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you use intuition to come up with the argument. then you use a proof to convince others.

No, you use intuition to come up with an idea the use that idea to write a proof for yourself. Intuition is invaluable as a starting point, but it's often misleading.

you're very naive. i think you're probably still an undergrad or maybe have never done serious math.

https://terrytao.wordpress.com/career-advice/there%E2%80%99s...

But the article agrees with me:

> The point of rigour is not to destroy all intuition; instead, it should be used to destroy bad intuition while clarifying and elevating good intuition.

In mathematics, intuition is the starting point and guiding principle, but never worth anything by itself without rigour.

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