# The smith.edu would not be sponsoring my further research on QM in any form… Yaaawn…

The smith.edu would not be sponsoring my further research on QM in any form. … Yaaawn…

Just saying it. Not that they ever did.

But that’s what I gather anyway. That they will neither be sponsoring nor be supporting my research. Nor take its findings in the true spirit of science.

I am neither happy nor unhappy about it. Just plain [yawningly] curious. … To be dealt with, some other time—these kind of American-borns. All these intellectual goons who take pleasure in tearing down my well-constructed thoughts, in evading the several virtues of my research.

Sometimes they do make me laugh.

But apart from being authentic intellectual goons, they—these Americans—also are very, very powerful. Laughing at them is not, really speaking, the right response. I mean it can be. But the response should not consist of laughing them away. They are very, very powerful. And, rich. And, goons.

In the meanwhile, check out two neat resources on QM, both free and seemingly written with unusually high degree of personal involvement with the writing project:

Sure enough, I’ve mentioned them both here before. But they need to be highlighted again. Especially, the second one. I will be assuming that you have gone through this paper when I return the next time. We will be dealing with this question that some (white, scholarly) goons (obviously) have no inkling of:

If the system wavefunction $\Psi$ is defined over a $3N$-dimensional abstract, mathematical configuration space in the mainstream QM, can a new theory have its wavefunction $\Psi$ defined over the physical $3$-dimensional space? If yes, how?

You know, there are those other Americans who find it shameful to ever reply to me. They think it is beneath them. Naturally, they are not grateful to me even if the errors of their own (analog vs. digital) or of their group (third law of thermodynamics) have been gently pointed out. Americans, after all. Prestigious Americans. Even their graves are going to be just that—impressive and prestigious.

Anyway, let’s leave their dirty souls with them, and focus on the third law of thermodynamics. Choose the correct answer: The third law of thermodynamics says that the absolute zero temperature:

(a) cannot exist in the universe.

(b) cannot be reached in any process.

If you know the correct answer and point it out to the prestigious Americans who have made a mistake about it, they don’t like it—no matter how indirectly and gently you do it.

How can you expect them to extend support to your QM research, let alone sponsor it? be it MA or CA?

Anyway, let me wind up…

A song I like:

(Hindi) “maanzee naiyaa DhoonDe kinaaraa”
Singer: Mukesh
Music: Laxmikant-Pyaarelaal
Lyrics: Anand Bakshi

[Again a song from my childhood days, a song for the soul.]

But tell me, really, aren’t all those Americans—white or black or brown or others, whether intellectuals or not… Aren’t they just plain goons? … Hasn’t that thought passed by you before? I mean, whether they are socially respectable or otherwise…

But why are they such goons?

/

# Learnability of machine learning is provably an undecidable?—part 3: closure

Update on 23 January 2019, 17:55 IST:

In this series of posts, which was just a step further from the initial, brain-storming kind of a stage, I had come to the conclusion that based on certain epistemological (and metaphysical) considerations, Ben-David et al.’s conclusion (that learnability can be an undecidable) is logically untenable.

However, now, as explained here [^], I find that this particular conclusion which I drew, was erroneous. I now stand corrected, i.e., I now consider Ben-David et al.’s result to be plausible. Obviously, it merits a further, deeper, study.

However, even as acknowledging the above-mentioned mistake, let me also hasten to clarify that I still stick to my other positions, especially the central theme in this series of posts. The central theme here was that there are certain core features of the set theory which make implications such as Godel’s incompleteness theorems possible. These features (of the set theory) demonstrably carry a glaring epistemological flaw such that applying Godel’s theorem outside of its narrow technical scope in mathematics or computer science is not permissible. In particular, Godel’s incompleteness theorem does not apply to knowledge or its validation in the more general sense of these terms. This theme, I believe, continues to hold as is.

Update over.

Gosh! I gotta get this series out of my hand—and also head! ASAP, really!! … So, I am going to scrap the bits and pieces I had written for it earlier; they would have turned this series into a 4- or 5-part one. Instead, I am going to start entirely afresh, and I am going to approach this topic from an entirely different angle—a somewhat indirect but a faster route, sort of like a short-cut. Let’s get going.

Statements:

Open any article, research paper, book or a post, and what do you find? Basically, all these consist of sentences after sentences. That is, a series of statements, in a way. That’s all. So, let’s get going at the level of statements, from a “logical” (i.e. logic-thoretical) point of view.

Statements are made to propose or to identify (or at least to assert) some (or the other) fact(s) of reality. That’s what their purpose is.

The conceptual-level consciousness as being prone to making errors:

Coming to the consciousness of man, there are broadly two levels of cognition at which it operates: the sensory-perceptual, and the conceptual.

Examples of the sensory-perceptual level consciousness would consist of reaching a mental grasp of such facts of reality as: “This object exists, here and now;” “this object has this property, to this much degree, in reality,” etc. Notice that what we have done here is to take items of perception, and put them into the form of propositions.

Propositions can be true or false. However, at the perceptual level, a consciousness has no choice in regard to the truth-status. If the item is perceived, that’s it! It’s “true” anyway. Rather, perceptions are not subject to a test of truth- or false-hoods; they are at the very base standards of deciding truth- or false-hoods.

A consciousness—better still, an organism—does have some choice, even at the perceptual level. The choice which it has exists in regard to such things as: what aspect of reality to focus on, with what degree of focus, with what end (or purpose), etc. But we are not talking about such things here. What matters to us here is just the truth-status, that’s all. Thus, keeping only the truth-status in mind, we can say that this very idea itself (of a truth-status) is inapplicable at the purely perceptual level. However, it is very much relevant at the conceptual level. The reason is that at the conceptual level, the consciousness is prone to err.

The conceptual level of consciousness may be said to involve two different abilities:

• First, the ability to conceive of (i.e. create) the mental units that are the concepts.
• Second, the ability to connect together the various existing concepts to create propositions which express different aspects of the truths pertaining to them.

It is possible for a consciousness to go wrong in either of the two respects. However, mistakes are much more easier to make when it comes to the second respect.

Homework 1: Supply an example of going wrong in the first way, i.e., right at the stage of forming concepts. (Hint: Take a concept that is at least somewhat higher-level so that mistakes are easier in forming it; consider its valid definition; then modify its definition by dropping one of its defining characteristics and substituting a non-essential in it.)

Homework 2: Supply a few examples of going wrong in the second way, i.e., in forming propositions. (Hint: I guess almost any logical fallacy can be taken as a starting point for generating examples here.)

Truth-hood operator for statements:

As seen above, statements (i.e. complete sentences that formally can be treated as propositions) made at the conceptual level can, and do, go wrong.

We therefore define a truth-hood operator which, when it operates on a statement, yields the result as to whether the given statement is true or non-true. (Aside: Without getting into further epistemological complexities, let me note here that I reject the idea of the arbitrary, and thus regard non-true as nothing but a sub-category of the false. Thus, in my view, a proposition is either true or it is false. There is no middle (as Aristotle said), or even an “outside” (like the arbitrary) to its truth-status.)

Here are a few examples of applying the truth-status (or truth-hood) operator to a statement:

• Truth-hood[ California is not a state in the USA ] = false
• Truth-hood[ Texas is a state in the USA ] = true
• Truth-hood[ All reasonable people are leftists ] = false
• Truth-hood[ All reasonable people are rightists ] = false
• Truth-hood[ Indians have significantly contributed to mankind’s culture ] = true
• etc.

For ease in writing and manipulation, we propose to give names to statements. Thus, first declaring

A: California is not a state in the USA

and then applying the Truth-hood operator to “A”, is fully equivalent to applying this operator to the entire sentence appearing after the colon (:) symbol. Thus,

Truth-hood[ A ] <==> Truth-hood[ California is not a state in the USA ] = false

Just a bit of the computer languages theory: terminals and non-terminals:

To take a short-cut through this entire theory, we would like to approach the idea of statements from a little abstract perspective. Accordingly, borrowing some terminology from the area of computer languages, we define and use two types of symbols: terminals and non-terminals. The overall idea is this. We regard any program (i.e. a “write-up”) written in any computer-language as consisting of a sequence of statements. A statement, in turn, consists of certain well-defined arrangement of words or symbols. Now, we observe that symbols (or words) can be  either terminals or non-terminals.

You can think of a non-terminal symbol in different ways: as higher-level or more abstract words, as “potent” symbols. The non-terminal symbols have a “definition”—i.e., an expansion rule. (In CS, it is customary to call an expansion rule a “production” rule.) Here is a simple example of a non-terminal and its expansion:

• P => S1 S2

where the symbol “=>” is taken to mean things like: “is the same as” or “is fully equivalent to” or “expands to.” What we have here is an example of an abstract statement. We interpret this statement as the following. Wherever you see the symbol “P,” you may substitute it using the train of the two symbols, S1 and S2, written in that order (and without anything else coming in between them).

Now consider the following non-terminals, and their expansion rules:

• P1 => P2 P S1
• P2 => S3

The question is: Given the expansion rules for P, P1, and P2, what exactly does P1 mean? what precisely does it stand for?

