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…

 

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“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.

 

My small contribution towards the controversies surrounding the important question of “1, 2, 3, …”

As you know, I have been engaged in writing about scalars, vectors, tensors, and CFD.

However, at the same time, while writing my notes, I also happened to think of the “1, 2, 3, …” controversy. Here is my small, personal, contribution to the same.


The physical world evidently consists of a myriad variety of things. Attributes are the metaphysically inseparable aspects that together constitute the identity of a thing. To exist is to exist with all the attributes. But getting to know the identity of a thing does not mean having a knowledge of all of its attributes. The identity of a thing is grasped, or the thing is recognized, on the basis of just a few attributes/characteristics—those which are the defining attributes (including properties, characteristics, actions, etc.), within a given context.

Similarities and differences are perceptually evident. When two or more concretely real things possess the same attribute, they are directly perceived as being similar. Two mangoes are similar, and so are two bananas. The differences between two or more things of the same kind are the differences in the sizes of those attribute(s) which are in common to them. All mangoes share a great deal of attributes between them, and the differences in the two mangoes are not just the basic fact that they are two separate mangoes, but also that they differ in their respective colors, shapes, sizes, etc.

Sizes or magnitudes (lit.: “bigness”) refer to sizes of things; sizes do not metaphysically exist independent of the things of which they are sizes.

Numbers are the concepts that can be used to measure the sizes of things (and also of their attributes, characteristics, actions, etc.).


It is true that sizes can be grasped and specified without using numbers.

For instance, we can say that this mango is bigger than that. The preceding statement did not involve any number. However, it did involve a comparative statement that ordered two different things in accordance with the sizes of some common attribute possessed by each, e.g., the weight of, or the volume occupied by, each of the two mangoes. In the case of concrete objects such as two mangoes differing in size, the comparative differences in their sizes are grasped via direct perception; one mango is directly seen/felt as being bigger than the other; the mental process involved at this level is direct and automatic.

A certain issue arises when we try to extend the logic to three or more mangoes. To say that the mango A is bigger than the mango B, and that the mango B is bigger than the mango C, is perfectly fine.

However, it is clear from common experience that the size-wise difference between A and B may not exactly be the same as the size-wise difference between B and C. The simple measure: “is bigger than”, thus, is crude.

The idea of numbers is the means through which we try to make the quantitative comparative statements more refined, more precise, more accurately capturing of the metaphysically given sizes.

An important point to note here is that even if you use numbers, a statement involving sizes still remains only a comparative one. Whenever you say that something is bigger or smaller, you are always implicitly adding: as in comparison to something else, i.e., some other thing. Contrary to what a lot of thinkers have presumed, numbers do not provide any more absolute a standard than what is already contained in the comparisons on which a concept of numbers is based.


Fundamentally, an attribute can metaphysically exist only with some definite size (and only as part of the identity of the object which possesses that attribute). Thus, the idea of a size-less attribute is a metaphysical impossibility.

Sizes are a given in the metaphysical reality. Each concretely real object by itself carries all the sizes of all its attributes. An existent or an object, i.e., when an object taken singly, separately, still does possess all its attributes, with all the sizes with which it exists.

However, the idea of measuring a size cannot arise in reference to just a single concrete object. Measurements cannot be conducted on single objects taken out of context, i.e., in complete isolation of everything else that exists.

You need to take at least two objects that differ in sizes (in the same attribute), and it is only then that any quantitative comparison (based on that attribute) becomes possible. And it is only when some comparison is possible that a process for measurements of sizes can at all be conceived of. A process of measurement is a process of comparison.

A number is an end-product of a certain mathematical method that puts a given thing in a size-wise quantitative relationship (or comparison) with other things (of the same kind).


Sizes or magnitudes exist in the raw nature. But numbers do not exist in the raw nature. They are an end-product of certain mathematical processes. A number-producing mathematical process pins down (or defines) some specific sense of what the size of an attribute can at all be taken to mean, in the first place.

Numbers do not exist in the raw nature because the mathematical methods which produce them themselves do not exist in the raw nature.

A method for measuring sizes has to be conceived of (or created or invented) by a mind. The method settles the question of how the metaphysically existing sizes of objects/attributes are to be processed via some kind of a comparison. As such, sure, the method does require a prior grasp of the metaphysical existents, i.e., of the physical reality.

However, the meaning of the method proper itself is not to be located in the metaphysically differing sizes themselves; it is to be located in how those differences in sizes are grasped, processed, and what kind of an end-product is produced by that process.

Thus, a mathematical method is an invention of using the mind in a certain way; it is not a discovery of some metaphysical facts existing independent of the mind grasping (and holding, using, etc.) it.

However, once invented by someone, the mathematical method can be taught to others, and can be used by all those who do know it, but only in within the delimited scope of the method itself, i.e., only in those applications where that particular method can at all be applied.


The simplest kind of numbers are the natural numbers: 1, 2, 3, \dots. As an aside, to remind you, natural numbers do not include the zero; the set of whole numbers does that.

Reaching the idea of the natural numbers involves three steps:

(i) treating a group of some concrete objects of the same kind (e.g. five mangoes) as not only a collection of so many separately existing things, but also as if it were a single, imaginary, composite object, when the constituent objects are seen as a group,

(ii) treating a single concrete object (of the same aforementioned kind, e.g. one mango) not only as a separately existing concrete object, but also as an instance of a group of the aforementioned kind—i.e. a group of the one,

and

(iii) treating the first group (consisting of multiple objects) as if it were obtained by exactly/identically repeating the second group (consisting of a single object).

