2001 : WHAT QUESTIONS HAVE DISAPPEARED?

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emeritus professor of the Philosophy department of the University of Calgary, Alberta Canada
monograph Goedel's theorem

Did Fermat's question, "is it true that there are no integers x, y, z and n, all greater than 2, such that x^n + y^n = z^n?", F? for short, raised in the 17th century, disappear when Andrew Wiles answered it affirmatively by a proof of Fermat's theorem F in 1995?

The answer is no.

The question F? can be explained to every child, but the proof of F is extremely sophisticated requiring techniques and results way beyond the reach of elementary arithmetic, thus raising the quest for conceptually simpler proofs. What is going on here, why do such elementary theorems require such intricate machinery for their proof? The fact of the truth of F itself is hardly of vital interest. But, in the wake of Goedel's incompleteness proof of 1931, F? finds it place in a sequence of elementary number theoretic questions for which there provably cannot exist any algorithmic proof procedure!

Or take the question D? raised by the gut feeling that there are more points on a straight line segment than there are integers in the infinite sequence 1,2,3,4,.... Before it can be answered the question what is meant by "more" must be dealt with. This done by the 18th Century's progress in the Foundations, D? became amenable to Cantor's diagonal argument, establishing theorem D. But this was by no means the end of the question!

The proof gave rise to new fields of investigation and new ideas. In particular, the Continuum hypothesis C?, a direct descendant of D? was shown to be "independent" of the accepted formal system of set theory. A whole new realm of questions sprang up; questions X? that are answered by proofs of independence, bluntly by: "that depends" ‹ on what you are talking about, what system you are using, on your definition of the word "is" and so forth. With this they give rise to comparative studies of systems without as well as with the assumption X added. Euclid's parallel axiom in geometry, is the most popular early example.

What about the question as to the nature of infinitesimal's, a question that has plagued us ever since Leibniz. Euler and his colleagues had used them with remarkable success boldly following their intuition. But in the 18th Century mathematicians became self conscious. By the time we were teaching our calculus classes by means of epsilon's, delta's and Dedekind cuts some of us might have thought that Cauchy, Weierstrass and Dedekind had chased the question away. But then along came logicians like Abraham Robinson with a new take on it with so-called non standard quantities ‹ another favorite of the popular science press.

Finally, turning to a controversial issue; the question of the existence of God can neither be dismissed by a rational "No" nor by a politically expedient "Yes". Actually as a plain yes-or-no question it ought to have disappeared long ago. Nietzsche, in particular, did his very best over a hundred years ago to make it go away. But the concept of God persists and keeps a maze of questions afloat, such as "who means what by Him"?, "do we need a boogie man to keep us in line"?, "do we need a crutch to hold despair at bay"? and so forth, all questions concerning human nature.

Good questions do not disappear, they mature, mutate and spawn new questions.

VERENA HUBER-DYSON, is a mathematician who taught at UC Berkeley in the early sixties, then at the U of Illinois' at Chicago Circle, before retiring from the University of Calgary. Her research papers on the interface between Logic and Algebra concern decision problems in group theory. Her monograph Goedel's theorem: a workbook on formalization is an attempt at a self contained interdisciplinary introduction to logic and the foundations of mathematics.