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]

 

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I am not participating in this FQXi competition

Actually, I had more or less reached this decision right by the middle of this month, but still, just because there was some time in hand, kept on indulging in aimless, useless, baseless, energy-less vacillations about it. Until it was the evening of August 31st, anyway.  … Yes, yes, I know there still is some more time in hand, even if only because India is ahead of USA in terms of the time of the time-zone. The time still in hand (about 15 hours) is enough to write a mere nine-page essay anyway. However, read on to get to know why I would still not be writing. …

The reason, ultimately, turns out to be grounded in the competition topic itself, viz., the question: “Which of our basic physical assumptions are wrong?”

Loosely speaking, the nature of this question makes it a “negative” topic, not a “positive” one. … That could take some explanation. …

OK. No. I am not a New Age idiot. So, I don’t mean to say that the reason I decided to keep away from this competition is because the topic invites some of kind of a criticism of others. That’s not what I mean by the adjective “negative” here.

What I mean by “negative,” here, is: the (logical) compliment of “the positive.” And, by “positive,” what I mean is: a direct existential kind of a statement. … More detailed explanation is called for.

OK. For example: “The Sun exists” is a positive statement. It directly captures a fact of reality. Another example: “The Sun shines brightly,” again, is a positive statement. Not primarily because the word “brightness” has a connotation of clarity, efficacy, joy, happiness, etc., all supposedly positive things, but simply because the adjective “brightly” does not logically negate the sense of the basic truth contained in the rest of that sentence.  On these lines, the statement: “A black hole sucks in everything, even light” also is a positive statement. Its purpose is to identify the nature of an existent rather than to deny, qualify, or question one.

In Ayn Rand’s way of putting it: The positive statements are of the kind: Existence is identity. Or: A is A. They are not concerned with what possibly might belong in the non-A. They are not concerned with the logical complement of A.

Alright. … What’s that to do with not participating in the essay competition?

I think I have already ended up hinting at this aspect, right during my last post.

The FQXi competition question does not ask: “What new, foundational thing have you got right?” It asks: Where did physicists got it wrong?

Answering the first question is always easier; the second one is not.

Physics is a science. It establishes its truth via a laborious process, one that proceeds through observations, analysis, hypotheses, experimentation, analysis of data, validation, integration, etc. And, it’s a cyclic or iterative process; the parts of it are interdependent. You don’t begin thinking of integration only after conducting the experiments; considerations of integration are part of the context of analysis and hypothesis generation as well. You don’t do analysis only before experimentation; you need it right during validation, too. So on and so forth. All in all, establishing a new truth requires a whole lot of real, hard, work. Naturally, the progress is slow. Naturally, therefore, there is very little to show in any annual/biennial essay contest, if at all one has produced anything in that time period. Naturally, therefore, it is very easy to identify what one does have to show, if one has it.

However, precisely due to the nature and extent of the hard-work involved, it is very difficult to even summarize what all things one does not have.

In a way, it’s a matter of teleology—and the crow-principle of epistemology. The goals are few. But they lead to a whole big “tree” of issues, factors, possibilities and considerations to manage. It’s always possible to tell the goals—and it’s even easier to tell the actually achieved goals from among them  (which are even fewer than the goals). But it’s always too complex to even indicate the extent and scope of the “tree”s directly involved in it: the teleological one, and the epistemological one (by which, here, I mean: the “tree” arising out of the basic meanings of the concepts, generalizations or issues at hand).

Beyond these complex trees, there also is that biggest of all considerations, when it comes to all matters concerning knowledge, viz., the consideration of integration. Integration of the new knowledge with the sum total of all of the rest of the knowledge.

Statements of goals and achievements are easier to make. But since the process of knowledge creation is so intricate, difficult and laborious, statements of what went wrong are far more difficult to make. The “A” is easy enough to identify if it involves the goals and the achievements; the “non-A” is far too voluminous for the mind to separately deal with.

That’s the basic reason why mere polemics never succeeds. Even if a polemics (the non-A) is objectively valid, and even if it’s done very neatly or sharply, the listener’s mind still is simply unable to hold on to much of it, unless the positive part—the “A” of it—is not explicitly identified.

So, you get the idea of why I couldn’t get much past that question.

Of course, as I stated in my last post, I could still have written something on some one important issue and simply dumped it. I did try. … As it so happens, I was participating in a LinkedIn discussion on the nature of randomness (“Is anything truly random?”). I wrote a lot there, and also got misunderstood—at least some, if not a lot. (Expectedly so.) So, it seemed like a good topic on which to write the FQXi essay. I did try along this direction.