The interplay between the concrete perception on the one hand and a more abstract, conceptual-level grasp of that perception on the other hand, occurs in each of the first two steps mentioned above. (Ayn Rand: “The ability to regard entities as mental units \dots” [^].)

In contrast, the synthesis of a new mental process that is suitable for making quantitative measurements, which means the issue in the third step, occurs only at an abstract level. There is nothing corresponding to the process of repetition (or for that matter, to any method of quantitative measurements) in the concrete, metaphysically given, reality.

In the third step, the many objects comprising the first group are regarded as if they were exact replicas of the concrete object from the second (singular) group.

This point is important. Primitive humans would use some uniform-looking symbols like dots (.) or circles (\bullet) or sticks (`|‘), to stand for the concrete objects that go in making up either of the aforementioned two groups—the group of the many mangoes vs. the group of the one mango. Using the same symbol for each occurrence of a concrete object underscores the idea that all other facts pertaining to those concrete objects (here, mangoes) are to be summarily disregarded, and that the only important point worth retaining is that a next instance of an exact replica (an instance of an abstract mango, so to speak) has become available.

At this point, we begin representing the group of five mangoes as G_1 = \lbrace\, \bullet\,\bullet\,\bullet\,\bullet\,\bullet\, \rbrace, and the single concretely existing mango as a second abstract group: G_2 = \lbrace\,\bullet\,\rbrace.


Next comes a more clear grasp of the process of repetition. It is seen that the process of repetition can be stopped at discrete stages. For instance:

  1. The process P_1 produces \lbrace\,\bullet\,\rbrace (i.e. the repetition process is stopped after taking \bullet once).
  2. The process P_2 produces \lbrace\,\bullet\,\bullet\,\rbrace (i.e. the repetition process is stopped after taking \bullet twice)
  3. The process P_3 produces \lbrace\,\bullet\,\bullet\,\bullet\,\rbrace (i.e. the repetition process is stopped after taking \bullet thrice)
    etc.

At this point, it is recognized that each output or end-product that a terminated repetition-process produces, is precisely identical to certain abstract group of objects of the first kind.

Thus, each of the P_1 \equiv \lbrace\,\bullet\,\rbrace, or P_2 \equiv \lbrace\,\bullet\,\bullet\,\rbrace, or  P_3 \equiv \lbrace\,\bullet\,\bullet\,\bullet\,\rbrace, $\dots$ is now regarded as if it were a single (composite) object.

Notice how we began by saying that P_1, P_2, P_3 etc. were processes, and then ended up saying that we now see single objects in them.

Thus, the size of each abstract group of many objects (the groups of one, of two, of three, of n objects) gets tied to a particular length of a terminated process, here, of repetitions. As the length of the process varies, so does the size of its output i.e. the abstract composite object.

It is in this way that a process (here, of repetition) becomes capable of measuring the size of the abstract composite object. And it does so in reference to the stage (or the length of repetitions) at which the process was terminated.

It is thus that the repetition process becomes a process of measuring sizes. In other words, it becomes a method of measurement. Qua a method of measurement, the process has been given a name: it is called “counting.”

The end-products of the terminated repetition process, i.e., of the counting process, are the mathematical objects called the natural numbers.


More generally, what we said for the natural numbers also holds true for any other kind of a number. Any kind of a number stands for an end-product that is obtained when a well-defined process of measurement is conducted to completion.

An uncompleted process is just that: a process that is still continuing. The notion of an end-product applies only to a process that has come to an end. Numbers are the end-products of size-measuring processes.

Since an infinite process is not a completed process, infinity is not a number; it is merely a short-hand to denote some aspect of the measurement process other than the use of the process in measuring a size.

The only valid use of infinity is in the context of establishing the limiting values of sequences, i.e., in capturing the essence of the trend in the numbers produced by the nature (or identity) of a given sequence-producing process.

Thus, infinity is a concept that helps pin down the nature of the trend in the numbers belonging to a sequence. On the other hand, a number is a product of a process when it is terminated after a certain, definite, length.

With the concept of infinity, the idea that the process never terminates is not crucial; the crucial thing is that you reach an independence  from the length of a sequence. Let me give you an example.

Consider the sequence for which the n-th term is given by the formula:

S_n = \dfrac{1}{n}.

Thus, the sequence is: 1, \dfrac{1}{2}, \dfrac{1}{3}, \dfrac{1}{4}, \dots.

If we take first two terms, we can see that the value has decreased, from 1 to 0.5. If we go from the second to the third term, we can see that the value has decreased even further, to 0.3333. The difference in the decrement has, however, dropped; it has gone from 1 - \dfrac{1}{2} = 0.5 to \dfrac{1}{2} - \dfrac{1}{3} =  0.1666666\dots. Go from the third to the fourth term, and we can see that while the value goes still down, and the decrement itself also has decreased, it has now become 0.08333 . Thus, two trends are unmistakable: (i) the value keeps dropping, but (ii) the decrement also becomes sluggish.  If the values were to drop uniformly, i.e. if the decrement were to stay the same, we would have immediately hit 0, and then gone on to the negative numbers. But the second factor, viz., that the decrement itself is progressively decreasing, seems to play a trick. It seems intent on keeping you afloat, above the 0 value. We can verify this fact. No matter how big n might get, it still is a finite number, and so, its reciprocal is always going to be a finite number, not zero. At the same time, we now have observed that the differences between the subsequent reciprocals has been decreasing. How can we capture this intuition? What we want to say is this: As you go further and further down in the sequence, the value must become smaller and ever smaller. It would never actually become 0. But it will approach 0 (and no number other than 0) better and still better. Take any small but definite positive number, and we can say that our sequence would eventually drop down below the level of that number, in a finite number of steps. We can say this thing for any given definite positive number, no matter how small. So long as it is a definite number, we are going to hit its level in a finite number of steps. But we also know that since n is positive, our sequence is never going to go so far down as to reach into the regime of the negative numbers. In fact, as we just said, let alone the range of the negative numbers, our sequence is not going to hit even 0, in finite number of steps.

