David Harriman’s “The Logical Leap”: Grade “A” (But a Qualified One, Not Straight)

1. Introductory

It is a pleasure to review David Harriman’s book: “The Logical Leap: Induction in Physics” [^]. The book forms a significant and welcome addition not only to the special history and philosophy of physics, but also to general philosophy proper.

[At 5000+ words, this review is long. You are warned.]

I shall not go into that usual, almost mechanical aspect of a book review whereby the reviewer feels obliged to narrate the contents of the book: e.g., how many chapters the book has, what chapter covers what topic, etc. All such information (and more!) is easily available at the book’s Web site as well as via book-previews at the booksellers’ Web sites.

What I instead propose to do here is to jot down an informal, rapidly written, and a decidedly blogsome (a word I just coined) review of the book, telling you what I found about it during, and immediately after finishing, my very first and rapid reading of this book. I am sure I will notice many things to a better depth and also much differently, at a later date. At the same time, I must say that though rapid, my reading has been careful enough that I can say that I have got a fairly good overall sense of the thesis of the book. On second thoughts, make it theses, i.e. thesis in the plural.

2. The Main Themes (or Threads of Theory) Developed in the Book

Usually books this small (~270 paperback pages) carry just one central thread or theme, at most two (in which case, the second theme oftentimes is minor). In contrast, this book is ambitious. It has several central themes, which may perhaps be clubbed together into four main categories given below. Note: The grouping scheme given below is mine, not of the author himself. The four themes (not all being of equal importance) are:

  1. To identify the broad philosophic nature of induction: to present a new theory of induction and to tie it with Ayn Rand’s existing theory of concepts. In short, to extend the Objectivist epistemology, being a theory of concepts to that of generalizations; to identify the nature of the starting material of induction, and to show how an inductive generalization necessarily follows from it and why; etc.
  2. To trace the threads of historical development of a few important discoveries of physics, in order to isolate how the principles of induction were actually being used in their work by path-breaking or great physicists like Galileo, Newton, Dalton, Maxwell, and others; to illustrate the author’s view of the interrelations between causality, theory and experiment; etc.
  3. To present some of the prominent ways in which the process of induction might get misapplied, i.e. the errors of inductions. Further, to show what happens when physicists altogether abandon induction, and to illustrate such errors or disintegrations via suitable examples from the history of physics
  4. To identify the broad, philosophic nature of physics and of mathematics. Here, the author not only tries to show how and why the famous “unreasonable effectiveness of mathematics in physics” is indeed “reasonable”. The author goes much farther and says that such an effectiveness is not only “reasonable” but also necessary; that it must directly follow from Rand’s theory of concepts, as interpreted and extended by him.

You can easily see that the scope of issues with which this book deals is broad. (BTW, ambition is a good thing, as far as I am concerned.)

3. How Does the Author Fare on the Main Themes, in an Overall Sense?

How does the author fare on these concerns of his? Before we begin looking at this question, let me first note something important.

A significant portion of this book has drawn not only on the published lectures and other materials by Dr. Leonard Peikoff, but also on Harriman’s in-depth interaction with Peikoff. Indeed, the book is a result of a collaboration between the two. The Ayn Rand Institute, I gather, had treated the writing of this book a major academic project for a considerable length of time (a decade, the author notes in the preface). Strictly speaking, Peikoff has written only an introduction to the book; formally, he is not a co-author. However, going by what the author has noted in the Preface, and also after going through the text, as far as I am concerned, the long and short of it is this: I consider Dr. Peikoff as an informal but definite second author of this book.

Now, let’s return to the question of how the book fares on all the above-mentioned ambitious themes that it aims to tackle. I will provide quite a few details below, but let me first give you my overall evaluation right away.

In this book, the author has performed exceedingly well, nay, brilliantly, on the first three counts, i.e. 1. through 3. in the above list. He demonstrates an extraordinarily virtuoso performance not only on the first and the third count, where Dr. Peikoff might be expected to have had a very direct influence on the writing, but also on the second count, where I guess, for all practical purposes, Harriman either would have been completely on his own, or otherwise can only be imagined to be the first and the main mover in the collaboration.

However, as far as the last theme, i.e. theme number4. above is concerned, the author has offered, what in my opinion is, a flawed thesis. I have issues with almost all aspects of it, not just an isolated one. In this post, I will try to only indicate why I think so; I will take up a more detailed look at this issue in a separate follow-up post some time in the indefinite future.