However, as soon as I finished my initial take on the “outline” part of it, I realized the essential dissonance. What I was addressing was an identification of the basic nature of randomness. But what I was supposed to be addressing was a survey of the physical assumptions that are not only wrong but also obvious enough in their wrongness as to be ripe to be acknowledged by the (astro)physicist community (at MIT) as indeed being wrong. Too much of a chasm in there. Too much of dissonance.

The essay indeed would have carried some (what I think are) good contents. Indeed,  I should sincerely try to bring it all together, give it some good, additional touches—specifically, the scholarly kind of touches (doing a real serious lit search etc.). And, then, submit it to a suitable journal (e.g. Foundations of Physics/AJP/The Physics Teacher/Whatever; not Nature/Science/PRL). However, just the fact that it can make for a good essay, doesn’t mean that it makes for a good essay for this competition.

Could anyone have written a good essay to address that particular question?

For the reasons indicated above, I don’t think anyone could have. There is just too much that is bad with the present-day physics. The situation is so (or such) bad that, as mentioned in my last post, in one’s attempts to condense it all, the first things to strike one happen to be all philosophic in nature. And, rational philosophic principles are not always well-known to people. So, you also have to face the issue of having to explain even simpler among the terms, as you go along.

Further, things have gone bad to such extent that not just the Platonic intrinsicism but even classical subjectivism looks brightly rational in comparison. For example, in the so-called Many Worlds Interpretation, originated in an American’s 1950’s PhD at Princeton, people don’t just create their own reality, as a classical subjectivist would have you believe. The Many Worlds Interpretation’s point is that no people are basically necessary to create world, not even a creator god (the idea of it). Physical processes, occurring in the universe, by themselves, are enough to spawn infinity of universes. Yes, you read it right. That’s the actual thesis. (And, idiots at places like Princeton, Berkeley, MIT, etc., have by now developed a tradition of taking such nonsense seriously.) So, in short, there indeed is a lot of nonsense out there. (It’s precisely because the situation is so bad philosophically that it also is so bad quantitatively.)

But there still is that nine page limit. So, you can’t possibly both write well and comprehensively enough to actually address the essay question. There are objective principles because of which an essay of the kind that the competition question demands, is impossible to write.

So, no, I don’t think that anyone else could possibly have a good essay that also actually addresses the actually posed question. They may have good essays that don’t address the question or bad essays that do. But they can’t possibly have both. Now, that’s something that looks like modern physics, doesn’t it, FQXi?

* * * * *   * * * * *   * * * * *

No “A Song I Like” section, once again. I still go jobless. Keep that in mind.

[This is initial draft, published on August 31, 2012, 6:31 PM, IST. May be I will make some minor corrections/updates later on]
[E&OE]

So, you think physicists got it wrong?

So, you think physicists got it wrong?

If so, why not tell them—or, even if they wouldn’t listen, at least to the world—what precisely it is?

The obvious reference is to the latest FQXi essay contest. The topic they have selected for this edition of the essay contest is:

“Which of Our Basic Physical Assumptions Are Wrong?”

For more details, see here [^]. Note that the last date of submitting your essays is August 31, 2012. There also is a chance that “proper” physicists may end up reading your essay; see the “Who is FQXi” page here [^].

However, in case you didn’t know about FQXi, also note that this is not the first time that they are conducting such an essay contest. Check out the winning essays from the earlier contests: 2008 (on “The Nature of Time”) [^], 2009 (on “What Is Ultimately Possible in Physics?”) [^], and 2010-11 (“Is Reality Digital or Analog?”) [^].

To read the essays already submitted for the current (still open) contest—and the ongoing public discussions on them—follow this URL [^].

* * * * *

In case you are curious to know my opinion of this essay contest, the reason why I didn’t participate in it so far, whether I would participate at least now, etc.:

Why I didn’t participate thus far. Well, there are different reasons for it, not a single one.

As to the very first contest, I would have liked to participate in it. However, I simply wasn’t even aware of the FQXi itself at that time. In fact, despite my fairly extensive browsing of physics-related sites, I didn’t come to know of the first edition of the essay contest any time before it was already over.

What would I have written for the first contest (“The Nature of Time”)?