To capture all these facts, viz.: (i) We will always go below the level any positive real number R, no matter how small R may be, in a finite number of steps, (ii) the number of steps n required to go below a specified R level would always go on increasing as R becomes smaller, and (iii) we will never reach 0 in any finite number of steps no matter how large n may get, but will always experience decrement with increasing n, we say that:

the limit of the sequence S_n as n approaches infinity is 0.

The word “infinity” in the above description crucially refers to the facts (i) and (ii), which together clearly establish the trend in the values of the sequence S_n. [The fact (iii) is incidental to the idea of “infinity” itself, though it brings out a neat property of limits, viz., the fact that the limit need not always belong to the set of numbers that is the sequence itself. ]


With the development of mathematical knowledge, the idea of numbers does undergo changes. The concept number gets more and more complex/sophisticated, as the process of measurement becomes more and more complex/sophisticated.

We can form the process of addition starting from the process of counting.

The simplest addition is that of adding a unit (or the number 1) to a given number. We can apply the process of addition by 1, to the number 1, and see that the number we thus arrive at is 2. Then we can apply the process of addition by 1, to the number 2, and see that the number we thus arrive at is 3. We can continue to apply the logic further, and thereby see that it is possible to generate any desired natural number.

The so-called natural numbers thus state the sizes of groups of identical objects, as measured via the process of counting. Since natural numbers encapsulate the sizes of such groups, they obviously can be ordered by the sizes they encapsulate. One way to see how the order 1, then 2, then 3, \dots, arises is to observe that in successively applying the process of addition starting from the number 1, it is the number 2 which comes immediately after the number $1$, but before the number 3, etc.

The process of subtraction is formed by inverting the process of addition, i.e., by seeing the logic of addition in a certain, reverse, way.

The process of addition by 1, when repeatedly applied to a given natural number, is capable of generating all the natural numbers greater than the given number. The process of subtraction by 1, when repeatedly applied to a given natural number, is capable of generating all the natural numbers smaller than the given number.

When the process of subtraction by 1 is applied right to the number 1 itself, we reach the idea of the zero. [Dear Indian, now you know that the idea of the number zero was not all that breath-taking, was it?]

In a further development, the idea of the negative numbers is established.


Thus, the concept of numbers develops from the natural numbers (1, 2, 3, \dots) to whole numbers (0, 1, 2, \dots) to integers (\dots, -2, -1, 0, 1, 2, \dots).

At each such a stage, the idea of what a number means—its definition—undergoes a definite change; at any such a stage, there is a well-defined mathematical process, of increasing conceptual complexity, of measuring sizes, whose end-products that idea of numbers represents.


The idea of multiplication follows from that of repeated additions; the idea of division follows from that of the repeated subtractions; the two process are then recognized as the multiplicative inverses of each other. It’s only then that the idea of fractions follows. The distinction between the rational and irrational fractions is then recognized, and then, the concept of numbers gets extended to include the idea of the irrational as well as rational numbers.

A crucial lesson learnt from this entire expansion of knowledge of what it means to be a number, is the recognition of the fact that for any well-defined and completed process of measurement, there must follow a certain number (and only that unique number, obviously!).


Then, in a further, distinct, development, we come to recognize that while some process must exist to produce a number, any well-defined process producing a number would do just as well.

With this realization, we then come to a stage whereby, we can think of conceptually omitting specifying any specific process of measurement.

We thus come to retain only the fact while some process must be specified, any valid process can be, and then, the end-product still would be just a number.

It is with this realization that we come to reach the idea of the real numbers.


The purpose of forming the idea of real numbers is that they allow us to form statements that would hold true for any number qua a number.


The crux of the distinction of the real numbers from any of the preceding notion of numbers (natural, whole, integers) is the following statement, which can be applied to real numbers, and only to real numbers—not to integers.

The statement is this: there is an infinity of real numbers existing between any two distinct real numbers R_1 and R_2, no matter how close they might be to each other.

There is a wealth of information contained in that statement, but if some aspects are to be highlighted and appreciated more than the others, they would be these:

(i) Each of the two numbers R_1 and R_2 are recognized as being an end-product of some or the other well-defined process.

The responsibility of specifying what precise size is meant when you say R_1 or R_2 is left entirely up to you; the definition of real numbers does not take that burden. It only specifies that some well-defined process must exist to produce R_1 as well as R_2, so that what they denote indeed are numbers.

A mathematical process may produce a result that corresponds to a so-called “irrational” number, and yet, it can be a definite process. For instance, you may specify the size-measurement process thus: hold in a compass the distance equal to the diagonal of a right-angled isoscales triangle having the equal sides of 1, and mark this distance out from the origin on the real number-line. This measurement process is well-specified even if \sqrt{2} can be proved to be an irrational number.

(ii) You don’t have to specify any particular measurement process which might produce a number strictly in between R_1 and R_2, to assert that it’s a number. This part is crucial to understand the concept of real numbers.

The real numbers get all their power precisely because their idea brings into the jurisdiction of the concept of numbers not only all those specific definitions of numbers that have been invented thus far, but also all those definitions which ever possibly would be. That’s the crucial part to understand.

