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

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

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


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

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

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

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


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

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

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

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

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

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


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

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

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

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

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


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

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

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

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

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

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


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

Reaching the idea of the natural numbers involves three steps:

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

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

and

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

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

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

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

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

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


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

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

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

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

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

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

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

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

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


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

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

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

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

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

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

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

S_n = \dfrac{1}{n}.

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

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

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

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

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


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

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

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

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

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

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

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

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


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

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


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

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


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

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

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

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


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


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

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

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

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

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

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

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

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

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

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

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


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

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

This idea that

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

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

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

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

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

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

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


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

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

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

Huh? Where’s the catch?

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


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

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

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

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

The real statement is this:

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

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


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

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

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

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

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

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


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


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

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

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

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


A Song I Don’t Like:

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

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

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

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

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

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


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

 

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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
Music: Madan Mohan
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.]