
Far From 'Infinitesimal': A Mathematical Paradox's Role In History - thejteam
http://www.npr.org/2014/04/20/303716795/far-from-infinitesimal-a-mathematical-paradoxs-role-in-history
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Nacraile
> 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
Right. Infinite is a procedure that can be taken to be a value (with the right
context.)

~~~
anaphor
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
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)

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monochromatic
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
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
> 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
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
> 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.)

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j2kun
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
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
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.

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drudru11
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.

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gohrt
> "What you have to build now is some space where dissent can be allowed,
> within _limits_ at least,"

Pun intended?

~~~
thret
With mathematicians, puns are always intended.

~~~
mjcohen
Especially if inadvertent.

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GregBuchholz
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
Thanks for that link. I too find the orthodoxy of infinity today a bit hard to
take.

