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Re: Bridgers' letter to MATH. TEACHER
Posted:
Sep 8, 1995 1:27 PM


"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.
 Michael Paul Goldenberg University of Michigan 310 E. Cross St. School of Education 4002 Ypsilanti, MI 48198 Ann Arbor, MI 481091259 (313) 4829585 (313) 7472244  "Truth is a mobile army of metaphors." Friedrich Nietzsche 



