Widely accepted mathematical results that were later shown to be wrong? | Hacker News Reader

Widely accepted mathematical results that were later shown to be wrong?

mathoverflow.net

3 pointsreverse8 comments

I think most folk just don't see scientific theories in the right way. Any theory, even the most basic and self-evident one, should be be understood as "the best current explanation to a question" instead of the absolute truth. That's how we can keep moving forward and making better theories.

Mathematical results aren't the same as scientific theories.

Note also that the word "theory" isn't used in the question.

Yes, mathematicians go use proofs and logical deductions instead of experimentation. But the idea still holds, whatever conclusion they reach should be taken as the best current answer instead of absolute immutable god-given truth.

I'd be interested to know how much experience you have of advanced math.

If you think I'm wrong feel free to correct me, I'm quite willing to learn from different points of view. Badly veiled ad hominems on the other hand do not seem productive.

It's genuinely not intended to be an ad hominem, because it's genuinely not an attack. I'm trying to place your comment in a context, because without a context I can't really make sense of it.

Quoting:

> mathematicians go use proofs and logical deductions instead of experimentation.

I don't understand this - what is the word "go" doing in there?

And mathematicians don't "use proofs and logical deductions instead of experimentation" because that implies there's a choice. There is no choice - if you want to say that in Euclidean Geometry, a right-angled triangle is always such that the square of the length of the hypotenuse is the sum of the squares of the lengths of the other two sides, that's not something that can ever be verified with experiments. Similarly, a prime of the form 4k+1 for k a non-negative integer is always the sum of two integer squares. You can verify small cases, but you can't know that it's always true based on experiments.

So there's no real choice. Even so, I think most mathematicians would wonder exactly what it is that you are saying - it's really not clear to me, for one. That's why I wanted to know the context, and how much experience you have of these things, so I can try to interpret what you've said, and work out what you might mean by it.

Context matters.

Instead of experimentation in contrast to the normal scientific method other fields use was comparison I was trying to make. I never implied choice or lack thereof. As far for the syntactical error I do apologies as English is not my first language and I sure that was the basis for your confusion.

> ... English is not my first language ...

I am always in awe of anyone who engages in complex conversations in a language other than their first. In that I commend you.

You said:

> Any theory, even the most basic and self-evident one, should be be understood as "the best current explanation to a question" instead of the absolute truth.

That's absolutely true in science. But as was pointed out, the linked article is not about science, it's about mathematics.

Editing lightly - please correct me if I'm mis-representing you:

> ... mathematicians use proofs and logical deductions instead of experimentation.

Yes.

> But the idea still holds, whatever conclusion they reach should be taken as the best current answer instead of absolute immutable god-given truth.

I think nearly every mathematician would agree that (a) There is a lot of mathematics that is a human construction; (b) There are mistakes, although not many; (c) There is a fundamental difference in the nature of the uncertainty in mathematical proofs as compared with scientific theories; (d) Most of mathematics is not "God-Given Truth", but it is not just "the best current answer."

This is why I wanted to know how much experience you have of advanced mathematics, I wanted to know where you are in this spectrum.

And I'm still not really sure of what point you're trying to make. It's obvious that mathematical proofs are fundamentally different from scientific theories, so you must be saying something else, perhaps something deeper. I just don't know what it is.

Perhaps I never will.

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