Re: Mathematics and science

Paul J Gans wrote:
bernardz <bernardz@xxxxxxxx> wrote:
On Nov 13, 12:35 am, b...@xxxxxxxxx (Robert Grumbine) wrote:
In article <fh4q6j$6l...@xxxxxxxxxxxxxxxxx>,
Paul J Gans <g...@xxxxxxxxx> wrote:



bernardZ<Berna...@xxxxxxxxxx> wrote:
I have almost finished a book "The fellowship by John Gribbin"
which is about the early history of the royal society so strictly
speaking this is a bit OT for this group.

It has an interesting scene. Edmond Halley, Christoper Wren and
Robert Hooke have a discussion in Jan 1684. By now, they know the
inverse- square law for gravity. They are all convinced that
Kepler is correct and the planets go in an elliptical orbit.
What they cannot prove is that inverse-square law must produce
elliptical orbits! They are not that good mathematicians.

So Halley goes to see the man regarded as the best mathematician
in Europe - Newton with the problem. In the meeting Halley asks
Newton what curve would a planet follow under the inverse-square
law. Newton replies an ellipse and he has the proof which he
sends in a paper shortly afterwards. Everyone of course is very
excited.

(By the way I have a read a very similar story with Leibniz as
the main character not Newton.)

Now my question why should anyone believe then that mathematics
holds the key to the answer and why should anyone at this point
believe the mathematical result.

When do we start to see mathematics entering science?

We believe the results of mathematics because most of mathematics,
classical mathematics at least, is all deductive. So to deny
the mathematical conclusions is to say that logic does not work.

Mathematics, being logic, has been used for many centuries. The
rise of western (post-Greek) mathematics comes from the attempt
to calculate the future dates of Easter. This, in the end, comes
down to calculating the orbit of the moon, a problem that dominated
the works of many mathematicians down to the 19th century.

But the basic answer is that whenever an investigation became
quantitative (think of Galileo and the pendulum) mathematics
becomes automatically involved.

Math became *consistently* involved with science in the 17th
century, though it is hard to give an exact date. Indeed, some
people (myself included) think that the physical sciences did
not really start *until* observations became mathematized.
Up to that point one had "descriptive" science. After that
one had "predictive" science which then was applicable to
engineering and other things.

I'll put some additional emphasis on Paul's _consistently_,
that also including continuously. In the west, the string of
mathematical involvement in science gets broken in the early
hundreds AD, if not earlier. The present run goes back to
Galileo and a bit farther (Copernicus, I'd say certainly,
somewhat farther arguably).

On the other hand, to the extent that the question is about
earliest intertwining of math and science, you can go back at
least to Eratosthenes and Hipparchos. Both also satisfy Paul's
mathematizing of observations. Eratothsenes took observations
to determine the circumference of the earth; Hipparchos backed
out the fact of the precession of the equinoxes from observations.
Aristotle also uses mathematical demonstration in explaining,
for example, observations of rainbows.

You could add to the list the Pythagorean music, allegedly the first
science law ever derived from maths.

However these ones here are descriptive.

What I am a talking about is using mathematics to prove a theory.

Maybe the problem is that until you get Galileo's mechanics and
Kepler's ellipses together, any mathematical study of the universe is
extremely complex with all its circles within circles and unprovable
with mathematics.

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.

You're going to have the Society for the History of Alchemy and Chemistry
( http://www.ambix.org/ ) challenge you :-)

Prior to that one had plenty of practical engineering. But
aside from geometry, it was driven by experience and trial
and error.

But 'engineering' is just as problematic a term as 'science'.

It was only after science became established that scientific
prediction began to lead engineering.

Oh, really? Did thermodynamics lead to the steam engine or was it the other
way around?

So I've always shuddered a bit when folks talk about medieval
science or classical science. I'd call the speculation
"natural philosophy" (Aristotle, et al.) or "practical
engineering".

It is needless to say that very few agree with me on this
definition of science. And my definition is only valid if
it helps in the understanding of what was and is going on
with invention. (Invention certainly is ancient. Very
ancient.)

Both the History of Science and the History of Technology are very well
developed as academic disciplines - but this seems to have passed you by :-)
--
John Briggs


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