I would have written about the nature of space before coming to that of time. If further curious, my position would have been in many ways quite similar to Ron Pisaturo’s [^]. A few asides: (i) I didn’t know about Pisaturo or his position at that time. (ii) Pisaturo, in his articles, addresses more points than I would have. In fact, some of these points existed only faintly on my radar; it was he, who, in addressing them, highlighted their existence/importance to me. (iii) Regarding the nature of space itself (not to mention other issues like the finitude or otherwise of the physical universe), my position was (and remains) independently arrived at. In fact, I found out via an exchange of a few emails with him that there could be some differences in our positions, may be even some essential ones, esp. at the level of details. In particular, it’s concerning whether space is a concept of mathematics, physics, or both. Now, as far as my own position is concerned, I had been jotting down my points in small pocket notebooks (the paper version!) that I usually carry around. I hope to find the time, and more importantly, the right frame of mind, to convert these into an essay. I would certainly like to do that, but only after I am more than halfway through writing my QM book. Which means: after about a year or more. Ok. Enough about the first contest.

For the immediate next contest (“What Is Ultimately Possible in Physics”), while the topic selection here was rather smart, I personally didn’t think that it was well focused enough. And so, in all probability, I wouldn’t have participated in it. However, it didn’t matter one way or the other because I happened to miss the deadline once again. (FQXi contests are not held periodically i.e. regularly.)

As to the last contest (“Is Reality Digital or Analog”), I thought that the topic was, at least at the very first sight, a bit frivolous. However, as it happened, what I thought of it on the second thoughts also didn’t matter anyway because I once again missed the deadline, this time round by just a few days or so.

I think there was some discussion on HBL on a related topic roughly around that time, and I, in fact had a subscription to the HBL at that time. That topic, I think, began with the discussion on whether 0.999999… equals 1.0 or not; then went a bit on to series and infinitesimals in a geometrical context; and then, another related thread made appearance: whether, as we go “all the way down,” does the physical stuff at that level have sharp boundaries or not. BTW, this is a far, far better way of formulating what the FQXi contest topic had merely hinted at. … My answers, without providing full justifications here: the stuff “all the way down” cannot have (infinitely) sharp boundaries, because infinity does not exist in the physical reality (HB had this same position); 0.9… does equal to 1.0, but only in the limiting sense—the former does not “go to” the latter. Here, surprisingly, HB differed from me, in the sense, he didn’t at least immediately agree with me; he kept quiet—perhaps was thinking about it. (In case you missed the reason why I might have found it surprising: the infinitesimal is nothing but the infinitely small. Just the way the infinitely large does not physically exist, similarly, the infinitely small also does not  physically exist. Both are mathematical concepts—concepts of methods.)

… Anyway, coming back to the FQXi contest, notice the difference that the FQXi topic had from that discussed at the HBL: both digital and analog are, primarily, mathematical concepts, not physical. The fact that they can be successfully applied to physical reality does not, by itself, make them physical. That’s the reason why I said that the terms in which the issue got discussed at HBL were better—the formulation there captured the essential issue more directly, in fact, quite explicitly. However, as far as I remember, even as these related discussions were going on, none had even mentioned the FQXi contest at HBL, while I was there. So, I missed that edition too.

So, this is the first time that I have run into a FQXi contest while there still is some time left for it.

Would I participate now?

As of today, frankly, I don’t know. … As you can see by now, as far as I am concerned, it’s the topic that matters more than anything else, actually.

Come to think of it, I am not afraid of putting even the inchoate among my thoughts, in an essay contest like this. And, that’s to a large extent because, I am most certainly not at all afraid of participating in it and also not winning anything—not even a fourth prize. One doesn’t enter an essay contest in order to win a prize (just the way one does not take an examination to score the highest marks/ranks). In case you are sufficiently idiotic to not get it, notice that what I said in the last line is not an argument against having prizes in contests (or taking them home if you win them). It is merely a way of highlighting the fact that prizes do not deterministically elicit better responses (just the way top examinations ranks do not necessarily always go to the best guy). (If you are not convinced, substitute “fatalistically” in place of “deterministically.”)

The main function of prizes is to attract publicity, and thereby, possibly increase one’s chances of finding, or reaching out to, the right people, the right minds. Prizes serve to attract a better audience rather than a better set of participants. That is, statistically speaking, of course.

You don’t necessarily have to win prizes in order to reach out to a better audience. (And, what’s a better audience, you ask? Obvious. It’s an audience that is itself capable, employs you, pays you, respects you, etc.—overall, values you on a rational basis.) That is the reason why participation matters more than winning.

