Recovering-ed/Recovered-ing

The general impression among philosophers of science and physicists alike is that maths is simple.

According to this viewpoint, maths may be—nay, even  must be—beautiful. But for all its complexity, speaking in the cultured tones, it is condemned to stay simple. The subtle shades of the evanescent feelings and emotions, say as captured by a piece of poetry or a work of fine art, they say, is not accessible to the hard, cold, “objective,” world of even of science in general, let alone the “world” of mathematics.

And, yes, as a matter of a plain truth, blogs must still be written, for the most part, using plain languages, for instance, in English! Not in mathematics.


Now, as far as I am concerned, I do seem to have sometime in the past much appreciated what folks such as these mean by those words.

But a subtle change took a root in my mind over the course of the last, ummmm…, 6–7 days, whose final culmination is what this post is all going to be about.


I mean to say, over the course of the past week or so, I seemed to be steadily recovering from my RSI (duly reported earlier on this very blog; see my last post).

Yesterday, the situation was that I seemed to have “fully” recovered from it.

And yet, as I was at it—I mean: at my poor keyboard—once again, I developed, you know, … a feeling. A feeling, now, near the base of my right-hand thumb. A feeling of a bit of a pain.

Now, given the really, really smart person that I am, I exactly knew what to do next: I stopped doing work, and ordered for me, through official channels [if you must be ever so curious], a new, more ergonomic, and < Rs. 500 keyboard. And then, I rested upon my newfound hint of an oncoming pain. [“Prevention is better than cure.”]

Then, sometime this late afternoon, as I was toying with the idea of slipping myself out of this sense of a highly diluted but nevertheless all-pervading boredom, I noticed that I cannot express myself at all. I mean to say: in plain English.

The “real truth” of the matter is this:

I think I have recovered—at least with all of today’s (and past few days’) boredom.

The thing also is: I think I have not recovered—not at least with that slight-ish pain, now appearing at the basal region of my right thumb.

Now, see, this is a situation that is so well-captured by maths in the following manner [but before going over that, may I remind you, for the nth time, that the proper spelling of the proper short-form of “mathematics” does naturally carry an ‘s’ at its end]:

Let w be defined as the wellness index. Then, states of well-/ill-ness can be easily expressed according to the following scheme:

  • Illness \Leftrightarrow -1.0 \leq w \leq 0.0
  • Recovered \Leftrightarrow w = 1.0
  • Recovering \Leftrightarrow 0.0 < w < 1.0

Simple enough a scheme, right?

So, now, the only question is: what English phrase do you use for the case which is captured by the expression: 0.999\dots \leq w < 1.0? especially if it also includes a time-evolution? a progress with the passage of time?

If you try to put it in English, referring to the above-mentioned points, there is no word in the English language (or any other “natural” language) to express this thought, this aspect of the actual reality (i.e. the condition of my typing hands). After all, “Recovered” does mean 1.0 but this number is not acceptable because of the use of the strong “<” sign.

As to “Recovering,” the 0.0 < w < 1.0 range, in this case, turns out to be of a rather Very Large Scale. In fact, as compared to the expression: “0.999\dots \leq w < 1.0“, it actually refers to an infinitely large Scale.

So, how do you express yourself in English, as far as that quoted expression, viz., “0.999\dots \leq w < 1.0,” goes?

After taking into account the time-evolution part of it, you would very naturally say something like: “recovering-ed” or “recovered-ing”. … You choose between the two.

Precisely both precisely are kind of usages that the Wren and Martin of my childhood times wouldn’t permit me, or any other child. (It’s not that I took the pair very seriously even back then, but the point is: I’ve come to know what painful book to quote when.)

And yet, the title usage is amply justified. As so well illustrated by the already established correspondence of maths and English.


And so, as I once again get back to typing [a lot]—but not on this (or any other) blog—what do you do in the meanwhile?