The crucial part is not the fact that there are an infinity of numbers lying between any two R_1 and R_2. In fact, the existence of an infinity of numbers is damn easy to prove: just take the average of R_1 and R_2 and show that it must fall strictly in between them—in fact, it divides the line-segment from R_1 to R_2 into two equal halves. Then, take each half separately, and take the average of its end-points to hit the middle point of that half. In the first step, you go from one line-segment to two (i.e., you produce one new number that is the average). In the next step, you go from the two segments to the four (i.e. in all, three new numbers). Now, go easy; wash-rinse-repeat! … The number of the numbers lying strictly between R_1 and R_2 increases without bound—i.e., it blows “up to” infinity. [Why not “down to” infinity? Simple: God is up in his heavens, and so, we naturally consider the natural numbers rather than the negative integers, first!]

Since the proof is this simple, obviously, it just cannot be the real meat, it just cannot be the real reason why the idea of real numbers is at all required.

The crucial thing to realize here now is this part: Even if you don’t specify any specific process like hitting the mid-point of the line-segment by taking average, there still would be an infinity of numbers between the end-points.


Another closely related and crucial thing to realize is this part: No matter what measurement (i.e. number-producing) process you conceive of, if it is capable of producing a new number that lies strictly between the two bounds, then the set of real numbers has already included it.

Got it? No? Go read that line again. It’s important.

This idea that

“all possible numbers have already been subsumed in the real numbers set”

has not been proven, nor can it be—not on the basis of any of the previous notions of what it means to be a number. In fact, it cannot be proven on the basis of any well-defined (i.e. specified) notion of what it means to be a number. So long as a number-producing process is specified, it is known, by the very definition of real numbers, that that process would not exhaust all real numbers. Why?

Simple. Because, someone can always spin out yet another specific process that generates a different set of numbers, which all would still belong only to the real number system, and your prior process didn’t cover those numbers.

So, the statement cannot be proven on the basis of any specified system of producing numbers.

Formally, this is precisely what [I think] is the issue at the core of the “continuum hypothesis.”

The continuum hypothesis is just a way of formalizing the mathematician’s confidence that a set of numbers such as real numbers can at all be defined, that a concept that includes all possible numbers does have its uses in theory of measurements.

You can’t use the ideas like some already defined notions of numbers in order to prove the continuum hypothesis, because the hypothesis itself is at the base of what it at all means to be a number, when the term is taken in its broadest possible sense.


But why would mathematicians think of such a notion in the first place?

Primarily, so that those numbers which are defined only as the limits (known or unknown, whether translatable using the already known operations of mathematics or otherwise) of some infinite processes can also be treated as proper numbers.

And hence, dramatically, infinite processes also can be used for measuring sizes of actual, metaphysically definite and mathematically finite, objects.

Huh? Where’s the catch?

The catch is that these infinite processes must have limits (i.e., they must have finite numbers as their output); that’s all! (LOL!).


It is often said that the idea of real numbers is a bridge between algebra and geometry, that it’s the counterpart in algebra of what the geometer means by his continuous curve.

True, but not quite hitting the bull’s eye. Continuity is a notion that geometer himself cannot grasp or state well unless when aided by the ideas of the calculus.

Therefore, a somewhat better statement is this: the idea of the real numbers is a bridge between algebra and calculus.

OK, an improvement, but still, it, too, misses the mark.

The real statement is this:

The idea of real numbers provides the grounds in algebra (and in turn, in the arithmetics) so that the (more abstract) methods such as those of the calculus (or of any future method that can ever get invented for measuring sizes) already become completely well-defined qua producers of numbers.

The function of the real number system is, in a way, to just go nuts, just fill the gaps that are (or even would ever be) left by any possible number system.


In the preceding discussion, we had freely made use of the 1:1 correspondence between the real numbers and the beloved continuous curve of our school-time geometry.

This correspondence was not always as obvious as it is today; in fact, it was a towering achievement of, I guess, Descartes. I mean to say, the algebra-ization of geometry.

In the simplest (1D) case, points on a line can be put in 1:1 correspondence with real numbers, and vice-versa. Thus, for every real number there is one and only one point on the real-number line, and for any point actually (i.e. well-) specified on the real number-line, there is one and only one real number corresponding to it.

But the crucial advancement represented by the idea of real numbers is not that there is this correspondence between numbers (an algebraic concept) and geometry.

The crux is this: you can (or, rather, you are left free to) think of any possible process that ends up cutting a given line segment into two (not necessarily equal) halves, and regardless of the particular nature of that process, indeed, without even having to know anything about its particular nature, we can still make a blanket statement:

if the process terminates and ends up cutting the line segment at a certain geometrical point, then the number which corresponds to that geometrical point is already included in the infinite set of real numbers.


Since the set of real numbers exhausts all possible end-products of all possible infinite limiting processes too, it is fully capable of representing any kind of a continuous change.


We in engineering often model the physical reality using the notion of the continuum.

Inasmuch as it’s a fact that to any arbitrary but finite part of a continuum there does correspond a number, when we have the real number system at hand, we already know that this size is already included in the set of real numbers.

Real numbers are indispensable to us the engineers—theoretically speaking. It gives us the freedom to invent any new mathematical methods for quantitatively dealing with continua, by giving us the confidence that all that they would produce, if valid, is already included in the numbers-set we already use; that our numbers-set will never ever let us down, that it will never ever fall short, that we will never ever fall in between the two stools, so to speak. Yes, we could use even the infinite processes, such as those of the calculus, with confidence, so long as they are limiting.

That’s the [theoretical] confidence which the real number system brings us [the engineers].


