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Just because Archimedes used infinitesimals doesn't mean he was anywhere close to inventing them, which would have required constructing the hyperreals and all

Just because Archimedes used infinitesimals doesn't mean he was anywhere close to inventing them, which would have required constructing the hyperreals and all of the necessary math to do that. His proofs depended on an informal notion of infinitesimals that doesn't resemble the modern definition at all.

pdonis 13y ago
> His proofs depended on an informal notion of infinitesimals that doesn't resemble the modern definition at all. To the extent this is correct, the same goes for Newton's proofs, but he was able to do calculus in a practical sense with them. However, I'm not sure what you say is correct; didn't Archimedes give a rigorous definition of infinite numbers as numbers that are greater than all natural numbers? (I.e., a number I is infinite if I > n for all natural numbers n.) And then didn't he define infinitesimals as numbers x such that x is not zero and 1 / x is infinite?
- psychometry 13y ago
  Archimedes had that conceptual argument, but the modern definition is based on a lot of math Archimedes didn't have, including model theory and the algebra of ordered fields.
  - pdonis 13y ago
    Yes, but do you need the modern definition to do calculus, in a practical sense? I don't think so. The way things actually went, practical computations using calculus, by Newton and others, came a couple of centuries, at least, before the modern rigorous foundations were developed. So if some other development had intervened, in the sort of alternate history postulated in the article, to prevent the practical computations from being taken further, that could have prevented the modern rigorous foundations of calculus from being developed. To put it another way, if other developments had not intervened, the next couple of centuries after Archimedes could possibly have seen something like our modern rigorous foundations for calculus being developed.
- JadeNB 13y ago
  > However, I'm not sure what you say is correct; didn't Archimedes give a rigorous definition of infinite numbers as numbers that are greater than all natural numbers? (I.e., a number I is infinite if I > n for all natural numbers n.) And then didn't he define infinitesimals as numbers x such that x is not zero and 1 / x is infinite? Without disagreeing, since I am not qualified to do so, I am surprised by this claim, which I had never heard before. Do you have a reference?
  - pdonis 13y ago
    See the Wikipedia page on infinitesimals: http://en.wikipedia.org/wiki/Infinitesimal http://en.wikipedia.org/wiki/Infinitesimal In particular the first paragraph in the section "History of the infinitesimal" and footnote 4 there.

pdonis 13y ago

> His proofs depended on an informal notion of infinitesimals that doesn't resemble the modern definition at all. To the extent this is correct, the same goes for Newton's proofs, but he was able to do calculus in a practical sense with them. However, I'm not sure what you say is correct; didn't Archimedes give a rigorous definition of infinite numbers as numbers that are greater than all natural numbers? (I.e., a number I is infinite if I > n for all natural numbers n.) And then didn't he define infinitesimals as numbers x such that x is not zero and 1 / x is infinite?

psychometry 13y ago
Archimedes had that conceptual argument, but the modern definition is based on a lot of math Archimedes didn't have, including model theory and the algebra of ordered fields.
- pdonis 13y ago
  Yes, but do you need the modern definition to do calculus, in a practical sense? I don't think so. The way things actually went, practical computations using calculus, by Newton and others, came a couple of centuries, at least, before the modern rigorous foundations were developed. So if some other development had intervened, in the sort of alternate history postulated in the article, to prevent the practical computations from being taken further, that could have prevented the modern rigorous foundations of calculus from being developed. To put it another way, if other developments had not intervened, the next couple of centuries after Archimedes could possibly have seen something like our modern rigorous foundations for calculus being developed.
JadeNB 13y ago
> However, I'm not sure what you say is correct; didn't Archimedes give a rigorous definition of infinite numbers as numbers that are greater than all natural numbers? (I.e., a number I is infinite if I > n for all natural numbers n.) And then didn't he define infinitesimals as numbers x such that x is not zero and 1 / x is infinite? Without disagreeing, since I am not qualified to do so, I am surprised by this claim, which I had never heard before. Do you have a reference?
- pdonis 13y ago
  See the Wikipedia page on infinitesimals: http://en.wikipedia.org/wiki/Infinitesimal http://en.wikipedia.org/wiki/Infinitesimal In particular the first paragraph in the section "History of the infinitesimal" and footnote 4 there.