Two recent posts about Heart of Darkness centered on strange pictures, mathematical pictures. The object of this post is to talk about such pictures, and others equally strange, if not more so. But I wish first to begin with an ordinary photograph, and to think a little, just a little, about how such photographs made and how we understand them.
An Ordinary Photograph
Here’s a photograph of the Empire State Building at night.
Figure 1: The Empire State Building
Though I don’t understand much about the inner workings of my camera or my computer, I nonetheless feel that I’ve got a pretty good understanding of how that picture came to be: I pointed my camera at the Empire State Building, snapped the shutter, ‘developed’ it on my Macintosh, and then uploaded it to the web.
At THIS level of analysis, the level I’m taking in this post, that’s OK. While we know that one can easily monkey with digital photos, I know that didn’t happen. How do I know that? Because I controlled every step of the process. I saw the Empire State Building that night, I took the picture, etc. No one or nothing interfered with the process.
You, of course, pretty much have to take my word on all that. You might suspect, for example, that I didn’t actually take a shot of the Empire State Building and a street lamp. Rather, you suspect, I combined two images. Such things are possible, and quite easy with digital technology. I assure you, I didn’t do that. But, really, that’s all I can do here and now. Offer assurances. I might be lying.
Something Very Small
Now look at this picture:
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| Figure 2 |
It’s not a photograph. It’s a line drawing. What’s it depict? We see two spirals connected by rods. Maybe it’s a design for a wrought-iron staircase.
Maybe. But you know it’s not. You know that it depicts (the structure of) a DNA molecule. That’s the illustration Watson and Crick used in the 1953 paper in Nature in which they announced this structure to the scientific world.
How do you know that? Have you seen a DNA molecule yourself, with your unaided eyes. No, you haven’t. They’re too small.
Perhaps you saw it through an optical microscope. No? That’s right, it’s too small for that. The wavelength of visible light is too long to resolve such small structures.
If you didn’t see it with your own eyes, then, how can you know what it is? Because you’ve seen other pictures and have been told that they represent the structure. And you believe what you’ve been told about those pictures; you have faith in the fundamental integrity of the social process needed to inform and support such pictures.
But, how did Watson and Crick come up with that image? What did they look at? They could no more see it with their eyes than you could see it with yours; and their optical telescopes are no better than yours. Well, yes, maybe they did have better optical microscopes. But no such microscope of whatever quality can see such things; that’s a matter of fundamental physics.
So how’d they “see” the DNA molecule? Alas, that’s not a story I understand very well. The process, I believe, involves beaming x-rays through crystallized DNA and onto a photographic emulsion. The emulsion is developed, yielding images like this one, from an article that accompanied the seminal Watson and Crick paper:
Figure 3: X-ray diffraction ‘smudges’
That doesn’t look anything like a double helix.
So where’d the double helix come from? Watson and Crick made it up. Not out of whole cloth, but they made it up. Knowing the properties of x-rays, they guessed / deduced / hypothesized that those kinds of smudges (in Figure 3) would be produced if the DNA crystal had the form of a double helix.
But just what IS it that has the form of a double-helix?
The DNA molecule, yeah.
But what is that? Is it two continuous ribbons, like in the diagram?
No, that’s just a conventional way of representing the overall arrangement of a bunch of atoms linked together into a very large molecule.
Atoms? You mean it’s made of atoms? What are they?
And THAT conversation could go on for awhile. My point is simply that that diagram, which looks so very physical – and is, in an abstract way – “rests” on a considerable body of empirical practice, x-ray diffraction imaging, and theory. Anyone with reasonable vision can see the diagram, but to understand what it represents, that’s a different matter.
Pictures of the Text
Look at this picture:
Figure 4: Paragraphs in Heart of Darkness, ordered by position in the text
If you’ve been following this blog for the past couple of weeks, you know what that represents, the lengths of paragraphs in a particular text, Heart of Darkness. The length of each bar is proportional to the length of the paragraph it represents, where paragraph length is measured by word count. The paragraphs are listed as they appear in the text, first to last, from left to right.
Now, that text is a perfectly perceptible object. You can hold a hardcopy version in your hands or you can view an electronic version on the screen. You can read it, and you can certainly note that some paragraphs are longer than others and that paragraphs of differing lengths appear together in the same stretch of text.
But you can’t really ‘see’ the whole text. Well, yes, you could rip the pages out of a book and put them all on the floor or tack them to a wall. Then you could literally see the whole text. But you couldn’t actually see the distribution of paragraph lengths. It’s too diffuse.
The graph abstracts everything away from the text—the individual characters are gone—except the length of the paragraphs. And that one factor is represented by the length of a line. Now that one factor can be displayed by looking at a bunch of lines.
This picture, too, is an abstract one. It’s perhaps not so ‘far’ from the underlying phenomenon as the DNA diagram is from the actual molecule, but it’s not so ‘close’ as the photo of the Empire State Building is to the building itself.
Now look at this graph:
Figure 5: Paragraphs in Heart of Darkness, ordered by length of paragraph
It depicts the same paragraphs as in the previous picture, but in a different order. This time I sorted them by length, with the long paragraphs at the left and the short ones to the right. Notice how we get a fairly smooth curve. Why? What’s that about? That curve might have been a straight line, or it might have been stair-steps, but it’s not. It is what’s called an exponential curve.
Again, why? Is that a question about how we like to organize information into paragraph-size chunks. That is, is it a question about the mind. Or is it a question about mere printing conventions, where paragraphing is unrelated to what’s said in the text but is just a matter of arranging ink on the page?
Surely it is the former. That is, the printing convention is subordinate to considerations of information presentation. Given that, again I ask, why? We does the mind do that?
We don’t know. That image depicts the merest trace of a mental process, a process that we cannot observe directly. Interpreted THAT way, the image is surely as remote from the basic underlying phenomenon as is the diagram of the DNA molecule.
In fact, it might be better to liken these two images—Figure 4, where we see paragraphs ordered as they are in the text, and Figure 5, where they’re ordered by size—to the photographic plates that record the raw data of x-ray diffraction imaging, as in Figure 3.
I have no idea what kind of mental-process diagram will correspond to Figure 2, the structural diagram for the DNA molecule.
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