• P1 => (P2) P S1 => S3 (P) S1 => S3 S1 S2 S1

In the above, we first take the expansion rule for P1. Then, we expand the P2 symbol in it. Finally, we expand the P symbol. When no non-terminal symbol is left to expand, we arrive at our answer that “P1” means the same as “S3 S1 S2 S1.” We could have said the same fact using the colon symbol, because the colon (:) and the “expands to” symbol “=>” mean one and the same thing. Thus, we can say:

• P1: S3 S1 S2 S1

The left hand-side and the right hand-side are fully equivalent ways of saying the same thing. If you want, you may regard the expression on the right hand-side as a “meaning” of the symbol on the left hand-side.

It is at this point that we are able to understand the terms: terminals and non-terminals.

The symbols which do not have any further expansion for them are called, for obvious reasons, the terminal symbols. In contrast, non-terminal symbols are those which can be expanded in terms of an ordered sequence of non-terminals and/or terminals.

We can now connect our present discussion (which is in terms of computer languages) to our prior discussion of statements (which is in terms of symbolic logic), and arrive at the following correspondence:

The name of every named statement is a non-terminal; and the statement body itself is an expansion rule.

This correspondence works also in the reverse direction.

You can always think of a non-terminal (from a computer language) as the name of a named proposition or statement, and you can think of an expansion rule as the body of the statement.

Easy enough, right? … I think that we are now all set to consider the next topic, which is: liar’s paradox.

The liar paradox is a topic from the theory of logic [^]. It has been resolved by many people in different ways. We would like to treat it from the viewpoint of the elementary computer languages theory (as covered above).

The simplest example of the liar paradox is , using the terminology of the computer languages theory, the following named statement or expansion rule:

• A: A is false.

Notice, it wouldn’t be a paradox if the same non-terminal symbol, viz. “A” were not to appear on both sides of the expansion rule.

To understand why the above expansion rule (or “definition”) involves a paradox, let’s get into the game.

Our task will be to evaluate the truth-status of the named statement that is “A”. This is the “A” which comes on the left hand-side, i.e., before the colon.

In symbolic logic, a statement is nothing but its expansion; the two are exactly and fully identical, i.e., they are one and the same. Accordingly, to evaluate the truth-status of “A” (the one which comes before the colon), we consider its expansion (which comes after the colon), and get the following:

• Truth-hood[ A ] = Truth-hood[ A is false ] = false           (equation 1)

Alright. From this point onward, I will drop explicitly writing down the Truth-hood operator. It is still there; it’s just that to simplify typing out the ensuing discussion, I am not going to note it explicitly every time.

Anyway, coming back to the game, what we have got thus far is the truth-hood status of the given statement in this form:

• A: “A is false”

Now, realizing that the “A” appearing on the right hand-side itself also is a non-terminal, we can substitute for its expansion within the aforementioned expansion. We thus get to the following:

• A: “(A is false) is false”

We can apply the Truth-hood operator to this expansion, and thereby get the following: The statement which appears within the parentheses, viz., the “A is false” part, itself is false. Accordingly, the Truth-hood operator must now evaluate thus:

• Truth-hood[ A ] = Truth-hood[ A is false] = Truth-hood[ (A is false) is false ] = Truth-hood[ A is true ] = true            (equation 2)

Fun, isn’t it? Initially, via equation 1, we got the result that A is false. Now, via equation 2, we get the result that A is true. That is the paradox.

But the fun doesn’t stop there. It can continue. In fact, it can continue indefinitely. Let’s see how.

If only we were not to halt the expansions, i.e., if only we continue a bit further with the game, we could have just as well made one more expansion, and got to the following:

• A: ((A is false) is false) is false.

The Truth-hood status of the immediately preceding expansion now is: false. Convince yourself that it is so. Hint: Always expand the inner-most parentheses first.

Homework 3: Convince yourself that what we get here is an indefinitely long alternating sequence of the Truth-hood statuses that: A is false, A is true, A is false, A is true

What can we say by way of a conclusion?

Conclusion: The truth-status of “A” is not uniquely decidable.

The emphasis is on the word “uniquely.”

We have used all the seemingly simple rules of logic, and yet have stumbled on to the result that, apparently, logic does not allow us to decide something uniquely or meaningfully.

Liar’s paradox and the set theory:

The importance of the liar paradox to our present concerns is this:

Godel himself believed, correctly, that the liar paradox was a semantic analogue to his Incompleteness Theorem [^].

Go read the Wiki article (or anything else on the topic) to understand why. For our purposes here, I will simply point out what the connection of the liar paradox is to the set theory, and then (more or less) call it a day. The key observation I want to make is the following:

You can think of every named statement as an instance of an ordered set.

What the above key observation does is to tie the symbolic logic of proposition with the set theory. We thus have three equivalent ways of describing the same idea: symbolic logic (name of a statement and its body), computer languages theory (non-terminals and their expansions to terminals), and set theory (the label of an ordered set and its enumeration).

As an aside, the set in question may have further properties, or further mathematical or logical structures and attributes embedded in itself. But at its minimal, we can say that the name of a named statement can be seen as a non-terminal, and the “body” of the statement (or the expansion rule) can be seen as an ordered set of some symbols—an arbitrarily specified sequence of some (zero or more) terminals and (zero or more) non-terminals.

Two clarifications:

• Yes, in case there is no sequence in a production at all, it can be called the empty set.
• When you have the same non-terminal on both sides of an expansion rule, it is said to form a recursion relation.

An aside: It might be fun to convince yourself that the liar paradox cannot be posed or discussed in terms of Venn’s diagram. The property of the “sheet” on which Venn’ diagram is drawn is, by some simple intuitive notions we all bring to bear on Venn’s diagram, cannot have a “recursion” relation.

Yes, the set theory itself was always “powerful” enough to allow for recursions. People like Godel merely made this feature explicit, and took full “advantage” of it.

Recursion, the continuum, and epistemological (and metaphysical) validity:

In our discussion above, I had merely asserted, without giving even a hint of a proof, that the three ways (viz., the symbolic logic of statements or  propositions, the computer languages theory, and the set theory) were all equivalent ways of expressing the same basic idea (i.e. the one which we are concerned about, here).

I will now once again make a few more observations, but without explaining them in detail or supplying even an indication of their proofs. The factoids I must point out are the following:

• You can start with the natural numbers, and by using simple operations such as addition and its inverse, and multiplication and its inverse, you can reach the real number system. The generalization goes as: Natural to Whole to Integers to Rationals to Reals. Another name for the real number system is: the continuum.
• You can use the computer languages theory to generate a machine representation for the natural numbers. You can also mechanize the addition etc. operations. Thus, you can “in principle” (i.e. with infinite time and infinite memory) represent the continuum in the CS terms.
• Generating a machine representation for natural numbers requires the use of recursion.

Finally, a few words about epistemological (and metaphysical) validity.

• The concepts of numbers (whether natural or real) have a logical precedence, i.e., they come first. The entire arithmetic and the calculus must come before does the computer-representation of some of their concepts.
• A machine-representation (or, equivalently, a set-theoretic representation) is merely a representation. That is to say, it captures only some aspects or attributes of the actual concepts from maths (whether arithmetic or the continuum hypothesis). This issue is exactly like what we saw in the first and second posts in this series: a set is a concrete collection, unlike a concept which involves a consciously cast unit perspective.
• If you try to translate the idea of recursion into the usual cognitive terms, you get absurdities such as: You can be your child, literally speaking. Not in the sense that using scientific advances in biology, you can create a clone of yourself and regard that clone to be both yourself and your child. No, not that way. Actually, such a clone is always your twin, not child, but still, the idea here is even worse. The idea here is you can literally father your own self.
• Aristotle got it right. Look up the distinction between completed processes and the uncompleted ones. Metaphysically, only those objects or attributes can exist which correspond to completed mathematical processes. (Yes, as an extension, you can throw in the finite limiting values, too, provided they otherwise do mean something.)
• Recursion by very definition involves not just absence of completion but the essence of the very inability to do so.

Closure on the “learnability issue”:

Homework 4: Go through the last two posts in this series as well as this one, and figure out that the only reason that the set theory allows a “recursive” relation is because a set is, by the design of the set theory, a concrete object whose definition does not have to involve an epistemologically valid process—a unit perspective as in a properly formed concept—and so, its name does not have to stand for an abstract mentally held unit. Call this happenstance “The Glaring Epistemological Flaw of the Set Theory” (or TGEFST for short).

Homework 5: Convince yourself that any lemma or theorem that makes use of Godel’s Incompleteness Theorem is necessarily based on TGEFST, and for the same reason, its truth-status is: it is not true. (In other words, any lemma or theorem based on Godel’s theorem is an invalid or untenable idea, i.e., essentially, a falsehood.)

Homework 6: Realize that the learnability issue, as discussed in Prof. Lev Reyzin’s news article (discussed in the first part of this series [^]), must be one that makes use of Godel’s Incompleteness Theorem. Then convince yourself that for precisely the same reason, it too must be untenable.