You listen to this song which I like…


A Song I Like:

(Marathi) “naval vartale ge maaye, ujaLalaa prakaashu…”
Lyrics: G. D. Madgulkar [Yes, that’s right, the words didn’t come to you from “sant dnyaaneshwara.”  [Yes, you further are wrong, “dnyaaneshwara” is never pronounced as “dnyaaneshwaraa,” let alone a “dnyaaneshwaraaaaaa.”]]
Singer: Asha Bhosale
Music: C. Ramchandra


[PS: May be I will streamline this post just a bit later tomorrow or the day after or so… .]

 

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Expanding on the procedure of expanding: Where is the procedure to do that?

Update on 18th June 2017:

See the update to the last post; I have added three more diagrams depicting the mathematical abstraction of the problem, and also added a sub-question by way of clarifying the problem a bit. Hopefully, the problem is clearer and also its connection to QM a bit more apparent, now.


Here I partly expand on the problem mentioned in my last post [^]. … Believe me, it will take more than one more post to properly expand on it.


The expansion of an expanding function refers to and therefore requires simultaneous expansions of the expansions in both the space and frequency domains.

The said expansions may be infinite [in procedure].


In the application of the calculus of variations to such a problem [i.e. like the one mentioned in the last post], the most important consideration is the very first part:

Among all the kinematically admissible configurations…

[You fill in the rest, please!]


A Song I Like:

[I shall expand on this bit a bit later on. Done, right today, within an hour.]

(Hindi) “goonji see hai, saari feezaa, jaise bajatee ho…”
Music: Shankar Ahasaan Loy
Singers: Sadhana Sargam, Udit Narayan
Lyrics: Javed Akhtar

 

I’m not…

“I’m not half the man I used to be.”

That’s what they all say when they become, say, mature. These days, the “they” includes me. Yep. That’s right. I have realized that I really am not half the man I used to be. … Let me count the ways…

I no longer write code every day.

I also no longer debug code every day.

I also no longer use C++ as my first programming language.

I also no longer read pop-science books on QM.

…You get the idea. …

But, actually, it’s become worse, much worse. I perhaps am no longer even a quarter of the man I used to be. …

The reason is, not only do I read through (all the pages of) the QM and maths text-books these days, I have in fact also begun going through some maths journals these days… And it doesn’t even stop there… I mean, I also have a recommendation for a maths journal, for you!

OK. Don’t be overly concerned about me. I am quite OK, that way…

About the only maths journal whose many issues I have browsed through (and have recommendation for) is the “SIAM Undergraduate Research Online (SIURO)” [^].

Do check it out. The published papers are available for free. The link to the published papers is right on the home page; it goes to here [^].

Neat… May be Americans have begun becoming smart these days or something…

[I forgot what it was that I was looking for when I stumbled across one of the papers in this journal… If I remember that paper, I will come back and mention it… [I told you, I’m not half the man I used to be…]]


A Song I Like:
(English) “Yesterday”
Band: The Beatles

[Anti-poignancy (or should it be un-poignancy?): The Beatles were in their early twenties when they penned these lyrics.]

[E&OE]

Mathematics—Historic, Contemporary, and Its Relation to Physics

The title of this post does look very ambitious, but in fact the post itself isn’t. I mean, I am not going to even attempt to integrate these diverse threads at all. Instead, I am going to either just jot down a few links, or copy-paste my replies (with a bit editing) that I had made at some other blogs.

 

1. About (not so) ancient mathematics:

1.1 Concerning calculus: It was something of a goose-bumps moment for me to realize that the historic Indians had very definitely gotten to that branch of mathematics which is known as calculus. You have to understand the context behind it.

Some three centuries ago, there were priority battles concerning invention of calculus (started by Newton, and joined by Liebniz and his supporters). Echoes of these arguments could still be heard in popular science writings as recently as when I was a young man, about three decades ago.

Against this backdrop, it was particularly wonderful that an Indian mathematician as early as some eight centuries ago had gotten to the basic idea of calculus.