A Song I Don’t Like:

[Here is a song I don’t like, didn’t ever like, and what’s more, I am confident, I would never ever like either. No, neither this part of it nor that. I don’t like any part of it, whether the partition is made “integer”-ly, or “real”ly.

Hence my confidence. I just don’t like it.

But a lot of Indian [some would say “retards”] do, I do acknowledge this part. To wit [^].

But to repeat: no, I didn’t, don’t, and wouldn’t ever like it. Neither in its 1st avataar, nor in the 2nd, nor even in an hypothetically \pi-th avataar. Teaser: Can we use a transcendental irrational number to denote the stage of iteration? Are fractional derivatives possible?

OK, coming back to the song itself. Go ahead, listen to it, and you will immediately come to know why I wouldn’t like it.]

(Hindi) “1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 \n …” [OK, yes, read the finite sequence before the newline character, using Hindi.]
Credits: [You go hunt for them. I really don’t like it.]


PS: As usual, I may come back and make this post even better. BTW, in the meanwhile, I am thinking of relying on my more junior colleagues to keep me on the track towards delivering on the promised CFD FDP. Bye for now, and take care…

 

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… .]

 

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]

 

The Other Clay Maths Problem

[Major updates to this post are now all complete. 2014.08.31 12:10 PM.]

Everybody knows about The Clay Maths Problem. There are claims, and then there are counter-claims. … First, there are some claims regarding The Clay Maths Problem. Then there are some claims going counter to them… And then, there also are claims about the claims about The Clay Maths Problem. Here is an example of a claim of the third kind.

If what Prof. Scott Aaronson often writes on his blog [^] about this issue is to be taken even semi-seriously, then he routinely receives something like [the particular estimates being mine] 2^{n^n} emails per week, all seeking his opinion about what he thinks of the 2^n arXiv article submissions per week claiming to have proved the P-vs-NP problem one way or the other, where n is a very, very large number; who knows, it might even be approaching \infty.

[Since this is a problem from theoretical computer science, for all estimates, the base has to be `2′; any thing else would be unacceptable. Aaronson is usually silent on precisely where the partition lies: whether the number of claims proving P = NP is statistically equal to those proving P != NP. However, he seems to hint that the two are equal. If so, then the CS-favorite number 2 would slip in once again, now as a divisor.]

The P-vs-NP problem is, thus, THE well-known Clay Maths Problem. Everyone knows about it.

Few people also know that there also is/was one more Clay Maths problem. … For example, they know that some decidedly crazy guy fooled them all—first, the mathematicians, and then, also himself. They—the mathematicians—accepted his solution, but he declined to accept the award, even the $1 million prize money that goes with it [^]. [Since the definition of a proper solution for the award is acceptance by mathematicians, it is easily conceivable that someone manages to fool them for two years and collects the prize.]

Relatively fewer people still know that as many as seven such million dollar problems were announced by the Clay Institute at the turn of the millennium.

As to the other problems, still fewer people ever bother to get past talking about the Yang-Mills problem. Even when it comes to this problem, as usual, they do not entertain any hope about seeing its resolution in the near future, where the “near” is left unquantified. But they all agree that it is a problem from mathematics—not physics.

What theoreticians agree on is always more interesting than what they disagree on. And, guess it was Ayn Rand who said it: also more dangerous.

Then, there are even fewer people who at all know anything about the Navier-Stokes problem—the mathematical version of it.

And, from my Web searches yesterday, there are very, very, very few people, at most a handful, who do something serious about it. This post is about them.

* * * * *  * * * * *   * * * * *

The first gentleman who purportedly continues to remain concerned about this Navier-Stokes problem is: Prof. Charles Fefferman. He should have no choice in the matter, I suppose, because it was he who wrote the official problem statement for Mr. Clay in the first place. Prof. Fefferman is an American mathematician [^].

The second man person to stay worried about this problem for as long a period as one entire month, was one maths professor from Lehigh, one Ms. Penny Smith [^]. She—an American—soon later retracted her solution [^]. Judging by the dates of the v1 and v5 versions on abstract page of her arXiv paper, the retraction took about half a month. (The number two, again!) Most of the ‘net discussions regarding her solution seem to have since then undergone a process of typographical mistake-making [^], or of plain vaporization [^]. Peter Woit, an American professor of Mathematics Physics Mathematics at Columbia, often better known on the ‘net for his proclivity to arbitrarily delete others’ replies on his blog even after having first having allowed their publication there, however, tenaciously holds on to some of that discussion [^]. (Here, I read only the first and the last still-published comments.)

The third person to stay sufficiently bothered about this problem so as to go to the extent of  writing a significant paper on it, seems to have been Prof. Mukhtarbay Otelbayev of Kazakhstan [^].

The first guy to venture discussing Otelbayev’s solution on the Mathematics StackExchange forum, chose to do so anonymously [^]. There must be something about the character of this problem that makes even people from Berlin, Germany, behave this way—writing anonymously—even on the Mathematics part of the StackExchange forum. Indeed, the question to strike this “Unknown” Berlin-based guy wasn’t the correctness or otherwise of Otelbayev’s solution; it was: Did Otelbayev solve the same problem as was posed by the Clay Institute? [^].

There were other follow-up discussions, but soon enough [within a month, of course] the author admitted that there was a mistake in his proof. The ‘net discussion on his proposal is still available [^]. [Professor Otelbayev is not an American.]

To my utter and great surprise, I also found (during an Internet search right this week) that in the meanwhile, there also has been none other than Terry Tao himself jumping into the frey issue. (No one calls him Professor Terence Tao. That’s exactly like how very, very few people, if any one, anywhere, ever, calls Prof. Aaronson by his last name and/or profession.)

