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SSJPabs
2005-Mar-04, 02:14 AM
So I was wondering, who do you all think should get the credit for Calculus? While we credit Newton for it (at least in standard academia) we actually use the notation for calculus that Leibniz developed.

Also if this should go in babbling, just feel free to toss it there.

crosscountry
2005-Mar-04, 02:26 AM
as long as it works :lol:

Bozola
2005-Mar-04, 02:35 AM
http://mally.stanford.edu/graphics/leibniz.gif

scourge
2005-Mar-04, 02:49 AM
Good question--I've heard two scenarios from different sources, and would like to know how it really happened. One source told me that the two men developed Calculus totally independently and essentially simultaneously...that it was an idea whose time had come. I've also heard implications that Newton became aware of Leibnitz's work, and quietly stole it. In either case, I recall that there was a competition to solve a mathematical problem afoot at the time, and this new form of math was essential to get the right answer. Nothing like a good prize competition to spur human ingenuity...

Disinfo Agent
2005-Mar-04, 11:04 AM
Contemporary historians of mathematics seem to agree that both developed calculus fairly independently.
The fact that they had markedly different approaches to it (fluxions for Newton, differentials for Leibniz) suggests the same.

kucharek
2005-Mar-04, 12:18 PM
Seems they really developed it independendly. And when they learnt of each other, it seems Leibniz handled the case better than Newton.

Funnily, Leibniz worked for the Duke of Braunschweig-Luneburg in Hanover, who later became King Georg I. of England. Despite Leibniz' pledges, he was left behind in Hanover, as the quarrel between Newton and Leibniz had become a matter of national pride.

Leibniz' notation proofed to be much more usuable than Newton's dot-notation and so, because those on the island stuck to Newton's, continental calculus flourished, the d outplayed the .

This changed, when in the early 19th centure, in Cambridge a society was formed "to introduce the principles of pure d-ism in opposition to the dot-age [pun intended] of the University".

Harald

jfribrg
2005-Mar-04, 02:28 PM
The book "A History of the calculus and its conceptual Development" (http://www.amazon.com/exec/obidos/tg/detail/-/0486605094/qid=1109945340/sr=8-1/ref=sr_8_xs_ap_i1_xgl14/104-4515275-3175935?v=glance&s=books&n=507846) gives a good, if overly drawn out, explanation of the whole matter. I voted for both, although this doesn't do justice to all the people who laid the groundwork. The development of the Calculus did not start nor end with Newton and Liebnitz. It would be another 200 years before all of the theoretical justification for the "method of fluxions" or differentials would be completed. It all depended on exactly the definition of a real number and a continuous function. If these definitions seem obvious, then try making sense of these 28 pages (http://www.badastronomy.com/phpBB/viewtopic.php?t=17953) :D

I do feel that Newton in his lifetime gave Liebnitz the shaft, although Liebnitz got his revenge from the grave when the English refused to use his superior differential notation and as a result, for the next 2 centuries were left far behind the rest of the continent in mathematical prowess.

kucharek
2005-Mar-04, 02:31 PM
I do feel that Newton in his lifetime gave Liebnitz the shaft, although Liebnitz got his revenge from the grave when the English refused to use his superior differential notation and as a result, for the next 2 centuries were left far behind the rest of the continent in mathematical prowess.

Are there any fields of mathematics where the mathematicians are looking for a better notation of their problems? As the d-ism vs dot-age example shows, a good notation is a powerful tool.

Harald

jfribrg
2005-Mar-04, 02:44 PM
I do feel that Newton in his lifetime gave Liebnitz the shaft, although Liebnitz got his revenge from the grave when the English refused to use his superior differential notation and as a result, for the next 2 centuries were left far behind the rest of the continent in mathematical prowess.

Are there any fields of mathematics where the mathematicians are looking for a better notation of their problems? As the d-ism vs dot-age example shows, a good notation is a powerful tool.

Harald

The problem is that nobody knows that the current notation is inferior until some genius improves upon it. Two examples that come to mind are Vectors and determinants. The equations of orbital mechanics are rather cumbersome unless you use vector notation. You get the same results, but it is much easier with vectors. The same holds true with determinants, although an example eludes me at the moment. I will check some of my math books when I get home and pad my post count with an example later.

