Different people think in different ways. Some people, such as Norbert Weiner, think visually and then translate into words; others, such as Von Neumann, think in words or algebras.
Weiner said once that he visualized lots of molecules; then wrote mathematics for the visualization. (So did the early Einstein.) Von Neumann did not visualize; he created ad hoc languages (algebras) and thought in them.
People are resistant to talking about how they think; I suspect it is because people feel stupid compared to those who come to mind when they think about thinking; they are not Von Neumann or Picasso.
(Also, as people think differently in different circumstances; and come to think differently over time. Incidentally, I spoke out loud to myself when learning to fly, but not in other circumstances.)
Here is an example of different thinking styles:
Paul Horwitz searches for artificially generated radio signals from beyond the solar system. At dinner once with Richard Stallman and me, he pointed out that a single-frequency signal is good candidate for further investigation. However, there are problems. Suppose an interstellar cloud is traveling across the line of sight: that spreads the frequency of the signal.
Richard Stallman asked why is that? So the three of us each stopped to think what would happen.
You can do this now: why would a cloud travelling perpendicular to a single-frequency radio beam widen the frequency of that transmission?
It turned out that each of us used a different method of thinking about the problem.
Imagine the rays coming from the source at a 30 degree angle to the right of the direct line from the source and then bouncing to your radio telescope receiver from a leftwards moving cloud; the frequency is blue shifted. Imagine rays bouncing off the leftwards moving cloud on the left side; the frequency is red shifted. All in all the frequency is widened.
Each of us solved the problem. My style is best for qualitative or grade-school understanding. It is easy to draw the picture and explain it a kid who visualizes and knows a little about the Doppler effect. (Describing the picture in words for a message like this is much more difficult, both for me writing it and for you reading it.)
Stallman's method is best for programming a computer to predict the widening. And as it happens, he is a very good programmer. (He wrote the integrated working environment in which I am writing.)
Horowitz' method is best for understanding electromagnetic signals in interstellar space -- his comment on clouds absorbing and then re-radiating the radiation suggest not only the Doppler shifted frequency changes with which we concerned here, but also differential absorption, other kinds of frequency shift, changes in temperature of the cloud, and the like -- a host of issues.
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