My surprise was not entirely baseless. Terry Tao is a Fields medallist [^]. It was plain inconceivable that someone who already is a Fields medallist, would directly take on a(ny) million [American-] dollar problem.

That, indeed, turns out to have been the actual case. Terry Tao didn’t directly tackle the Clay Maths problem itself. See the Simons Foundation’s original coverage here [^], or the San Francisco-based Scientific American’s copy-paste job, here [^]. What Terry instead did is to pose a similar, and related, problem, and then solved it [^].

The “Unknown” Berlin-er mentioned above, was absolutely on the right track. It’s one thing to pose a problem. It’s another thing to pose it well.

That’s the light in which you might want to examine both the Clay Institute’s formulation of the problem and Tao’s recent efforts concerning it.  Realize, Terry didn’t solve the original problem. He solved another, well-posed, problem. And, as to the well-posed-ness of the original Clay Maths problem, there has been another notable effort.

Challenging the well posed-ness of the Clay Maths NS problem seems to have been the track adopted by Prof. Claes Johnson [^] for quite some time by now—several years or so.

Johnson is not a pure mathematician, but a fluid dynamicist. In fact, he is a computational fluid dynamicist, who has actually worked on some practical fluid dynamical problems [^][^]. He seems to be an interesting fellow. Despite having an h-Index of 52 [^], he has written against the climate-warming alarmism [(.pptx) ^][^]. Also see his response after having been selected for the Prandtl medal [^].

There is no Simons Foundation coverage on Johnson’s work. Naturally, any coverage by the Scientific American is plain inconceivable—especially if Johnson is going to write about his positions in springs and summers.

Johnson raises the issue of whether the Clay Maths formulation is well-posed or not. In simple words, can it at all be solved (by any one, ever) or not—whether the problem is in principle amenable to a solution or not. In case you don’t know, the “well-posed-ness” is a technical concept from mathematics [^].

Yes, that way, the issue of whether a problem is well-posed or not, does mean something like: “do you yourself know what you are asking, or not,” but the sense in which Hadamard meant it was certainly a bit more refined.

The distinction of the well-posed vs. ill-posed applies specifically to the solution of differential equations, and it means something like the following:

If you are going to throw a ball so as to hit a distant target (i.e., technically, a two-point BV problem for the second-order differential equation i.e. Newton’s 2nd law), you have the following choice: within appropriate limits, you can select the initial angle for the parabolic trajectory of the ball, in which case you have no choice about its initial speed—the horizontal distance to cover, together with the initial angle, would fix the value of the initial speed with which the ball must be thrown if it is to hit the target. (It would fix how tall the parabola should be, given the initial slope and a fixed horizontal distance.) Alternatively, you can choose the initial speed for the ball, in which case you have no longer have a choice about the initial angle. If hitting the target is your objective, you cannot arbitrarily specify both the auxiliary conditions: the initial angle and the initial speed, at the initial point. The nature of the differential equation is such that specifying both the auxiliary conditions at the same time at the same point renders this differential equation problem ill-posed. That, probably, is the simplest conceivable example of what it means for a problem to be ill-posed or well-posed.

As Hadamard pointed out, a differential equation problem, to be well-posed, must fulfill three conditions: (a) a solution must exist, (b) the solution must be unique, (c) the solution must change continuously with data (i.e., auxiliary conditions, i.e., the boundary and initial conditions).

Sometimes, the solution exists, but is not unique. For example, the diffusion equation problem is well-posed in the forward time direction but not in the reverse, in general. The diffusion process tends to smoothen out any initial sharpness. For example, if you place a drop of ink in the shape of a square on a blotting paper, it soon spreads and becomes a big, thin blot, growing ever rounder and rounder in shape as time passes by. Therefore, the information about the initial shape of the blot gets smeared all over the domain in such a way that starting from this later, bigger and roundish shape, and then going back in time following the diffusion equation, you cannot uniquely recover the initial shape of the blot. Whether the initial ink blot is square or hexagon, they both become round during diffusion. The resulting round shape doesn’t hold enough clue as to the number and locations of the sharp corners in the initial shape. In other words, the information about the initial sharpness is immediately lost during the diffusion process. And so, you can’t uniquely say whether it was a square or a hexagon: both (and infinity of other shapes) are possibilities. [As an aside, I do have some objections to this logic of the diffusion equation, but more on it, in a separate blog post, some time later.] So, the forward diffusion problem is well-posed, but the reverse one is not—no unique solution exists for the latter.

There also are other considerations for well-posed and ill-posed problems, which are more complicated. They refer to the continuous dependence of the solution to the auxiliary data. The auxiliary data, for the time-marching problems like diffusion and fluid flow, crucially means: the initial data.

Thus, the additional relevant consideration concerning the NS system has to do with the smoothness or otherwise of the initial velocity field. Johnson and a colleague rightly point in a paper [(.PDF) ^], that:

“If a vanishingly small perturbation can have a major effect on a solution, then the solution (or problem) is illposed [in the Hadamard sense], and in this case the solution may not carry any meaningful information and thus may be meaningless from both mathematical and applications points of view.

In this perspective it is remarkable that the issue of wellposedness does not appear in the formulation of the Millenium Problem. The purpose of this note is to seek an explanation of this fact, which threatens to make the problem formulation itself illposed in the sense that a resolution is either trivial or impossible.”

I agree. Do see the paper in original.

While Johnson and colleague’s technical paper may be out of the reach of many people—and in any case, at many places in the later half, it certainly is beyond my reach—the first half of the paper as well as Johnson’s blog entries are simple and clear enough to be understood even by any graduate engineer. See the series of his blog posts on this topic, here [^]. For ease of reference, the Clay Maths official problem description is here [(.PDF) ^].