(The Olympics participants usually get it right—and most of the humanities folks, never do. For example, consider: If it were to be just a matter of exceeding one’s own past performances, or to see the limits of one’s abilities, why not go to a secluded place, exceed your abilities to your heart’s content, and then, never let anyone else get even the wind of it? Ditto, even if your motivation is less exalted, and consists solely of exceeding others’ abilities—beating others. Here, suppose that there are just the two of you, you and your opponent (or the ten of you, or ten teams), and suppose that you (or your team) win (wins) over (all) your opponent(s)—but strictly under the condition that no one else ever gets to know of it. None. You continue to know that you exceeded your past records, or that you beat others, but there is no audience for it, no better consequences to follow in your own life, out of it… And, now, also consider a contrasting scenario: What if you do get to connect to the right kind of an audience even if you don’t win a contest in which you participate. What would it be like? Here, I am tempted to speculate: It would be just like any of our (India’s) sports teams, especially, our cricket team. … So, either way, it is the participation that matters more than the winning. QED, nah?)

So, the idea of participating in a contest like FQXi is quite OK by me. So, coming back to the topic for the current edition of this contest:

As soon as I read about the contest (which was something like the last week or so), I got the sense that the topic selection was, once again, rather smart—but also that the topic was a bit too open-ended, though probably not too broad. Reading through the vast variety of the essays that people have submitted so far only confirmed, in a way, this apprehension of mine.

If a well-informed physicist friend were to ask me in an informal but serious chat the question  of the topic (“Which of our basic physical assumptions are wrong?”), the first set of things to strike me would have been rather philosophical in nature. But then, this is not an essay contest in philosophy as such, though, I guess, certain parts of philosophy clearly are not out of place here—in fact, dealing, as the contest does, with questioning the foundations, philosophy of physics, and also relevant principles from general philosophy, are clearly welcome. However, the main part that philosophy can play here would be limited to identifying the broad context; the essay cannot be concerned with expounding philosophy itself, as such.  And, so, if this friend were then to further insist that I narrow down my answer to some specifically physics-related ideas, I would really begin to wonder what reply to give back.

That, in particular, was the position in which I found myself for the past few days.

I found that, if I have to think of some issues or ideas specifically from physics to answer that question, I could easily think of not just one or two but at least five-six issues, if not ten or more of them. And, I found that I could not really pick out one over the other without also substantially involving general philosophy as well—comparing and contrasting these issues in the light of philosophy, i.e. using philosophy to put every one of them in a common broad context embracing them all—in which case, it would become (at least a small) book and cease to be just an essay i.e. an article.

So, honestly speaking, this essay contest has, in a way, foxed me.

In a way, it had become a challenge for me to see if I could find just one or two issues out of all those numerous issues. Without there being adequate space to put all of those issues in context, treating just one or two of them would come to mean, I thought, that I consider the selected issues to be at least more pertinent if not more foundational than the others left out of the essay. And, there, I realized, my home-work is not yet well done. I don’t have a very clear idea as to why I should pick out this issue over that one. That’s why, at least as far as I am concerned, the essay topic had, in fact, become a challenge to me.

I then decided to see if I could challenge the challenge (!). Namely, what if I pick up a few issues almost at random, and write something about them, without thereby necessarily implying that these selected issues must be taken as the hierarchically more foundational/at the core/important than the others? Would it then be possible for me to write something?

BTW, there would have been another point against participating, which no longer matters: Sometimes, the discussion at FQXi seemed to digressed too much into inconsequential matters. Submitting an essay is to commit to having discussions. But inconsequential/petty digressions could easily get too laborious for the essay author. Here, however, I have noticed that as the contest and its management matures, the degree of such largely pointless digressions seems to be going down. I think you now can more easily ignore the issues/folks you don’t want to tackle/answer, especially so if you really don’t care much about winning the prize. So, that’s another point in favor of participating.

So, the churning in my mind regarding the topic, regarding whether to participate in it or not, is still going on, even as I write this blog post. However, I think I am getting increasingly inclined towards the idea of writing something anyway and dumping it there. … Let me see if I can do something along that line. And, the only way to see whether that is doable or not is to actually sit down and start writing something. I will do that. … If something “sensible” comes out it, you will see me submitting an entry by August 31. If not, here is a promise: I will at least share a bit from whatever that I wrote (and decided not to submit for the contest), here at my personal blog, and possibly also other blogs/public fora.

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No “A Song I Like” section, once again. I still go jobless. Keep that in mind.

[Minor corrections and updates were made on July 23, 2012.]
[E&OE]

Where the Mind Stops—Not!