Note, I will not write much here about the first three (good) aspects. The reason is: there is just too much of great, and new, material concerning these aspects in this book. If I pick up one, I would be doing injustice to others. And, in a first reading, it really is not possible to develop a comprehensive sense of what is outstanding in such a rich array of ideas. I therefore will leave most of the good part alone in this review.

In other words, don’t try to judge the relative volumes of the good and the flawed in this book, going by how many words I allocate to each in this review.

4. The Main (Non-Negligible) Theoretical Flaw Present in the Book

Now, returning to what I think are the flaws of the book:

The author (i.e. the author-duo) believes that physics is inherently mathematical. I find that proposition very troublesome, full-stop.

Now, an alternative view, wrong again in my opinion, but arguably much less wrong than the view presented by the author, would consist of reversing the order and saying: “mathematics is inherently physical.” Now, of course, this second view, too, is plain wrong—in my opinion.

The reason for both the statements being wrong is this: the two subject areas, viz. physics and mathematics, simply cannot be connected via any kind of an inheritance relationship—forward or reverse. Stronger: It’s Rationalistic to suppose that the two can be. The objection remains even if you try to extend the meaning of the term “inherently” in some sense.

In other words, despite first noting that this book has astonishingly great virtues, in addition, what I am saying is also this: The author of this book does take an outright wrong position as far as the nature of the relationship between physics and mathematics is concerned.

But then, the author doesn’t just stop there. True to his integrating spirit (otherwise a very admirable trait), the author attempts to justify that wrong position of his by appeal to what I consider to be a mistaken interpretation of Rand’s epistemology, viz., her measurement-omission principle. I will provide the details later, but if you must have something right away (and if you already know Objectivist epistemology): the author confuses the quantitative measurements with the numerical ones, effectively treating the second as if it was not only coextensive but also synonymous with the first. As expected, he also confuses the qualitative and the quantitative. And similarly for the related aspects.

Thus, it is this fourth theme of the book which is truly troublesome for me. There also are some other minor flaws—the passages where he uses the words “approximate,” or (physical) “quantity” for example, or the place where he tries to relate numerical measurements and consciousness. But, to my mind, these are either only secondary or relatively minor.

Please note what I am saying concerning this fourth theme. What I am saying is not that the author has a good thesis but has simply not explained it well enough. I am also not saying that he has a potentially good thesis but left out some important aspects of it in the present book. No. I am saying neither.

My position is stronger: As far as the fourth themes is concerned, the very thesis that the author assumes, explicitly puts forth, explains, and tries to integrate with all of the rest of his arguments and positions throughout this book, is fundamentally untenable. The position he takes is wrong! Further, to the best of my knowledge, such a position also would not be justifiable on the basis of Ayn Rand’s epistemology (ITOE, 2nd edition)!!

Ok. So that’s one major flaw. Anything else to be noted in an overall sense? Any other significant +ves and -ves?

Let me jot down a few, more or less completely randomly and on the fly. (I might write a better deliberated and hopefully more concise review later.)

5. This Book Has Tremendous Virtues, Too!

As I said before, this book has enormous virtues. Let me list a few:

(5.a) It offers a new philosophic solution to the age-old problem of validating induction.

Yes, what this book offers is a new solution. So, the book does represent a step forward, a very important one. However, I wouldn’t call it epochal. To my mind, the truly game-changing—and therefore epoch-marking—book was Ayn Rand’s ITOE.

But, yes, even if not epochal, the book still is really great. I would call it: path-breaking and englightening. The book truly illuminates and enlightens all the essential aspects of the problem of induction—and of the new philosophic solution it offers.

Let me give you just one example of the kind of sub-issues that have to be addressed in forming a new philosophic solution. In this book, the author states, and convincingly explains, a new sub-proposition: “Discovery is proof.”

To form a new solution is to non-contradictorily add to the existing (i.e. Ayn Rand’s) Objectivist epistemology. The author certainly succeeds in doing that. Here is an example of one sub-issue that the author tackles: Concepts are what make induction possible and necessary.

Even regarding the theme where he has failed in a broad manner (i.e. in the theme concerning physics and mathematics), the author presents many wonderful ideas. Here is one:

“Ideas within a mind are an inseparable part of a total cognitive state in a way that is not true of physical bodies.”

