How many numbers are there in the real number system?

Post updated on 2018/04/05, 19:25 HRS IST:

See the sections added, as well as the corrected and expanded PDF attachment.


As usual, I got a bit distracted from my notes-taking (on numbers, vectors, tensors, CFD, etc.), and so, ended up writing a small “note” on the title question, in a rough-and-ready plain-text file. Today, I converted it into a LaTeX PDF. The current version is here: [^].

(I may change the document contents or its URL without informing in advance. The version “number” is the date and time given in the document itself, just below the title and the author name.)

(However, I won’t disappoint those eminent scholars who are interested in tracing my intellectual development. I will therefore keep the earlier, discarded, versions too, for some time. Here they are (in the later-to-earlier order): [^][^][ ^ ].)


This PDF note may look frivolous, and in some ways it is, but not entirely:

People don’t seem to “get” the fact that any number system other than the real number system would be capable of producing a set consisting of only distinct numbers.

They also don’t easily “get” the fact that the idea of having a distinct succession numbers is completely different from that of a continuum of them, which is what the real number system is.

The difference is as big as (and similar to) that between (the perceptually grasped) locations vs. (the perceptually grasped) motions. I guess it was Dr. Binswanger who explained these matters in one of his lectures, though he might have called them “points” or “places” instead of ”locations”. Here, as I recall, he was explaining from what he had found in good old Aristotle: An object in motion is neither here (at one certain location) nor there (in another certain location), Aristotle said; it’s state is that it is in motion. The idea of a definite place does not apply to objects in motion. That was the point Dr. Binswanger was explaining.

In short, realize where the error is. The error is in the first two words of the title question: “How many”. The phrase “how many” asks you to identify a number, but an infinity (let alone an infinity of infinity of infinity …) cannot be taken as a number. There lies the contradiction.


BTW, if you are interested, you may check out my take on the concept of space, covered via an entire series of (long) posts, some time ago. See the posts tagged “space”, here [^]


When they (the mathematicians, who else?) tell you that there are as many rational fractions as there are natural numbers, that the two infinities are in some sense “equal”, they do have a valid argument.

But typical of the modern-day mathematicians, they know, but omit to tell you, the complete story.

Since I approach mathematics (or at least the valid foundational issues in maths) from (a valid) epistemology, I can tell you a more complete story, and I will. At least briefly, right here.

Yes, the two infinities are “equal.” Yes, there are as many rational fractions as there are natural numbers. But the densities of the two (over any chosen finite interval) are not.

Take the finite interval [1.0, 101.0). There are 100 number of distinct natural numbers in them. The size of the finite interval, measured using real numbers, also is 100.o. So the density of the natural numbers over this interval is: 1.0.

But the density of the rational fractions over the same interval is far greater. In fact it is so greater that no number can at all be used to identify its size: it is infinite. (Go, satisfy yourself that this is so.)

So, your intuition that there is something wrong to Cantor’s argument is valid. (Was it he who began all this business of the measuring the “sizes” of infinite sets?)

Both the number of natural numbers and the number of rational fractions are infinities, and these infinities are of the same order, too. But there literally is an infinite difference between their local densities over finite intervals. It is  this fact that the “smart” mathematicians didn’t tell you. (Yes, you read it here first.)

At the same time, even if the “density” over the finite interval when the interval is taken “in the gross” (or as a whole) is infinite, there still are an infinite number of sub-intervals that aren’t even touched (let alone exhausted) by the infinity of these rational fractions, all of them falling only within that [1.0, 101.0) interval. Why? Because, notice, we defined the interval in terms of the real numbers, that’s why! That’s the difference between the rational fractions (or any other number-producing system) and the real numbers.


May be I will write another quick post covering some other distractions in the recent times as well, shortly. I will add the songs section at that time, to that (upcoming) post.

Bye for now.