The issue was highlighted by Prof. Abinandanan at the blog nanpolitan, here [^]. It was based on an article by Prof. Biman Nath that had appeared in the magazine Frontline [^]. My replies can be found at Abi’s post. I am copy-pasting my replies here. I am also taking the opportunity to rectify a mistake—somehow, I thought that Nath’s article appeared in the Hindu newspaper, and not in the Frontline magazine. My comment (now edited just so slightly):

A few comments:

0. Based on my earlier readings of the subject matter (and I have never been too interested in the topic, and so, it was generally pretty much a casual reading), I used to believe that the Indians had not reached that certain abstract point which would allow us to say that they had got to calculus. They had something of a pre-calculus, I thought.

Based (purely) on Prof. Nath’s article, I have now changed my opinion.

Here are a few points to note:

1. How “jyaa” turned to “sine” makes for a fascinating story. Thanks for its inclusion, Prof. Nath.

2. Aaryabhata didn’t have calculus. Neither did Bramhagupta [my spelling is correct]. But if you wonder why the latter might have laid such an emphasis on the zero about the same time that he tried taking Aaryabhata’s invention further, chances are, there might have been some churning in Bramhagupta’s mind regarding the abstraction of the infinitesimal, though, with the evidence available, he didn’t reach it.

3. Bhaaskara II, if the evidence in the article is correct, clearly did reach calculus. No doubt about it.

He did not only reach a more abstract level, he even finished the concept by giving it a name: “taatkaalik.” Epistemologically speaking, the concept formation was complete.

I wonder why Prof. Nath, writing for the Frontline, didn’t allocate a separate section to Bhaaskara II. The “giant leap” richly deserved it.

And, he even got to the max-min problem by setting the derivative to zero. IMO, this is a second giant leap. Conceptually, it is so distinctive to calculus that even just a fleeting mention of it would be enough to permanently settle the issue.

You can say that Aaryabhata and Bramhagupta had some definite anticipation of calculus. And you can’t possible much more further about Archimedes’ method of exhaustion either. But, as a sum total, I think, they still missed calculus per say.

But with this double whammy (or, more accurately, the one-two punch), Bhaaskara II clearly had got the calculus.

Yes, it would have been nice if he could have left for the posterity a mention of the limit. But writing down the process of reaching the invention has always been so unlike the ancient Indians. Philosophically, the atmosphere would generally be antithetical to such an idea; the scientist, esp. the mathematician, may then be excused.

But then, if mathematicians had already been playing with infinite series with ease, and were already performing the calculus of finite differences in the context of these infinite series, even explicitly composing verses about their results, then they can be excused for not having conceptualized limits.

After all, even Newton initially worked only with the fluxion and Leibniz with the infinitesimal. The modern epsilon-delta definition still was some one–two centuries (in the three–four centuries of modern science) in the coming.

But when you explicitly say “instantaneous,” (i.e. after spelling out the correct thought process leading to it), there is no way one can say that some distance had yet to be travelled to reach calculus. The destination was already there.

And as if to remove any doubt still lingering, when it comes to the min-max condition, no amount of merely geometric thinking would get you there. Reaching of that conclusion means that the train had not already left the first station after entering the calculus territory, but also that it had in fact gone past the second or the third station as well. Complete with an application from astronomy—the first branch of physics.

I would like to know if there are any counter-arguments to the new view I now take of this matter, as spelt out above.

4. Maadhava missed it. The 1/4 vs. 1/6 is not hair-splitting. It is a very direct indication of the fact that either Maadhava did a “typo” (not at all possible, considering that these were verses to be by-hearted by repetition by the student body), or, obviously, he missed the idea of the repeated integration (which in turn requires considering a progressively greater domain even if only infinitesimally). Now this latter idea is at the very basis of the modern Taylor series. If Maadhava were to perform that repeated integration (and he would be a capable mathematical technician to be able to do that should the idea have struck him), then he would surely get 1/6. He would get that number, even if he were not to know anything about the factorial idea. And, if he could not get to 1/6, it’s impossible that he would get the idea of the entire infinite series i.e. the Taylor series, right.

5. Going by the content of the article, Prof. Nath’s conclusion in the last paragraph is, as indicated above, in part, non-sequitur.