See the kind of tightly-knit formulations the author presents? The book teems over with such gems. (If you didn’t find the position that discovery is proof surprising, then probably you might not consider the other examples I gave here worthy of admiration either. In such a case, neither the book nor the present review is meant for you.)

(5.b) It is an exceptionally well-written book.

It has been written with exceptional clarity and lucidity. The book remains lucid through and through.

However, sometimes, the style of expression tends to get a bit too much “Californian,” sort of like “spacey”. This is especially noticeable when the author gets into the details of the history of physics, the place where he begins unraveling the various complex threads of reasoning present in Galileo’s work vs. those in the others, and how Newton differed from them. In this part, the text does seem to lose its grip a bit; it might even become boring to some. However, that’s just the stylistic part. The important point is, even in such passages, the writeup itself remains unusually clear in exposition.

The book is also very well-structured. However, the flow of the writing is such that I doubt if a casual reader would really spot the underlying structuring, upon first reading. I noticed the structuring aspect only after I began scribbling my margin notes and underlying sentences for that mathematics-physics issue.

(5.c) The book also carries some rare anecdotes, stories or other treasure items dug up from history of physics.

The author obviously has taken pains to go through a lot of original material. In the end, he comes up with some bits that not only illustrates his fresh philosophic positions, also help correct some wide-spread or popular myths/legends/anecdotes. Invariably, these also show physicists in a better light than what tradition has allowed them. Harriman considers great physicists his heroes, and it shows through his writing.

For example, did you know that Ben Franklin had taken the care to wear an insulating glove before flying his famous kite in that rainy storm? that it was a carefully planned experiment, preceded by a long chain of conceptual analysis, and with some clearly anticipated results? The popular accounts typically show Franklin as a blind adventurer who turned lucky in not getting electrocuted in that thunderstorm—and the characterization subtly helps build the myth that discoveries, esp. great discoveries, are made by blind chance. Harriman senses the looming theoretical danger right in advance, and proceeds to overcome it by supplying the right facts after a thorough digging through the history.

Let me directly cite a passage (taken at random) to show how, in his writing, the author combines authentic history and in-depth remarks about physics.

Most engineers know that it was Count Rumford who first observed that during boring of guns, the metal would get hot, and that this observation was one of the earliest steps that eventually led to the formulation of the laws thermodynamics. Now here’s how accurately and succinctly the author describes the development:

“In the eighteenth century, heat was widely believed to be a fluid (called “calloric”) that flowed from hot to cold bodies. At the end of the century, however, two experiments provided strong evidence that heat was not a substance, but rather some internal motion of the matter comprising the bodies.

In 1798, Count Rumford (aka Benjamin Thompson) was supervising the manufacture of cannons at the military arsenal in Munich. He was surprised by the enormous amount of heat generated in the process of boring the cannons, and he decided to investigate the phenomenon. He placed a brass gun barrel in a wooden box containing cold water and bored it with a blunt steel drill. After about two and one-half hours, the water began to boil. The apparently inexhaustible supply of heat led Rumford to reject the caloric theory. As he put it, “Anything which any insulated body, or system of bodies, can continue to furnish without limitation, cannot possibly a material substance.”

There are many such stories in the book.

6. Scope for Improvements

Now, let me note down a few points where there is scope for improvement.

(6.a) For a book that claims the heritage of Ayn Rand’s ITOE and aims to further that legacy, a chapter-wise summary is conspicuous by its absence . A summary is an objective necessity for a book like this, dealing, as it does, with a complex subject: presenting a new theory in epistemology, and integrating it with a completely fresh look, in its light, at the history of physics. For a matter this complex, it would be easy for the novice reader to get lost in the trees and fail to see the forest. Even if not “perfect,” a “next-to-perfect” summary is a necessity. The author should seriously consider writing a chapter-by-chapter summary document, and uploading a PDF/doc file of it at the book site/blog.

(6.b) A more direct (perhaps point-wise comparative) account of how the author is adding to Mills’ principles, would have been helpful. It’s obvious to me that Mills couldn’t have had a sound theory of concepts, and therefore, of generalizations. Mills must have worked with blunt tools. But did he at least address, in whatever form, the idea that integration was necessary to induction? the crucial role of the conceptual frame-work, and the reasons thereof, that Harriman puts forth?

