5 ms·

Far From 'Infinitesimal': A Mathematical Paradox's Role In History

42 points by thejteam 12y ago

monochromatic 12y ago
Some interesting history, but that last section seems entirely unwarranted: > What might have happened if the Jesuits and Hobbes had won out? What if the infinitesimal had been successfully stamped out everywhere? > "I think things would have been very different," Alexander muses. "I think if they had won, then it would have been a much more hierarchical society. In a world like that, there would not be room for democracy, there would not be room for dissent." > And more materially, he says, we might not have all the modern fruits of this kind of math. "Modern science, modern technology, and everything from your cell phone to this radio station to airplanes and cars and trains — it is all fundamentally dependent on this technique of infinitesimals." It's pretty unthinkable that we still wouldn't have calculus at this point. And the conclusion about democracy feels very handwavy.
- jjoonathan 12y ago
  Yep. Ctrl+F "compact" > Not found Modern calculus doesn't depend on infinitesimals. The concept of "compactness" is the technical solution that lets you formulate calculus without them -- it's typically only taught to math majors because infinitesimals are less awkward to do algebra with: we can now prove that the shortcut works, so why bother with the long way unless you have good reason? They allude to this fact: > Today, mathematicians have found ways to answer that question so that modern calculus is rigorous and reliable. but they bury this scant acknowledgement behind the linkbaity overstated conclusion it contradicts: > Modern science, modern technology, and everything from your cell phone to this radio station to airplanes and cars and trains — it is all fundamentally dependent on this technique of infinitesimals Besides, many of the big-name ancient Greek philosophers used inconsistent definitions of infinity or assumed properties of infinity to arrive at ridiculous paradoxes and conclusions. They look utterly silly to someone with the slightest bit of modern mathematical training in the notion of infinity, not unlike Newton and his alchemy look to a modern chemist. The Jesuits' misgivings about infinitesimals were entirely understandable in the context of wanting to avoid the same fate (not to mention wasting their time).
  - JadeNB 12y ago
    > Modern calculus doesn't depend on infinitesimals. The concept of "compactness" is the technical solution that lets you formulate calculus without them -- it's typically only taught to math majors because infinitesimals are less awkward to do algebra with: we can now prove that the shortcut works, so why bother with the long way unless you have good reason? They allude to this fact: This seems to be an unusual claim, especially since the real numbers, the traditional domain of calculus, are not compact. To be sure, they are locally compact, but I'm not sure that I would say that this is "what makes calculus work". That honour seems better to belong to completeness.
    - jjoonathan 12y ago
      Completeness gives you limits, the simplest workable definitions of the derivative and integral, and the differentiability of many important classes of functions. It gets you 3/4 of the way there. It also doesn't fundamentally require any particularly sophisticated definitions of infinity, even though in a modern curriculum this is usually when infinity is first properly discussed. Yes, we stick the infinity symbol below many a limit, but re-phrase the limit into epsilon-delta language and the infinity disappears along with the associated philosophical difficulty. The extra 1/4 is where the trouble starts. Proving that, e.g. piecewise continuous functions are integrable, is by far the most philosophically complicated bit of elementary calculus. It's the place where the problem of breaking a uncountably infinite domain into pieces and putting it back together again enters the picture. Compactness is the modern answer. The historical mechanisms for rigorously solving the problem (that I've seen) look more or less equivalent to proving the compactness (or "almost" compactness) of their domain. > This seems to be an unusual claim, especially since the real numbers, the traditional domain of calculus, are not compact. R might be the traditional domain of derivatives, but it's certainly not the traditional domain of integrals. Integrals are an important part of calculus, so I don't see how you can claim that R is the traditional domain of calculus.
      - JadeNB 12y ago
        > Completeness gives you limits, the simplest workable definitions of the derivative and integral, and the differentiability of many important classes of functions. Nit-picking can be continued endlessly, but, as a final salvo, the definitions of limits, derivatives, and integrals don't depend on completeness (which is a good thing in the first case, since the (uniform-space, as opposed to order-theoretic) notion of completeness depends on that of limits). As you say, the existence of certain limits and integrals needs completeness. (I don't know off the top of my head any derivatives that one needs completeness to compute—rather nice consequences of derivatives, like that only constant functions have 0 derivative—but that's probably my ignorance, rather than a genuine lack.)