6. But yes, I, too, very eagerly look forward to what Prof. Nath has to say subsequently on this and related issues.

But as far as the issues such as the existence of progress only in fits here and there, and indeed the absence of a generally monotonously increasing build-up of knowledge (observe the partial regression in Bramhagupta from Aaryabhat, or in Maadhav from Bhaaskar II), I think that philosophy as the fundamental factor in human condition, is relevant.

7. And, oh, BTW, is “Matteo Ricci” a corrupt form of the original “Mahadeva Rishi” [or “Maadhav Rishi”] or some such a thing? … May Internet battles ensue!

1.2 Concerning “vimaan-shaastra” and estimating \pi: Once again, this was a comment that I made at Abi’s blog, in response to his post on the claims concerning “vimaan-shaastra” and all, here[^]. Go through that post, to know the context in which I wrote the following comment (reproduced here with a bit of copy-editing):

I tend not to out of hand dismiss claims about the ancient Indian tradition. However, this one about the “Vimaan”s and all does seem to exceed even my limits.

But, still, I do believe that it can also be very easy to dismiss such claims without giving them due consideration. Yes, so many of them are ridiculous. But not all. Indeed, as a less noted fact, some of the defenders themselves do contradict each other, but never do notice this fact.

Let me give you an example. I am unlike some who would accept a claim only if there is a direct archaeological evidence for it. IMO, theirs is a materialistic position, and materialism is a false premise; it’s the body of the mind-body dichotomy (in Ayn Rand’s sense of the terms). And, so, I am willing to consider the astronomical references contained in the ancient verses as an evidence. So, in that sense, I don’t dismiss a 10,000+ old history of India; I don’t mindlessly accept 600 BC or so as the starting point of civilization and culture, a date so convenient to the missionaries of the Abrahamic traditions. IMO, not every influential commentator to come from the folds of the Western culture can be safely assumed to have attained the levels obtained by the best among the Greek or enlightenment thinkers.

And, so, I am OK if someone shows, based on the astronomical methods, the existence of the Indian culture, say, 5000+ years ago.

Yet, there are two notable facts here. (i) The findings of different proponents of this astronomical method of dating of the past events (say the dates of events mentioned in RaamaayaNa or Mahaabhaarata) don’t always agree with each other. And, more worrisome is the fact that (ii) despite Internet, they never even notice each other, let alone debate the soundness of their own approaches. All that they—and their supporters—do is to pick out Internet (or TED etc.) battles against the materialists.

A far deeper thinking is required to even just approach these (and such) issues. But the proponents don’t show the required maturity.

It is far too easy to jump to conclusions and blindly assert that there were material “Vimaana”s; that “puShpak” etc. were neither a valid description of a spiritual/psychic phenomenon nor a result of a vivid poetic imagination. It is much more difficult, comparatively speaking, to think of a later date insertion into a text. It is most difficult to be judicious in ascertaining which part of which verse of which book, can be reliably taken as of ancient origin, which one is a later-date interpolation or commentary, and which one is a mischievous recent insertion.

Earlier (i.e. decades earlier, while a school-boy or an undergrad in college etc.), I tended to think the very last possibility as not at all possible. Enough people couldn’t possibly have had enough mastery of Sanskrit, practically speaking, to fool enough honest Sanskrit-knowing people, I thought.

Over the decades, guess, I have become wiser. Not only have I understood the possibilities of the human nature better on the up side, but also on the down side. For instance, one of my colleagues, an engineer, an IITian who lived abroad, could himself compose poetry in Sanskrit very easily, I learnt. No, he wouldn’t do a forgery, sure. But could one say the same for every one who had a mastery of Sanskrit, without being too naive?

And, while on this topic, if someone knows the exact reference from which this verse quoted on Ramesh Raskar’s earlier page comes, and drops a line to me, I would be grateful. http://www.cs.unc.edu/~raskar/ . As usual, when I first read it, I was impressed a great deal. Until, of course, other possibilities struck me later. (It took years for me to think of these other possibilities.)

BTW, Abi also had a follow-up post containing further links about this issue of “vimaan-shaastra” [^].

