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Topic: Re: Bridgers' letter to MATH. TEACHER
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Michael Paul Goldenberg

Posts: 7,041
From: Ann Arbor, MI
Registered: 12/3/04
Re: Bridgers' letter to MATH. TEACHER
Posted: Sep 8, 1995 1:27 PM
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"It will seem not a little paradoxical to ascribe a great
importance to observations even in that part of mathematical sciences which
is usually called Pure Mathematics, since the current opinion is that
observations are restricted to physical objects that make impression on the
senses. As we must refer the numbers to the pure intellect alone, we can
hardly understand how observations and quasi-
experiments can be of use in investigating the nature of the numbers. Yet,
in fact, as I shall show here with very good reasons, the properties of the
numbers known today have been mostly discovered by observation, and
discovered long before their truth has been confirmed by rigid
demonstrations. There are even many properties of numbers with which we are
well acquainted, but which we are not yet able to prove; only observations
have led us to their knowledge. Hence we see that in the theory of numbers,
which is still very imperfect, we can place our highest hopes in
observations; they will lead us continually to new properties which we
shall endeavor to prove afterwards. The kind of knowledge which is
supported only by observations and is not yet proved must be carefully
distinguished from the truth; it is gained by induction, as we usually say.
Yet we have seen cases in which mere induction led to error. Therefore, we
should take great care not to accept as true such properties of the numbers
which we have discovered by observation and which are supported by
induction alone. INDEED, WE SHOULD USE SUCH A DISCOVERY AS AN OPPORTUNITY
TO INVESTIGATE MORE EXACTLY THE PROPERTIES DISCOVERED AND TO PROVE OR
DISPROVE THEM; IN BOTH CASES WE MAY LEARN SOMETHING USEFUL." (emphasis
added).

-Leonhard Euler, OPERA OMNIA, ser. 1, vol.2, p. 459, Specimen de usu
observationum in mathesipura

quoted in Polya, MATHEMATICS AND PLAUSIBLE REASONING, Vol. I: INDUCTION AND
ANALOGY IN MATHEMATICS, p.3.


I think the above passage says far more articulately than I ever could why
we need to value a wider range of thinking processes than we often do in
the mathematics classroom, especially at the K - 12 levels.

As to the risk of misinterpretation by hurried or careless readers, I can
only say that when one writes in hopes of being understood and one has done
a fair and honest job of presenting one's ideas, there is no accounting for
what will be done with one's words by careless readers.

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|Michael Paul Goldenberg
|University of Michigan 310 E. Cross St.
|School of Education 4002 Ypsilanti, MI 48198
|Ann Arbor, MI 48109-1259 (313) 482-9585
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|"Truth is a mobile army of metaphors."
|Friedrich Nietzsche
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