- pdonis 12y ago
  > It's pretty unthinkable that we still wouldn't have calculus at this point. I'm not so sure. Several ancient Greeks, notably Archimedes, came tantalizingly close to the theory of infinitesimals. If that had panned out, we could have had calculus two thousand years sooner. Since we didn't, a few hundred years' more delay in a slightly altered history doesn't seem all that unlikely. (I agree that the stuff about democracy is unwarranted, though.)
  - PeterWhittaker 12y ago
    Not to mention ideas that bore striking parallels with QM. It's been a while and I cannot recall the book that suggested this to me, so I have no hope of providing a reference, but after Democritus and the atom, there were arguments around the might-bes and maybes of atomic time and space. I love my life and times, I really do, but there are times I wonder what might have been had the theory-and-practice Greeks prevailed over the copy-and-perfect Romans. Of course that whole Dark Ages thing was a bit of a bitch as well, but.... (Bought Infinitesimals just before posting, looking forward to bedtime reading tonight!) (Yes, geek. Why do you ask? :->)
    - judk 12y ago
      Are you thinking perhaps of Scott Aaronson's Quantum Computing since Democritus? Even if not, that's a great book, free online.
      - PeterWhittaker 12y ago
        No, a much older book that was much more on the history of mathematics. My undergrad was in physics and I recognized a lot of parallels between modern math and QM on the one hand and what the Greeks were doing on the other. Aaronson's book is in my pile. I love his work, but that for popular audiences and the more technical stuff. While we're here: Read Valiant's Evolvability.
  - psychometry 12y ago
    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.
gohrt 12y ago
> "What you have to build now is some space where dissent can be allowed, within limits at least," Pun intended?
- thret 12y ago
  With mathematicians, puns are always intended.
  - mjcohen 12y ago
    Especially if inadvertent.
Nacraile 12y ago
> You can keep on dividing forever, so every line has an infinite amount of parts. But how long are those parts? If they're anything greater than zero, then the line would seem to be infinitely long. And if they're zero, well, then no matter how many parts there are, the length of the line would still be zero. Sigh. There is no paradox here, except to those who fundamentally fail to grasp the concept of infinity. Infinity is not a number. Multiplying length by infinity is a type error. This is nonsensical, not paradoxical. I don't buy the hand-waving argument that this somewhat arcane debate had so much impact on the course of history. Obviously nobody can prove it true or false, because the alternative outcome is unknowable. I'm inclined to ignore unfalsifiable speculation.
- goldenkey 12y ago
  Right. Infinite is a procedure that can be taken to be a value (with the right context.)
  - anaphor 12y ago
    It annoys me when people try to argue that (0.99...) != 1 because "infinity goes on forever, it's not a number!" though, without understanding the difference between a representation of a number and the number itself.
    - GregBuchholz 12y ago
      Of course that argument only applies when talking about the standard real number system. http://arxiv.org/pdf/1007.3018.pdf?origin=publication_detail http://arxiv.org/pdf/1007.3018.pdf?origin=publication_detail http://arxiv.org/pdf/0811.0164.pdf http://arxiv.org/pdf/0811.0164.pdf ...of course you can make a pretty good case that uncountable, uncomputable, unnameable, unknowable "real" numbers don't exist. http://arxiv.org/abs/math/0404335 http://arxiv.org/abs/math/0404335 ...(the impatient should jump to chapter 5). And maybe also: http://web.maths.unsw.edu.au/~norman/views2.htm http://web.maths.unsw.edu.au/~norman/views2.htm
- GregBuchholz 12y ago
deleted 12y ago
[deleted]
j2kun 12y ago
It's a bit sad to me that people who write about mathematics to a general audience often have to stoop down so far below the mathematics that their writing becomes a bad caricature. They're literally forced to talk about "dividing a line in half" as quickly and as sloppily as possible so they can get to the thing they actually want to talk about, which is the history and people involved. There may be no other way to write such a piece, or it could be that the people writing it are just not well versed enough in mathematics. Whatever it is, it makes me sad.
- marktangotango 12y ago
  Dividing a line in half resulting in an infinite set of lines that can thus be divided into an infinite set of lines defines the problem quite nicely. How would you prefer to see the concepts of the continuum and uncountable vs countable infinite presented?
  - j2kun 12y ago
    As stated it's not a problem with the continuum and uncountable vs countable (which did not exist in the historical setting mentioned!), but a problem of measurement.
GregBuchholz 12y ago
Here's another treatment: http://web.maths.unsw.edu.au/~norman/papers/Ordinals.pdf http://web.maths.unsw.edu.au/~norman/papers/Ordinals.pdf
- drudru11 12y ago
  Thanks for that link. I too find the orthodoxy of infinity today a bit hard to take.
drudru11 12y ago
Funny thing is... they stopped using infinitesimals when teaching calculus in most US math courses. I think they should be brought back into the curriculum.

