# Wanna try this one? (Even if not on QM, it does involve complex numbers, randomness, …)

So, you know all there is to know about QM, I mean things like complex numbers, randomness, and all that, don’t you?

Or at least, you have read all there is to read about, say, QM, complex numbers, their amplitudes, probability, randomness, and all that, haven’t you?

If so, I have a question for you. Let me see how you approach it. …

… As far as this question is concerned, it doesn’t matter whether the answer is right or wrong. Not for this question, and not to me anyway. It’s the approach you take that would be really interesting—at least to me, and at least for this question…

So, ok? Ready? Here we go:

The question is:

What is meant by randomness? Can you give me an example of a random (or at least a pseudo-random) sequence of complex numbers? How would you convince me that it in fact is random (or at least pseudo-random), whatever you mean by that term?

I will wait for a while for answers to come in [less likely], or for people to post entries on their blogs, or for the media to post articles dealing with this aspect [more likely], but without mentioning anything about me, of course [certainly]! … As to me: I will run any answers you try here. … Further, as to others’ blogs/media articles: If I find them OK, or even just plain interesting, then I will even copy-paste those answers (or at least excerpts) here, and provide links to them.

Then, I will come back and give you my answer, after a while.

… One of the reasons it might take at least a week for me to come back (on this question) is: we once again are set to get busy at work. (Yes, we will be working on this week-end.)

The other reason is: I would really like to wait for a while, and let you try the question.

When I come back with my answer, I will also add the usual songs-section.

Bye for now, and take care…

# “The spiritual heritage of India”

I wrote a few comments at Prof. Scott Aaronson’s blog, in response to his post of the title: “30 of my favorite books”, here [^].

Let me give you the links to my comments: [^], [^], [^] and [^].

Let me reproduce the last one of my four comments, with just so slight bit of editing. [You know I couldn’t have resisted the opportunity, right?]:

Since I mentioned the “upnishad”s above (i.e. here [ ^]), and as far as this topic is concerned, since the ‘net is so full of the reading material on this topic which isn’t so suitable for this audience, let me leave you with a right kind of a reco.

If it has to be just one short book, then the one which I would pick up is this:

Swami Prabhavananda (with assistance of Frederick Manchester), “The Spiritual Heritage of India,” Doubleday, New York, 1963.

A few notes:

1. The usual qualifications apply. For instance, I of course don’t agree with everything that has been said in the book. And, more. I may not even agree that a summary of something provided here is, in fact, a good summary thereof.

2. I read it very late in life, when I was already past my 50. Wish I had laid my hands on it, say, in my late 20s, early 30s, or so. I simply didn’t happen to know about it, or run into a copy, back then.

3. Just one more thing: a tip on how to read the above book:

First, read the Preface. Go through it real fast. (Reading it faster than you read the newspapers would be perfectly OK—by me).

Then, if you are an American who has at least a smattering of a knowledge about Buddhism, then jump directly on to the chapter on Jainism. (Don’t worry, they both advocate not eating meat!) And, vice-versa!!

If you are not an American, or,  if you have never come across any deeper treatment on any Indian tradition before, then: still jump on to the chapter on Jainism. (It really is a very good summary of this tradition, IMHO.)

Then, browse through some more material.

Then, take a moment and think: if you have appreciated what you’ve just read, think of continuing with the rest of the text.

Else, simple: just call it a book! (It’s very inexpensive.)

No need to add anything, but looking at the tone of the comments (referring to the string “Ayn Rand”) that got generated on this above-mentioned thread, I find myself thinking that, may be, given my visitor-ship pattern (there are more Americans hits today to my blog than either Indian or British), I should explain a bit of a word-play which I attempted in that comment (and evidently, very poorly—i.e. non-noticeably). It comes near the end of my above-quoted reply.

“Let’s call it a day” is a very neat British expression. In case you don’t know its meaning, please look it up on the ‘net. Here’s my take on it (without looking it up):

Softly folding away a day, with a shade of an anticipation that a day even better might be about to arrive tomorrow, and so, softly reminding yourself that you better leave the party or the function for now, so as to be able to get ready for yet another party, yet another function, some other day, later…

A sense of something like that, is implied by that expression.

I just attempted a word-play, and so, substituted “book” for the “day”.

Anyway, good night. Do read my last post, the document attached to it, and the links therefrom.

Bye for now.

Oh, yes! There is a song that’s been playing off-and-on at the back of my mind for some time. Let me share it with you.

A Song I Like:

(Hindi) “dil kaa diyaa jala ke gayaa…”
Lyrics: Majrooh Sultaanpuri
Singer: Lata Mangeshkar
Music: Chitragupt

[PS: The order of the listing of the credits, once again, is completely immaterial here.]

Anyway, enjoy the song, and the book mentioned in the quotes (and hopefully, also my past few posts and their attachments)… I should come back soon, with a maths-related puzzle/teaser/question. … Until then, take care and bye!

# 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.

# To the “subscribers” of this blog

This post is being written entirely on-the-fly.

I have over some period of time observed that far too many of the subscribers of this blog (may be more than half of them) actually are/should be fake accounts.

But, as you perhaps might know, I have been, say, “follow-up”ed for a somewhat longer length of a time—and, with those “followers” never having to have had created any email account any-where, to be able to “follow-up” on me—in real life, too.

So, being “follow-up”ed, but without causing immediate trouble in my immediate life/surroundings, was a bit of a curiosity for me, and so, I tolerated them—these recent email IDs, so to speak.

No, not with a sense of amusement, but with that of keeping them, as they say, “under observation.”

Anyway, as to the non-authentic ones:

I invite these “subscribers” to get themselves off, silently if they prefer, but very certainly—and, very immediately.

[Yes, they may “post” their “protests” in the forms that are able to more silently hit me in ways more than just a few postings here and there on the ‘net. I don’t care, any longer.

Neither about these account “creators”, nor even about those who are (and were) skeptical about what such forms could possibly be—even if I wrote about such forms honestly.]

But for those among my “subscribers” who are willingly to unsubscribe from this blog, I shall give them a time-period, of until:

$2 \text{April\ } 2018 - \epsilon \text{\ IST\ }$

where ($\epsilon \rightarrow 0$) is: what even an idiot who has never studied beyond XI/XII science would be able to tell them—or, should be.

In other words, the Fool’s Day is their last day, as far as this blog of mine is concerned.

In other words, I “promise” to grant them a personal pardon that if they do wind themselves up, off my blog, in the due time-limit.

… No, I don’t expect them to do that. …

But if not, I shall do the latter for them.

[… No, never ‘been afraid of an extra bit of a work, ever in my life. …]

A Song I Like:

(Hindi) “kitnaa pyaaraa waadaa hai in matwaali aankhon kaa…”
Music: R. D. Burman