Context and Motivation:
The title question of this post has been lingering in my mind for quite some time—actually, years (nay, decades). Some two decades ago or so, I thought I had reached some good understanding of it. But then, some of the discussion at a recent iMechanica thread “A point and a particle” [^] seemed to suggest otherwise. The issue again got raised, in a somewhat indirect manner, in relation to this comment [^] on yet another iMechanica thread today. In between, there also were a few message exchanges that I had at HBL last year, not all of which made it to the published HBL exchange. There, too, my own position was at odds with that of Dr. Harry Binswanger, an Objectivist professor of philosophy (and the way he sometimes describes himself, an amateur scientist).
The essential difference is this: People seem to think, for example, that:
(i) you can take a small but finite line-segment, subject it to an infinitely long limiting process, and what you get in the end is a point; or,
(ii) as the chord of a circle is systematically made ever smaller by bringing its two end-points closer, even as always keeping them on the circle, eventually, the circle, in comparison with the straight-chord, seems to get flattened out so much that eventually, in an infinite process, it becomes indistinguishable from a straight-line, and so, the circular arc becomes the chord (which is the same as saying that the chord becomes the arc); or,
(iii) a particle’s geometry is fully described by a point; etc.
All of these examples, in some way, touch on the title question. For instance, since a point does not have any parts, and if in an infinite process a line goes to a point, then, obviously, an infinitesimal cannot have parts. And so on…
Now, I seem to disagree with the views expressed by people, as above. I also think that some of the basic confusions arising in quantum mechanics (e.g. those concerning the quantum spin) in part arise out of this issue.
[Therefore, an immediate declaration: If someone gets a better idea of what QM really is like, after reading this post, thank me, and also, regardless of that and more importantly: give me appropriate and explicit intellectual credit. To my knowledge, the topic has not been treated so directly and in the following way anywhere else before.]
Consider an arbitrary but “nice” enough a function: . Consider two points and lying on the curve but a finite distance apart. The slope of the line-segment is given by: , where the subscript put on indicates that this is a finite-distance case. As you know, there is an infinity of points in between the end-points of any finite line-segment.
To determine the slope of the curve at the point , we take the limit of the ratio as the distance between and approaches zero. In symbolic terms: , where is supposed to be the slope of the curve at the point .
Clarifications—The Idea of Slope:
The italicized parts in the above statement are important. Firstly, it is implicitly (and somewhat blithely) assumed that a curve can have a slope, which can be approximated by that of a line-segment such as . Secondly, it is even more implicitly (and even more blithely) assumed that there exists something such as a slope at a point. Let’s examine both a bit more closely.
What does the notion of slope mean? The extreme case of and the pathological case of apart, what the notion basically means is that you are going to either gain or lose your current height as you travel (in some direction).
Notice that immediately implicit here is the idea of there being two different locations whose heights are being compared! You cannot define slope without there being two distinct reference points. Hence, you also should not use the term in those contexts where only one reference point is given. If so, how can we speak of a slope at a point?
Realize that the above objection applies as much to the points lying on a straight line-segment as those on a curved line-segment. Even single points on straight lines cannot, strictly speaking, can be said to have a slope—only the straight line-segment, as a whole (or any finite parts of it) may be said to have one. If so, what does it mean when we speak of a slope at a point?
My answer: Primarily, it means nothing! It’s just a loose way of putting things. What it really means is the entire limiting process, and the result of it (if there is any valid result coming off that limiting process).
The slope of a line at a point (whether that line is straight or curved, it does not matter) is just the definite “tendency” shown in the trends of the actual slopes of all the small but finitely long straight line-segments in the close neighbourhood of the given point. You cannot speak of a slope at a point in any other terms. Not even for straight-lines. Straight lines just happen to be a special case wherein all the slope values are the same, and so, determining the trend is a trivial matter. Yet, the principle of having to make a reference to an actual trend of certain property displayed by a definitely ordered sequence of finitely long segments, in an appropriate limiting process, does remain there. It is only in this sense that lines can have slopes at various points. Ditto, for the curved lines.
Clarifications—A Line “Going” “to” a Point:
Now, there is something even funnier. At least in applied science and engineering, we often speak of the above kind of a limiting process, in terms like the following:
Take close to , as close as possible. In the limit, as the length of “goes” “to” “zero,” the slope of the segment “goes” “to” “the slope of the curve” “at” ““.
