Feynman the Babylonian
golem.ph.utexas.edu
golem.ph.utexas.edu
- notations have usability. Each one makes it easier or harder to do certain things. One notation might be good at one task, and poor at another; whereas another notation vice versa, even though both are equally "true". Notations also make it harder or easier to think/reason about certain things, in the way that language tends to shape our thoughts (not determine them, Dr. Whorf).
- notations are similar or different to related concepts and notations, making some connections easier to see, and some harder.
- notations generalize differently, having different suggestions for the "obvious" next step.
A particular notation is then just a shadow of the truth - a one dimensional projection, capturing only a plane of the unseen reality. It seems that mathematicians and scientists routinely invent new notations for new domains (not necessarily good ones). The advantages of different notations is what Alan Kay was talking about when he said that "point of view is worth 80 IQ points". Notation is a form of point of view.
It was based on a few things from the past like how smart you had to be in Roman times to multiply two numbers together; only geniuses did it. We haven't gotten any smarter, we've just changed our representation system. We think better generally by inventing better representations; that's something that we as computer scientists recognize as one of the main things that we try to do http://ecotopia.com/webpress/futures.htm
"By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and, in effect, increases the mental power of the race. Before the introduction of the Arabic notation, multiplication was difficult, and the division even of integers called into play the highest mathematical faculties. Probably nothing in the modern world would have more astonished a Greek mathematician than to learn that ... a large proportion of the population of Western Europe could perform the operation of division for the largest numbers. This fact would have seemed to him a sheer impossibility ... Our modern power of easy reckoning with decimal fractions is the almost miraculous result of the gradual discovery of a perfect notation. [...] By the aid of symbolism, we can make transitions in reasoning almost mechanically, by the eye, which otherwise would call into play the higher faculties of the brain. [...] It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilisation advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle -- they are strictly limited in number, they require fresh horses, and must only be made at decisive moments."
--from An Introduction to Mathematics, 1911