Disinfo Agent
2005-Mar-04, 02:48 PM
I voted for both, although this doesn't do justice to all the people who laid the groundwork. The development of the Calculus did not start nor end with Newton and Liebnitz.
Very true. Here's another more recent book that tells the story of the creation of calculus, although it's about mathematics in general: A History of Mathematics: An Introduction (http://www.amazon.com/exec/obidos/tg/detail/-/0321016181/qid=1109947421/sr=1-5/ref=sr_1_5/104-0669509-8751119?v=glance&s=books), by Victor J. Katz.


The same holds true with determinants, although an example eludes me at the moment. I will check some of my math books when I get home and pad my post count with an example later.
The cross product formula for 3-dimensional vectors, and the formula for the curl of a 3-dimensional vector field, are much easier to grasp with determinant notation, although purists frown at those formulas since they "aren't really determinants".

SSJPabs
2005-Mar-04, 06:02 PM
Thanks for indulging me with discussion. I first learned about this calculus discrepency in a 17th Century philosophy class where as part of the general overview of Leibniz at the beginning this was mentioned. My instructor also went on to say that when Leibniz went to I guess Cambrige to argue this, Newton was the chair of the department so it was of course, no surprise at the end result. Maybe I'll pick up one of these books to read later on when I have more time even if I don't strictly speaking, known calculus very well.

Normandy6644
2005-Mar-04, 06:14 PM
I've always heard that Newton came up with it first, but Leibniz' notation was superior. I voted for Newton simply because of the time issue, though I'm sure you could really argue either side.

Incidentally, I think there is increasing evidence that Archimedes at least knew about infinitessimals, or something related. There was a recently discovered document of his that showed he was able to calculate an infinite sum. Also, Kepler came really close was he was investigating the volumes of various solids.

Disinfo Agent
2005-Mar-04, 06:35 PM
Archimedes used infinitesimals as an informal heuristic, as did other authors after him, and probably most ancient Greek mathematicians. However, he was not able to put them on a solid theoretical framework, and for that reason he never used them in rigorous proofs.

Kepler used infinitesimals to solve particular problems, but Newton and Leibniz were the first to come up with a general method for the manipulation of infinitesimal quantities.

nixiel
2005-Oct-20, 08:50 PM
All I have to say is that, while both came up with the idea, Liebnitz's notation alone has proven significantly more useful.
(My calculus teacher pulled up this thread on the projector at the end of class. My honor compelled me to vote :p)

<3 to the Math,
Kate

George
2005-Oct-20, 10:37 PM
There is a summary here (http://www.angelfire.com/md/byme/mathsample.html).

It's my understanding Newton had the basics down sooner than Leibniz. I thought much, much sooner, but maybe not.

Newton worked with limits and practical applications; Leibniz more abstract.
Newton, apparently, was the first to state the fundimental theory, and first to apply integration and differentiation.

Since the two wrote to each other, Leibniz was accused of plagarism, but exonerated after his death.

It's a tough call. Leibniz was actually first to publish (1684) with an explanation in 1686; vs. Newton's 1687 publication.

snarkophilus
2005-Oct-21, 06:52 AM
Are there any fields of mathematics where the mathematicians are looking for a better notation of their problems? As the d-ism vs dot-age example shows, a good notation is a powerful tool.


Probably the best (and best known) example of the usefulness of good notation is the development of Feynman diagrams. Things were very ugly in the physics world before that, with very complicated math required to solve simple problems. Now we can just draw a bunch of arrows and see results almost immediately.

I would really like to see shorthand notations for a lot of values that can't be solved analytically, but whose numerical approximations are accurately known. There are a lot of integrals that come up all the time, and the answer is always the same, but there are no standard names for those values. They're not as useful as pi, perhaps, but I still think that they deserve some recognition. (Something like integral( e^(-x^2) ) from 0 to infinity is a good example.)

For calculus, I think that credit should be shared by more than just those two. Like pretty much every other discovery (relativity, for instance), there was a lot of existing knowledge just sitting there already, waiting to be formally put together. Kepler should get a little credit. Archimedes probably should, too. And many people after Newton and Leibnitz should also be recognized.