 

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Some suggested time-pass (including ideas for Python scripts involving vectors and tensors)

Actually, I am busy writing down some notes on scalars, vectors and tensors, which I will share once they are complete. No, nothing great or very systematic; these are just a few notings here and there taken down mainly for myself. More like a formulae cheat-sheet, but the topic is complicated enough that it was necessary that I have them in one place. Once ready, I will share them. (They may get distributed as extra material on my upcoming FDP (faculty development program) on CFD, too.)

While I remain busy in this activity, and thus stay away from blogging, you can do a few things:


1.

Think about it: You can always build a unique tensor field from any given vector field, say by taking its gradient. (Or, you can build yet another unique tensor field, by taking the Kronecker product of the vector field variable with itself. Or, yet another one by taking the Kronecker product with some other vector field, even just the position field!). And, of course, as you know, you can always build a unique vector field from any scalar field, say by taking its gradient.

So, you can write a Python script to load a B&W image file (or load a color .PNG/.BMP/even .JPEG, and convert it into a gray-scale image). You can then interpret the gray-scale intensities of the individual pixels as the local scalar field values existing at the centers of cells of a structured (squares) mesh, and numerically compute the corresponding gradient vector and tensor fields.

Alternatively, you can also interpret the RGB (or HSL/HSV) values of a color image as the x-, y-, and z-components of a vector field, and then proceed to calculate the corresponding gradient tensor field.

Write the output in XML format.


2.

Think about it: You can always build a unique vector field from a given tensor field, say by taking its divergence. Similarly, you can always build a unique scalar field from a vector field, say by taking its divergence.

So, you can write a Python script to load a color image, and interpret the RGB (or HSL/HSV) values now as the xx-, xy-, and yy-components of a symmetrical 2D tensor, and go on to write the code to produce the corresponding vector and scalar fields.


Yes, as my resume shows, I was going to write a paper on a simple, interactive, pedagogical, software tool called “ToyDNS” (from Toy + Displacements, Strains, Stresses). I had written an extended abstract, and it had even got accepted in a renowned international conference. However, at that time, I was in an industrial job, and didn’t get the time to write the software or the paper. Even later on, the matter kept slipping.

I now plan to surely take this up on priority, as soon as I am done with (i) the notes currently in progress, and immediately thereafter, (ii) my upcoming stress-definition paper (see my last couple of posts here and the related discussion at iMechanica).

Anyway, the ideas in the points 1. and 2. above were, originally, a part of my planned “ToyDNS” paper.


3.

You can induce a “zen-like” state in you, or if not that, then at least a “TV-watching” state (actually, something better than that), simply by pursuing this URL [^], and pouring in all your valuable hours into it. … Or who knows, you might also turn into a closet meteorologist, just like me. [And don’t tell anyone, but what they show here is actually a vector field.]


4.

You can listen to this song in the next section…. It’s one of those flowy things which have come to us from that great old Grand-Master, viz., SD Burman himself! … Other songs falling in this same sub-sub-genre include, “yeh kisine geet chheDaa,” and “ThanDi hawaaein,” both of which I have run before. So, now, you go enjoy yet another one of the same kind—and quality. …


A Song I Like:

[It’s impossible to figure out whose contribution is greater here: SD’s, Sahir’s, or Lata’s. So, this is one of those happy circumstances in which the order of the listing of the credits is purely incidental … Also recommended is the video of this song. Mona Singh (aka Kalpana Kartik (i.e. Dev Anand’s wife, for the new generation)) is sooooo magical here, simply because she is so… natural here…]

(Hindi) “phailee huyi hai sapanon ki baahen”
Music: S. D. Burman
Lyrics: Sahir
Singer: Lata Mangeshkar


But don’t forget to write those Python scripts….

Take care, and bye for now…

 

Stress is defined as the quantity equal to … what?

Update on 01 March 2018, 21:27, IST: I had posted a version of this post also at iMechanica, which led to a bit of a very interesting interaction there [^] too. Check it out, if you want… Also see my today’s post concerning the idea of stress, here [^].


In this post, I am going to note a bit from my personal learning history. I am going to note what had happened when a clueless young engineering student that was me, was trying hard to understand the idea of tensors, during my UG years, and then for quite some time even after my UG days. May be for a decade or even more….

