[SomeStupidPoint]

The end result of mathematics is precision; the inception of an idea, not so much. Indeed, if you look at the early papers on a topic, they are -- for lack of a better term -- "fuzzy" and very often contain technical mistakes. The art of mathematics -- which it shares with all arts -- is to make the leap in understanding and then carve out something to show others the same path in a saner manner, using technical skill. But that initial leap very often doesn't take place in a purely technical framework, and is fuzzy and imprecise. Usually, it takes many drafts of an idea until a suitable technical framework is found.

"Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere."

-- W. S. Anglin

https://en.wikiquote.org/wiki/Mathematics#A

[_qc3o]

But you see how you are making my point. Mathematicians as a bunch are very creative and precise at the same time. Going back to my point about the article setting up a false dichotomy. Mathematicians study formal systems which are as precise a thing as humans have managed to make so far. Even if their initial exploration of new territory is not precise it is always in the context of formal systems.

I think the problem is the article is trying to put vaguely defined terms on some spectrum and the tension he is talking about is just an artefact of his own special mapping. Really there is no tension and this mapping is not canonical because the terms are not well-defined. So comparing accuracy, creativity, etc. on a single spectrum doesn't make sense. Mathematics being an existence proof that you can do both things at the same time.

[TheOtherHobbes]

Not sure why you're getting downvoted. I had the same reaction. It's like comparing cheese with bedsprings.

The most powerful ideas in science are concise, unexpected, predictive, and mathematically rigorous.

Creativity generates the unexpected. It opens up new spaces for exploration.

But it's easy to confuse it with mimicry, which is formulaic repetition of existing practices in existing spaces that may or may not have useful outcomes.

Einstein's development of SR and GR was creative. Weber's gravity waves claims were exploring a space that Einstein created, not generating a new space.

There's nothing wrong with mimicry - it's an essential process in human culture. We think of the arts as creative, but in fact most art is made by somewhat modified mimicry of existing tropes, not by outstanding originality.

Original creativity is much rarer and very different phenomenon. It expands human experience into spaces that weren't previously accessible at all.

[SomeStupidPoint]

It's because you and the person you're replying to have the tail wagging the dog when it comes to being precise.

Powerful ideas are made precise, but they don't spring into being that way -- their inception is usually a fuzzy mess of analogy and hunch. Similarly, if we had an idea in mathematics that didn't fit in a formal system, we wouldn't just bin it, we'd change the formal system so we could implement it. (It's not like we have a formal system, or even a formal system we think implements all the things one should. Formalisms research is a major ongoing topic only ~200 years old.)

The precision and formality are the final product, but almost no ideas would be had if we required that they met our standards for formality when they were first conceived.

Creativity happens without being precise, and precision is usually a sign that the creative portion of the work has moved elsewhere.