As to Newton publishing after Leibnitz, that's probably just because Newton hated publishing, and probably would never have published anything if people hadn't constantly pushed him to do so.

George
2005-Oct-21, 12:40 PM
As to Newton publishing after Leibnitz, that's probably just because Newton hated publishing, and probably would never have published anything if people hadn't constantly pushed him to do so.
Yes. The story of Hooke's bet with Halley (and a 3rd person) regarding Kepler's elliptical orbits is very interesting, if not amusing. Supposedly, Halley asked Newton why planetary orbits were elliptical, and Newton said it was becasue he had calculated them. However, Newton could not find his work. He was encouraged to get busy writting. The result was Principia. At least, that's how I heard it.

Apparently, Newton's work went back to 1666 (where he had notes from his legendary apple event), though he did not publish Principia till 1687.

Grey
2005-Oct-21, 01:21 PM
There are a lot of integrals that come up all the time, and the answer is always the same, but there are no standard names for those values. They're not as useful as pi, perhaps, but I still think that they deserve some recognition. (Something like integral( e^(-x^2) ) from 0 to infinity is a good example.)Isn't this one just the square root of pi over two?

Nowhere Man
2005-Oct-21, 01:31 PM
So I was wondering, who do you all think should get the credit for Calculus? Credit? How about "Blame" instead? Calcuseless was the most miserable 4 quarters I had in college, not to mention precalcuseless in high school before, and a quarter of diffy scr..., er, Q, afterwards. It was required for my major (CS) and I've never used it since.

Fred

hhEb09'1
2005-Oct-21, 02:25 PM
Credit? How about "Blame" instead? Calcuseless was the most miserable 4 quarters I had in college, not to mention precalcuseless in high school before, and a quarter of diffy scr..., er, Q, afterwards. It was required for my major (CS) and I've never used it since.I haven't got much out of those hours and hours of square dancing, track and field, biology lab, and yearbook staff either :)

Are there any fields of mathematics where the mathematicians are looking for a better notation of their problems? As the d-ism vs dot-age example shows, a good notation is a powerful tool.Einstein wasn't a mathematician, but there's the Einstein tensor notation...

Eroica
2005-Oct-21, 04:10 PM
I'd vote for Archimedes. He was so far ahead of his time, and we probably have only a small percentage of his works...

Disinfo Agent
2005-Oct-21, 06:03 PM
There is a summary here.Your link isn't working.


It's my understanding Newton had the basics down sooner than Leibniz. I thought much, much sooner, but maybe not.

Newton worked with limits and practical applications; Leibniz more abstract.Careful. In a sense, both of them worked with limits; in another, neither of them did. The notion of limit was only formalised by Cauchy, in the next century.

It's true that Newton wrote a passage where he was clearly grasping for a notion of "limit", but he never made use of it in his work (although his physical interpretation of fluxion has some similarities with the idea of limit), and his definiton is, at best, vague.

snarkophilus
2005-Oct-22, 09:01 AM
Isn't this one just the square root of pi over two?

Heh heh yes... a poor example. You know what I meant. :)

rahuldandekar
2005-Oct-22, 09:36 AM
Well, I guess, if there are two similar independent discoverers, not very distant in time, then we can credit both of them. Probably one of the ancients had got it, but since we do not know whether or not he had, there's no use for idle speculation.

Disinfo Agent
2005-Oct-22, 02:50 PM
Heh heh yes... a poor example. You know what I meant. :)Well, I don't. You wrote:


I would really like to see shorthand notations for a lot of values that can't be solved analytically, but whose numerical approximations are accurately known. There are a lot of integrals that come up all the time, and the answer is always the same, but there are no standard names for those values. They're not as useful as pi, perhaps, but I still think that they deserve some recognition. (Something like integral( e^(-x^2) ) from 0 to infinity is a good example.Did you mean the error function (http://mathworld.wolfram.com/Erf.html)? Because that one does have a shorthand notation, and a standard name!

worzel
2005-Oct-22, 07:00 PM
I first discovered calculus, and so did my wife!

snarkophilus
2005-Oct-23, 08:07 AM
Well, I don't. You wrote:

Did you mean the error function (http://mathworld.wolfram.com/Erf.html)? Because that one does have a shorthand notation, and a standard name!

