The role of conjecture in mathematics & Seldom Asked Questions [SAQ’s]For many people mathematics is a closed canon – a subject that is complete and ready to be taught by those who “know” mathematics to those who do not. Others, with a somewhat more enlightened view, think of mathematics as a subject that makes progress by a continuing succession of theorems and proofs. Indeed, the way mathematics is taught in many of our high schools and colleges would seem to support that view. But where do the theorems that mathematicians work to prove come from? Who formulates them? How did people ever think up the idea that · any even number can be written as the sum of two prime numbers, or that · the altitudes of any triangles meet at a point, or that · there is no integer greater than n = 2 that satisfies the equation an + bn = cn where a, b and c are positive integers? All of these statements started as conjectures[1]. How do they arise in minds of their inventors? I claim that they come about because the people who devise these conjectures can “mess around” with numbers, or triangles, or equations and notice patterns. To the people who are doing the “messing around” these patterns suggest regularities of behavior of numbers, triangles, equations and other mathematical objects that may (or may not) be true beyond the cases they have inspected. Conjectures lead to theorems and theorems are sometimes proven to be true (and sometimes proven to be false). It seems that using graphical representations of functions makes it easier to pose questions that lead students and teachers to conjecture about patterns of behavior of functions. Unfortunately, we don’t often pose such questions to our students. Not doing so, deprives them of a possible opportunity to invent, create and explore in the domain of mathematics. For me the making and exploring of conjectures is the “heart and soul” of making mathematics. I have tried to construct these exploratory environments with accompanying questions and conundrums in the hope that might you might be provoked to formulate conjectures of your own. I do this in the hope that you will encourage your students to do likewise. I call such questions “Seldom Asked Questions” or SAQs. [1] The first conjecture is known as the Goldbach conjecture and has never been proven to be true, although there are no known counterexamples. The second and third conjectures have been proven although it took more than two hundred and fifty years to prove the third one, which is known as Fermat’s Last Theorem Judah L. Schwartz
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