Notation as a Tool of Thought (1979)

It's easy to think of notation like shell expansions, that all you're doing is replacing expressions with other expressions. But it goes much deeper than that. Once my professor explained how many great discoveries are often paired with new notation. That new notation signifies "here's a new way to think about this problem". And that many unsolved problems today will give way to powerful notation.

As mentioned elsewhere, the right notation[0] allowed us to discover new juggling patterns that had never been done[1][3]. I freely admit that current mathematical notation has problems, but most of the proposed reforms seem to lose the predictive and creative power, becoming mere notation and nothing more. The problem is that without experiencing that extra dimension it's impossible to see that, and you can't experience that extra dimension without investing the time and effort to learn mathematics to a…

Does the notion of suggestability extend to the notion of "creative breakage"? The latter is an important feature of a mathematical notation. For example the dy/dx notation in calculus naturally leads to inquisitive thinking. Can you multiply by dx? It looks like a division but it isnt, really, except that a soon as you get past Calc 101 you are spraying dys and dxs all over. the notation is just incredibly suggestive. Another example is exponents where m^n is introduced…

In Physics and Mathematics it is widely accepted that having an efficient notation is of paramount importance when thinking about a given problem. A good example for this are Maxwell's equations: https://en.wikipedia.org/wiki/Maxwell%27s_equations Using the vector differential operator notation those equations neatly fit on a napkin, whereas Maxwell's original equations (written in Integral form) would cover several pages. In fact, looking at his original manuscript always fills me with awe for the man as he had…