All the words put in the scare-quotes (“”) are important.
What does it mean for a length of a straight line-segment to go to zero? It means: and are coincident—i.e. they are one and the same point. (There is no such a thing as two different points occupying the same point—it’s either two names for the same mathematical object, or a contradiction in terms.)
So, can a slope have a curve? The very idea is meaningless outside the context of a limiting process. Yes, you may gain or lose height as you traverse the curve, sure. But does it mean that the curve has a slope? Nope. Not unless your context has the right limiting process in it.
Clarifications—Points, Lines, and the Nature of Limiting Processes:
Now, a bit about the nature of limiting process.
Realize that there is a fundamental difference between a point and a line. (For our purposes, both may be taken as given axiomatically, as abstractions of the locations and the edges of the actually existing objects. That there also is suggested an infinite process in reaching such abstractions is a subtle point that we choose to ignore for the time being.)
The units of a point and a line are different. You cannot compare a point and a line in any commensurate manner whatsoever, full-stop. (Incommensurability is quite frequent in mathematics, more often than what most people realize.)
A line segment may be put in one:one correspondence with an (orderly) infinite set of points, and in this way, it may abstractly be seen to consist of points. However, realize that infinity does not exist. The one:one correspondence process, should you wish to conduct one in actuality, will never terminate, and hence, you will never get a line starting from points, or vice versa: a point, starting from a line. Incidentally, that’s just another way of realizing that a line is incommensurate with a point. Then how is it that we can talk meaningfully of such a process?
What we mean when we talk of a line as being made of an infinity of points is this:
Take a finite line-segment, say from the point to . Take a point lying on it. Find the finite lengths of from each of its end-points. (Aside: It is here that the defining processes of a point, a line, etc. that we have chose to ignore in this post, create some tricky issues. We will deal with them later, in another post.)
Now, take a sub-segment from any of the two end-points to the middle point (whose location, in the general case, is arbitrary; it need not exactly divide the segment into two equal halves.) Suppose we take the sub-segment . Now, conduct a limiting process by reducing the size of , while holding fixed. (BTW, observe that every limiting process involves holding something the same even as varying something else.) Making the sub-segment monotonically smaller in size means that the end-point of the segment in the reduced size corresponding to , say, monotonically increasingly gets closer to . But, it never quite reaches .
The only case in which could reach is if it is coincident with—i.e. is the same point as—. However, in this case, there cannot be two distinct end-points left to serve as the end-points of the diminishing sub-segment, and hence, no sub-segment left to speak of.
Hence, we have to say that the point never quite reaches —not even in this infinite limiting process. The most crucial point of the logic is already thus given. The rest is a bit of house-keeping so that even if we revise the entire description here by expressing a point via a limiting process, the essential logic as spelt about remains unaffected.
Now, repeat the process for another, distinct, point , lying on the same original line-segment. Since and are not one and the same point, and since the “getting closer” process for any arbitrary sub-part of the line-segment cannot terminate for either of them, and further, since both lie on the same original finite segment and thereby enjoy an ordering relation between them (e.g. that etc.), we must conclude that there must be an infinity of points corresponding to any arbitrarily given point . Just make coincident with (or the same as) , and the inevitable conclusion follows, namely, that there must be an infinity of such processes for them to span all the distinct points lying over the entire original line-segment.
The existence of this infinity of such “getting closer” processes is what we actually mean when we say a line is “made of” an infinity of points.
Emphatically, it does not mean that a point and a line are commensurate. It only means that the endpoints of a line can be made as close to a given point lying on that line as you wish. That’s all.
Clarifications—An Infinitesimal of a Finite:
Now, we are ready to tackle the idea of infinitesimal.
An infinitesimal of a line-segment is an imaginary projection of the result that would be had if a line-segment were to be made ever smaller in a limiting infinite (i.e. definite but unterminating) process.
Notice that we didn’t jump directly to what the term “infinitesimal” means in a general sense. We simply made a statement in respect of the infinitesimal of a line-segment. This distinction is important. The reason is that there is no such thing as a general infinitesimal!