(6.c) There are no equations in this book! Equations could have served as a final encapsulating device, after arriving at the end of certain discussions. Not as a substitute for the preceding thought, but only at the culmination of a thread. And, even there, not as a substitute to a conceptual statement, but by way of condensation into mathematical terms. Perhaps, also using the then existing historical notation itself, in addition to the modern notation. The absence of equations is a rather curious omission—especially given the author’s (wrong) view that physics is inherently mathematical.

(6.d) As far as the printing-layout of the book goes, the page headers alternately carry the book title and the author’s name. But, this way, the current chapter number is not easily accessible to the reader. Though a very minor matter, it does become something of a major inconvenience while referring to the book-end references. Consider an example. Suppose I wanted to look up what reference no., say, [17] was, at some point in the main text. Keeping one finger in the main text, I would turn to the References section at the end of the book, only to realize that I had forgotten what was the number of the chapter that I was into. Now, to find the chapter number, I had to turn all the way back to the front of the book, but in two steps: first, to the current page, to find the page number I was at, and then to the Contents section in the beginning, to find the chapter number (by interpolation)! Then, I had once again go back to the end of the book to look up the referred work—provided I had managed to remember the reference number in this process! (It was [17] in this case.) Plain inconvenience!!

… Every one knows it’s you who wrote the book, Dave, so drop your name from the headers section and replace it by chapter numbers during the next print revision. And, it’s not at all a bad idea to give also the chapter titles in the References section also. Simple things, but someone needs to look into them.

7. How “Big” Is the Main Theoretical Flaw? Why Does It Matter? What Grade to Assign This Book?

OK, so you can now see that, overall, there are many great virtues to this book, one important flaw—but virtually no other significant flaw as far as the contents or the main thesis of the book goes.

So, it’s now time to look into what that aforementioned major flaw does to the overall evaluation of the book.

Usually, I have little or no difficulty in forming judgments about any book that carries as much clarity and as many virtues as this one does—it’s a straight “A”. But then, this book did present me with a real difficulty.

The trouble, essentially, was this: The flawed part is not “cut-able”.

This book is not written in a compartmentalized manner (which, otherwise, is a virtue). So, contents-wise, the flaw is not restricted to just one or two isolated chapters or sections. The author has weaved-in the wrong view with so much other material, sub-themes, too. The flaw is weaved-in throughout the entirety of the book.

It therefore was very hard for me to come up with a final judgment in terms of a letter-grade: Should I give this book an “A”? Does it really deserve an “A”? Shouldn’t the grade reflect the important flaw especially because it’s all woven-in, via a “B” grade?

If I gave the book an “A”, I would run the risk of supporting what I consider is a wrong and misleading view. In my opinion, a lot of chaos in theoretical physics (modern or classical, but especially the modern physics) can be directly or indirectly attributed to this wrong view of mathematics and physics.

And, apart from that abstract considerations of evaluating a book, I do have something personally important and practical matter at the stake too. It is this (erroneous) view of physics which, I think, might allow people to dismiss my research (or indeed any research such as mine) out of the hand, via the rhetoric: “But where is the mathematics for your theory, Ajit? I see no equations.” The implication being: Is it really physics you are talking about?

Now, it is useless to tell such people that no suitable mathematical notation has yet been evolved to directly capture the physics theory I am formulating and putting forth. … The only alternative is to use an infinity of integrals… Take a moment to imagine how bad that would be, to use as a notation… And, another matter. If I use that notation, I also run the risk that the actual physics of my theory would get confused with, say, Feynman’s approach, i.e., with what has already been said (which, in my opinion, is a wrong physics theory too).

Note, my theory does have quantitative relationships, but no equations, not yet. Even then, it is possible to point out, right away, that there would be definite, even if small, differences in the transient dynamics of photon propagation—and similar differences will remain in a broader theory of electron propagation too. Making this prediction is possible. And, it is possible to build computer simulations that clearly bring out the difference too.

Yet, Harriman’s position makes it impossible to distinguish between a superficial or incompetent attempt at theorization in physics, and a proper theory of physics that does not have equations. Such a position conforms to the conventional wisdom; see Wigner, Feynman (“The Character of Physical Law”), Hamming, Narlikar, et al. Such people can, and do, critique (if not outright dismiss) my work on the count of a perceived lack of “rigour!” Why? Conceivably only because it’s not “mathematical” enough.