[In this update of my blog post, I have edited a lot around here—from the meaning of the well-posed-ness, to what constitutes Johnson’s position. Some of my writing in the very first version of this post simply was some draft in (very rapid) progress, and so didn’t summarize Johnson’s position well. But then, I had noted that I was going to come back and edit this whole post. In particular, in the following couple of updates, I have deleted the line to the effect that Johnson meant that a blow-up won’t occur in the NS, as if this were to be his final, unqualified position, which, of course, it is not. ]

So,  in summary, Johnson has been repeatedly pointing out some important considerations regarding this problem, for a long time. A summary post could be this one, his latest [^].

And then, tarries along Terry Tao, correctly poses and seemingly correctly solves a problem that is teasingly near the original Clay Maths problem. He shows that a blow-up does occur—but only in his system, not necessarily in the NS system.

Terry, obviously, is interested in only teasing his reader, but not yet quite willing to jump into the… [Ahem.] … You see, otherwise, he could have easily dressed up the same result in different terms [say, in fuller clothes] without making any reference to the Clay Institute or its problem. But he does.

So, that’s what Terry Tao does. He wants to be seen as both addressing and not addressing the Clay Maths Problem. [He is an American.]

And, of course, even though Terry Tao responds at his blog to many, many people, he curiously doesn’t at all respond to a well-established European professor with definitely impressive credentials, like Johnson [^]. [Interestingly, though, Tao also does not delete Johnson’s replies once he publishes them. [Tao also is an Australian—he is a dual citizen.]] But the lack of response in such a context takes the matter closer to “tantalizing” than plain “teasing.” At least to someone like me, it does.

The issue is not whether the particular arguments that Prof. Johnson forwards are in themselves general or powerful enough to settle by themselves the Clay Maths Problem, or not.

The real issue is: the broader, valid and extremely relevant point regarding well-posed-ness which he repeatedly raises. Professor Tao should have responded to that. … You simply can’t go on beating [dancing] around the bush [pole], you know! [Professor Tao works in California, USA.]

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Ok, enough of “cross-referencing.”

But if you ask me what my take on this whole issue is, let me stick my neck out and say (at least) this much:

I think that:

(i) Prof. Johnson’s broader point, regarding the well-posed-ness is absolutely relevant, and unless it is addressed in a forthright manner by the Clay Institute, this Millenium problem essentially remains muddy, and

(ii)  Prof. Tao’s line of thought—the idea of fluid logic gates and machines to be built out of the ideal fluid, and the fluid computer, etc., etc., etc.—would eventually be regarded as not at all relevant to the very core of resolving the NS issue. Tao at best clarifies some abstract part of the nature of the problem, and this part, IMO, isn’t going to be relevant—not to the NS problem itself. His best contribution concerns the energy cascade across the scales, but he offers absolutely no insight as to how that cascade might respond in the actual NS problem settings.

Let me put the second point in a round-about and enormously hand-waving kind of way: If Tao’s current work is at all relevant to the NS problem, then it means that the mathematical community’s standards are sufficiently lax that a solution to the P-vs-NP problem could also be had without the mathematical community first coming to agree with some new and explicit clarity about some of the particular details about how the continuum hypothesis is to be interpreted in that context. But the second is not going to happen, simply because it is THE Clay Maths Problem. Ergo, Tao’s current work… QED.

On the other hand, to repeat, the points that Johnson raises are relevant, and they will stay relevant.

And, of course, all that is quite apart from another, “related” issue: even if there is or isn’t a blow-up in the NS system (and even if the Clay prize gets awarded for “proving” it either way (i.e. for someone getting the mathematicians to agree with him for two consecutive years)), the real issue would still remain: whether the Navier-Stokes system happens to be a good model for real fluids or not.

Just for the starters, as every one knows, at appropriate Knudsen numbers, the no-slip condition is no good. If you are going to complicate the auxiliary conditions to such an extent that the complexity of their effects exceeds that of the basic governing differential equation, then, you sure gotta ask of what good use your basic differential equation is, in the first place. In physics and engineering, we adopt the differential equation paradigm only because it has an epistemological value: it helps reduce cognitive load. If the bells and whistles are going to weigh a ton, how do you expect a cart powered by a toy-spring (or a 12 V 0.4 A Watt stepper motor) to get off to any start?

Or, for another matter, if what happens at the small, local, scale is extraordinarily different from what happens at the large, global, scale, or if pathological connections exists between the two scales, then, you have gotta ask some time: Why elevate some scale-based parameters that only give kicks to the mathematicians but does not make life any simpler to anyone else? Why stick to this continuum-based description for all the scales, in the first place? After all, even simplest phenomena like droplet formation or coalescence are in any case beyond the reach of this “basic,” “fundamental,” etc. etc. Navier-Stokes formalism anyway. Why elevate a theory of such obvious flaws to such a high pedestal? Why keep such a narrow mind that it can’t even deal with some of the simplest phenomena of nature? And, if you must do that, then why not take just one more step and declare another million dollar prize for some latest parlour game? What ultimately does distinguish mathematics as we know it, from the parlour games?

But, of course, we need not go all that far, really speaking. In many ways, the NS system is of enormous practical importance. All that we need to do is to bring some astute observations regarding differential equations, into the problem formulation.

Absent that one, no one has proved either a blow-up, or its absence, in the NS system, thus far. Not even Terry Tao. And, he must know that that’s because of the ill-posed-ness of the problem formulation itself. That’s why he must be choosing to remain silent to Johnson’s query, here [^]. Remember, he both works and does not work on the Clay Maths problem?