[Yes, Betteridge’s law [^] holds.]

Other remarks:

Remark 1:

As “asymptotical” pointed out at the relevant Reddit thread [^], the authors themselves say, in another paper posted at arXiv [^] that

While this case may not arise in practical ML applications, it does serve to show that the fundamental definitions of PAC learnability (in this case, their generalization to the EMX setting) is vulnerable in the sense of not being robust to changing the underlying set theoretical model.

What I now remark here is stronger. I am saying that it can be shown, on rigorously theoretical (epistemological) grounds, that the “learnability as undecidable” thesis by itself is, logically speaking, entirely and in principle untenable.

Remark 2:

Another point. My preceding conclusion does not mean that the work reported in the paper itself is, in all its aspects, completely worthless. For instance, it might perhaps come in handy while characterizing some tricky issues related to learnability. I certainly do admit of this possibility. (To give a vague analogy, this issue is something like running into a mathematically somewhat novel way into a known type of mathematical singularity, or so.) Of course, I am not competent enough to judge how valuable the work of the paper(s) might turn out to be, in the narrow technical contexts like that.

However, what I can, and will say is this: the result does not—and cannot—bring the very learnability of ANNs itself into doubt.

Phew! First, Panpsychiasm, and immediately then, Learnability and Godel. … I’ve had to deal with two untenable claims back to back here on this blog!

… Code! I have to write some code! Or write some neat notes on ML in LaTeX. Only then will, I guess, my head stop aching so much…

Honestly, I just downloaded TensorFlow yesterday, and configured an environment for it in Anaconda. I am excited, and look forward to trying out some tutorials on it…

BTW, I also honestly hope that I don’t run into anything untenable, at least for a few weeks or so…

…BTW, I also feel like taking a break… May be I should go visit IIT Bombay or some place in konkan. … But there are money constraints… Anyway, bye, really, for now…

A song I like:

Music: Sooraj (the pen-name of “Shankar” from the Shankar-Jaikishan pair)
Lyrics: Ramesh Anavakar

[Any editing would be minimal; guess I will not even note it down separately.] Did an extensive revision by 2019.01.21 23:13 IST. Now I will leave this post in the shape in which it is. Bye for now.

# Learnability of machine learning is provably an undecidable?—part 1

Update on 23 January 2019, 17:55 IST:

In this series of posts, which was just a step further from the initial, brain-storming kind of a stage, I had come to the conclusion that based on certain epistemological (and metaphysical) considerations, Ben-David et al.’s conclusion (that learnability can be an undecidable) is logically untenable.

However, now, as explained here [^], I find that this particular conclusion which I drew, was erroneous. I now stand corrected, i.e., I now consider Ben-David et al.’s result to be plausible. Obviously, it merits a further, deeper, study.

However, even as acknowledging the above-mentioned mistake, let me also hasten to clarify that I still stick to my other positions, especially the central theme in this series of posts. The central theme here was that there are certain core features of the set theory which make implications such as Godel’s incompleteness theorems possible. These features (of the set theory) demonstrably carry a glaring epistemological flaw such that applying Godel’s theorem outside of its narrow technical scope in mathematics or computer science is not permissible. In particular, Godel’s incompleteness theorem does not apply to knowledge or its validation in the more general sense of these terms. This theme, I believe, continues to hold as is.

Update over.

This one news story has been lying around for about a week on my Desktop:

Lev Reyzin, “Unprovability comes to machine learning,” Nature, vol. 65, pp. 166–167, 10 January 2019 [^]. PDF here: [^]

(I’ve forgotten how I came to know about it though.) The story talks about the following recent research paper:

Ben-David et al., “Learnability can be undecidable,” Nature Machine Intelligence, vol. 1, pp. 44–48, January 2019 [^]. PDF here: [^]

I don’t have the requisite background in the theory of the research paper itself, and so didn’t even try to read through it. However, I did give Reyzin’s news article a try. It was not very successful; I have not yet been able to finish this story yet. However, here are a few notings which I made as I tried to progress through this news story. The quotations here all come from from Reyzin’s news story.

Before we begin, take a moment to notice that the publisher here is arguably the most reputed one in science, viz., the Nature publishing group. As to the undecidability of learnability, its apparent implications for practical machine learning, artificial intelligence, etc., are too obvious to be pointed out separately.

“During the twentieth century, discoveries in mathematical logic revolutionized our understanding of the very foundations of mathematics. In 1931, the logician Kurt Godel showed that, in any system of axioms that is expressive enough to model arithmetic, some true statements will be unprovable.”

Is it because Godel [^] assumed that any system of axioms (which is expressive enough to model arithmetic) would be based on the standard (i.e. mathematical) set theory? If so, his conclusion would not be all that paradoxical, because the standard set theory carries, from an epistemological angle, certain ill-conceived notions at its core. [BTW, throughout this (short) series of posts, I use Ayn Rand’s epistemological theory; see ITOE, 2e [^][^].]

To understand my position (that the set theory is not epistemologically sound), start with a simple concept like “table”.

According to Ayn Rand’s ITOE, the concept “table” subsumes all possible concrete instances of tables, i.e., all the tables that conceivably exist, might have ever existed, and might ever exist in future, i.e., a potentially infinite number of concrete instances of them. Ditto, for any other concept, e.g., “chair.” Concepts are mental abstractions that stand for an infinite concretes of a given kind.

Now, let’s try to run away from philosophy, and thereby come to rest in the arms of, say, a mathematical logician like Kurt Godel [^], or preferably, his predecessors, those who designed the mathematical set theory [^].

The one (utterly obvious) way to capture the fact that there exist tables, but only using the actual terms of the set theory, is to say that there is a set called “tables,” and that its elements consist of all possible tables (i.e., all the tables that might have existed, might conceivably exist, and would ever conceivably exist in future). Thus, the notion again refers to an infinity of concretes. Put into the terms of the set theory, the set of tables is an infinite set.

OK, that seems to work. How about chairs? Once again, you set up a set, now called “chairs,” and proceed to dump within its braces every possible or conceivable chair.

So far, so good. No trouble until now.

The trouble begins when you start applying operators to the sets, say by combining them via unions, or by taking their intersections, and so on—all that Venn’s diagram business [^]. But what is the trouble with the good old Venn diagrams, you ask? Well, the trouble is not so much to the Venn diagrams as it is to the basic set theory itself:

The set theory makes the notion of the set so broad that it allows you to combine any sets in any which way you like, and still be able to call the result a meaningful set—meaningful, as seen strictly from within the set theory.

Here is an example. You can not only combine (take union of) “tables” and “chairs” into a broader set called “furniture,” you are also equally well allowed, by the formalism of the set theory, to absorb into the same set all unemployed but competent programmers, Indian HR managers, and venture capitalists from the San Francisco Bay Area. The set theory does not by itself have anything in its theoretical structure, formalism or even mathematical application repertoire, using which it could possibly so much as raise a finger in such matters. This is a fact. If in doubt, refer to the actual set theory ([^] and links therein), take it strictly on its own terms, in particular, desist mixing into it any extra interpretations brought in by you.

Epistemology, on the other hand, does have theoretical considerations, including explicitly formulated rules at its core, which together allow us to distinguish between proper and improper formulations of concepts. For example, there is a rule that the concrete instances being subsumed under a concept must themselves be conceptually commensurate, i.e., they must possess the same defining characteristics, even if possibly to differing degrees. Epistemology prevents the venture capitalists from the San Francisco Bay Area from being called pieces of furniture because it clarifies that they are people, whereas pieces of furniture are inanimate objects, and for this crucial reason, the two are conceptually incommensurate—they cannot be integrated together into a common concept.

To come back to the set theory, it, however, easily allows for every abstractly conceivable “combination” for every possible operand set(s). Whether the operation has any cognitive merit to it or not, whether it results into any meaningful at all or not, is not at all a consideration—not by the design of the set theory itself (which, many people suppose, is more fundamental to every other theory).

So—and get this right—calling the collection of QC scientists as either politicians or scoundrels is not at all an abuse of the mathematical structure, content, and meaning of the set theory. The ability to take an intersection of the set of all mathematicians who publish papers and the set of all morons is not a bug, it is very much a basic and core feature of the set theory. There is absolutely nothing in the theory itself which says that the intersection operator cannot be applied here, or that the resulting set has to be an empty set. None.

Set theory very much neglects the considerations of the kind of a label there is to a set, and the kind of elements which can be added to it.

More on this, later. (This post has already gone past 1000 words.)

The songs section will come at the conclusion of this (short) series of posts, to be completed soon enough; stay tuned…

# Some running thoughts on ANNs and AI—1

Go, see if you want to have fun with the attached write-up on ANNs [^] (but please also note the version time carefully—the write-up could change without any separate announcement).

The write-up is more in the nature of a very informal blabber of the kind that goes when people work out something on a research blackboard (or while mentioning something about their research to friends, or during brain-storming session, or while jotting things on the back of the envelop, or something similar).

