Zyzzyva will be interested in this one.
My personal feeling is that it would have lots of knock-on effects, but is hard to do: arabic numerals are so patently superior to roman ones that it's hard to see how this could be achieved with any kind of peaceful east-west contact whatsoever.
India in general tended to be pretty advanced in
Mathematics. Not that I mean to denigrate B von A- I think this is an interesting and fresh POD but I just took exception to the idea that mathematical developments of this sort were a product of the Enlightenment. The Enlightenment was a very specific reaction to the European context.
Yeah, but it's still a Hero of Alexandria / Chinese woodcut press scenario: whatever else you can say about the Enlightenment, it took these ideas and ran like hell with them.
One thought. The ancient Greeks managed a pretty sophisticated mathematical understanding without a zero. Including the use of pi.
Ehhh, their understanding of pi was pretty sketchy at best. Well, maybe not Archimedes', but he accomplished amazing things
despite the numbers he was shackled with, not because of them.
Brother!
Arabic numerals themselves do not save time; it is the creation of a method of digital representation. More specifically, it requires a concrete way of representing rational numbers (i.e., numbers that can be represented as the ratio of two integers).
For example, using some modern notation, we can represent any finite decimal as the ratio of two numbers:
.1 = I/X
.2 = I/V
.3 = I/III
.4 = II/V
.5 = I/II
.99 = XCIX/C
22/7 = XXII/VII
and so on and so forth. It is from this basis we can do most of our needed calculations.
You're looking at this too much as a formalist and not enough from a historical perspective. Roman numerals
suffice but they're an absolute bitch to work with. I suppose you
can do anything decimals can do using just roman numerals and... continued fraction representation, I guess?... but it would be an
ungodly pain in the neck. Math would be technically possible but would go nowhere - witness Archimedes: he was as smart as Newton, certainly as smart as Leibniz, and he got 90% of the way to inventing calculus in the 220s BC - but he
didn't, because the disaster of a number system he had to use just had too much overhead to use properly.
Yes and no; what evolved from the Enlightenment was the search for full mathematical formalism and simplification. It wasn't until the late 19th and early 20th centuries that such formalism was fully realized (mostly through the efforts of
Hilbert and his ilk, although the list of mathematicians who contributed to the rise of formalism is a long one). Such searches originated from attempts at the removal of
Euclid's Fifth Axiom, which eventually lead to the creation of Hyperbolic Geometry. Once hyperbolic geometry was developed fully, it became apparent that the underlying basis of mathematics needed to be explored.
Huh? No, that was the big thing of the 19th C. Between the invention of the calculus and about 1820 or so nobody did formalism: that came about because of all the problems people using, eg, naive "infinitesimal" calculus ran into. Claiming that the quest for formalism existed before then is reading far too much direction into history to be plausible.
Mostly this is due to historical factors; people sought not to overturn Euclid, but to vindicate his axiomatic system by constructing such a system that the fifth axiom is unneeded (and thus the entirety of the universe is founded on Euclidean Geometry, which is nice and easy to comprehend). With the developments in
absolute geometry from the 17th to 19th century, mathematics began probing its underlying foundations with renewed vigor.
Where are you getting these dates from? Saccheri didn't publish until 1733, and there was a century of dead space between him and Lobachevsky. Again (except the doesn't-really-count-cuz-he-backed-off Saccheri) it's all 19th C.
This area is probably the most difficult to probe; with a sufficiently rigorous mathematical base (namely those axioms of
set theory that allow us to arrive at the reals [We'll just use
ZFC since that's the most commonly taught axioms]), we can arrive at the real numbers and thus calculus. Although calculus development is twofold;
Newton developed calculus mostly from physical observations.
Liebniz approached the subject from a much more abstract (that is not to say that he approached the subject with sufficient rigor).
Of course he didn't approach it with sufficient rigour, the
concept of sufficient rigour in modern terms didn't exist until Cauchy!
...Huh, I meet another mathematician and immediately begin tearing him a new one. The nature of the internet, I suppose.
The history is true; I was arguing from the reverse (i.e., establishing our axioms, we arrive at calculus). My argument more explicitly stated is simply that we will arrive at this point once we have the proper set of axioms, regardless of however we choose to represent numbers and concepts. Of course, reaching that point is a separate discussion, and an interesting history.
But it's really, really, implausible; with the exception of a couple of really modern fields almost all math starts out with some naive version and doesn't get formalized until it starts to run unto problems as a result. You can't just run that backwards: like how it's
possible to build single-wing planes with 1880s tech but it didn't happen because they needed the experience creating grotesque kites-with-engines first before they could start paring the design down.
I have studied the continuum, as well as real analysis. I use Weierstraß formulation all the time, for the obvious reasons. Although I don't so much "take it for granted" as "it will be a waste of my time to use anything else, and a waste of breath justifying this other method." Also, it is what everyone else uses (in classes), so why bother utilizing anything else (and taking extra time that never is available)?
Because if the Wright brothers had had to build a plane that could carry 250 people across the Atlantic in six hours from the first go, we'd still be using Zeppelins.
All true. The history of these developments should be taught within math classes, but it is rarely thus. I bothered reading these histories; I failed to be sufficiently thorough in my explanation (another mathematical capital crime, unless its a proof by intimidation
).
Haha! My status as resident mathie (historian) is still intact!
True, but neither does any integral representation (like Arabic numerals). It takes a more nuanced approach to develop this knowledge (like the convergence of rational series), but such developments do not appear to be dependent on the number system implemented, rather the number of estoric examinations of the behavior of certain mathematical objects. After all, once the Romans can construct s = SUM(i=0,infinity, 1/(2^n) ), then they can observe that s-> 2.
But they won't, because doing so in the Roman system is far far far more trouble than it's worth, and generalizing to, eg, cauchy sequences or whatever is such a leap as to be laughable. Besides, even the concept of using a symbol for a variable was not really common practice until the 1500s or so, IIRC (Diophantus did it, but only for the number being solved for and thus only for one number at a time. Plus nobody else followed his lead for a millennia or so).
I think you've been seriously let down by the lack of math history courses at your university. (So have I, really, I just read a hell of a lot about it.) The fact that nowadays we can develop all of this stuff so directly and abstractly does not mean that anyone could have at any time. We have the benefit of centuries of earlier work that enable us to see why, eg, the epsilon-delta definition of continuity is so much better than the intuitive version, and the mindset that developed out of that history, which is quite a recent development - if you tried to teach a modern calc-101-for-math-majors course in 1800, you'd be laughed out of town; hell, there are parts that would seem awkward and useless in 1880. Not because you're not teaching real math, but just because it's so dependent on modern mathematical mentality that it would seem bizarre and pointless. I'm a platonist; I think the conclusions math comes to are true - but it's still based on an enormous pile of contingent historical development, and you can't just kick that out from under it and claim "the conclusions are true, so they could still be developed".
...Er, by which I mean welcome, fellow mathie.