But, in case you missed it, I do want to highlight my question again: Do you know the reference from which this verse quoted by Ramesh Raskar (now a professor at MIT Media Lab) comes? If yes, please do drop me a line.

 

2. An inspiring tale of a contemporary mathematician:

Here is an inspiring story of a Chinese-born mathematician who beat all the odds to achieve absolutely first-rank success.

I can’t resist the temptation to insert my trailer: As a boy, Yitang Zhang could not even attend school because he was forced into manual labor on vegetable-growing farms—he lived in the Communist China. As a young PhD graduate, he could not get a proper academic job in the USA—even if he got his PhD there. He then worked as an accountant of sorts, and still went on to solve one of mathematics’ most difficult problems.

Alec Wilkinson writes insightfully, beautifully, and with an authentic kind of admiration for man the heroic, for The New Yorker, here [^]. (H/T to Prof. Phanish Suryanarayana of GeorgiaTech, who highlighted this article at iMechanica [^].)

 

3. FQXi Essay Contest 2015:

(Hindi) “Picture abhi baaki nahin hai, dost! Picture to khatam ho gai” … Or, welcome back to the “everyday” reality of the modern day—modern day physics, modern day mathematics, and modern day questions concerning the relation between the two.

In other words, they still don’t get it—the relation between mathematics and physics. That’s why FQXi [^] has got an essay contest about it. They even call it “mysterious.” More details here [^]. (H/T to Roger Schlafly [^].)

Though this last link looks like a Web page of some government lab (American government, not Indian), do check out the second section on that same page: “II Evaluation Criteria.” The main problem description appears in this section. Let me quote the main problem description right in this post:

The theme for this Essay Contest is: “Trick or Truth: the Mysterious Connection Between Physics and Mathematics”.

In many ways, physics has developed hand-in-hand with mathematics. It seems almost impossible to imagine physics without a mathematical framework; at the same time, questions in physics have inspired so many discoveries in mathematics. But does physics simply wear mathematics like a costume, or is math a fundamental part of physical reality?

Why does mathematics seem so “unreasonably” effective in fundamental physics, especially compared to math’s impact in other scientific disciplines? Or does it? How deeply does mathematics inform physics, and physics mathematics? What are the tensions between them — the subtleties, ambiguities, hidden assumptions, or even contradictions and paradoxes at the intersection of formal mathematics and the physics of the real world?

This essay contest will probe the mysterious relationship between physics and mathematics.

Further, this section actually carries a bunch of thought-provocative questions to get you going in your essay writing. … And, yes, the important dates are here [^].

Now, my answers to a few questions about the contest:

Is this issue interesting enough? Yes.

Will I write an essay? No.

Why? Because I haven’t yet put my thoughts in a sufficiently coherent form.

However, I notice that the contest announcement itself includes so many questions that are worth attempting. And so, I will think of jotting down my answers to these questions, even if in a bit of a hurry.

However, I will neither further forge the answers together in a single coherent essay, nor will I participate in the contest.

And even if I were to participate… Well, let me put it this way. Going by Max Tegmark’s and others’ inclinations, I (sort of) “know” that anyone with my kind of answers would stand a very slim chance of actually landing the prize. … That’s another important reason for me not even to try.

But, yes, at least this time round, many of the detailed questions themselves are both valid and interesting. And so, it should be worth your while addressing them (or at least knowing what you think of them for your answers). …

As far as I am concerned, the only issue is time. … Given my habits, writing about such things—the deep and philosophical, and therefore fascinating things, the things that are interesting by themselves—have a way of totally getting out of control. That is, even if you know you aren’t going to interact with anyone else. And, mandatory interaction, incidentally, is another FQXi requirement that discourages me from participating.

So, as the bottom-line: no definitive promises, but let me see if I can write a post or a document by just straight-forwardly jotting down my answers to those detailed questions, without bothering to explain myself much, and without bothering to tie my answers together into a coherent whole.

Ok. Enough is enough. Bye for now.

[May be I will come back and add the “A Song I Like” section or so. Not sure. May be I will; may be I won’t. Bye.]

[E&OE]