There certainly were, and are likely to be even today, many students like [the past] me. So, in the further description, I will use the term “we.” Obviously, the “we” here is the collegial “we,” perhaps even the pedagogical “we,” but certainly neither the pedestrian nor the royal “we.”


What we would like to understand is the idea of tensors; the question of what these beasts are really, really like.

As with developing an understanding of any new concept, we first go over some usage examples involving that idea, some instances of that concept.

Here, there is not much of a problem; our mind easily picks up the stress as a “simple” and familiar example of a tensor. So, we try to understand the idea of tensors via the example of the stress tensor. [Turns out that it becomes far more difficult this way… But read on, anyway!]

Not a bad decision, we think.

After all, even if the tensor algebra (and tensor calculus) was an achievement wrought only in the closing decade(s) of the 19th century, Cauchy was already been up and running with the essential idea of the stress tensor right by 1822—i.e., more than half a century earlier. We come to know of this fact, say via James Rice’s article on the history of solid mechanics. Given this bit of history, we become confident that we are on the right track. After all, if the stress tensor could not only be conceived of, but even a divergence theorem for it could be spelt out, and the theorem even used in applications of engineering importance, all some half a century before any other tensors were even conceived of, then developing a good understanding of the stress tensor ought to provide a sound pathway to understanding tensors in general.

So, we begin with the stress tensor, and try [very hard] to understand it.


We recall what we have already been taught: stress is defined as force per unit area. In symbolic terms, read for the very first time in our XI standard physics texts, the equation reads:

\sigma \equiv \dfrac{F}{A}               … Eq. (1)

Admittedly, we had been made aware, that Eq. (1) holds only for the 1D case.

But given this way of putting things as the starting point, the only direction which we could at all possibly be pursuing, would be nothing but the following:

The 3D representation ought to be just a simple generalization of Eq. (1), i.e., it must look something like this:

\overline{\overline{\sigma}} = \dfrac{\vec{F}}{\vec{A}}                … Eq. (2)

where the two overlines over \sigma represents the idea that it is to be taken as a tensor quantity.

But obviously, there is some trouble with the Eq. (2). This way of putting things can only be wrong, we suspect.

The reason behind our suspicion, well-founded in our knowledge, is this: The operation of a division by a vector is not well-defined, at least, it is not at all noted in the UG vector-algebra texts. [And, our UG maths teachers would happily fail us in examinations if we tried an expression of that sort in our answer-books.]

For that matter, from what we already know, even the idea of “multiplication” of two vectors is not uniquely defined: We have at least two “product”s: the dot product [or the inner product], and the cross product [a case of the outer or the tensor product]. The absence of divisions and unique multiplications is what distinguishes vectors from complex numbers (including phasors, which are often noted as “vectors” in the EE texts).

Now, even if you attempt to “generalize” the idea of divisions, just the way you have “generalized” the idea of multiplications, it still doesn’t help a lot.

[To speak of a tensor object as representing the result of a division is nothing but to make an indirect reference to the very operation [viz. that of taking a tensor product], and the very mathematical structure [viz. the tensor structure] which itself is the object we are trying to understand. … “Circles in the sand, round and round… .” In any case, the student is just as clueless about divisions by vectors, as he is about tensor products.]

But, still being under the spell of what had been taught to us during our XI-XII physics courses, and later on, also in the UG engineering courses— their line and method of developing these concepts—we then make the following valiant attempt. We courageously rearrange the same equation, obtain the following, and try to base our “thinking” in reference to the rearrangement it represents:

\overline{\overline{\sigma}} \vec{A} = \vec{F}                  … Eq (3)

It takes a bit of time and energy, but then, very soon, we come to suspect that this too could be a wrong way of understanding the stress tensor. How can a mere rearrangement lead from an invalid equation to a valid equation? That’s for the starters.

