We have encountered the question of how to extend our formal calculus to take account of operator variables.

In the days when I scribbled these things on the backs of computer punchcards, the first thing I tried was drawing big loopy script characters, placing some inside the loops of others.  Lower case alphas, betas, gammas, deltas, and so on worked best.  Graphics like these conveyed the idea that a character-shaped boundary drawn around another space can be viewed as absent or present depending on whether the formal value of the character is unmarked or marked.  The same idea can be conveyed by attaching characters directly to the edges of graphs.

Here is how we might suggest an algebraic expression of the form where the absence or presence of the operator depends on the value of the algebraic expression the operator being absent whenever is unmarked and present whenever is marked.

It was obvious from the outset this sort of tactic would need a lot of work to become a usable calculus, especially when it came time to feed those punchcards back into the computer.

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