> 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.
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.
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).
> 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.)
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.
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?
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.
[+] [-] Nacraile|12 years ago|reply
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.
[+] [-] GregBuchholz|12 years ago|reply
http://en.wikipedia.org/wiki/Hyperreal_number
http://en.wikipedia.org/wiki/Surreal_number
[+] [-] goldenkey|12 years ago|reply
[+] [-] monochromatic|12 years ago|reply
> 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|12 years ago|reply
> 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).
[+] [-] pdonis|12 years ago|reply
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.)
[+] [-] j2kun|12 years ago|reply
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|12 years ago|reply
[+] [-] unknown|12 years ago|reply
[deleted]
[+] [-] drudru11|12 years ago|reply
[+] [-] gohrt|12 years ago|reply
Pun intended?
[+] [-] thret|12 years ago|reply
[+] [-] GregBuchholz|12 years ago|reply
http://web.maths.unsw.edu.au/~norman/papers/Ordinals.pdf
[+] [-] drudru11|12 years ago|reply