The current situation, in many, many ways, is something like this:

Suppose that mathematicians are busy building castles out of thin air (say, debating endlessly about whether the d’Almbert paradox is mathematically consistent or not). Suppose that the working epistemology of the culture has reached such a low level that no one can tell if the theories of thermodynamics and EM are consistent within themselves, let alone with each other. So, a rich guy steps in and declares a big mathematical prize for someone proving whether the Rayleigh-Jean blow-up really follows from the Maxwell system or not. Predictably, there is a flurry of activity, and thus there follow a few mathematicians who can’t get it right. And then, there steps in a young, brilliant mathematician, and declares that he can prove that a blow-up cannot occur but that his proof is limited only to a subcategory of the classical EM fields: the ones that are averaged in some sense. And then comes along some industrial physicist (i.e. actually a theoretical engineer, who has got a job declaring himself an applied mathematician). The industrial physicist points out that the mathematician’s argument can also be taken to imply the exactly opposite conclusion: namely, that these average fields must necessarily lead to the ultraviolet catastrophe. The mathematician chooses not to respond. The audience claps for the mathematician, and falls dead silent for the industrial physicist.

And, no one thinks of instituting a new prize that would reward such efforts as of building a new theory of physics which shows how to prevent the blow-up, even if doing so would involve breaking away from the clutches of some deeply held assumptions about the physical nature of reality, even if such a break-away is only a desperate last measure.

Sure, the situation is not exactly analogous. But if you go through the history of QM, and see how no physicist (or mathematician) ever left others’ valid queries unanswered—if you see how, on the contrary, they rapidly and openly communicated if not collaborated with each other—and then, if you see the kind of hype and blogging practices currently going on in our times, you will begin to see some pattern, and if not that, then at least some semblance, somewhere.

And, you will dearly feel something like—what? wistfulness?—about the good old times now so distant from ours: the times when the rational culture of science had already taken strong roots and it still was mostly a free, application-driven enterprise (the studies of cavity radiation, leading to the first physics Nobel, were sponsored precisely so as to help produce brighter bulbs more cheaply, for better business profits); the times when the state control of science was barely in its nascent stages or altogether absent. (Check out the history of Income Tax on Google, for instance. And, remember, that one—the Income Tax—is only for the starters, as far as the means of the state control of science goes.)

Guess I have made the most important two points which I had.

[Guess my major editing is over, except perhaps for a typo here and there. This long week-end for the Ganesh Chaturthee ends today, and from tomorrow, I will be back to my heavy class-room teaching duties—i.e. away from blogging. (In fact, factoring in the preparations for lectures, I would be into my heavy teaching duty starting right this afternoon.) So, bye for now.]

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A Song I Like:
(Hindi) “purvaiyya leke chali…”
Music: Ravindra Jain
Singers: Lata Mangeshkar and Shailendra Singh
Lyrics: ? (Ravindra Jain?)

[E&OE]

Yo—2: The eminent bumpiness of the “non-analytic” mathematics

[Do check out the Update on 19th August 2014 near the end of this post.]


The “Yo versions” of a blog are so comfy to write. … And, what’s more, they can also instantly lend you an aura of the mathematical respectability—whether you [I mean the blogger himself] really understand[s] any of what’s going on or not. (But then, what do you expect from a “Yo version” anyway?)

Hmm…. Anyway, here is yet another “Yo” version of a blog post:

The last time, we touched upon the Taylor series and the function:
f(x) = e^{-1/x^2} for x \neq 0 and
f(x) = 0 for x = 0.

Getting a bit more complex complicated, have you ever heard of the so-called bump function?

The prominent (“canonical”) example of a bump function reads like this:

f(x) = e^{-\dfrac{1}{1-x^2}} for \vert x \vert < 1 and f(x) = 0 otherwise.

And, it really looks like a bump on the road; check out the graphs on Wiki [^] or the Wolfram MathsWorld [^].

This function is infinitely differentiable—but that fact still does not make it analytic [^]. Its support is not just bounded but also compact—but its Fourier transform isn’t so. … And that’s where I had initially found it interesting. … But then, as it turns out, thinking about the issue via the bump functions and all is what I now find to be, way (way) too complicated for me—my purposes [^].

And, yet, mathematicians call such functions mollifiers [^].

If this be a mollifier to a mathematician, then what would be an aggravator to him?

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In the meanwhile, after a bit of a college-wide “re-org,” I have been asked to increase my teaching load, and therefore further engage an additional undergraduate course on Thermodynamics. (We have two shifts for the UG programs, and so, effectively, I am teaching three courses now: Heat Transfer to two shifts and Thermodynamics to one shift.) … Obviously, therefore, even Yo-n posts should come out only once in a while during the rest of this semester… It’s just that we’ve just had a long week-end this time round, and so, I could slip this one in. … Otherwise, it’s all teaching, teaching and teaching. … All to UG students!


Update on 2014.08.19

I have to link up to a great post related to this topic (the Taylor series, the bump function, and in fact also the Lagrangian and the Taylor polynomials, etc.) by David Lowry-Duda [^]. David is a PhD student in [ahem!] maths, at Brown [^]. I came to know of his blog only after publishing this post. Also, if you wish, do check out another great post of his on Intuitive Introduction to Calculus [^]. … Ok, more, later.


 

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A Song I Like:
(Hindi) “haay re tere chanchal nainwaa…”
Music: Chitragupta
Singers: Lata Mangeshkar and Mahendra Kapoor
Lyrics: Majrooh Sultanpuri

[E&OE]