A “song” I don’t like:

“Credits”: Go, figure [^]. E.g., here [^]. Yes, the video too is (very strongly) recommended.

Update on 05 October 2018 10:31 IST:

Psychic attack on 05 October 2018 at around 00:40 IST (i.e. the night between 4th and 5th October, IST).

# Absolutely Random Notings on QM—Part 1: Bohr. And, a bad philosophy making its way into physics with his work, and his academic influence

TL;DR: Go—and keep—away.

I am still firming up my opinions. However, there is never a harm in launching yet another series of posts on a personal blog, is there? So here we go…

Quantum Mechanics began with Planck. But there was no theory of quanta in what Planck had offered.

What Planck had done was to postulate only the existence of the quanta of the energy, in the cavity radiation.

Einstein used this idea to predict the heat capacities of solids—a remarkable work, one that remains underappreciated in both text-books as well as popular science books on QM.

The first pretense at a quantum theory proper came from Bohr.

Matter, esp. gases, following Dalton, …, Einstein, and Perin, were made of distinct atoms. The properties of gases—especially the reason why they emitted or absorbed radiation only at certain distinct frequencies, but not at any other frequencies (including those continuous patches of frequencies in between the experimentally evident sharp peaks)—had to be explained in reference to what the atoms themselves were like. There was no other way out—not yet, not given the sound epistemology in physics of those days.

Thinking up a new universe still was not allowed back then in science let alone in physics. One still had to clearly think about explaining what was given in observations, what was in evidence. Effects still had be related back to causes; outward actions still had to be related back to the character/nature of the entities that thus acted.

The actor, unquestionably by now, was the atom. The effects were the discrete spectra. Not much else was known.

Those were the days were when the best hotels and restaurants in Berlin, London, and New York would have horse-driven buggies ushering in the socially important guests. Buggies still was the latest technology back then. Not many people thus ushered in are remembered today. But Bohr is.

If the atom was the actor, and the effects under study were the discrete spectra, then what was needed to be said, in theory, was something regarding the structure of the atom.

If an imagined entity sheer by its material/chemical type doesn’t do it, then it’s the structure—its shape and size—which must do it.

Back then, this still was regarded as one of the cardinal principles of science, unlike the mindless opposition to the science of Homeopathy today, esp. in the UK. But back then, it was known that one important reason that Calvin gets harassed by the school bully was that not just the sheer size of the latter’s matter but also that the structure of the latter was different. In other words: If you consumed alcohol, you simply didn’t take in so many atoms of carbon as in proportion to so many atoms of hydrogen, etc. You took in a structure, a configuration with which these atoms came in.

However, the trouble back then was, none had have the means to see the atoms.

If by structure you mean the geometrical shape and size, or some patterns of density, then clearly, there was no experimental observations pertaining to the same. The only relevant observation available to people back then was what had already been encapsulated in Rutherford’s model, viz., the incontestable idea that the atomic nucleus had to be massive and dense, occupying a very small space as compared to an atom taken as a whole; the electrons had to carry very little mass in comparison. (The contrast of Rutherford’s model of c. 1911 was to the earlier plum cake model by Thomson.)

Bohr would, therefore, have to start with Rutherford’s model of atoms, and invent some new ideas concerning it, and see if his model was consistent with the known results given by spectroscopic observations.

What Bohr offered was a model for the electrons contained in a nuclear atom.

However, even while differing from the Rutherford’s plum-cake model, Bohr’s model emphatically lacked a theory for the nature of the electrons themselves. This part has been kept underappreciated by the textbook authors and science teachers.

In particular, Bohr’s theory had absolutely no clue as to the process according to which the electrons could, and must, jump in between their stable orbits.

The meat of the matter was worse, far worse: Bohr had explicitly prohibited from pursuing any mechanism or explanation concerning the quantum jumps—an idea which he was the first to propose. [I don’t know of any one else originally but independently proposing the same idea.]

Bohr achieved this objective not through any deployment of the best possible levels of scientific reason but out of his philosophic convictions—the convictions of the more irrational kind. The quantum jumps were obviously not observable, according to him, only their effects were. So, strictly speaking, the quantum jumps couldn’t possibly be a part of his theory—plain and simple!

But then, Bohr in his philosophic enthusiasm didn’t stop just there. He went even further—much further. He fully deployed the powers of his explicit reasoning as well as the weight of his seniority in prohibiting the young physicists from even thinking of—let alone ideating or offering—any mechanism for such quantum jumps.

In other words, Bohr took special efforts to keep the young quantum enthusiasts absolutely and in principle clueless, as far as his quantum jumps were concerned.

Bohr’s theory, in a sense, was in line with the strictest demands of the philosophy of empiricism. Here is how Bohr’s application of this philosophy went:

1. This electron—it can be measured!—at this energy level, now!
2. [May be] The same electron, but this energy level, now!
3. This energy difference, this frequency. Measured! [Thank you experimental spectroscopists; hats off to you, for, you leave Bohr alone!!]
4. OK. Now, put the above three into a cohesive “theory.” And, BTW, don’t you ever even try to think about anything else!!

Continuing just a bit on the same lines, Bohr sure would have said (quoting Peikoff’s explanation of the philosophy of empiricism):

1. [Looking at a tomato] We can only say this much in theory: “This, now, tomato!”
2. Making a leeway for the most ambitious ones of the ilk: “This *red* tomato!!”

Going by his explicit philosophic convictions, it must have been a height of “speculation” for Bohr to mumble something—anything—about a thing like “orbit.” After all, even by just mentioning a word like “orbit,” Bohr was being absolutely philosophically inconsistent here. Dear reader, observe that the orbit itself never at all was an observable!

Bohr must have in his conscience convulsed at this fact; his own philosophy couldn’t possibly have, strictly speaking, permitted him to accommodate into his theory a non-measurable feature of a non-measurable entity—such as his orbits of his electrons. Only the allure of outwardly producing predictions that matched with the experiment might have quietened his conscience—and that too, temporarily. At least until he got a new stone-building housing an Institute for himself and/or a Physics Nobel, that is.

Possible. With Herr Herr Herr Doktor Doktor Doktor Professor Professors, anything is possible.

It is often remarked that the one curious feature of the Bohr theory was the fact that the stability of the electronic orbits was postulated in it, not explained.

That is, not explained in reference to any known physical principle. The analogy to the solar system indeed was just that: an analogy. It was not a reference to an established physical principle.

However, the basically marvelous feature of the Bohr theory was not that the orbits were stable (in violation of the known laws of electrodynamics). It was: there at all were any orbits in it, even if no experiment had ever given any evidence for the continuously or discontinuously subsequent positions electrons within an atom or of their motions.

So much for originator of the cult of sticking only to the “observables.”

What Sommerfeld did was to add footnotes to Bohr’s work.

Sommerfeld did this work admirably well.

However, what this instance in the history of physics clearly demonstrates is yet another principle from the epistemology of physics: how a man of otherwise enormous mathematical abilities and training (and an academically influential position, I might add), but having evidently no remarkable capacity for a very novel, breakthrough kind of conceptual thinking, just cannot but fall short of making any lasting contributions to physics.

“Math” by itself simply isn’t enough for physics.

What came to be known as the old quantum theory, thus, faced an impasse.

Under Bohr’s (and philosophers’) loving tutorship, the situation continued for a long time—for more than a decade!

A Song I Like:

(Marathi) “sakhi ga murali mohan mohi manaa…”
Music: Hridaynath Mangeshkar
Singer: Asha Bhosale
Lyrics: P. Savalaram

PS: Only typos and animals of the similar ilk remain to be corrected.

# 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 $1$st avataar, nor in the $2$nd, 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…

# Causality. And a bit miscellaneous.

0. I’ve been too busy in my day-job to write anything at any one of my blogs, but recently, a couple of things happened.

1. I wrote what I think is a “to read” (if not a “must read”) comment, concerning the important issue of causality, at Roger Schlafly’s blog; see here [^]. Here’s the copy-paste of the same:

1. There is a very widespread view among laymen, and unfortunately among philosophers too, that causality requires a passage of time. As just one example: In the domino effect, the fall of one domino leads to the fall of another domino only after an elapse of time.

In fact, all their examples wherever causality is operative, are of the following kind:

“If something happens then something else happens (necessarily).”

Now, they interpret the word then’ to involve a passage of time. (Then, they also go on to worry about physics equations, time symmetry, etc., but in my view all these are too advanced considerations; they are not fundamental or even very germane at the deepest philosophical level.)

2. However, it is possible to show other examples involving causality, too. These are of the following kind:

“When something happens, something else (necessarily) happens.”

Here is an example of this latter kind, one from classical mechanics. When a bat strikes a ball, two things happen at the same time: the ball deforms (undergoes a change of shape and size) and it “experiences” (i.e. undergoes) an impulse. The deformation of the ball and the impulse it experiences are causally related.

Sure, the causality here is blatantly operative in a symmetric way: you can think of the deformation as causing the impulse, or of the impulse as causing the deformation. Yet, just because the causality is symmetric here does not mean that there is no causality in such cases. And, here, the causality operates entirely without the dimension of time in any way entering into the basic analysis.

