inventor of calculus
__
Newton or Leibniz??
G. A. Edgar
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--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/
Dirk Van de moortel
"__" <duswl...@hotmail.com> wrote in message news:208930b4-0cd5-429a...@q39g2000hsf.googlegroups.com...
> who is the most important inventor of calculus?
>
> Newton or Leibniz??
What's better, HP or TI ?
Sorry, G. A, couldn't resist.
Cheers,
Dirk Vdm
G.E. Ivey
John Bailey
wrote:
>who is the most important inventor of calculus?
>
>Newton or Leibniz??
Having just read Stewart's The Courtier and the Heretic: Leibniz,
Spinoza, and the Fate of God in the Modern World ;
http://www.amazon.com/Courtier-Heretic-Leibniz-Spinoza-Modern/dp/0393058980
Leibniz comes across as snobbish, sycophantic, manipulative, and
wishy-washy. However:
from: galenet.galegroup.com
"Both men had independently developed a complex system of mathematical
calculation called calculus, although each used different symbols and
notations. Leibniz's notation, however, was considered superior and
came to be preferred over that of Newton, causing a bitter controversy
between the two. Their conflict quickly became a matter of national
pride, and English scientists refused to accept Leibniz's version.
Unfortunately this stubbornness kept the English from significantly
contributing to mathematics for nearly a century."
Newton developed calculus on his way to a theory of gravitation but
apparently, his direct contribution to calculus was superceded by
Leibniz's.
Depending on how you parse your question, the answer changes.
Who was THE MOST IMPORTANT .... answer: Newton based on his multiple
contributions to science.
most important INVENTOR OF CALCULUS answer: Liebniz ignoring his
foibles and looking at his direct effect on mathematics.
The question is, would anyone in the 17th century have had a need to
comprehend calculus were is not for trying to understand Newton's
theory of gravity? (Similarly, Einstein created the market for
tensors)
math...@hotmail.com.cut
wrote:
>who is the most important inventor of calculus?
>
>Newton or Leibniz??
define "most important" !
Dave L. Renfro
> The question is, would anyone in the 17th century
> have had a need to comprehend calculus were is not
> for trying to understand Newton's theory of gravity?
Calculus was used/needed for much more than Newton's theory
of gravity. Take a look through the table of contents
of L'Hopital's 1696 calculus text:
http://gallica.bnf.fr/ark:/12148/bpt6k205444w
Dave L. Renfro
Herman Rubin
>Newton or Leibniz??
Nobody knows, and without a time machine, it will
not be known. Our present notation is essentially
that of Leibniz. However, Newton made more use of it.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hru...@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
porky_...@my-deja.com
Leibniz's notation, however, was considered superior and
> came to be preferred over that of Newton, causing a bitter controversy
> between the two.
and it *is* superior and highly suggestive and that's why it is being
used for all the formal manipulations in Calculus, the way it's
currently being taught. Furthermore, the notions of differential (and
differential form) puts Leibniz's notation on a solid mathematical
ground, without requirement to 'infinitesimals'.
Ross A. Finlayson
Infinite descent, the partition of a polygon into arbitrarily small
polygons "exactly approximating" the area of a circle (integration in
polar with differential phase angle) has been around since Archimedes.
I recommend Carl Boyer's history of the development of the calculus,
there is much more to the fabric of the notions besides an epiphany of
Newton or Leibniz.
You'll notice it's not a finite descent. Correspondingly, the
differential is not finite nor zero.
Regards,
Ross
--
Finlayson Consulting
porky_...@my-deja.com
> You'll notice it's not a finite descent. Correspondingly, the
> differential is not finite nor zero.
>
?????
Differential, dy = f'(x) dx, is a linear function, defined at some
fixed point x.
Dave L. Renfro
>> You'll notice it's not a finite descent. Correspondingly,
>> the differential is not finite nor zero.
porky_...@my-deja.com wrote:
> ?????
>
> Differential, dy = f'(x) dx, is a linear function,
> defined at some fixed point x.
Yes, but Finlayson's semi-random juxtaposition of
mathy-philosophical sounding words makes it all so
much more mysterious!
One day (it's almost true now), clicking on a random
sci.math post will be like using the postmodern
generator at
http://www.elsewhere.org/pomo/
Dave L. Renfro
Dave L. Renfro
> One day (it's almost true now), clicking on a random
> sci.math post will be like using the postmodern
> generator at
>
> http://www.elsewhere.org/pomo/
Ooops! I forgot about all the hit-and-run posts for
solutions manuals in sci.math. (At least Ross stays
around and participates. Given that some of what he
said about the history of calculus was actually on
target, that was probably a low blow on my part in
my previous post, but I couldn't resist.)
Dave L. Renfro
Ross A. Finlayson
> Ross A. Finlayson wrote (in part):
>
> >> You'll notice it's not a finite descent. Correspondingly,
> >> the differential is not finite nor zero.
> > ?????
>
> > Differential, dy = f'(x) dx, is a linear function,
> > defined at some fixed point x.
>
> Yes, but Finlayson's semi-random juxtaposition of
> mathy-philosophical sounding words makes it all so
> much more mysterious!
