Far From 'Infinitesimal': A Mathematical Paradox's Role In History
npr.org
npr.org
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:
> 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.
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.
Pun intended?