assume that you understand logic:-

Do you accept that A is not non A ?

ontologicalrealist

ontologicalrealist,

LOL!

No, I don’t accept that A is non-A.

–Ajit

[E&OE]

I did not ask you about whether you accept that A is non-A.

I asked you whether you accept that A is not non A ?

Please check again my question and your answer.

Wow OR (i.e. ontologicalrealist), wow!

Usually, I don’t make reading mistakes like that. But I here, I evidently did. Thanks for catching it. … I canveryeasily make mistakes while writing, but usually not, while reading.Anyway, the reason I here said “wow” was not so much for catching my mistake. It was because I found your question real cool! … Your question is way more subtle than it sounds.

Let me think about how to write my answer better for a little time, and then come back to you, say in a day or two; I will probably put up my answer as a separate post on this blog.

I actually began writing an answer on the fly, but then found it not satisfactory… In the process, I also grasped the real nature of the issue, and explaining that one—writing it down in the best possible manner—is going to take a bit of thinking. That’s why I am asking for time…

But to give you the gist of my answer right away:

You have to define your terms. Usually, engineers and scientists take the terms A and non-A in the (mathematical) set-theoretical sense; but I can now see that when Aristotle (and later Ayn Rand) used such terms, they meant them in the broader, philosophical sense. To give an unambiguous answer to your question

is, therefore, to spell out one’s position on the issue of whether one takes the mathematical theory of sets as more fundamental than the theory of universals (or concepts), or not. Onthis, broader issue, I regards concepts as more fundamental than the sets of the set theory.Thanks (really) once again, for raising that question (and for persisting with it!). I will come back ASAP.

Best,

–Ajit

[E&OE]

Thanks, it takes nobility of character to acknowledge one’s own mistake.

Please take as much time as you need before replying.

You seem to have understood the importance of this question.

Cool, OR, thanks! The post is online already at: https://ajitjadhav.wordpress.com/2016/10/22/on-whether-a-is-not-non-a/

–Ajit

[E&OE]

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