Here is another example, now from QM: When a quantum particle is measured at a point of space, its wavefunction collapses. Here, you can say that the measurement operation causes the wavefunction collapse, and you can also say that the wavefunction collapse causes (a definite) measurement. Treatments on QM are full of causal statements of both kinds.

3. There is another view, concerning causality, which is very common among laymen and philosophers, viz. that causality necessarily requires at least two separate objects. It is an erroneous view, and I have dealt with it recently in a miniseries of posts on my blog; see https://ajitjadhav.wordpress.com/2017/05/12/relating-the-one-with-the-many/.

4. Notice, the statement “when(ever) something happens, something else (always and/or necessarily) happens” is a very broad statement. It requires no special knowledge of physics. Statements of this kind fall in the province of philosophy.

If a layman is unable to think of a statement like this by way of an example of causality, it’s OK. But when professional philosophers share this ignorance too, it’s a shame.

5. Just in passing, noteworthy is Ayn Rand’s view of causality: http://aynrandlexicon.com/lexicon/causality.html. This view was basic to my development of the points in the miniseries of posts mentioned above. … May be I should convert the miniseries into a paper and send it to a foundations/philosophy journal. … What do you think? (My question is serious.)

Thanks for highlighting the issue though; it’s very deeply interesting.

Best,

–Ajit

3. The other thing is that the other day (the late evening of the day before yesterday, to be precise), while entering a shop, I tripped over its ill-conceived steps, and suffered a fall. Got a hairline crack in one of my toes, and also a somewhat injured knee. So, had to take off from “everything” not only on Sunday but also today. Spent today mostly sleeping relaxing, trying to recover from those couple of injuries.

This late evening, I naturally found myself recalling this song—and that’s where this post ends.

4. OK. I must add a bit. I’ve been lagging on the paper-writing front, but, don’t worry; I’ve already begun re-writing (in my pocket notebook, as usual, while awaiting my turn in the hospital’s waiting lounge) my forth-coming paper on stress and strain, right today.

OK, see you folks, bye for now, and take care of yourselves…

A Song I Like:

(Hindi) “zameen se hamen aasmaan par…”
Singer: Asha Bhosale and Mohammad Rafi
Lyrics: Rajinder Krishan

# Relating the One with the Many

0. Review and Context: This post is the last one in this mini-series on the subject of the one vs. many (as understood in the context of physics). The earlier posts in this series have been, in the chronological and logical order, these:

1. Introducing a very foundational issue of physics (and of maths) [^]
2. The One vs. the Many [^]
3. Some of the implications of the “Many Objects” idea… [^]
4. Some of the implications of the “One Object” idea… [^]

In the second post in this series, we had seen how a single object can be split up into many objects (or the many objects seen as parts of a single object). Now, in this post, we note some more observations about relating the One with the Many.

The description below begins with a discussion of how the One Object may be separated into Many Objects. However, note that the maths involved here is perfectly symmetrical, and therefore, the ensuing discussion for the separation of the one object into many objects also just as well applies for putting many objects together into one object, i.e., integration.

In the second and third posts, we handled the perceived multiplicity of objects via a spatial separation according to the varying measures of the same property. A few remarks on the process of separation (or, symmetrically, on the process of integration) are now in order.

1. The extents of spatial separation depends on what property you choose on the basis of which to effect the separation:

To begin with, note that the exact extents of any spatial separations would vary depending on what property you choose for measuring them.

To take a very “layman-like” example, suppose you take a cotton-seed, i.e. the one with a soft ball of fine cotton fibres emanating from a hard center, as shown here [^]. Suppose if you use the property of reflectivity (or, the ability to be seen in a bright light against a darker background), then for the cotton-seed, the width of the overall seed might come out to be, say, 5 cm. That is to say, the spatial extent ascribable to this object would be 5 cm. However, if you choose some other physical property, then the same object may end up registering quite a different size. For instance, if you use the property: “ability to be lifted using prongs” as the true measure for the width for the seed, then its size may very well come out as just about 1–2 cm, because the soft ball of the fibres would have got crushed to a smaller volume in the act of lifting.

In short: Different properties can easily imply different extensions for the same distinguished (or separated)“object,” i.e., for the same distinguished part of the physical universe.

2. The One Object may be separated into Many Objects on a basis other than that of the spatial separation:

Spatial attributes are fundamental, but they don’t always provide the best principle to organize a theory of physics.

The separation of the single universe-object into many of its parts need not proceed on the basis of only the “physical” space.

It would be possible to separate the universe on the basis of certain basis-functions which are defined over every spatial part of the universe. For instance, the Fourier analysis gives rise to a separation of a property-function into many complex-valued frequencies (viz. pairs of spatial undulations).

If the separation is done on the basis of such abstract functions, and not on the basis of the spatial extents, then the problem of the empty regions vaporizes away immediately. There always is some or the other “frequency”, with some or the other amplitude and phase, present at literally every point in the physical universe—including in the regions of the so-called “empty” space.

However, do note that the Fourier separation is a mathematical principle. Its correspondence to the physical universe must pass through the usual, required, epistemological hoops. … Here is one instance:

Question: If infinity cannot metaphysically exist (simply because it is a mathematical concept and no mathematical concept physically exists), then how is it that an infinite series may potentially be required for splitting up the given function (viz. the one which specifies the variations the given property of the physical universe)?

Answer: An infinite Fourier series cannot indeed be used by way of a direct physical description; however, a truncated (finite) Fourier series may be.

Here, we are basically relying on the same trick as we saw earlier in this mini-series of posts: We can claim that what the truncated Fourier series represents is the actual reality, and that that function which requires an infinite series is merely a depiction, an idealization, an abstraction.

3. When to use which description—the One Object or the Many Objects:

Despite the enormous advantages of the second approach (of the One Object idea) in the fundamental theoretical physics, in classical physics as well as in our “day-to-day” life, we often speak of the physical reality using the cruder first approach (the one involving the Many Objects idea). This we do—and it’s perfectly OK to do so—mainly because of the involved context.

The Many Objects description of physics is closer to the perceptual level. Hence, its more direct, even simpler, in a way. Now, note a very important consideration:

The precision to used in a description (or a theory) is determined by its purpose.

The purpose for a description may be lofty, such as achieving fullest possible consistency of conceptual interrelations. Or it may be mundane, referring to what needs to be understood in order to get the practical things done in the day-to-day life. The range of integrations to be performed for the day-to-day usage is limited, very limited in fact. A cruder description could do for this purpose. The Many Objects idea is conceptually more economical to use here. [As a polemical remark on the side, observe that while Ayn Rand highlighted the value of purpose, neither Occam nor the later philosophers/physicists following him ever even thought of that idea: purpose.]

However, as the scope of the physical knowledge increases, the requirements of the long-range consistency mandate that it is the second approach (the one involving the One Object idea) which we must adopt as being a better representative of the actual reality, as being more fundamental.

Where does the switch-over occur?

I think that it occurs at a level of those physics problems in which the energetics program (initiated by Leibnitz), i.e., the Lagrangian approach, makes it easier to solve them, compared to the earlier, Newtonian approach. This answer basically says that any time you use the ideas such as fields, and energy, you must make the switch-over, because in the very act of using such ideas, implicitly, you are using the One Object idea anyway. Which means, EM theory, and yes, also thermodynamics.

And of course, by the time you begin tackling QM, the second approach becomes simply indispensable.

A personal side remark: I should have known better. I should have adopted the second approach earlier in my life. It would have spared me a lot of agonizing over the riddles of quantum physics, a lot of running in loops over the same territory (like a dog chasing his own tail). … But it’s OK. I am glad that at least by now, I know better. (And, engineers anyway don’t get taught the Lagrangian mechanics to the extent physicists do.)

A few days ago, Roger Schlafly had written a nice and brief post at his blog saying that there is a place for non-locality in physics. He had touched on that issue more from a common-sense and “practical” viewpoint of covering these two physics approaches [^].

Now, given the above write-up, you know that a stronger statement, in fact, can be made:

As soon as you enter the realm of the EM fields and the further development, the non-local (or the global or the One Object) theories are the only way to go.

A Song I Like:

[When I was a school-boy, I used to very much like this song. I would hum [no, can’t call it singing] with my friends. I don’t know why. OK. At least, don’t ask me why. Not any more, anyway 😉 .]

(Hindi) “thokar main hai meri saaraa zamaanaa”
Singer: Kishore Kumar
Music: R. D. Burman
Lyrics: Rajinder Krishan

OK. I am glad I have brought to a completion a series of posts that I initiated. Happened for the first time!

I have not been able to find time to actually write anything on my promised position paper on QM. … Have been thinking about how to present certain ideas better, but not making much progress… If you must ask: these involve entangled vs. product states—and why both must be possible, etc.

So, I don’t think I am going to be able to hold the mid-2017 deadline that I myself had set for me. It will take longer.