The way people use language, changes.

In the mid- and late-1990s, when the Internet was new, when blogs had yet to become widespread, when people would often use their own Web sites (or the feedback forms and “guestbooks” at others’ Web sites) to express their own personal thoughts, opinions and feelings—in short, when it still was Web 1.0—one would often run into expressions of the title sort. For example: XYZ is a very great course—NOT! XYZ university has a very great student housing—NOT! XYZ is a very cute product—NOT! … You get the idea—you really do! (NO not!!)… That’s the sense in which the title of this post is to be taken.

For quite some time, I had been thinking of a problem, a deceptively simple problem, from engineering sciences and mechanics. Actually, it’s not a problem, but a way of modeling problems.

Consider a body or a physical object, say a piece of chalk. Break it into two pieces. Easy to do so physically? … Fine. Now, consider how you would represent this scenario mathematically. That is the problem under consideration. … Let me explain further.

The problem would be a mere idle curiosity but for the fact that it has huge economic consequences. I shall illustrate it with just two examples.

Example 1: Consider hot molten metal being poured in sand molds, during casting. Though “thick,” the liquid metal does not necessarily flow very smoothly as it runs everywhere inside the mold cavity. It brushes against mold-walls, splashes, and forms droplets. These flying droplets are more effective than the main body of the molten metal in abrading (“scrubbing”) the mold-walls, and thereby dislodging sand particles off the mold walls. Further, the droplets themselves both oxidize fast, and cool down fast. Both the oxidized and solidified droplets, and the sand particles abraded or dislodged by the droplets, fall into the cooling liquid metal. Due to oxidized layer the solidified droplets (or due to the high melting point of silicates, the sand particles) do not easily remelt once they fall into the main molten metal. The particles remain separate, and thus get embedded into the casting, leading to defective castings. (Second-phase particles like oxidized droplets and sand particles adversely affect the mechanical load-carrying capacity of the casting, and also lead to easier corrosion.) We need the flow here to be smooth, not so much because laminar flow by itself is a wonderful to have (and mathematically easier to handle). We need it to remain smooth mainly in order to prevent splashing and to reduce wall-abrasion. The splashing part involves separation of a contiguous volume of liquid into several bodies (the main body of liquid, and all the splashed droplets). If we can accurately, i.e. mathematically, model how droplets separate out from a liquid, we would be better equipped to handle the task of designing the flow inside a mold cavity.

Example 2: Way back in mid-1980s, when I was doing my MTech at IIT Madras, I had already run into some report which had said that the economic losses due to unintended catastrophic fractures occurring in the US alone were estimated to be some $5 billion annually. … I quote the figure purely from my not-so-reliable memory. However, even today, I do think that the quoted figure seems reasonable. Just consider just one category of fractures: the loss of buildings and human life due to fractures occurring during earthquakes. Fracture mechanics has been an important field of research for more than half a century by now. The process of fracture, if allowed to continue unchecked, results in a component or an object fragmenting into many pieces.

It might surprise many of you (in fact, almost anyone who has not studied fluid mechanics or fracture mechanics) that there simply does not exist any good way to mathematically represent this crucial aspect of droplets formation or fracture: namely, the fact of one body becoming several bodies. More accurately, no one so far (at least to my knowledge) has ever proposed a neat mathematical way to represent such a simple physical fact. Not in any way that could even potentially prove useful in building a better mechanics of fluids or fracture.

Not very surprising. After all, right since Newton’s time, the ruling paradigm of building mathematical models has been: differential equations. Differential equations necessarily assume the existence of a continuum. The region over which a given differential equation is to be integrated, may itself contain holes. Now, sometimes, the existence of holes in a region of space by itself leads to some troubles in some areas of mechanics; e.g., consider how the compatibility criteria of elasticity lose simplicity once you let a body carry holes. Yet, these difficulties are nothing once you theoretically allow the original single body to split apart into two or more fragments. The main difficulty is the following:

A differential equation is nothing but an equation defined in terms of differentials. (That is some insight!) In the sense of its usage in physics/engineering, a differential equation is an equation defined over a differential element. A differential element (or an infinitesimal) is a mathematical abstraction. It begins with a mathematically demarcated finite piece of a continuum, and systematically takes its size towards zero. A “demarcated finite piece” here essentially means that it has boundaries. For example, for a 1D continuum, there would be two separate points serving as the end-points of the finite piece. Such a piece is, then, subjected to the mathematical limiting process, so as to yield a differential element. To be useful, the differential equation has to be integrated over the entire region, taking into consideration the boundary and initial values. (The region must be primarily finite, and it usually is so. However, sometimes, through certain secondary mathematical considerations and tricks involving certain specific kinds of boundary conditions, we can let the region to be indefinitely large in extent as well.)

