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## Main Question or Discussion Point

I would like the perspective of either a physicist or a mathemtician. Please indicate which you are before posting.

When learning mathematics, is it better to follow a strictly rigorous approach? By this I mean such things as being exposed to Calculus via Spivak, Complex variables by Ahlorfs and so on. Or rather that an intuitive approach (with some theory) be employed, with the likes of say Salas for Calc and Fisher for Complex then re-learned by the pure counterparts.

With the rigorous approach I see the benefit of being exposed to everything at once. This lets you take the math more seriously, and leaves a more lasting impression. You will not ask "why this is so" because everything is proved from the ground up. Not to mention all the extra practice you get with proofs and theory, which should prepare you for upper texts which follow that format. On the downside however, you lack much or all intuition. You may not even be able to apply your knowledge to a very simple problem. In a lot of cases, you may not be able to appretiate the theory and be put off by a dull approach. And as I've learned with Linear Algebra, the theorems don't stick at all!

Obviously with an applied approach you will learn more rapidly and get to fundamental results. But are you just learning bad habits?

The reason I ask is because I am debating whether to enrol in pure math courses, or in the applied ones. In the end I intend to learn the pure math, but is it better if I've had applied exposure? That way I can appretiate the theory more? Or should I start the right way from the start?

Spivak sure was fun when I already knew Calc 1... but would it have been fun without...

When learning mathematics, is it better to follow a strictly rigorous approach? By this I mean such things as being exposed to Calculus via Spivak, Complex variables by Ahlorfs and so on. Or rather that an intuitive approach (with some theory) be employed, with the likes of say Salas for Calc and Fisher for Complex then re-learned by the pure counterparts.

With the rigorous approach I see the benefit of being exposed to everything at once. This lets you take the math more seriously, and leaves a more lasting impression. You will not ask "why this is so" because everything is proved from the ground up. Not to mention all the extra practice you get with proofs and theory, which should prepare you for upper texts which follow that format. On the downside however, you lack much or all intuition. You may not even be able to apply your knowledge to a very simple problem. In a lot of cases, you may not be able to appretiate the theory and be put off by a dull approach. And as I've learned with Linear Algebra, the theorems don't stick at all!

Obviously with an applied approach you will learn more rapidly and get to fundamental results. But are you just learning bad habits?

The reason I ask is because I am debating whether to enrol in pure math courses, or in the applied ones. In the end I intend to learn the pure math, but is it better if I've had applied exposure? That way I can appretiate the theory more? Or should I start the right way from the start?

Spivak sure was fun when I already knew Calc 1... but would it have been fun without...

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