For the same reasons, may be I will be blogging less… Or, who knows, may be I will write very short general notings here and there…

Bye for now and take care…

# Some of the implications of the “Many Objects” idea…

0. Context and Review:

This post continues from the last one. In the last post, we saw that the same perceptual evidence (involving two moving grey regions) can be conceptually captured using two entirely different, fundamental, physics ideas.

In the first description, the perceived grey regions are treated as physical objects in their own right.

In the second description, the perceived grey regions are treated not as physical objects in their own right, but merely as distinguishable (and therefore different) parts of the singleton object that is the universe (the latter being taken in its entirety).

We will now try to look at some of the implications that the two descriptions naturally lead to.

1. The “Many Objects” Viewpoint Always Implies an In-Principle Empty Background Object:

To repeat, in the first description, the grey regions are treated as objects in their own right. This is the “Many Objects” viewpoint. The universe is fundamentally presumed to contain many objects.

But what if there is one and only one grey block in the perceptual field? Wouldn’t such a universe then contain only that one grey object?

Not quite.

The fact of the matter is, even in this case, there implicitly are two objects in the universe: (i) the grey object and (ii) the background or the white object.

As an aside: Do see here Ayn Rand’s example (in ITOE, 2nd Edition) of how a uniform blue expanse of the sky by itself would not even be perceived as an object, but how, once you introduce a single speck of dust, the perceptual contrast that it introduces would allow perceptions of both the speck and the blue sky to proceed. Of course, this point is of only technical importance. Looking at the real world while following the first description, there are zillions of objects evidently present anyway.

Leaving aside the theoretically extreme case of a single grey region, and thus focusing on the main general ideas: the main trouble following this “Many Objects” description is twofold:

(i) It is hard to come to realize that something exists even in the regions that are “empty space.”

(ii) Methodologically, it is not clear as to precisely how one proceeds from the zillions of concrete objects to the singleton object that is the universe.  Observe that the concrete objects here are physical objects. Hence, one would look for a conceptual common denominator (CCD) that is narrower than just the fact that all these concrete objects do exist. One would look for something more physical by way of the CCD, but it is not clear what it could possibly be.

2. Implications of the “Many Objects” Viewpoint for Causality:

In the first description, there are two blocks and they collide. Let’s try to trace the consequences of such a description for causality:

With the supposition that there are two blocks, one is drawn into a temptation of thinking along the following lines:

the first block pushes on the second block—and the second block pushes on the first.

Following this line of thought, the first block can be taken as being responsible for altering the motion of the second block (and the second, of the first). Therefore, a certain conclusion seems inevitable:

the motion of the first block may be regarded as the cause, and the (change in) the motion of the second block may be regarded as the effect.

In other words, in this line of thought, one entity/object (the first block) supplies, produces or enacts the cause, and another entity/object (the second block) suffers the consequences, the effects. following the considerations of symmetry and thereby abstracting a more general truth (e.g. as captured in Newton’s third law), you could also argue that that it is the second object that is the real cause, and the first object shows only effects. Then, abstracting the truth following the consideration of symmetry, you could say that

the motion (or, broadly, the nature) of each of the two blocks is a cause, and the action it produces on the other block is an effect.

But regardless of the symmetry considerations or the abstractness of the argument that it leads to, note that this entire train of thought still manages to retain a certain basic idea as it is, viz.:

the entity/actions that is the cause is necessarily different from the entity/actions that is the effect.

Such an idea, of ascribing the cause and the effect parts of a single causal event (here, the collision event) to two different object not only can arise in the many objects description, it is the most common and natural way in which the very idea of causality has come to be understood. Examples abound: the swinging bat is a cause; the ball flying away is the effect; the entities to which we ascribe the cause and the effect are entirely different objects. The same paradigm runs throughout much of physics. Also in the humanities. Consider this: “he makes me feel good.”

Every time such a separation of cause and effect occurs, logically speaking, it must first be supposed that many objects exist in the universe.

It is only on the basis of a many objects viewpoint that the objects that act as causes can be metaphysically separated, at least in an event-by-event concrete description, from the objects that suffer the corresponding effects.

3. Implications of the “Many Objects” Viewpoint, and the Idea of the “Empty” Space:

Notice that in the “many objects” description, no causal role is at all played by those parts of the universe that are “empty space.” Consider the description again:

The grey blocks move, come closer together, collide, and fly away in the opposite directions after the collision.

Notice how this entire description is anchored only to the grey blocks. Whatever action happens in this universe, it is taken by the grey blocks. The empty white space gets no metaphysical role whatsoever to play.

It is as if any metaphysical locus standi that the empty space region should otherwise have, somehow got completely sucked out of itself, and this locus standi then got transferred, in a way overly concentrated, into the grey regions.

Once this distortion is allowed to be introduced into the overall theoretical scheme, then, logically speaking, it would be simple to propagate the error throughout the theory and its implication. Just apply an epistemologically minor principle like Occam’s Razor, and the metaphysical suggestion that this entire exercise leads to is tantamount to this idea:

why not simply drop the empty space out of any physical consideration? out of all physics theory?

A Side Remark on Occam’s Principle: The first thing to say about Occam’s Principle is that it is not a very fundamental principle. The second thing to say is that it is impossible to state it using any rigorous terms. People have tried doing that for centuries, and yet, not a single soul of them feels very proud when it comes to showing results for his efforts. Just because today’s leading theoretical physics love it, vouch by it, and vigorously promote it, it still does not make Occam’s principle play a greater epistemological role than it actually does. Qua an epistemological principle, it is a very minor principle. The best contribution that it can at all aspire to is: serving as a vague, merely suggestive, guideline. Going by its actual usage in classical physics, it did not even make for a frequently used epistemological norm let alone acted as a principle that would necessarily have to be invoked for achieving logical consistency. And, as a mere guideline, it is also very easily susceptible to misuse. Compare this principle to, e.g., the requirement that the process of concept formation must always show both the essentials: differentiation and integration. Or compare it to the idea that concept-formation involves measurement-omission. Physicists promote Occam’s Principle to the high pedestal, simply because they want to slip in their own bad ideas into physics theory. No, Occam’s Razor does not directly help them. What it actually lets them do, through misapplication, is to push a wedge to dislodge some valid theoretical block from the well-integrated wall that is physics. For instance, if the empty space has no role to play in the physical description of the universe [preparation of misapplication], then, by Occam’s Principle [the wedge], why not take the idea of aether [a valid block] out of  physics theory? [which helps make physics crumble into pieces].

It is in this way that the first description—viz. the “many objects” description—indirectly but inevitably leads to a denial of any physical meaning to the idea of space.

If a fundamental physical concept like space itself is denied any physical roots, then what possibly could one still say about this concept—about its fundamental character or nature? The only plausible answers would be the following:

That space must be (a) a mathematical concept (based on the idea that fundamental ideas had better be physical, mathematical or both), and (b) an arbitrary concept (based on the idea that if there is no hard basis of the physical reality underlying this concept, then it can always be made as soft as desired, i.e. infinitely soft, i.e., arbitrary).

If the second idea (viz., the idea that space is an arbitrary human invention) is accorded the legitimacy of a rigorously established truth, then, in logic, anyone would be free to bend space any which way he liked. So, there would have to be, in logic, a proliferation in spaces. The history of the 19th and 20th centuries is nothing but a practically evident proof of precisely this logic.

Notice further that in following this approach (of the “many objects”), metaphysically speaking, the first casualty is that golden principle taught by Aristotle, viz. the idea that a literal void cannot exist, that the nothing cannot be a part of the existence. (It is known that Aristotle did teach this principle. However, it is not known if he had predecessors, esp. in the more synthetic, Indic, traditions. In any case, the principle itself is truly golden—it saves one from so many epistemological errors.)

Physics is an extraordinarily well-integrated a science. However, this does not mean that it is (or ever has been) perfectly integrated. There are (and always have been) inconsistencies in it.

The way physics got formulated—the classical physics in particular—there always was a streak of thought in it which had always carried the supposition that there existed a literal void in the region of the “gap” between objects. Thus, as far as the working physicist was concerned, a literal void could not exist, it actually did. “Just look at the emptiness of that valley out there,” (said while standing at a mountain top). Or, “look at the bleakness, at the dark emptiness out there between those two shining bright stars!” That was their “evidence.” For many physicists—and philosophers—such could be enough of an evidence to accept the premise of a physically existing emptiness, the literal naught of the philosophers.

Of course, people didn’t always think in such terms—in terms of a literal naught existing as a part of existence.

Until the end of the 19th century, at least some people also thought in terms of “aether.”

The aether was supposed to be a massless object. It was supposed that “aether” existed everywhere, including in the regions of space where there were no massive objects. Thus, the presence of aether ensured that there was no void left anywhere in the universe.

However, as soon as you think of an idea like “aether,” two questions immediately arise: (i) how can aether exist even in those places where a massive object is already present? and (ii) as to the places where there is no massive object, if all that aether does is to sit pretty and do nothing, then how is it really different from those imaginary angels pushing on the planets in the solar system?