But a more important consideration is this one: Any quantity must be definable via an equation that follows the following format:

the quantiy being defined, and nothing else but that quantity, as appearing on the left hand-side
=
some expression involving some other quantities, as appearing on the right hand-side.

Let’s call this format Eq. (4).

Clearly, Eq. (3) does not follow the format of Eq. (4).

So, despite the rearrangement from Eq. (2) to Eq. (3), the question remains:

How can we define the stress tensor (or for that matter, any tensors of similar kind, say the second-order tensors of strain, conductivity, etc.) such that its defining expression follows the format given in Eq. (4)?


Can you answer the above question?

If yes, I would love to hear from you… If not, I will post the answer by way of an update/reply/another blog post, after some time. …

Happy thinking…


A Song I Like:
(Hindi) “ye bholaa bhaalaa man meraa kahin re…”
Singers: Kishore Kumar, Asha Bhosale
Music: Kishore Kumar
Lyrics: Majrooh Sultanpuri


[I should also be posting this question at iMechanica, though I don’t expect that they would be interested too much in it… Who knows, someone, say some student somewhere, may be interested in knowing more about it, just may be…

Anyway, take care, and bye for now…]

Blog-Filling—Part 3

Note: A long Update was added on 23 November 2017, at the end of the post.


Today I got just a little bit of respite from what has been a very tight schedule, which has been running into my weekends, too.

But at least for today, I do have a bit of a respite. So, I could at least think of posting something.

But for precisely the same reason, I don’t have any blogging material ready in the mind. So, I will just note something interesting that passed by me recently:

  1. Catastrophe Theory: Check out Prof. Zhigang Suo’s recent blog post at iMechanica on catastrophe theory, here [^]; it’s marked by Suo’s trademark simplicity. He also helpfully provides a copy of Zeeman’s 1976 SciAm article, too. Regular readers of this blog will know that I am a big fan of the catastrophe theory; see, for instance, my last post mentioning the topic, here [^].
  2. Computational Science and Engineering, and Python: If you are into computational science and engineering (which is The Proper And The Only Proper long-form of “CSE”), and wish to have fun with Python, then check out Prof. Hans Petter Langtangen’s excellent books, all under Open Source. Especially recommended is his “Finite Difference Computing with PDEs—A Modern Software Approach” [^]. What impressed me immediately was the way the author begins this book with the wave equation, and not with the diffusion or potential equation as is the routine practice in the FDM (or CSE) books. He also provides the detailed mathematical reason for his unusual choice of ordering the material, but apart from his reason(s), let me add in a comment here: wave \Rightarrow diffusion \Rightarrow potential (Poisson-Laplace) precisely was the historical order in which the maths of PDEs (by which I mean both the formulations of the equations and the techniques for their solutions) got developed—even though the modern trend is to reverse this order in the name of “simplicity.” The book comes with Python scripts; you don’t have to copy-paste code from the PDF (and then keep correcting the errors of characters or indentations). And, the book covers nonlinearity too.
  3. Good Notes/Teachings/Explanations of UG Quantum Physics: I ran across Dan Schroeder’s “Entanglement isn’t just for spin.” Very true. And it needed to be said [^]. BTW, if you want a more gentle introduction to the UG-level QM than is presented in Allan Adam (et al)’s MIT OCW 8.04–8.06 [^], then make sure to check out Schroeder’s course at Weber [^] too. … Personally, though, I keep on fantasizing about going through all the videos of Adam’s course and taking out notes and posting them at my Web site. [… sigh]
  4. The Supposed Spirituality of the “Quantum Information” Stored in the “Protein-Based Micro-Tubules”: OTOH, if you are more into philosophy of quantum mechanics, then do check out Roger Schlafly’s latest post, not to mention my comment on it, here [^].

The point no. 4. above was added in lieu of the usual “A Song I Like” section. The reason is, though I could squeeze in the time to write this post, I still remain far too rushed to think of a song—and to think/check if I have already run it here or not. But I will try add one later on, either to this post, or, if there is a big delay, then as the next “blog filler” post, the next time round.