I did not mean that. I simply chose a bad example (because I am bad at choosing examples). I mean that there are a number of constants (or other functions) that don't have names, but which show up with some frequency. I suppose you can invent names for them as they arise, but I always hate doing that because anyone who later looks at my work might have his own, different, name for it.

Now that I think of it, what I really want is a nice big database of functions/constants and common names and abbreviations for them. That way, when someone writes a book and wants to call his constant for bimolecular combination k, he can look in the database and say, "Hey! A gazillion people have already used this letter before, and the context in which I'm using it is ambiguous! I will use k sub b instead! Oh dear... that is usually the Boltzmann constant. How about k sub bc? Perfect!"

Really, most functions don't have analytic solutions to their integrals. A lot of the simple problems have named solutions (like the error function), but anything with more than a couple of terms usually does not. So it would be nice to have a good, consistent way to name all of those functions and the constants you get when you integrate on [0,1], [0,infinity), (-infinity, infinity), etc, as the need for fast lookup of those values dictates.

(When I see erf(x), I always -- always -- think 2.7182... * radius * f(x). They should re-name that one. Just for my sake. I've long wanted to see someone find a nice analytic form for that one, too... I know it can't be done with sqrt/sin/cos/etc, but maybe with some Watson-esque analysis something that didn't require numerical methods could be found.)

(Some Watson for the math geek in you: http://mathworld.wolfram.com/WatsonsTripleIntegrals.html )

hhEb09'1
2005-Oct-23, 05:04 PM
Now that I think of it, what I really want is a nice big database of functions/constants and common names and abbreviations for them.I think those sort of lists exist. Maybe you're just not as familiar with the literature of the particular emphasis--you didn't recognize the error function, after all.

(When I see erf(x), I always -- always -- think 2.7182... * radius * f(x). They should re-name that one.It must drive you crazy when someone says "hi"
Just for my sake.:)

snarkophilus
2005-Oct-24, 12:39 AM
I think those sort of lists exist. Maybe you're just not as familiar with the literature of the particular emphasis--you didn't recognize the error function, after all.


Perhaps (probably), but even with spherical coordinates, which is a pretty elementary concept, there can be confusion as to what the symbols phi and theta mean. It depends on the author, and it can be very confusing.

Even with Cartesian coordinate systems you'll see some people use the y axis as up and the z axis as depth, while others use z as up and y as depth. It doesn't usually matter, but sometimes it does, and there should be a standard, or else a standard method of denoting which system is being used.




It must drive you crazy when someone says "hi":)


You know it! ;)

George
2005-Oct-24, 01:15 PM
Your link isn't working.
Sorry. I fixed it and it still was a problem. Oddly, it would work through Google but not from here. I tried at various times and two days, too. However, it still could have been my computer for some reason. Anyway, it seems to work now.

Here (http://www.angelfire.com/md/byme/mathsample.html) again is the link that made mention of limits. In googling elsewhere I found little else regarding the limits issue and suspect your assesment is correct.


BTW, WELCOME (belated) Nixiel! :clap:

01101001
2006-Sep-23, 04:18 AM
Leibniz butter cookies (http://www.bende.com/ecommerce/shopdetails.cfm?p=139&cat=20) or Fig Newtons (http://www.nabiscoworld.com/./Brands/ProductInformation.aspx?BrandKey=newtons&Site=1&Product=4400000527)?

rahuldandekar
2006-Sep-23, 12:54 PM
Hahahaha... the Newtons seem better intuitively.

Ivan Viehoff
2006-Sep-27, 02:15 PM
Neither of them. Basic integral calculus goes back to the Greeks, specifically Eudoxus, around 200BC, and later developed by Archimedes. Infinitesimals and differential equations were studied as early as 500AD in India.

Moving on a millenium, a textbook called the Yuktibhasa written in 1530 in Malayalam (the main language of the SW Indian state of Kerala) contains material on convergence (generally attributed to Cauchy three centuries later), differentiation, integration, "Taylor" series expansions, etc. It was written by a man called Jyesthadeva, but is a compendium of the work of many others who preceded him.