You can have infinitesimals of (finite) lines, surfaces, volumes, etc. Or, of quantities that, essentially, are some kind of densities of some other quantities which have been defined in a “wholesale” manner over finite lines (or surfaces, volumes, etc.). But you cannot have infinitesimals “in general,” as such.
Infinitesimals not only acquire their meaning only in some definite kind of an infinite limiting process, but they also do so only in reference to the certain finite thing (and its associated properties) which is being subjected to that process. A process without an input or an output is a contradiction in terms. An infinitesimal can only result when you begin in the first place with a finite.
Since an infinitesimal must always refer to its input finite thing (be it a length, a surface, etc. or a density variable defined with respect to these), therefore, it must always carry some units—which are the same as that of the finite thing.
The “infinitesimal-izing” process (to coin a new word!) does not touch the units of the finite thing, and hence, neither does the end-result of that process—even if the result be only via an imaginary projection. Thus, the infinitesimal of a line always retains the units of, say, , and that of a surface, , etc.
The above precisely is the reason why we can “cancel out” with where the first expression is a product of lengths, and the second one is an area—and wherein all the quantities are infinitesimals. Infinitesimals have units; equations formulated in infinitesimal terms must follow the law of dimensional homogeneity.
Clarifications—Can Infinitesimals Have Parts?
Now, having examined the nature of infinitesimals to (hopefully) sufficient extent, we are finally ready to answer the title question: “Can an infinitesimal have parts?”
I will not directly answer the question in yes or no terms. My answer should be obvious to you by now. (If not, kick yourself a couple of times, and proceed to read further or, equally well, abandon this blog forever.)
First, observe that it is only a finite line-segment which, when subject to an infinitesimal-izing process, becomes an infinitesimal.
Apart from its two end-points, you can always take a third point lying on that finite segment such that it divides the segment into two (not necessarily equal) parts. Say, . Now, observe that as you take to an infinitesimally small quantity, you also thereby subject and to the same infinitesimal-izing process such that the equation holds as a result. (The reason we can directly put this relation in this way is that the rates with which each becomes small is identical. In contrast, the area gets smaller at a rate faster than that of the length—another way of seeing that an infinitesimal always has dimensions i.e. units.)
Now, returning back to today’s discussion. At iMechanica, I have raised a couple of points:
(i) Do we define stress in relation to a plane? Or do we do so in relation to a thin plate made infinitesimally small? The difference, now you can see, is this: a plane has no thickness. But a plate does. Its thickness has the units of length, which can’t be made zero. Hence the question.
(ii) Is the elemental cube (used for defining variations in stress, say to the first order) have a finite length? Or is it (or can it be) infinitesimal?
Once again, I will not provide a direct answer to these questions. However, I will leave you with a very very obvious clue (apart from what all I have mentioned above)—but one, which, nevertheless, raises further curious issues. These are essentially nothing but the same as the issues we have chosen to ignore today—what are points? lines? surfaces? do they exist? Anyway, the clue, presently, is the following.
Take a brick. You can always make its size ever smaller in a limiting process so as to get an infinitesimal Cartesian volume element. Agreed? OK.
Now, take a pack of playing cards. Subject it to a similar limiting process. And, ask yourself the above two questions. The answer(s) should be obvious!! (As to the tricky part: Ask yourself: Can you assume zero thickness in between two adjacent playing cards in the same pack? Your answer to the question of whether stress is defined in relation to a plane or an infinitesimally thin plate, will in part differ depending on how you answer this question!)
[PS: I think I might edit this post a bit. If I do so, I will also note down any major change (e.g. that of the logic or of hierarchical precedence, etc.) that I make. For instance, I am not at all happy with the way I have explained the idea of “an infinity of points in a line, even though a line never goes to a point.” That part hasn’t at all come out well. I expect to make changes there—or, may be, perhaps, write another post to once again give a try to that part. … Hey, after all, this is not a paper on mathematics—just a blog post, OK? 🙂 ]
[A side note: I know that the limit notation as rendered here on the Web page does not look nice, but that’s an issue primarily with the WordPress support of LaTeX. I am not going to hack around with \dfrac etc. just to get the \lim look nice here!]
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A Song I Like:
(Hindi) “dil beqaraar saa hai…”
Singer: Lata Mangeshkar (I like her version better than Rafi’s)
Lyrics: Majrooh Sultanpuri