Re:apples and oranges

["Donald E. Simanek" <dsimanek@eagle.lhup.edu>]

Exellent observations, Larry, at last this thread is getting down to
basics. I could certainly multiply my blood count times the number of
hairs on my head and call it the "Simanek constant". The constant would be
a well-defined physical quantity, with units, dimensions, etc. But is it
meaningful in science? One can't know until one finds and verifies a law
using that constant. If someone found that the Simanek constant predicted
a person's likelihood of getting gout, to 10% certainty, the constant
would become scientifically meaningful, at least empirically. If someone
found out a mechanism linking those two measurables to the biological
process which cause gout, and that it fit into some larger theory of human
metabolism, it would be meaningful on another level. But I doubt that
either will happen. 

	-- Donald

......................................................................
Dr. Donald E. Simanek                            Office: 717-893-2079
Professor of Physics                                FAX: 717-893-2048
Lock Haven University of Pennsylvania, Lock Haven, PA. 17745
dsimanek@eagle.lhup.edu                 http://www.lhup.edu/~dsimanek
......................................................................

On Wed, 28 Oct 1998, L. R. Cartwright (Larry) wrote:

> Years ago I acquired a math major and certification, and was taught that
> math teachers (except at the most elementary levels) shouldn't be thinking
> of apples and oranges;  they should be thinking about numbers in an
> abstract and generalized manner.
>
> It happens that in our chosen base-10 number system multiplication turns
> out to behave the same as repetitive addition for many numbers, but only
> for those integers and rationals which have a finite number of digits. The
> characterization breaks down for infinitely repeating rationals and is
> virtually meaningless for irrationals.  There comes a point when one needs
> to put away one's apples-and-oranges perspective and look at things in a
> more sophisticated way.  Beyond that point, thinking in terms of apples
> and oranges gets to be artificial and pointless, and gets in the way of
> understanding what is really happening.  The math we learned in elementary
> school does have its limits of usefulness.
>
> Just as in physics there is a point when algebra and geometry fail to
> accurately describe what is observed. At that point we have to be able to
> give up our elementary mathematical preconceptions, and deal with the new
> and more powerful mathematics of the calculus that enables us to
> understand and communicate what is occurring.
>
> BTW, I don't think there is any mathematical *prohibition* against adding
> apples and oranges.  Only the *admonition* that adding apples and oranges
> is not a simple operation and needs to be done with great caution.  You
> can add or multiply any different things you want to, but only within the
> constraints of the laws of number theory; and mathematics like physics
> doesn't provide any guarantees that the outcome will be meaningful.
>
> Best wishes,
>
> Larry