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mathoverflow.net
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Much worse than this heuristic is the “official” definition of a function X→YX→Y as a subset of X×YX×Y satisfying various axioms. – Amritanshu Prasad Oct 20 ‘16 at 18:05

@LSpice […] My point is that sometimes functions should be thought of as formulas, rather than as relations of a special kind, and this is something that mathematicians often do (for example when we talk about formal power series or polynomials as functions). – Amritanshu Prasad Nov 4 ‘16 at 8:55

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That’s because on formulas differentiation is nice and integration is hard, but on computable functions differentiation is hard and integration is nice. In theory, we have a denotational semantics between formulas that functions that should transport these notions back-and-forth, but we really really don’t. There are tons and tons of papers in computer algebra which basically boil down to this massive gulf between abstract analysis (the study of functions given by properties) and concrete analysis (study of functions given by formulas). – Jacques Carette Mar 13 ‘10 at 3:50

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“You can’t add apples and oranges.”

False. You can in the free abelian group generated by an apple and an orange. As Patrick Barrow says, “A failure of imagination is not an insight into necessity.”

Of course, one can debate whether this would be adding or multiplying the apple and orange. If it’s just a group rather than a ring or field, then it’s undefined what the (single) operation is supposed to be, as additive and multiplicative structures are both groups (and group-theoretically, there is no difference).

The `apple + orange` reference … it leads nicely to this shameless plug of my fiction work.

“We almost died in there, Farisa,” It was Runar. “Remember? We were a the square root of a c— hair away from being orc food.”

“If ‘c— hair’ means a small quantity,” Farisa said, “an eepsila between zero and one, then taking the square root of it makes it bigger. So that’s actually more than a c— hair.”

“Why does no one ever let eepsila be less than zero?”

“This is getting really analytical,” Farisa said.

“Farisa the punisher,” Mazie said.