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Comment by Pacabel | Hacker News Reader

Yes, "mathematics" is a relatively narrow concept, even if it embodies a great deal of knowledge.

When it comes to mathematics, the exact form of the outcome isn't very important. Maybe it's a valid proof in one case, maybe it's a specific number in another case.

It's the degree of rigor that's important. Rigor is what separates mathematics from other fields of study.

Thinking and reasoning about abstract concepts aren't without value, of course. But if this is done without an extreme degree of rigor, it probably should not be considered to be mathematics.

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Something tells me you've never sat in on an algebraic topology class where the teacher draws a donut and a pair of pants on the board, along with a line or two, and calls it a proof. A key characteristic of mathematics is that it can be made rigorous, but we certainly don't work in full rigor day to day. A look at Principia Mathematica should demonstrate simply why that would be infeasible.

You just blew my mind. I now want to learn everything I can about algebraic topology.

Q: How can you tell there is an topologist in the cafeteria?

A: Because they can't tell their coffee cup and donut apart.

I up voted this because it made me laugh, even though I don't understand.

Fundamentally they both have only one hole through them, so under the right sort of deformations you can turn one shape into the other: They have the same topology.

As a practicing mathematician, I disagree completely. Much mathematics is done with the understanding that the ideas can be made rigorous, and the point of educating people in mathematics is to allow them to craft arguments and revise when they find holes. One can and should introduce rigor gradually over time, and get the benefits of learning to reason and doing mathematics.

When it comes to mathematics, the exact form of the outcome is extremely important. Proofs that are aesthetically pleasing are better than proofs that are not. Proofs that glean insight are better than proofs that do not. In fact, most people don't care about whether a theorem is true unless the proof provides sufficient insights.

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