How To Read A Book- A Classic Guide to Intelligent Reading - Mortimer J. Adler, Charles Van Doren [124]
The first three propositions in Book I of the Elements are all problems of construction. Why is this? One answer is that the constructions are needed in the proofs of the theorems.
This is not apparent in the first four propositions, but we can see it in the fifth proposition, which is a theorem. It states that in an isosceles triangle ( a triangle with two equal sides ) the base angles are equal. This involves the use of Proposition 3, for a shorter line is cut off from a longer line. Since Proposition 3, in tum, depends on the use of the construction in Proposition 2, while Proposition 2 involves Proposition 1, we see that these three constructions are needed for the sake of Proposition 5.
How to Read Science and Mathematics 263
Constructions can also be interpreted as serving another purpose. They bear an obvious similarity to postulates; both constructions and postulates assert that geometrical operations can be performed. In the case of the postulates, the possibility is assumed; in the case of the propositions, it is proved. The proof, of course, involves the use of the postulates. Thus, we might wonder, for example, whether there is really any such thing as an equilateral triangle, which is defined in Definition 20. Without troubling ourselves here about the thorny question of the existence of mathematical objects, we can at least see that Proposition 1 shows that, from the assumption that there are such things as straight lines and circles, it follows that there are such things as equilateral triangles.
Let us return to Proposition 5, the theorem about the equality of the base angles of an isosceles triangle. When the conclusion has been reached, in a series of steps involving reference to previous propositions and to the postulates, the proposition has been proved. It has then been shown that if something is true ( namely, the hypothesis that we have an isosceles triangle ) , and if some additional things are valid ( the definitions, postulates, and prior propositions ) , then something else is also true, namely, the conclusion. The proposition asserts this if-then relationship. It does not assert the truth of the hypothesis, nor does it assert the truth of the conclusion, except when the hypothesis is true. Nor is this connection between hypothesis and conclusion seen to be true until the proposition is proved. It is precisely the truth of this connection that is proved, and nothing else.
Is it an exaggeration to say that this is beautiful? We do not think so. What we have here is a reaUy logical exposition of a reaUy limited problem. There is something very attractive about both the clarity of the exposition and the limited nature of the problem. Ordinary discourse, even very good philosophical discourse, finds it difficult to limit its problems in this way. And the use of logic in the case of philosophical problems is hardly ever as clear as this.
264 HOW TO READ A BOOK
Consider the difference between the argument of Proposition 5, as outlined here, and even the simplest of syllogisms, such as the following:
All animals are mortal;
All men are animals;
Therefore, all men are mortal.
There is something satisfying about that, too. We can treat it as though it were a piece of mathematical reasoning. Assuming that there are such things as animals and men, and that animals are mortal, then the conclusion follows with the same certainty as the one about the angles of the triangle. But the trouble is that there really are animals