10 ms·

I don't like this attitude at all. Euclid, Archimedes, Newton, Leibniz, Riemann, and many others did a fine job correctly calculating areas, volumes, and so on

I don't like this attitude at all. Euclid, Archimedes, Newton, Leibniz, Riemann, and many others did a fine job correctly calculating areas, volumes, and so on without need of measure theory. Why is it so important to avoid the problem laid out by Banach-Tarski on the first go through the concept? We, as a species, did not even realize such a problem existed until about 1920, and yet we still managed to build stuff based on what we knew.

wolfgke 11y ago
> I don't like this attitude at all. Euclid, Archimedes, Newton, Leibniz, Riemann, and many others did a fine job correctly calculating areas, volumes, and so on without need of measure theory. The history of mathematics is a history of ever-increasing standards for proofs. "Proofs" that, say, Euler wrote down that were perfectly valid for the standards of their time would probably not accepted in todays standard. I personally also think it's plausible that today's proof standard in "typical" math papers will not be acceptable in 100 years anymore, when they perhaps will additionally have to be computer-checkable (and computer-checked).
- yaks_hairbrush 11y ago
  The issue of proof is very different from the issue of learning. I thought it was the latter being discussed.
  - wolfgke 11y ago
    Mathematics is about proofs. Learning mathematics thus means learning about proofs.
    - yaks_hairbrush 11y ago
      I do not agree with your first statement. Mathematics is about developing and studying tools to solve problems. Proofs are only one aspect of that.
      - wolfgke 11y ago
        > Mathematics is about developing and studying tools to solve problems. There clearly exist methods for solving problems that have nothing to do with mathematics. This should falsify your thesis.
        possibility 11y ago
        I think maybe you made a mistake, because the exchange reads like this: A: Knives are for eating. B: Eating is only a part of it. Knives are for handling food. A: You can handle food with a fork. Your claim is false. Specifically, "math is for solving problems" does not imply "problems are only solved by math".

wolfgke 11y ago

> I don't like this attitude at all. Euclid, Archimedes, Newton, Leibniz, Riemann, and many others did a fine job correctly calculating areas, volumes, and so on without need of measure theory. The history of mathematics is a history of ever-increasing standards for proofs. "Proofs" that, say, Euler wrote down that were perfectly valid for the standards of their time would probably not accepted in todays standard. I personally also think it's plausible that today's proof standard in "typical" math papers will not be acceptable in 100 years anymore, when they perhaps will additionally have to be computer-checkable (and computer-checked).

yaks_hairbrush 11y ago
The issue of proof is very different from the issue of learning. I thought it was the latter being discussed.
- wolfgke 11y ago
  Mathematics is about proofs. Learning mathematics thus means learning about proofs.
  - yaks_hairbrush 11y ago
    I do not agree with your first statement. Mathematics is about developing and studying tools to solve problems. Proofs are only one aspect of that.
    - wolfgke 11y ago
      > Mathematics is about developing and studying tools to solve problems. There clearly exist methods for solving problems that have nothing to do with mathematics. This should falsify your thesis.
      - possibility 11y ago
        I think maybe you made a mistake, because the exchange reads like this: A: Knives are for eating. B: Eating is only a part of it. Knives are for handling food. A: You can handle food with a fork. Your claim is false. Specifically, "math is for solving problems" does not imply "problems are only solved by math".