I haven’t written on QM for some time, and today I found myself wondering a bit about the title question.
When it comes to concepts, especially those of the physical sciences, we always carry some visual images concerning them.
No, the meaning of a concept isn’t the image—that would be an erroneous view of concept. Concepts are necessarily abstract. However, since sense-percepts indeed are both the beginning material as well as the ultimate foundation of concepts, it seems obvious to me that we should also have some kind of a sensory-perceptual data associated with concepts in an “informal” sort of a way. The data serve a certain psycho-epistemological function, viz., that of helping you recall the meaning of a concept, say, with great “vividness.”
Definitions are there. They do give identity to concepts. But most of the concepts that we use—in our daily life, but even more so in the physical sciences—are at a rather high (or even very high) level. They are far removed away from the direct sensory-perceptual data lying underneath them. Due to this distance from the perceptual data, definitions for most concepts themselves are abstract, too.
Definitions tell you that which is denoted by a concept. However, there also are other means that the mind uses in recalling and correctly using concepts. An important means here is the mental imagery: say a prominent picture, a sound, a schematic diagram, or even an instance of a kinesthetic sense—or a cluster of all these. They get associated with a concept, and their use gives you not only of a sense of the various underlying layers of meaning of that concept, but also the connections that a given concept has with the other concepts. I don’t know whether I am using a rigorously correct word or not, but at least for my personal usage, I call such things the connotations of a concept.
With many thinkers, esp. Objectivists, there is this tendency to look down upon connotations. I think this is wrong. If you are going to substitute connotations for denotations, then, of course, it’s a significant error; it is bad. But what if you don’t?
Realize, connotations (in the sense I use the term) themselves do not equate to mere “feelings.” It’s not as if denotations equal to Reason and connotations to Feelings. No. [In fact such a position would be Rationalistic.]
What I call connotations are not some generalized, difficult-to-verbalize, and background sort of vague feelings that occupy your feelings-sphere when you consider a concept. They instead are very specific items of imagery, of some perceptual data. The subconscious seems to work more efficiently when you involve these items. Especially if the concepts are abstract, if they are at a high level.
Let me give you some examples.
Since maths always is fully abstract, it’s inevitable that our minds would use the connotative imagery to even greater extent than in the other sciences.
Consider the concept: “derivative.” The first thing that comes to my mind when I begin to think of this concept is that std. XI graph of a curve, a point on that curve, and a series of chords approaching the tangent to the curve at that point. Everything that I have ever thought of “derivative” or “differential” is tied to this diagram, an image. [Indeed, since the chord approaches the tangent only from one side, every time I sit down to consider the concept of derivative, I still get an uncomfortable feeling about that asymmetry—the tangent isn’t approached from both the sides. I feel a bit comfortable even today, after all these 35+ years.]
Now, take a moment to consider what that imagery for “derivative” is like. The first thing to realize here is that this image is not a instance of a direct sense-perception. In nature, you never see a tangent or a series of chords. For that matter, you don’t even see a 1D curve. All you ever see are the 3D objects and their perceived limits (or extensions), which themselves are idealized as surfaces are curves. Thus, the connotative imagery itself consists of an abstract diagram. Yet, it helps you concretize the concept.
Recently, I was talking to a couple of mathematicians. [Yes, I am talkative. I can talk with any one—even mathematicians!] The issue was pretty abstract, even though we were talking mostly at the “physical intuitive” level. We were arguing at the blackboard [in actuality it was a whiteboard] from many different points of view, and we were doing the argumentative exchange fairly rapidly. So, inevitably, we were picking up only the highlights of an idea of a concept—just those bare tidbits that would be enough for the other person get the gist of how the argument from your side was progressing. For instance, here is what I once said during the discussion: “…Now, as far as the variational calculus goes, that’s not a problem with me [i.e. for the problem I was considering]… You see…” I rapidly drew XY axes, a step function, a flat line at the mid-height, rapidly hatched the area only under the step function. Then without pointing out to anything specific, I just said, “Both these areas are equal, and so, I am home free!” They understood. Even if I had never in fact pointed out the second area!
Clearly, not just me, but they, too, were using these connotative images. Else, communications would have been impossible.
That reminds of something… A girl had once [more than two decades ago] articulately told me, complete with her suave Mumbai accent, that she was not good in communications—with an emphasis on “communications.” That way, many, many people, have told me the same thing—in fact far too many people for my liking. But for some odd reason, this particular instance with this girl has stayed in my mind. My instinctive reaction back then was—the one which I didn’t share with her—that probably her problem was in understanding [anything straight], not in communicating whatever it was that she did understand. If she were only to “get it right,” it was very obvious to me, that she would have absolutely no problem in articulating it. An articulate dumb is an easy possibility; it’s not a contradiction in terms—even though I was (relatively more) new to the phenomenon back then.
But getting back to this recent discussion, if I myself were to make use of these “physical intuitive” imageries, then it would have been perfectly OK—I am an engineer. But the point is, at one point in discussion, in thinking aloud, one of these mathematicians themselves said something like: “So, when you integrate, you are going get this quantity [i.e. an expression he had written on the board] under the integral sign.” Then, in the same flow, he added without any distinct pause, still continuing to think aloud, still not addressing the line to anyone in particular: “You know integration—sum of areas under the curve. And so, …”
Clearly, even in his professional mathematical work, when it came to exploring a new path, [even if that path was only in a known territory], he wasn’t using either the formal epsilon-delta definition or the idea of the anti-derivative or the fundamental theorem of calculus. He was using a finite sum of finite number of finite areas under a curve. He would sure formalize his argument later on, and that’s when these beasts of formalization would come in. But in actually working out that new path, he was using only the simple connotative imagery.
We all always do.
So, as the thought of QM came up to me during my “purpose-less” kind of an idle arm-chair wondering on this fine monsoon evening, while comfortably sipping a cup of coffee at home, I happened to ask myself: what imagery do I really use when I say “QM”?
By “QM,” I meant, first and foremost, the concept itself. Not the implications of the findings of this field of science, but “quantum mechanics” as the idea.
I answered the question to myself immediately, of course. [How else could such imagery be of any use, in the first place?] I wanted to write about that question today. Instead, in explaining the meaning of imagery and connotations, I have ended up writing so much [about 1500 words] that I must now split this intended post into two parts. Accordingly, this writeup now becomes the part 1 of a (hopefully only) 2-part series of posts.
I will come to QM in the second part, hopefully soon enough. In the meanwhile, think about what your answer to that same question is like. [Yes, critical takes are perfectly welcome, too. Especially if they are sarcastic.]
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A Song I Like:
(Western popular) “The day before you came”