1 The Birth of Science - MSRI

2.2 Euclid’s Hypothetic-Deductive Method 33
could not prove everything without causing the proof of any statement to
involve an infinite regression. Second, paradoxes such as Zeno’s and the
impasse found by the Pythagoreans had made apparent the high degree of
subtlety of concepts such as space, time, and infinity, and of the relations
between discrete and continuous magnitudes; it had also shown how inadequate
everyday language is for dealing with such questions. Finally,
there was the question of the unclear relationship between the concepts of
mathematics and the real world.
In Euclid’s Elements we can see for the first time the solution to these
problems, which was reached by establishing mathematics as a scientific
theory — more precisely, by explicitly defining the theory’s entities (circles,
right angles, parallel lines, and so on) in terms of a few fundamental
entities (such as points, lines, and planes) 28 and by listing the statements
about such entities that must be accepted without proof.
In the Elements there are five statements of this type, called “postulates”
(): 29
1. [One can] draw a segment from any point to any point.
2. [One can] continuously extend a segment to a line.
3. [One can] draw a circumference with arbitrary center and radius.
4. All right angles are equal to one another.
5. If a line transversal to two lines forms with them in the same halfplane
internal angles whose sum is less than two right angles, the
two lines meet in that half-plane.
Any other statement regarding geometric entities can and should be accepted
as true only if it can be supported by a proof (), that is, a page 61
chain of logical implications that starts from the postulates (and the “common
notions”) and leads to the given statement. This method is known to
anyone who has studied mathematics in high school (at least that was the
case until a while ago), because it was inherited by modern mathematics.
Note the privileged role played, from the postulates on, by lines and circles.
The reason for this choice is clear: these two entities play a special
role because they are the mathematical models of what can be drawn with
ruler and compass. Euclidean geometry arises explicitly as the scientific
theory of the objects that can be drawn with ruler and compass. Euclid’s
28 In the Elements even these fundamental entities are “defined”, and the presence of these “definitions”
(which are mere tautologies or purely illustrative statements) appears to contradict the
thesis of the present discussion. This important question will be the subject of Section 10.14, where
we will be able to study it in light of the material contained in intervening chapters.
29 There are also five “common notions”, that is, statements that are not about the specific entities
of geometry. However, the authenticity of the “common notions” has often been contested. See, for
example, [Euclid/Heath], vol. 1, pp. 221 ff.
Revision: 1.12 Date: 2003/01/10 06:11:21