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 is bigger than the mango , and that the mango is bigger than the mango , is perfectly fine.
However, it is clear from common experience that the size-wise difference between and may not exactly be the same as the size-wise difference between and . 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: , , , . 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,
(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 ” [^].)
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 () 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 , and the single concretely existing mango as a second abstract group: .
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:
- The process produces (i.e. the repetition process is stopped after taking once).
- The process produces (i.e. the repetition process is stopped after taking twice)
- The process produces (i.e. the repetition process is stopped after taking thrice)
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 , or , or , $\dots$ is now regarded as if it were a single (composite) object.
Notice how we began by saying that , , 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 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 -th term is given by the formula:
Thus, the sequence is: .
If we take first two terms, we can see that the value has decreased, from to . If we go from the second to the third term, we can see that the value has decreased even further, to . The difference in the decrement has, however, dropped; it has gone from to . 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 . 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 , 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 value. We can verify this fact. No matter how big 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 . But it will approach (and no number other than ) 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 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 , in finite number of steps.
To capture all these facts, viz.: (i) We will always go below the level any positive real number , no matter how small may be, in a finite number of steps, (ii) the number of steps required to go below a specified level would always go on increasing as becomes smaller, and (iii) we will never reach in any finite number of steps no matter how large may get, but will always experience decrement with increasing , we say that:
the limit of the sequence as approaches infinity is .
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 . [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 ) to a given number. We can apply the process of addition by , to the number , and see that the number we thus arrive at is . Then we can apply the process of addition by , to the number , and see that the number we thus arrive at is . 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 , then , then , , arises is to observe that in successively applying the process of addition starting from the number , it is the number which comes immediately after the number $1$, but before the number , 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 , 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 , 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 is applied right to the number 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 () to whole numbers () to integers ().
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 and , 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 and 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 or 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 as well as , 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 , and mark this distance out from the origin on the real number-line. This measurement process is well-specified even if 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 and , 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 and . In fact, the existence of an infinity of numbers is damn easy to prove: just take the average of and and show that it must fall strictly in between them—in fact, it divides the line-segment from to 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 and 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 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 () case, points on a line can be put in 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 st avataar, nor in the nd, nor even in an hypothetically -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…