Re: Mathematics and science

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.

ex; (Paul knows this, but for others) consider Newtonian
dynamics and the question of where that thrown rock will fall.
The calculus method is to start with a rock at position X_0
(capitals denote vector quantities, lower case are scalars) with
velocity V_0. Then apply F = -mG to it (G = -gZ, g = 9.81 m/s^2)
and use governing equation mX'' = F (' meaning derivative in time).
Straightforward (also incorrect due to the presence of wind resistance)
and we solve our equations for V and X as functions of time, and
then interrogate them (algebraically) for what value x will have when
z returns to zero (ground level).

On the other hand, one can take the physics class which is not
calculus-based and know that the x component of velocity is unaffected
by gravity, so x = vx_o*t, that the vertical component = v_o - gt, and
that the elevation is = z_o + v_o*t - 1/2 g t^2 -- all more or less easily
observable quantities, which, again, you solve algebraically (or by
other means known much longer back). You can fancy it up by thinking
in terms of initial angle of the trajectory and looking for the angle
which maximizes distance. (45 degrees).

But now try to allow for wind resistance. The medieval empiricist
did so. The non-calculus based student would do so as easily. But
the calculus/differential equation method runs flat into the problem
of 'what is F'. Correction terms can be added to the equation -- _if_
you knew what they were. But how do you know what they are? ...
Fundamentally, you go out and empirically observe them, same as the
non-calculus method. You have a minor plus on the calculus-based
approach, in that you deduce an aerodynamic law that might apply with
some generality. But ... it doesn't really apply very generally. Load
a different size and shape rock into your catapult than the one the
computations were for, and the modern aerodynamicist is not fundamentally
better off than his medieval counterpart.


To digress, but not really, to internal combustion engines. I read
my grandfather's 1937 internal combustion engine textbook one summer.
One of the very striking things about it, to me, was that there was
exceedingly little that was not fundamentally known in 1937 that is now
in 2007 engines. Most of the differences between engines then and now
had nothing to do with principles of design. Rather, it was that the
designer back then couldn't count on the machine tolerances being as
fine as desired, the fuel could not be counted upon to be what was
desired (knocking, ...), the timing had to be provided mechanically
and in a fault-tolerant way (vs. electronically for parts now, with
less need for fault-tolerance), and on through a fairly lengthy list.
But, give a 1930s enginer his magic wand to produce the engine he
wanted to, given materials and methods _now_ available, and his engine
wouldn't have been much worse or different than modern. (Minus not
knowing about computer control, which truly is new, but makes far less
difference than being able to count on the materials performing as
desired.) The result would have been quite a bit different than what
he actually did do, given the constraints he worked under.

And such is engineering -- making something useful with
presently available materials/processes/knowledge/...
--
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
.

Relevant Pages

  • =?UTF-8?B?UmU6INmB2LHZiNi0INmF2KzZhdmI2LnZhyDaqdin2YXZhCDaqdiq2KfYqNmH2KfbjCDZhQ==?= =?UTF-8
    ... Statistical Applications in Process Control (Quality and Reliability ... Copyright © 2007 by Blackwell Publishing ISBN 9781405125192 ... Environmental Engineering, 3-Volume Set, 6th edition,[by Nelson L. ... Boundary Element Methods in Manufacturing (Oxford Engineering Science ...
    (sci.math.num-analysis)
  • 2008 ebooks share viewer
    ... Trade and Technology Policies: ... Engineering for Concrete Dams: Design, Performance, and Research ... Science in the Twenty-first Century. ... Statistics for Health Policy ...
    (sci.med.nutrition)
  • ebooks share list 30
    ... Trade and Technology Policies: ... Engineering for Concrete Dams: Design, Performance, and Research ... Science in the Twenty-first Century. ... Statistics for Health Policy ...
    (sci.med.nutrition)
  • How many lumps for first principles, was Re: instructors solutions manual for Materials Science and
    ... instructor solution manual for Advanced Digital Design with the ... instructor solution manual for Advanced Engineering Mathematics 3rd ed ... instructor solution manual for Calculus - Early Transcendentals, ...
    (rec.bicycles.tech)
  • ebooks share lits 62
    ... Rice Science and Technology Food Science and Technology (Marcel ... Erythropoietin in Clinical Applications: An International Perspective ... Kinetics and Catalysis in Microheterogeneous Systems Surfactant ... Selection and Application Mechanical Engineering ...
    (sci.med.nutrition)