Re: Mathematics and science

In article <fhg19j$gq$2@xxxxxxxxxxxxxxxxx>,
Paul J Gans <gans@xxxxxxxxx> wrote:
Robert Grumbine <bobg@xxxxxxxxx> wrote:
In article <fhcm2q$dud$1@xxxxxxxxxxxxxxxxx>,
Paul J Gans <gans@xxxxxxxxx> wrote:
bernardz <bernardz@xxxxxxxx> wrote:
On Nov 13, 2:50 pm, Paul J Gans <g...@xxxxxxxxx> wrote:
bernardz<berna...@xxxxxxxx> wrote:

That's correct. Some folks, myself included, claim that there
was no "real" science until Galileo's time. Of course one
can argue about the exact date, but it is approximately
correct.

Prior to that one had plenty of practical engineering. But
aside from geometry, it was driven by experience and trial
and error.
It was only after science became established that scientific
prediction began to lead engineering.

Practical engineering is older then history. There is a marriage
between philosophy and this that makes science.

But modern science is different to anything the Greeks came up with.
To take the example previous quoted, if the world is a sphere the
Greek calculation is correct about its size, it proves no explanation.
Surely there is something different in some people saying we have a
theory on mechanics, we also have a theory on gravity, now if we take
these two theories in a mathematical equation we can show this ..... and
if we show this it proves this......Then when we do get a result we submit
it to a peer review system based on experimental proof.

This to me shows quite a fundamental different view of the world to
anything the Greeks had and if it is not Greek then where did it come
from?

I'll generally agree. Much of the distinction comes from the
development of calculus. Calculus is very different than
geometry or algebra. Consider (x - 17) = 0. Here x = 17
forever. Algebra is about calculating constant values.

Calculus is different. If I write dx/dt = 2, then x is an
ever changing value. If I know what x was at 2 AM on 12 November,
2007, then I know what value x has at any time. past or future.

This is why calculus is so important in science. With it,
one can compute where a thrown rock will fall or how long it
will take to empty a tank of water with a small hole in its
bottom.

Hm. Well, certainly a lot of what I do professionally would
be impossible without calculus. But ... there's a lot that is
scientific (or at least was when it was being done originally)
that is about constant values. Even a lot of what is _now_ done
by calculus needn't be done that way.

I wasn't arguing that there is no science without calculus,
though I might not have made that clear. What I intended to
show was that calculus opened up a whole new ball game.

I need to write more clearly. ;-(

?Frowny face with uplifted eyebrow?

Not so much a matter of your clarity as my taking a chance
to ramble. Some more:

Zero (and, lesser, systematic algebra) made computation a more
feasible thing, and problems which were matters of computation
became more doable (and, hence, interesting). Geometric solutions
existed before that (and after), but only permit attack of certain
problems. So that's one sea change in science/engineering/mathematics.

Cartesian (analytic) geometry changed the face of mathematics
and science. I hadn't realized just how much until reading
Galileo and continually reaching, mentally, for a coordinate plane.
Even though everything from Euclid remained true, and no new
truths were invented by the coordinate plane, it permitted (forced)
thinking about the entities involved very differently, which
was very helpful to inventing calculus. y = f(x) just doesn't
make much sense in Euclidean geometry with its focus on point,
line, plane, circle; but y = f(x) invites thinking about how
much change you'd get in y for a change in x. (As far as I know,
though, it didn't affect engineering much.)

Calculus, actually taking that step of being concerned about
changes in one thing in terms of changes in another, certainly
was a sea change itself. It also turns out that nature is
fond of working in terms of changes, rather than in terms of
values, so that we've gotten extremely accustomed to turning to
calculus to solve our math/science/engineering problems. Yet ...
Newton didn't nearly as much as we would; much of the Principia was
Euclidean. It took quite some time to calculus-ize science.

It was in reading _Two New Sciences_ that I started to
appreciate just how much our mathematical approach had changed
(by way of analytic geometry and calculus). But it also
showed me how much can be done without either.


--
Robert Grumbine http://www.radix.net/~bobg/ Science faqs and amateur activities notes and links.
Sagredo (Galileo Galilei) "You present these recondite matters with too much
evidence and ease; this great facility makes them less appreciated than they
would be had they been presented in a more abstruse manner." Two New Sciences
.

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