Thus, there really is a lot at stake practically speaking too.

And, apart from my own case, I think there has been a huge waste of human effort that has actually occurred due to emptiness of thoughts resulting out of pursuing mathematics while attempting to do physics. In future, I will illustrate this assertion of mine by taking a prominent example: in reference to what Feynman was happilypreoccupied stating in his book, misleading people in his effervescent and charming style. Dirac’s was one very peculiar case; he could (only once) simply fiddle around with equations to produce some good physics. Such cases are peculiar but not unreasonable. And, more importantly, they are not anti-thetical to my position.

So, lionizing the role of mathematics in physics is not all that a new insight; it has been an idle past-time of many theoreticians, of both the Platonists in philosophy, and deductivists in physics.

And then, if a brand new book arrives on the scene, takes the position of primacy of induction, offers a great theory concerning induction, and also commits the same error of lionizing mathematics in physics, the risk multiplies many times over.

Thus, giving an “A” carries a real risk.

On the other hand, if I gave the book a “B”, I would end up insulting so much that is actually so valuable, nay, so astounding good, and yes, also so fresh i.e. new. New, even as a philosophic theory. This aspect of the book really needs as much praise—and, practically speaking, also as much of sales and circulation—as it can get.

I tried to weigh the matter for about a week or two by now. Actually, for more time. I became aware of the flaw of the book rather early into my reading. In fact, I had sensed that there could be an issue of this sort as soon as I had gone over the contents (index) of the book way back in May/June 2010 or so, and I had noted such an apprehension of mine in one of my iMechanica posts too.

Anyway, after weighing the matter enough, finally, I have decided to give the book a qualified “A”. Yes, it’s an “A”, but a qualified one. Though the final grade is an “A,” there is a footnote to it saying that the author(s) has (have) failed in an important and non-negligible respect.

Honestly, this failure of theirs has really surprised me—esp. given Peikoff’s insights into philosophic errors, esp. of Rationalistic thought processes, and given the fact of his supervision of this project.

But still, a failure is a failure. So, to conclude this matter, it’s only a qualified “A”: the authors manage to barely slip into the “A” grade, and only via grace marks.

(BTW, this long passage about what grade to choose, again, demonstrates the falsehood of hasty quantification—a possibly new fallacy I just coined. Two prominent examples: 1. Attempting to derive a single number (ordinal or cardinal) for what actually is a composite issue. 2. Attempting to assign equal probabilities to all possible outcomes, esp., assuming that tossing a fair coin illustrates anything fundamental.)

I think in a future post on this topic, I will provide the reasons why I think that the matter I call flaw really is a flaw, and also will give some details of the places in the book where the author reveals that flaw.

Please do note: Even if I write such a post next, my intention is not to keep discussing the flaws of the book. Nope. Let alone character assassination (of the book or of the author(s)), even polemics as such does not hold much interest for me. On the other hand, I think I do have something positive to develop, present, and defend: What I think is the correct view of mathematics, physics, and their interrelations. I will write when I find the time to do so; no specific promises as to when.

8. The Controversy Concerning This Book and Dr. John McCaskey

Incidentally, if you have been waiting for my opinion of the recent McCaskey-related controversy, let me begin by saying this: I cannot say that I have gone through his review at Amazon.com because, despite starting afresh some 4–5 times, I simply could not finish reading his comment even once. He writes very boringly. So please consider this entire comment of mine as based on an incomplete knowledge of his publicly available (and network-wise accessible) comments. I have also similarly tried to go through the emails exchange he had with Harriman, and have failed in finishing reading it.

On the above basis, what I have to say about the matter is this: I wouldn’t have had any problem if the concerns raised by McCaskey were actually to be in any way significant to a critical appraisal of the book. However, I do have a problem with McCaskey because nothing of what he writes regarding this issue in fact is in any way significant or germane to the book. And, I have no desire to debate this position of mine with any one. (I will delete comments to this post asking or nudging me towards this end.) But yes, I do want to write about this matter once, and thereby finish it off from my concerns.