Ignorant of Indian developments, even in Europe N&L can't claim too much. Fermat worked out the basic stuff that most of us learn as calculus at school: method for location of maxima and minima, how to integrate a polynomial function.

Newton and Leibniz's main contribution (for which they can claim independent discovery) is the Fundamental Theorem of Calculus, the one that says differentiation and integration are inverses of each other. What this did was facilitate indirect methods of calculating integrals, thus giving rise to the great calculating machine of integral calculus. This was the breakthrough needed to expand the field, and greatly increased the usefulness of calculus to people needing to make calculations. That is the sense in which they are considered the fathers of calculus. Given that Leibniz notation was far more useful for that purpose, and British mathematics floundered for the next couple of centuries due to the folly of trying to be Newtonian rather than Leibnizian, I think Leibniz can claim the greater practical contribution. This does not detract from Newton, whose claim to fame lies through his far greater contribution to physics.

hhEb09'1
2006-Sep-27, 02:23 PM
Neither of them.

::snip::

Newton and Leibniz's main contribution (for which they can claim independent discovery) is the Fundamental Theorem of Calculus, Another way of saying that though, is that they invented calculus. There's a reason that it is called the fundamental theorem :)

rahuldandekar
2006-Sep-27, 02:27 PM
I agree, Ivan. But the main thing, I think is independent discovery. The Indians, Newton, Leibniz... all discovered calculus independently.

The Leibniz notation is a gem. Newton's method, though, is more intuitive, or so I feel. But note that both Newton or Leibniz failed to make calculus rigorous. The infinitesimals confused scientists for generations.

Jim
2006-Sep-27, 02:50 PM
When I started taking calculus, the general opinion was that Newton "invented" it, though I never heard the term "credit" associated with it... "blame," yes.

hhEb09'1
2006-Sep-27, 02:56 PM
When I started taking calculus, the general opinion was that Newton "invented" it, though I never heard the term "credit" associated with it... "blame," yes.In other words, you didn't receive any credit for the class? :)

Jim
2006-Sep-27, 04:48 PM
In other words, you didn't receive any credit for the class?

You're joking, but you reminded me...

My first two years of college were at (what is now) UT Arlington before transfering to The UT. At Arlington, I took a semester titled Differential Equations. When I transferred, they wouldn't give me credit for the DE course (which I had passed, so there!) because UT didn't "have an equivalent."

They told me to take Calculus II... which proved to be exactly the same as the DE course I had taken.

Well, I passed it, too, with the same grade. I probably could have improved my grade, but I was mad and bored and cut class a lot.

I blame Newton.

01101001
2006-Sep-27, 05:17 PM
Leibniz butter cookies (http://www.bende.com/ecommerce/shopdetails.cfm?p=139&cat=20) or Fig Newtons (http://www.nabiscoworld.com/./Brands/ProductInformation.aspx?BrandKey=newtons&Site=1&Product=4400000527)?

Some say the Fig Newton is named after Isaac Newton, but most attribute its etymology to the city of Newton, Massachusetts.

Leibniz cookies, though, are named after the mathematician -- if the package is to be believed. Oh, yeah, I've got some in my hands. Tres cool. You see, I discovered them a few days ago during a trip through a local Chinese supermarket. Maybe Germans or Europeans, also, are very familiar with them. They are not found in typical California supermarkets, but the Chinese market had 'em -- right in the cookies-of-the-world aisle. The Chinese seem to be big on fancy imported cookies. I doubt most USAns have heard of Leibniz biscuits.

I laughed when I saw them and grabbed a package, thinking it would be fun to serve them, along with Fig Newtons, at my next meeting of nerd friends and joke about it.

I get home and read the package and the joke's apparently on me.


The Leibniz Story
In 1891 Hermann Bahlsen, the company's founder, created these butter biscuits and named them after Gottfried Wilhelm Leibniz, the most famous citizen of his hometown of Hannover. [...] Leibniz is one of the oldest trade names in business today.

Still haven't tried them. They must be pretty good, though, if Bahlsen's been making these butterkeks for 115 years. I'm saving the package for a special geek-occasion. I'll let you know then which I prefer.