>
> One day (it's almost true now), clicking on a random
> sci.math post will be like using the postmodern
> generator at
>
> http://www.elsewhere.org/pomo/
>
> Dave L. Renfro
If you would, please define "mathy." Is it akin to "foodie", or
"truthy", in the sense of "afficionado" or so?
I'm sincere, I think there are true and currently nonstandard notions
of the "infinite" (as non-well-founded) and "infinitesimal" (as
particularly applicable) that are exclusive in their consideration
from a relatively non-applied notion of mathematical infinity
currently in vogue, "transfinite cardinals", where infinitesimals are
"Cholera bacilli", invisible wee beasties that cause disease, where
the Fraenkel is noted as describing reliance on transfinite cardinals
as an illness.
Ross
--
Finlayson Consulting
Ross A. Finlayson
You'll notice you have a differential, dy, defined in terms of another
differential, dx, in a combination of the Leibnizian (dy/dx) and
Newtonian (f') notations for the one-dimensional derivative.
Leibniz' notation for the integral, eg S f(x) dx, has the summation of
the infinitely many infinitesimal width evaluations of the function,
there. Leibniz' notation of the differential as ratio is useful in
algebraic cancellation.
The "naked" or "raw" differential, by itself, eg dx, dsigma, dOmega,
is a regular participant in much that is applied: a differential,
generally with respect to a vector basis, infinitesimal, patch or
element.
Ross
--
Finlayson Consulting
porky_...@my-deja.com
> > Differential, dy = f'(x) dx, is a linear function, defined at some
> > fixed point x.
>
> You'll notice you have a differential, dy, defined in terms of another
> differential, dx, in a combination of the Leibnizian (dy/dx) and
> Newtonian (f') notations for the one-dimensional derivative.
>
Seems like you aren't that familiar with the notion of differential,
and this is not the place to go through discussion of the meaning of
dy = f('x) dx,
or, more generally,
dy = f_{x_1} dx_1 + f_{x_2} dx_2 + ... .
In any case, f'(x) (or f_{x_}) is stands for derivative (in fact it
isn't even Newton notation, Newton used dot on a top), dy is a linear
function, defined at a fixed point x, and dx is dependent variable,
the same as \Delta x. The fine point is that we can carefully reason
that if both dy and dx are the variables, depending on some other,
'unknown' variable, say t, then starting with
dy = y'(t) dt
and
dx = x'(t) dt,
we'll end up with exactly the same formula,
dy = y'(x) dx.
Thus we can always think of both dy and dx as independent variables,
and write
dy = y'(x) dx
rather than
dy = y'(x) \Delta x.
So, giving the expression
dy = y'(x) dx
we can interpret either dy as dependent variable, dx as independent,
or both dy and dx as dependent on some unknown variable. Furthermore,
we can 'pick and choose' which one is dependent and which one is
independent (or treat both as independent). So having
dy = y'(x) dx
we can write
dy
-- = f'(x)
dx
which matches the Leibniz notation for derivatives, but we no longer
have to think of either dy or dx as 'infinitesimals', or we can also
write
dx 1
-- = ---
dy f'(x)
Again, that type of manipulation isn't "formal manipulation with
infinitesimals which happens to give the correct answer", but well-
defined expression, once we fixed the value of x. The thorough
discussion of differentials and the role in Calculus, esp. in
multivariable Calculus setting, can be found in books on Advanced
Calculus, such as W. Kaplan. Unfortunately, many standard Calculus
texts skim treat the notion of differential rather lightly. Of course
if you got stuck with the notion of dy/dx as defined by Leibniz, well
- as I've said, this is not the time and the place to argue. Over and
that.
Ross A. Finlayson
That's not unreasonable, except for that there is still the definition
of the differential dy in terms of the parameter variable t, or where
t is implicit, in terms of the variable x. Whether or not x and y are
vectors, of real or complex or hypercomplex numbers, in the modern
definition instead of one-dimensional, as a consequence of the nonzero
components of x forming a basis of values in the space containing all
possible values of x, dy is defined in terms of dx.
The writing of f' to indicate the first derivative, then f'' then
variously f''' or f^{(3)} and etcetera, is often referred to as the
Newton notation for a derivative, where I concur that Newton's
original notation was f dot, f double-dot etcetera, where in modern
dynamical systems those refer to derivatives in terms of a parameter t
for time, dx/dt, (d)^2 x / (dt)^2, etcetera, in functional operators,
as it was then in single dimensional real variables, where continuum
analysis is widely used for the modeling of physical systems.
The algebraic cancellation of equal product terms on either side of
the algebraic equality, as a consequence of the "chain rule", that (dy/
dx) (dx/dt) = dy/dt, for suitably nice y(x) and x(t), is at the
surface a formal manipulation of the cancellation of redundant product
terms about the quotient bar (chain rule), reordering differential
terms in ratio and cancelling equal differential terms in ratio. So,
it is "formal manipulation with differentials that happens to give the
right answer", and the differential element or patch or bit, or
plainly "differential", is neither finite nor zero, so it's
infinitesimal.
Ross
--
Finlayson Consulting