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I don't think that analogy holds at all. A better analogy is comparing some boolean function written purely in terms of AND, OR, XOR, etc with its corresponding circuit diagram. From the diagram, the information flow is much more obvious, and common constructions (building adders, as a random example) can be visualised. The diagrammatics give a different intuition on the problem than purely symbolic algebra.

Of course, both the formula and the diagram encode exactly the same information. However when attempting to prove something about a formula, it can be very helpful to translate the formula into a diagram and use the diagram to guide algebraic manipulations (manipulations which perhaps we would not have known to make, if it weren't for the diagram).

I know that Feynman diagrams are a quite famous example - even if results do not use them directly, the intuition they give on a problem are indispensable. There are other examples in pure mathematics where new proofs have been given not in terms of diagrams, but the development of suitable diagrams have enabled people to come up with a proof.

There’s nothing wrong with using graphical intuition, my point is that nobody ever uses graphical notations to compute. For example, Feynman diagrams are a great way to tell you what to compute, but then the computation itself is always done in standard notation.

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