42 points by thejteam 12y ago

Some interesting history, but that last section seems entirely unwarranted: > What might have happened if the Jesuits and Hobbes had won out? What if the infinitesimal had been successfully stamped out everywhere? > "I think things would have been very different," Alexander muses. "I think if they had won, then it would have been a much more hierarchical society. In a world like that, there would not be room for democracy, there would not be room for dissent." > And more materially, he says, we might not have all the modern fruits of this kind of math. "Modern science, modern technology, and everything from your cell phone to this radio station to airplanes and cars and trains — it is all fundamentally dependent on this technique of infinitesimals." It's pretty unthinkable that we still wouldn't have calculus at this point. And the conclusion about democracy feels very handwavy.

jjoonathan 12y ago
Yep. Ctrl+F "compact" > Not found Modern calculus doesn't depend on infinitesimals. The concept of "compactness" is the technical solution that lets you formulate calculus without them -- it's typically only taught to math majors because infinitesimals are less awkward to do algebra with: we can now prove that the shortcut works, so why bother with the long way unless you have good reason? They allude to this fact: > Today, mathematicians have found ways to answer that question so that modern calculus is rigorous and reliable. but they bury this scant acknowledgement behind the linkbaity overstated conclusion it contradicts: > Modern science, modern technology, and everything from your cell phone to this radio station to airplanes and cars and trains — it is all fundamentally dependent on this technique of infinitesimals Besides, many of the big-name ancient Greek philosophers used inconsistent definitions of infinity or assumed properties of infinity to arrive at ridiculous paradoxes and conclusions. They look utterly silly to someone with the slightest bit of modern mathematical training in the notion of infinity, not unlike Newton and his alchemy look to a modern chemist. The Jesuits' misgivings about infinitesimals were entirely understandable in the context of wanting to avoid the same fate (not to mention wasting their time).
- JadeNB 12y ago
  > Modern calculus doesn't depend on infinitesimals. The concept of "compactness" is the technical solution that lets you formulate calculus without them -- it's typically only taught to math majors because infinitesimals are less awkward to do algebra with: we can now prove that the shortcut works, so why bother with the long way unless you have good reason? They allude to this fact: This seems to be an unusual claim, especially since the real numbers, the traditional domain of calculus, are not compact. To be sure, they are locally compact, but I'm not sure that I would say that this is "what makes calculus work". That honour seems better to belong to completeness.
  - jjoonathan 12y ago
    Completeness gives you limits, the simplest workable definitions of the derivative and integral, and the differentiability of many important classes of functions. It gets you 3/4 of the way there. It also doesn't fundamentally require any particularly sophisticated definitions of infinity, even though in a modern curriculum this is usually when infinity is first properly discussed. Yes, we stick the infinity symbol below many a limit, but re-phrase the limit into epsilon-delta language and the infinity disappears along with the associated philosophical difficulty. The extra 1/4 is where the trouble starts. Proving that, e.g. piecewise continuous functions are integrable, is by far the most philosophically complicated bit of elementary calculus. It's the place where the problem of breaking a uncountably infinite domain into pieces and putting it back together again enters the picture. Compactness is the modern answer. The historical mechanisms for rigorously solving the problem (that I've seen) look more or less equivalent to proving the compactness (or "almost" compactness) of their domain. > This seems to be an unusual claim, especially since the real numbers, the traditional domain of calculus, are not compact. R might be the traditional domain of derivatives, but it's certainly not the traditional domain of integrals. Integrals are an important part of calculus, so I don't see how you can claim that R is the traditional domain of calculus.
    - JadeNB 12y ago
      > Completeness gives you limits, the simplest workable definitions of the derivative and integral, and the differentiability of many important classes of functions. Nit-picking can be continued endlessly, but, as a final salvo, the definitions of limits, derivatives, and integrals don't depend on completeness (which is a good thing in the first case, since the (uniform-space, as opposed to order-theoretic) notion of completeness depends on that of limits). As you say, the existence of certain limits and integrals needs completeness. (I don't know off the top of my head any derivatives that one needs completeness to compute—rather nice consequences of derivatives, like that only constant functions have 0 derivative—but that's probably my ignorance, rather than a genuine lack.)
pdonis 12y ago
> It's pretty unthinkable that we still wouldn't have calculus at this point. I'm not so sure. Several ancient Greeks, notably Archimedes, came tantalizingly close to the theory of infinitesimals. If that had panned out, we could have had calculus two thousand years sooner. Since we didn't, a few hundred years' more delay in a slightly altered history doesn't seem all that unlikely. (I agree that the stuff about democracy is unwarranted, though.)
- PeterWhittaker 12y ago
  Not to mention ideas that bore striking parallels with QM. It's been a while and I cannot recall the book that suggested this to me, so I have no hope of providing a reference, but after Democritus and the atom, there were arguments around the might-bes and maybes of atomic time and space. I love my life and times, I really do, but there are times I wonder what might have been had the theory-and-practice Greeks prevailed over the copy-and-perfect Romans. Of course that whole Dark Ages thing was a bit of a bitch as well, but.... (Bought Infinitesimals just before posting, looking forward to bedtime reading tonight!) (Yes, geek. Why do you ask? :->)
  - judk 12y ago
    Are you thinking perhaps of Scott Aaronson's Quantum Computing since Democritus? Even if not, that's a great book, free online.
    - PeterWhittaker 12y ago
      No, a much older book that was much more on the history of mathematics. My undergrad was in physics and I recognized a lot of parallels between modern math and QM on the one hand and what the Greeks were doing on the other. Aaronson's book is in my pile. I love his work, but that for popular audiences and the more technical stuff. While we're here: Read Valiant's Evolvability.
- psychometry 12y ago
  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.