Since the basic definition of the differential element itself refers to a continuum, i.e. to a continuous region of space, this entire paradigm requires that cuts or holes not existing initially in the region cannot at all be later introduced. A hole is, as I said above, mostly acceptable in mathematical physics. However, the hole cannot grow so as to actually severe a single contiguous region of space into two (or more) separate regions. A cut cannot be allowed to run all the way through. The reason is: (i) either the differential element spanning the two sides of the cut must be taken out of the model—which cannot be done under the differential equations paradigm, (ii) or the entire model must be rejected as being invalid.

Thus, no cut—no boundary—can be introduced within a differential element. A differential element may be taken to end on a boundary, in a sense. However, it can never be cut apart. (This, incidentally, is the reason why people fall silent when you ask them the question of one of my previous posts: can an infinitesimal carry parts?)

You can look at it as a simple logical consistency requirement. If you model anything with differential elements (i.e. using the differential equations paradigm), then, by the logic of the way this kind of mathematics has been built and works, you are not allowed to introduce a cut into a continuum and make fragments out of it, later on.  In case you are wondering about a logically symmetrical scenario: no, you can also not join two continua into one—the differential equation paradigm does not allow you to do that either. And, no, topology does not lead to any actual progress with this problem either. Topology only helps define some aspects of the problem in mathematically precise terms. But it does not even address the problem I am mentioning here.

Such a nature of continuum modeling is indeed was what I had once hinted at, in one of my comments at iMechanica [^]. I had said (and none contradicted me at that forum for it) that:

As an aside, I think in classical mathematics there is no solution to this issue, and there cannot be—you simply cannot model a situation like “one thing becomes two things” or “two infinitesimally close points become separated by a finite distance” within any continuum theory at all…

In other words, this is a situation where, if one wishes to think about it in mathematical terms, one’s mind stops.

Or does it?

Today, I happened to idly go over these thoughts once again. And then, a dim possibility of appending a NOT appeared.

The reason I say it’s a dim possibility is because: (i) I haven’t yet carefully thought it through; (ii) and so, I am not sure if it really does not carry philosophic inconsistencies (philosophy, here, is to be rather taken in the sense of philosophy of science, of physics and mathematics); (iii) I already know enough to know that this possibility would not in any way help at least that basic fracture mechanical problem which I have mentioned above; and (iv) I think an application simpler than the basic problem of fracture mechanics, should be possible—with some careful provisos in place. May be, just may be. (The reason I am being so tentative is that the idea struck me only this afternoon.)

I still need to go over the matter, and so, I will not provide any more details about that dim possibility, right here, right today. However, I think I have already provided a sufficiently detailed description of the problem (and the supposed difficulty about it) that, probably, anyone else (trained in basic engineering/physics and mathematics) could easily get it.

So, in the meanwhile, if you can think of any solution—or even a solution approach—that could take care of this problem, drop me a line or add a comment.  … If you are looking for a succinct statement of the problem out of this (as usual) verbose blog-post, then take the above-mentioned quote from my iMechanica comment, as the problem statement. … For years (two+ decades) I thought no solution/approach to that problem was possible, and even at iMechanica, it didn’t elicit any response indicating otherwise. … But, now, I think there could perhaps be a way out—if I am consistent by basic philosophic considerations, that is. It’s a simple thing, really speaking, a very obvious one too, and not at all a big deal… However, the point is, now the (or my) mind no longer comes to a complete halt when it comes to that problem…

Enough for the time being. I will consider posting about this issue at iMechanica after a little while. … And, BTW, if you are in a mode to think very deeply about it, also see something somewhat related to this problem, viz., the 2011 FQXi Essay Contest (and what its winners had to say about that problem): [^]. Though related, the two questions are a bit different. For the purpose of this post, the main problem is the one I mentioned above. Think about it, and have fun! And if you have something to say about it, do drop me a line! Bye for now!!

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A Song I Like
(Hindi) “nahin nahin koee tumasaa hanseen…”
Singers: Kishore Kumar, Asha Bhosale
Music: Rajesh Roshan
Lyrics: Anand Bakshi

[PS: Perhaps, a revision to fix simple errors, and possibly to add a bit of content here and there, is still due.]
[E&OE]