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A poignant moment arises in Stand and Deliver, a film about a group of students at Garfield High School in East Los Angeles whose lives are changed by their mathematics teacher, Jaime Escalante. He has finally begun to connect with them. With wide smiles they are following his lead in repeating a standard algebraic rule: "A negative times a negative equals a positive." After several rounds of this chant there is a pregnant pause, after which Mr. Escalante—with an equally wide smile—asks, "Why?"
The camera then cuts to another scene, leaving the question unanswered. Some in the viewing audience may very well wish that the scene had continued, having remembered the rule but forgotten the explanation. Indeed, why does a negative times a negative equal a positive? Most people have accepted this maxim even if they never heard a good explanation for it. After all, the laws of algebra have the same status as the laws that came down on a tablet from Mount Sinai, don't they?
Well, no, at least not according to Alberto Martínez, whose Negative Math: How Mathematical Rules Can Be Positively Bent is at once scholarly and readable. He argues that many of the rules of algebra could have been otherwise. The adopted ones came about by a variety of factors, and don't necessarily offer the best resource for dealing with every kind of scientific inquiry.
This attitude may seem odd: many thinkers view numerical theorems as independent of human construction. In the words of Martin Gardner, "If two dinosaurs met two others in a forest clearing, there would have been four dinosaurs there—even though the beasts were too stupid to count and there were no humans around to watch."
Interestingly, this opinion of mathematical independence may not be so widely held with respect to the theorems of geometry. The Kantian view that our minds are equipped with a category of (Euclidean) space—a category that exists inside us as a condition of knowledge—likely dissipated with the advent of non-Euclidean geometries in the mid 1800s. But a general conviction that our minds have a category of quantity—necessary for the perception of real things—seems to hold sway with the general public.
When the absolute truth of geometry fell from grace, mathematicians became bolder in thinking that deviant algebraic constructs—hitherto deemed as nonsensical because they did not correspond with geometric ideas—could seriously be considered. In this spirit Martínez, although steering clear of abstract philosophical discussions, wants to show that even well-established algebraic axioms might be altered, especially those that do not fully match our experience.
The first part of Martínez's book contains a sweeping survey of the historical development of algebraic systems, concentrating on the problems of dealing with negative quantities. Some readers may be surprised to learn that serious discussion of negative numbers spanned several centuries—from at least Girolamo Cardano's publication of Ars Magna (Great Art) in 1545, to Augustus De Morgan's On the Study and Difficulties of Mathematics, published in 1831. Even into the 1880s, imaginary numbers (quantities dealing with the square root of minus one) had dubious status at Cambridge University.
Martínez writes in an informal style: "If later tonight you go to sleep and tomorrow somehow forget everything else in these pages, then at least please remember this." He treats his readers to quotations from PhD theses, textbooks, journal articles, and letters, not mention remarks from many central mathematical and philosophical figures (Berkeley, Cauchy, Comte, d'Alembert, Dedekind, Einstein, Euler, Gauss, Hamilton, Kant, Leibniz, Newton, Russell, and Wallis, among others), all of whom had something to say about negative numbers.
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