gohrt 12y ago

> "What you have to build now is some space where dissent can be allowed, within limits at least," Pun intended?

thret 12y ago
With mathematicians, puns are always intended.
- mjcohen 12y ago
  Especially if inadvertent.

Nacraile 12y ago

> You can keep on dividing forever, so every line has an infinite amount of parts. But how long are those parts? If they're anything greater than zero, then the line would seem to be infinitely long. And if they're zero, well, then no matter how many parts there are, the length of the line would still be zero. Sigh. There is no paradox here, except to those who fundamentally fail to grasp the concept of infinity. Infinity is not a number. Multiplying length by infinity is a type error. This is nonsensical, not paradoxical. I don't buy the hand-waving argument that this somewhat arcane debate had so much impact on the course of history. Obviously nobody can prove it true or false, because the alternative outcome is unknowable. I'm inclined to ignore unfalsifiable speculation.

goldenkey 12y ago
Right. Infinite is a procedure that can be taken to be a value (with the right context.)
- anaphor 12y ago
  It annoys me when people try to argue that (0.99...) != 1 because "infinity goes on forever, it's not a number!" though, without understanding the difference between a representation of a number and the number itself.
  - GregBuchholz 12y ago
    Of course that argument only applies when talking about the standard real number system. http://arxiv.org/pdf/1007.3018.pdf?origin=publication_detail http://arxiv.org/pdf/1007.3018.pdf?origin=publication_detail http://arxiv.org/pdf/0811.0164.pdf http://arxiv.org/pdf/0811.0164.pdf ...of course you can make a pretty good case that uncountable, uncomputable, unnameable, unknowable "real" numbers don't exist. http://arxiv.org/abs/math/0404335 http://arxiv.org/abs/math/0404335 ...(the impatient should jump to chapter 5). And maybe also: http://web.maths.unsw.edu.au/~norman/views2.htm http://web.maths.unsw.edu.au/~norman/views2.htm
GregBuchholz 12y ago

deleted 12y ago

[deleted]

j2kun 12y ago

It's a bit sad to me that people who write about mathematics to a general audience often have to stoop down so far below the mathematics that their writing becomes a bad caricature. They're literally forced to talk about "dividing a line in half" as quickly and as sloppily as possible so they can get to the thing they actually want to talk about, which is the history and people involved. There may be no other way to write such a piece, or it could be that the people writing it are just not well versed enough in mathematics. Whatever it is, it makes me sad.

marktangotango 12y ago
Dividing a line in half resulting in an infinite set of lines that can thus be divided into an infinite set of lines defines the problem quite nicely. How would you prefer to see the concepts of the continuum and uncountable vs countable infinite presented?
- j2kun 12y ago
  As stated it's not a problem with the continuum and uncountable vs countable (which did not exist in the historical setting mentioned!), but a problem of measurement.

GregBuchholz 12y ago

Here's another treatment: http://web.maths.unsw.edu.au/~norman/papers/Ordinals.pdf http://web.maths.unsw.edu.au/~norman/papers/Ordinals.pdf

drudru11 12y ago
Thanks for that link. I too find the orthodoxy of infinity today a bit hard to take.

drudru11 12y ago

Funny thing is... they stopped using infinitesimals when teaching calculus in most US math courses. I think they should be brought back into the curriculum.