Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: Re:apples and oranges



Dr Newbolt emphasizes an important feature of our mathematical
manipulations and models:
We may begin by defining a set of mathematical entities and defining
certain operations on, between and among them. It often happens that we
then find it useful to extend the population of interest to a wider kind
of mathematical entity and that the operations already defined do not
enjoy the transparent, intuitive meaning from which they took birth
(within the original, restricted set). When this happens we find it
calculationally useful to DEFINE the result of these operations, when the
larger set is involved, in such a way that our "rules of algebra" will be
preserved and be useful to calculational models involving the larger set.
We generalize the definition of an operation so that it will apply
usefully to a wider set than that which gave it birth.

For example, we might begin by defining the meanings and algebra of
trigonometric functions simply from the geometry of a right triangle. We
soon find it useful to extend all of this to angles larger than 90 deg.
We do this by DEFINING the meaning of sin(T), etc, for T>90 deg so that
we can carry over the same (generalized)identities, algebra, etc which
arose from right triangle geometry for only acute angles. The
generalization of the desired algebra motivates the definition, in the
interest of usefulness.

The same happens when we extend the algebra of the natural numbers to
include operations involving zero, negative numbers, irrationals,
complex, etc. Again, the heart of the issue is that this is a matter of
a freely chosen human creation, with usefulness as the determining
motivation (of course, within the bounds of logical consistency).

The realization that science is a humanly invented description of
reality, motivated by empirical and conceptual usefulness as opposed to
"truth", frees the learner from the nagging "mystery" lurking in such
matters.

-Bob

Bob Sciamanda
Physics, Edinboro Univ of PA (ret)
trebor@velocity.net
http://www.velocity.net/~trebor
-----Original Message-----
From: Dr. William Newbolt <NewboltW@madison.acad.wlu.edu>
To: phys-l@atlantis.uwf.edu <phys-l@atlantis.uwf.edu>
Date: Wednesday, October 28, 1998 5:14 PM
Subject: Re:apples and oranges


I seem to be doing a lot of musing these days, but in
reference to this thread I have been thinking about the way
the concept of multiplication has grown in my own mind. I
can still remember when I learned that 1/2 of 10 was the
same as 1/2 multiplied by 10. It may be obvious, but when
I learned it I was delighted. Then I knew how to find out
what 1/2 or 1/3 was.

Many of the jobs that use multiplication have little
residue of repeated addition in them. F = ma is a good
example. The concepts of polynomials, powers,
and proportion, in which multiplication is foremost do not
seem to partake of the repeated addition idea. I think we
need to say that the study of multiplication as repeated
multiplication may be useful at some level in the learning
process, but at some point we need to mature to learn what
multiplication really is. WBN
Barlow Newbolt
Department of Physics and Engineering
Washington and Lee University
Lexington, VA 24450

There is something fascinating about science. One
gets such wholesale returns of conjecture out of
such trifling investments of fact.
Mark Twain
Telephone and Phone Mail: 540-463-8881
Fax: 540-463-8884
e-mail: NewboltW@madison.acad.wlu.edu