Hard questions, these two. None could have satisfactorily answered these two questions. … In fact, as far as I know, none in the history of physics has ever even raised the first question! And therefore, the issue of whether, in the history of thought, there has been any satisfactory answer provided to it or not, cannot even arise in the first place.

It is the absence of satisfactory answers to these two questions that has really allowed Occam’s Razor to be misapplied.

By the time Einstein arrived, the scene was already ripe to throw the baby out with the water, and thus he could happily declare that thinking in terms of the aether concept was entirely uncalled for, that it was best to toss it into in the junkyard of bad ideas discarded in the march of human progress.

The “empty” space, in effect, progressively got “emptier” and “emptier” still. First, it got replaced by the classical electromagnetic “field,” and then, as space got progressively more mathematical and arbitrary, the fields themselves got replaced by just an abstract mathematical function—whether the spacetime of the relativity theory or the $\Psi$ function of QM.

4. Implications of the “Many Objects” Viewpoint and the Supposed Mysteriousness of the Quantum Entanglement:

In the “many objects” viewpoint, the actual causal objects are many. Further, this viewpoint very naturally suggests the idea of some one object being a cause and some other object being the effect. There is one very serious implication of this separation of cause and effect into many, metaphysically separate, objects.

With that supposition, now, if two distant objects (and metaphysically separate objects always are distant) happen to show some synchronized sort of a behavior, then, a question arises: how do we connect the cause with the effect, if the effect is observed not to lag in time from the cause.

Historically, there had been some discussion on the question of “[instantaneous] action at a distance,” or IAD for short. However, it was subdued. It was only in the context of QM riddles that IAD acquired the status of a deeply troubling/unsettling issue.

5. Miscellaneous:

5.1

Let me take a bit of a digression into philosophy proper here, by introducing Ayn Rand’s ideas of causality at this point [^]. In OPAR, Dr. Peikoff has clarified the issue amply well: The identity or nature of an entity is the cause, and its actions is the effect.

Following Ayn Rand, if two grey blocks (as given in our example perceptual field) reverse their directions of motions after collision, each of the two blocks is a cause, and the reversals in the directions of the same block is the effect.

Make sure to understand the difference in what is meant by causality. In the common-sense thinking, as spelt out in section 2. of this post, if the block A’ is the cause, then the block B’ is the effect (and vice versa). However, according to Ayn Rand, if the block A’ is the cause, then the actions of this same block A’ are the effect. It is an important difference, and make sure you know it.

Thus, notice, for the time being, that in Ayn Rand’s sense of the terms, the principle of causality actually does not need a multiplicity of objects.

However, notice that the causal role of the “empty” space continues to remain curiously unanswered even after you bring Ayn Rand’s above-mentioned insights to bear on the issue.

5.2:

The only causal role that can at all be ascribed to the “empty” space, it would seem, is for it to continuously go on “monitoring” if a truly causal body—a massive object—was impinging on itself or not, and if such a body actually did that, to allow it to do so.

In other words, the causal identity of the empty space becomes entirely other-located: it summarily depends on the identity of the massive objects. But the identity of a given object pertains to what that object itself is—not to what other objects are like. Clearly, something is wrong here.

In the next post, we shall try to trace the implications that the second description (i.e. The One Object) leads to.

A Song I Like:

(Hindi) “man mera tujh ko maange, door door too bhaage…”
Singer: Suman Kalyanpur
Music: Kalyanji Anandji
Lyrics: Indivar

[PS: May be an editing pass is due…. Let me see if I can find the time to come back and do it…. Considerable revision done on 28 April 2017 12:20 PM IST though no new ideas were added; I will leave the remaining grammatical errors/awkward construction as they are. The next post should get posted within a few days’ time.]

# On whether A is not non-A

This post has its origin in a neat comment I received on my last post [^]; see the exchange starting here: [^].

The question is whether I accept that A is not non-A.

My answer is: No, I do not accept that, logically speaking, A is not non-A—not unless the context to accept this statement is understood clearly and unambiguously (and the best way to do that is to spell it out explicitly).

Another way to say the same thing is that I can accept that “A is not non-A,” but only after applying proper qualifications; I won’t accept it in an unqualified way.

Let me explain by considering various cases arising, using a simple example.

The Venn diagram:

Let’s begin by drawing a Venn diagram.

Draw a rectangle and call it the set $R$. Draw a circle completely contained in it, and call it the set $A$. You can’t put a round peg to fill a rectangular hole, so, the remaining area of the rectangle is not zero. Call the remaining area $B$. See the diagram below.

Case 1: All sets are non-empty:

Assume that neither $A$ nor $B$ is empty. Using symbolic terms, we can say that:
$A \neq \emptyset$,
$B \neq \emptyset$, and
$R \equiv A \cup B$
where the symbol $\emptyset$ denotes an empty set, and $\equiv$ means “is defined as.”

We take $R$ as the universal set—of this context. For example, $R$ may represent, say the set of all the computers you own, with $A$ denoting your laptops and $B$ denoting your desktops.

I take the term “proper set” to mean a set that has at least one element or member in it, i.e., a set which is not empty.

Now, focus on $A$. Since the set $A$ is a proper set, then it is meaningful to apply the negation- or complement-operator to it. [May be, I have given away my complete answer right here…] Denote the resulting set, the non-A, as $A^{\complement }$. Then, in symbolic terms:
$A^{\complement } \equiv R \setminus A$.
where the symbol $\setminus$ denotes taking the complement of the second operand, in the context of the first operand (i.e., “subtracting” $A$ from $R$). In our example,
$A^{\complement } = B$,
and so:
$A^{\complement } \neq \emptyset$.
Thus, here, $A^{\complement }$ also is a proper (i.e. non-empty) set.

To conclude this part, the words “non-A”, when translated into symbolic terms, means $A^{\complement }$, and this set here is exactly the same as $B$.

To find the meaning of the phrase “not non-A,” I presume that it means applying the negation i.e. the complement operator to the set $A^{\complement }$.

It is possible to apply the complement operator because $A ^{\complement } \neq \emptyset$. Let us define the result of this operation as $A^{\complement \complement}$; note the two $^{\complement}$s appearing in its name. The operation, in symbols becomes:
$A^{\complement \complement} \equiv R \setminus A^{\complement} = R \setminus B = A$.
Note that we could apply the complement operator to $A$ and later on to $A^{\complement}$ only because each was non-empty.

As the simple algebra of the above simple-minded example shows,
$A = A^{\complement\complement}$,
which means, we have to accept, in this example, that A is not non-A.

Remarks on the Case 1:

However, note that we can accept the proposition only under the given assumptions.

In  particular, in arriving at it, we have applied the complement-operator twice. (i) First, we applied it to the “innermost” operand i.e. $A$, which gave us $A^{\complement}$. (ii) Then, we took this result, and applied the complement-operator to it once again, yielding $A^{\complement\complement}$. Thus, the operand for the second complement-operator was $A^{\complement}$.

Now, here is the rule:

Rule 1: We cannot meaningfully apply the complement-operator unless the operand set is proper (i.e. non-empty).

People probably make mistakes in deciding whether A is not non-A, because, probably, they informally (and properly) do take the “innermost” operand, viz. $A$, to be non-empty. But then, further down the line, they do not check whether the second operand, viz. $A^{\complement}$ turns out to be empty or not.

Case 2: When the set $A^{\complement}$ is empty:

The set $A^{\complement}$ will be empty if $B = \emptyset$, which will happen if and only if $A = R$. Recall, $R$ is defined to be the union of $A$ and $B$.

So, every time there are two mutually exclusive and collectively exhaustive sets, if any one of them is made empty, you cannot doubly apply the negation or the complement operator to the other (nonempty) set.

Such a situation always occurs whenever the remaining set coincides with the universal set of a given context.

In attempting a double negation, if your first (or innermost) operand itself is a universal set, then you cannot apply the negation operator for the second time, because by Rule 1, the result of the first operator comes out as an empty set.

The nature of an empty set:

But why this rule that you can’t negate (or take the complement of) an empty set?

An empty set contains no element (or member). Since it is the elements which together impart identity to a set, an empty set has no identity of its own.

As an aside, some people think that all the usages of the phrase “empty set” refers to the one and the only set (in the entire universe, for all possible logical propositions involving sets). For instance, the empty set obtained by taking an intersection of dogs and cats, they say, is exactly the same empty set as the one obtained by taking an intersection of cars and bikes.

I reject this position. It seems to me to be Platonic in nature, and there is no reason to give Plato even an inch of the wedge-space in this Aristotlean universe of logic and reality.

As a clarification, notice, we are talking of the basic and universal logic here, not the implementation details of a programming language. A programming language may choose to point all the occurrences of the NULL string to the same memory location. This is merely an implementation choice to save on the limited computer memory. But it still makes no sense to say that all empty C-strings exist at the same memory location—but that’s what you end up having if you call an empty set the empty set. Which brings us to the next issue.

If an empty set has no identity of its own, if it has no elements, and hence no referents, then how come it can at all be defined? After all, a definition requires identity.