Why is McCaskey so boring? I mean, he gives this detail, then that detail, then comes up with this advice to the author (Harriman), then that advice… But none of it goes anywhere even near to that which actually are the core concerns of the book! See the four themes I have given above, and see if McCaskey is directly concerned with any of these! Indeed, some of what he writes does not even make plain sense to me—whether related to the book or otherwise.

Then, I also factor in these facts: McCaskey has a PhD from Stanford and teaches there; before that education, he had co-founded/worked for a company which was founded by a President of the Stanford University; they both made millions or billions (in US $ terms) during the Internet boom of the late 1990s, and apparently managed to retain significant part of that money after the bubble burst. Ok.

I have worked in the SF Bay Area, right during those Internet boom and bust times, and I have a kind of sense (not a psychic sense but just plain ordinary sense) of what these asshole Americans, esp. Californians, esp. Bay Areans, esp. “successful” ones are like. A very definite sense.

One of their chief characteristic is best described using an analogy: they are very well adept in administering the Chinese punishment, in a mental manner. Allow me to explain.

These SF Bay Areans in particular (and Americans in general), esp. the techy ones, can summon an infinity of patience and keep creating and raising “issues” after “issues” in a decidedly polite manner (which often, but not necessarily always, follows political correctness). They can, and do, raise a lot of technically correct but actually meaningless “issues”. They can keep spawning threads after meaningless threads—and they do. The “drops” keep falling on the head of the intended victim… These Bay Arean (Californian, American…) assholes can keep doing it all until the automatic/biological/at least low-level functions of the opponent have to give away, and the victim cracks down.

Now, while I would always stand by the above description, let me hasten to add that I realize that such need not actually have been the case with McCaskey’s treatment of the book and of its author. What I mean to point out is this: use of such tactics, at least in a mild form, is very very easy for me to imagine. Even in this context. If this last interpretation is not valid, then I have to wonder: what else could be the point of raising all those meaningless or irrelevant historical details—to a colleague? I mean whether McCaskey wanted to mount that Chinese punishment sort of attack on Harriman or not, in any case, most of McCaskey’s stated concerns are far too detailed, at a few isolated “leaves” of a tree and not at its root or stem, and, come to think of it, inconsequential or  actually completely worthless as far the main thesis, the main subject matter, the overall nature, and the main thrust of this book is concerned.

And, though I have nothing personal for or against McCaskey, still, yes, I do think that Peikoff did a good thing by getting that kind of a pitter-pitt-patt, pitter-pitt-patt sort of nuisance (aiming for the conformance to the “accepted” etc. sort of history, no less!) out of the way.

Now, don’t get me wrong. Unlike McCaskey, I do have a lot of issues with Dr. Leonard Peikoff himself, too, among others, especially concerning some of the things such as reincarnation etc. which he labels “arbitrary”. Yet, in the matter concerning the McCaskey etc. matter, I think that Peikoff did the right thing.

… How I wish Peikoff had a similar perspicuity when it came to taking a proper view of the inter-relations between physics and mathematics—which will be our topic next time (i.e. the next time I write on this subject).

9. To Conclude

In the meanwhile: Go ahead, buy it. Buy it for yourself, and also ask your library to order a copy. If you are American/British/Canadian/French/Australian/…/Christian etc.: Consider giving it as a gift this holiday season (starting with this Thanks Giving week-end). And, while doing so, also add a mental note: Ajit Jadhav has some issues with what is presented in the book concerning the nature and relation of physics and mathematics, but not with the fundamental thesis concerning induction—with a virtuoso exposition of a magnificent theory.

This book would have made for a great deal for the reader even at a price like Rs. 2000/-. At about Rs. 600/-, it is a windfall—even if the author, IMO, doesn’t get some of the things right. So, go ahead, buy it—and read and re-read it. You will profit from it. That’s the bottom-line.

Version History:
— v.1.0 Began writing November 25, 2010, 10:00 PM; finished the same night; minor changes made and published on the blog on November 27, 2010, ~6:38 PM.
— V.1.1 Added paragraph titles and a little more matter on November 28, 2010, ~7:15 PM. One more minor revision is due.

* * * * *   * * * * *   * * * * *

A Song I Like

(Marathi): “kuNi jaal kaa, saangaal kaa…”
Lyrics: “Anil” (A. R. Deshpande)
Singer: Vasantrao Deshpande
Music: Yashwant Deo

[PS: Revised Nov. 28. One more minor revision is due.]



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