294 10. EUCLIDEAN GEOMETRY
FIGURE 15. How do we exclude the possibility that two lines per-
pendicular to the same line may intersect each other?In Book 1 Aristotle describes how to organize the study of a subject, looking
for all the attributes and subjects of both of the terms that are to appear in a
syllogism. The subject-attribute relation is mirrored in modern thought by the
notion of elements belonging to a set. The element is the subject, and the set it
belongs to is defined by attributes that can be predicated of all of its elements and
no others. Just as sets can be elements of other sets, Aristotle said that the same
object can be both a subject and a predicate. He thought, however, that there were
some absolute subjects (individual people, for example) that were not predicates of
anything and some absolute predicates (what we call abstractions, such as beauty)
that were never the subject of any proposition. 16 Aristotle says that the postulates
appropriate to each subject must come from experience. If we are thorough enough
in stating all the attributes of the fundamental terms in a subject, it will be possible
to prove certain things and state clearly what must be assumed.
In Book 2 he discusses ways in which reasoning can go wrong, including the
familiar fallacy of "begging the question" by assuming what is to be proved. In this
context he offers as an example the people who claim to construct parallel lines.
According to him, they are begging the question, starting from premises that cannot
be proved without the assumption that parallel lines exist. We may infer that there
were around him people who did claim to show how to construct parallel lines, but
that he was not convinced. It seems obvious that two lines perpendicular to the
same line are parallel, but surely that fact, so obvious to us, would also be obvious
to Aristotle. Therefore, he must have looked beyond the obvious and realized that
the existence of parallel lines does not follow from the immediate properties of lines,
circles, and angles. Only when this realization dawns is it possible to see the fallacy
in what appears to be common sense. Common sense—that is, human intuition—
suggests what can be proved: If two perpendiculars to the same line meet on one
side of the line, then they must meet on the other side also, as in Fig. 15. Indeed,
Ptolemy did prove this, according to Proclus. But Ptolemy then concluded that
two lines perpendicular to the same line cannot meet at all. "But," Aristotle would
have objected, "you have not proved that two lines cannot meet in two different
points." And he would have been right: the assumptions that two lines can meet in
only one point and that the two sides of a line are different regions (not connected
to each other) are equivalent to assuming that parallel lines exist.
16 In modern set theory it is necessary to assume that one cannot form an infinite chain of sets a,
6, c,... such that 6 e a, c å b,.... That is, at the bottom of any particular element of a set, there
is an "atom" that has no elements.