The answer is: Structurally speaking, an empty set acquires its meaning—its identity—“externally;” it has no “internally” generated identity.

The only identity applicable to an empty set is an abstract one which gets imparted to it externally; the purpose of this identity is to bring a logical closure (or logical completeness) to the primitive operations defined on sets.

For instance, intersection is an operator. To formally bring closure to the intersection operation, we have to acknowledge that it may operate over any combination of any operand sets, regardless of their natures. This range includes having to define the intersection operator for two sets that have no element in common. We abstractly define the result of such a case as an empty set. In this case, the meaning of the empty set refers not to a result set of a specific internal identity, but only to the operation and the disjoint nature the operands which together generated it, i.e., via a logical relation whose meaning is external to the contents of the empty set.

Inasmuch as an empty set necessarily includes a reference to an operation, it is a concept of method. Inasmuch as many combinations of various operations and operands can together give rise to numerous particular instances of an empty set, there cannot be a unique instance of it which is applicable in all contexts. In other words, an empty set is not a singleton; it is wrong to call it the empty set.

Since an empty set has no identity of its own, the notion cannot be applied in an existence-related (or ontic or metaphysical) sense. The only sense it has is in the methodological (or epistemic) sense.

Extending the meaning of operations on an empty set:

In a derivative sense, we may redefine (i.e. extend) our terms.

First, we observe that since an empty set lacks an identity of its own, the result of any operator applied to it cannot have any (internal) identity of its own. Then, equating these two lacks of existence-related identities (which is where the extension of the meaning occurs), we may say, even if only in a derivative or secondary sense, that

Rule 2: The result of an operator applied to an empty set again is another empty set.

Thus, if we now allow the complement-operator to operate also on an empty set (which, earlier, we did not allow), then the result would have to be another empty set.

Again, the meaning of this second empty set depends on the entirety of its generating context.

Case 3: When the non-empty set is the universal set:

For our particular example, assuming $B = \emptyset$ and hence $A = R$, if we allow complement operator to be applied (in the extended sense) to $A^{\complement}$, then

$A^{\complement\complement} \equiv R \setminus A^{\complement} = R \setminus (R \setminus A) = R \setminus B = R \setminus (\emptyset) = R = A$.

Carefully note, in the above sequence, the place where the extended theory kicks in is at the expression: $R \setminus (\emptyset)$.

We can apply the $\setminus$ operator here only in an extended sense, not primary.

We could here perform this operation only because the left hand-side operand for the complement operator, viz., the set $R$ here was a universal set. Any time you have a universal set on the left hand-side of a complement operator, there is no more any scope left for ambiguity. This state is irrespective of whether the operand on the right hand-side is a proper set or an empty set.

So, in this extended sense, feel free to say that A is not non-A, provided A is the universal set for a given context.

To recap:

The idea of an empty set acquires meaning only externally, i.e., only in reference to some other non-empty set(s). An empty set is thus only an abstract place-holder for the result of an operation applied to proper set(s), the operation being such that it yields no elements. It is a place-holder because it refers to the result of an operation; it is abstract, because this result has no element, hence no internally generated identity, hence no concrete meaning except in an abstract relation to that specific operation (including those specific operands). There is no “the” empty set; each empty set, despite being abstract, refers to a combination of an instance of proper set(s) and an instance of an operation giving rise to it.

Exercises:

E1: Draw a rectangle and put three non-overlapping circles completely contained in it. The circles respectively represent the three sets $A$, $B$, $C$, and the remaining portion of the rectangle represents the fourth set $D$. Assuming this Venn diagram, determine the meaning of the following expressions:

(i) $R \setminus (B \cup C)$ (ii) $R \setminus (B \cap C)$ (iii) $R \setminus (A \cup B \cup C)$ (iv) $R \setminus (A \cap B \cap C)$.

(v)–(viii) Repeat (i)–(iv) by substituting $D$ in place of $R$.

(ix)–(xvi) Repeat (i)–(viii) if $A$ and $B$ partly overlap.

E2: Identify the nature of set theoretical relations implied by that simple rule of algebra which states that two negatives make a positive.

A bit philosophical, and a form better than “A is not non-A”:

When Aristotle said that “A is A,” and when Ayn Rand taught its proper meaning: “Existence is identity,” they referred to the concepts of “existence” and “identity.” Thus, they referred to the universals. Here, the word “universals” is to be taken in the sense of a conceptual abstraction.

If concepts—any concepts, not necessarily only the philosophical axioms—are to be represented in terms of the set theory, how can we proceed doing that?

(BTW, I reject the position that the set theory, even the so-called axiomatic set theory, is more fundamental than the philosophic abstractions.)

Before we address this issue of representation, understand that there are two ways in which we can specify a set: (i) by enumeration, i.e. by listing out all its (relatively concrete) members, and (ii) by rule, i.e. by specifying a definition (which may denote an infinity of concretes of a certain kind, within a certain range of measurements).

The virtue of the set theory is that it can be applied equally well to both finite sets and infinite sets.

The finite sets can always be completely specified via enumeration, at least in principle. On the other hand, infinite sets can never be completely specified via enumeration. (An infinite set is one that has an infinity of members or elements.)

A concept (any concept, whether of maths, or art, or engineering, or philosophy…) by definition stands for an infinity of concretes. Now, in the set theory, an infinity of concretes can be specified only using a rule.

Therefore, the only set-theoretic means capable of representing concepts in that theory is to specify their meaning via “rule” i.e. definition of the concept.

Now, consider for a moment a philosophical axiom such as the concept of “existence.” Since the only possible set-theoretic representation of a concept is as an infinite set, and since philosophical axiomatic concepts have no antecedents, no priors, the set-theoretic representation of the axiom of “existence” would necessarily be as a universal set.

We saw that the complement of a universal set is an empty set. This is a set-theoretic conclusion. Its broader-based, philosophic analog is: there are no contraries to axiomatic concepts.

For the reasons explained above, you may thus conclude, in the derivative sense, that:

“existence is not void”,

where “void” is taken as exactly synonymous to “non-existence”.

The proposition quoted in the last sentence is true.

However, as the set theory makes it clear and easy to understand, it does not mean that you can take this formulation for a definition of the concept of existence. The term “void” here has no independent existence; it can be defined only by a negation of existence itself.

You cannot locate the meaning of existence in reference to void, even if it is true that “existence is not void”.

Even if you use the terms in an extended sense and thereby do apply the “not” qualfier (in the set-theoretic representation, it would be an operator) to the void (to the empty set), for the above-mentioned reasons, you still cannot then read the term “is” to mean “is defined as,” or “is completely synonymous with.” Not just our philosophical knowledge but even its narrower set-theoretical representation is powerful enough that it doesn’t allow us doing so.

That’s why a better way to connect “existence” with “void” is to instead say:

“Existence is not just the absence of the void.”

The same principle applies to any concept, not just to the most fundamental philosophic axioms, so long as you are careful to delineate and delimit the context—and as we saw, the most crucial element here is the universal set. You can take a complement of an empty set only when the left hand-side operator is a universal set.

Let us consider a few concepts, and compare putting them in the two forms:

• from “A is not non-A”
• to “A is not the [just] absence [or negation] of non-A,” or, “A is much more than just a negation of the non-A”.

Consider the concept: focus. Following the first form, a statement we can formulate is:

“focus is not evasion.”

However, it does make much more sense to say that

“focus is not just an absence of evasion,” or that “focus is not limited to an anti-evasion process.”

Both these statements follow the second form. The first form, even if it is logically true, is not as illuminating as is the second.

Exercises:

Here are a few sentences formulated in the first form—i.e. in the form “A is not non-A” or something similar. Reformulate them into the second form—i.e. in the form such as: “A is not just an absence or negation of non-A” or “A is much better than or much more than just a complement or negation of non-A”. (Note: SPPU means the Savitribai Phule Pune University):

• Engineers are not mathematicians
• C++ programmers are not kids
• IISc Bangalore is not SPPU
• IIT Madras is not SPPU
• IIT Kanpur is not SPPU
• IIT Bombay is not SPPU
• The University of Mumbai is not SPPU
• The Shivaji University is not SPPU

[Lest someone from SPPU choose for his examples the statements “Mechanical Engg. is not Metallurgy” and “Metallurgy is not Mechanical Engg.,” we would suggest him another exercise, one which would be better suited to the universal set of all his intellectual means. The exercise involves operations mostly on the finite sets alone. We would ask him to verify (and not to find out in the first place) whether the finite set (specified with an indicative enumeration) consisting of {CFD, Fluid Mechanics, Heat Transfer, Thermodynamics, Strength of Materials, FEM, Stress Analysis, NDT, Failure Analysis,…} represents an intersection of Mechanical Engg and Metallurgy or not.]

A Song I Like:

[I had run this song way back in 2011, but now want to run it again.]

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

[But I won’t disappoint you. Here is another song I like and one I haven’t run so far.]

(Hindi) “baaghon mein bahaar hain…”
Music: S. D. Burman [but it sounds so much like R.D., too!]