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).
> 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.
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.
> 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.)
> 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?
I don't know where you got the idea that compactness is in any way relevant to the formulation of calculus. Compactness is a property of topological spaces that, to an approximation, is a generalization of sets being finite or infinite. For example, with the discrete topology, a set is compact iff it is finite. There are many related notions of compactness. In R^n, a set is compact (and sequentially compact) iff it is closed and bounded.
Compactness is important for some ideas related to calculus, but it's not related to formulating calculus. For example, if a continuous function maps from a compact space to R, then it achieves a maximum/minimum (this can be seen of a generalization that there is always a maximum/minimum of a finite set of real numbers, but not necessarily for an infinite set).
The word infinitesimals is also a tricky word to use. To a mathematician, an infinitesimal would probably mean an algebraic object that formalizes the idea of a number smaller than any positive real number. This is not what is taught in calculus or analysis classes, and is only relevant for non-standard developments of calculus.
The winner in the modern formulation in calculus is the "epsilon-delta" formulation of limits; that's what's taught in both calculus classes (at least to an extent) and analysis classes. The weird thing is that calculus is stuck with Leibniz's notation, which does, in a sense, refer to infinitesimals. I think that's what you're really thinking of (rather than compactness) as how you can formulate calculus without infinitesimals. The thing is that, save notation, this is how calculus is taught today.
> I don't know where you got the idea that compactness is in any way relevant to the formulation of calculus.
I'm referring to the existence proofs for simple integrals. While you can certainly formulate the proofs without literal compactness, I have yet to see a proof that accomplishes this without invoking a strategy with such a degree of conceptual similarity to those using compactness that I cannot, in good faith, call it a fundamentally different approach.
> Compactness is important for some ideas related to calculus, but it's not related to formulating calculus.
I'd consider existence proofs for integrals pretty darn important to the formulation of calculus.
> The word infinitesimals is also a tricky word to use. To a mathematician, an infinitesimal would probably mean an algebraic object that formalizes the idea of a number smaller than any positive real number.
Yes...
> [The use of formal infinitesimals] is not what is taught in calculus or analysis classes, and is only relevant for non-standard developments of calculus.
Yes, that's why I said it was ridiculous for the author to claim that infinitesimals were fundamental to the development of calculus.
> The winner in the modern formulation in calculus is the "epsilon-delta" formulation of limits
Prove that continuous functions on [0,1] are Riemann integrable using epsilon-delta limits but without using compactness (or anything that I could reasonably point to and claim "that's compactness, you just called it something different").
> I think that's what you're really thinking of [is Leibniz's notation]
Why would you think that? By "infinitesimal" I mean, to use your words, "an algebraic object that formalizes the idea of a number smaller than any positive real number". You assumed that I meant something different, even though you were able to define precisely what the word meant. Why?
> 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.)
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!)
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.
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.
> 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?
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.
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.
> 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?
> 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.
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.
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?
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.
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.
32 comments
[ 2.8 ms ] story [ 72.3 ms ] thread> 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.
> 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).
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.
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.
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.)
I don't know where you got the idea that compactness is in any way relevant to the formulation of calculus. Compactness is a property of topological spaces that, to an approximation, is a generalization of sets being finite or infinite. For example, with the discrete topology, a set is compact iff it is finite. There are many related notions of compactness. In R^n, a set is compact (and sequentially compact) iff it is closed and bounded.
Compactness is important for some ideas related to calculus, but it's not related to formulating calculus. For example, if a continuous function maps from a compact space to R, then it achieves a maximum/minimum (this can be seen of a generalization that there is always a maximum/minimum of a finite set of real numbers, but not necessarily for an infinite set).
The word infinitesimals is also a tricky word to use. To a mathematician, an infinitesimal would probably mean an algebraic object that formalizes the idea of a number smaller than any positive real number. This is not what is taught in calculus or analysis classes, and is only relevant for non-standard developments of calculus.
The winner in the modern formulation in calculus is the "epsilon-delta" formulation of limits; that's what's taught in both calculus classes (at least to an extent) and analysis classes. The weird thing is that calculus is stuck with Leibniz's notation, which does, in a sense, refer to infinitesimals. I think that's what you're really thinking of (rather than compactness) as how you can formulate calculus without infinitesimals. The thing is that, save notation, this is how calculus is taught today.
I'm referring to the existence proofs for simple integrals. While you can certainly formulate the proofs without literal compactness, I have yet to see a proof that accomplishes this without invoking a strategy with such a degree of conceptual similarity to those using compactness that I cannot, in good faith, call it a fundamentally different approach.
> Compactness is important for some ideas related to calculus, but it's not related to formulating calculus.
I'd consider existence proofs for integrals pretty darn important to the formulation of calculus.
> The word infinitesimals is also a tricky word to use. To a mathematician, an infinitesimal would probably mean an algebraic object that formalizes the idea of a number smaller than any positive real number.
Yes...
> [The use of formal infinitesimals] is not what is taught in calculus or analysis classes, and is only relevant for non-standard developments of calculus.
Yes, that's why I said it was ridiculous for the author to claim that infinitesimals were fundamental to the development of calculus.
> The winner in the modern formulation in calculus is the "epsilon-delta" formulation of limits
Prove that continuous functions on [0,1] are Riemann integrable using epsilon-delta limits but without using compactness (or anything that I could reasonably point to and claim "that's compactness, you just called it something different").
> I think that's what you're really thinking of [is Leibniz's notation]
Why would you think that? By "infinitesimal" I mean, to use your words, "an algebraic object that formalizes the idea of a number smaller than any positive real number". You assumed that I meant something different, even though you were able to define precisely what the word meant. Why?
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.)
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? :->)
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.
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?
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.
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?
http://en.wikipedia.org/wiki/Infinitesimal
In particular the first paragraph in the section "History of the infinitesimal" and footnote 4 there.
Pun intended?
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.
http://arxiv.org/pdf/1007.3018.pdf?origin=publication_detail
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
...(the impatient should jump to chapter 5). And maybe also:
http://web.maths.unsw.edu.au/~norman/views2.htm
http://en.wikipedia.org/wiki/Hyperreal_number
http://en.wikipedia.org/wiki/Surreal_number
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.
http://web.maths.unsw.edu.au/~norman/papers/Ordinals.pdf