ABSTRACT
Many elementary mathematics books with an audience similar to that of this book have a section on logic, the logic of Aristotle.1 Thus a triplet of statements called a syllogism, soon appears, such as:
(Assumption) Hypothesis : All humans are mortal. (8.1)
(Assumption) Hypothesis : All mathematicians are human. (8.2)
Conclusion : All mathematicians are mortal. (8.3)
The syllogism is then illustrated with some so-called set theoretic diagram:
mortals humans mathematicians
FIGURE 8.1: Diagram of a Syllogism
The set (box) of mathematicians is contained in the set (box) of humans is contained in the set (box) of mortals. Thus by the “transitivity of containment,” an assumption in itself, we get that the set (box) of mathematicians is contained in the set (box) of mortals, cf. Figure 8.1. Like most mathematics applied to an actual situation, Aristotle’s logic is
an abstract idealization, an oversimplification of reality at best-a mistake at worst.2 I do not propose tossing out Aristotelian/sharp logic, in fact we will use sharp logic in creating the proofs of II. Such logic will remain a useful
1The Greek philosopher Aristotle (384 B.C.–322 B.C.) was a student of Plato and teacher of Alexander the Great. 2Which came first the chicken or the egg? This question becomes tractable if you abandon the implicit oversimplification.
