There’s a pathological concern for correctness in mathematics. Creativity is seldom about correctness, and anyway, both are targeting categorically different ideas.

If your professor says, “For fun, construct a non quasi-coherent sheaf of ideals” and the immediate response is that “There is a correct formulation of this, created by someone else, that I can refer to.” You’ve just commited intellectual suicide, and have ruined an opportunity to engage creatively. This matters, as discovery in mathematics is not too far off from invention.

Creation in mathematics is harder than creation in physical space. For one, your innate topology is already compatible with the space in question, so anything created by you will remain compatible. In some sense, you are an emebedded subobject. But in mathematics, neither you nor your creations are granted such a luxury. There is a typed-logic that needs to be checked, and its structure needs to be compatiable with the operations of the underlying topological space (and remember, everything is a topological space, or can be realized as one).

It’s of course similar to creating, using or speaking a language, which is an underlying structure, and a grammar (rules) that need to be adhered to for the object of your construction to be meaningful. And of course, gibberish is an object, and thus a creation, but it’s more of an immersed object than anything intrisically useful.

Immersion is necessarily the first step.

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