[Update on 23 Nov. 2017 09:25 AM IST: Added the Song I Like section; see below]

OK, that’s it! … Will catch you at some indefinite time in future here, bye for now and take care…


A Song I Like:

(Western, Instrumental) “Theme from ‘Come September'”
Credits: Bobby Darin (?) [+ Billy Vaughn (?)]

[I grew up in what were absolutely rural areas in Maharashtra, India. All my initial years till my 9th standard were limited, at its upper end in the continuum of urbanity, to Shirpur, which still is only a taluka place. And, back then, it was a decidedly far more of a backward + adivasi region. The population of the main town itself hadn’t reached more than 15,000 or so by the time I left it in my X standard; the town didn’t have a single traffic light; most of the houses including the one we lived in) were load-bearing structures, not RCC; all the roads in the town were of single lanes; etc.

Even that being the case, I happened to listen to this song—a Western song—right when I was in Shirpur, in my 2nd/3rd standard. I first heard the song at my Mama’s place (an engineer, he was back then posted in the “big city” of the nearby Jalgaon, a district place).

As to this song, as soon as I listened to it, I was “into it.” I remained so for all the days of that vacation at Mama’s place. Yes, it was a 45 RPM record, and the permission to put the record on the player and even to play it, entirely on my own, was hard won after a determined and tedious effort to show all the elders that I was able to put the pin on to the record very carefully. And, every one in the house was an elder to me: my siblings, cousins, uncle, his wife, not to mention my parents (who were the last ones to be satisfied). But once the recognition arrived, I used it to the hilt; I must have ended up playing this record for at least 5 times for every remaining day of the vacation back then.

As far as I am concerned, I am entirely positive that appreciation for a certain style or kind of music isn’t determined by your environment or the specific culture in which you grow up.

As far as songs like these are concerned, today I am able to discern that what I had immediately though indirectly grasped, even as a 6–7 year old child, was what I today would describe as a certain kind of an “epistemological cleanliness.” There was a clear adherence to certain definitive, delimited kind of specifics, whether in terms of tones or rhythm. Now, it sure did help that this tune was happy. But frankly, I am certain, I would’ve liked a “clean” song like this one—one with very definite “separations”/”delineations” in its phrases, in its parts—even if the song itself weren’t to be so directly evocative of such frankly happy a mood. Indian music, in contrast, tends to keep “continuity” for its own sake, even when it’s not called for, and the certain downside of that style is that it leads to a badly mixed up “curry” of indefinitely stretched out weilings, even noise, very proudly passing as “music”. (In evidence: pick up any traditional “royal palace”/”kothaa” music.) … Yes, of course, there is a symmetrical downside to the specific “separated” style carried by the Western music too; the specific style of noise it can easily slip into is a disjointed kind of a noise. (In evidence, I offer 90% of Western classical music, and 99.99% of Western popular “music”. As to which 90%, well, we have to meet in person, and listen to select pieces of music on the fly.)

Anyway, coming back to the present song, today I searched for the original soundtrack of “Come September”, and got, say, this one [^]. However, I am not too sure that the version I heard back then was this one. Chances are much brighter that the version I first listened to was Billy Vaughn’s, as in here [^].

… A wonderful tune, and, as an added bonus, it never does fail to take me back to my “salad days.” …

… Oh yes, as another fond memory: that vacation also was the very first time that I came to wear a T-shirt; my Mama had gifted it to me in that vacation. The actual choice to buy a T-shirt rather than a shirt (+shorts, of course) was that of my cousin sister (who unfortunately is no more). But I distinctly remember she being surprised to learn that I was in no mood to have a T-shirt when I didn’t know what the word meant… I also distinctly remember her assuring me using sweet tones that a T-shirt would look good on me! … You see, in rural India, at least back then, T-shirts weren’t heard of; for years later on, may be until I went to Nasik in my 10th standard, it would be the only T-shirt I had ever worn. … But, anyway, as far as T-shirts go… well, as you know, I was into software engineering, and so….

Bye [really] for now and take care…]