The Methodology of Mathematics Is Deduction

In a widely republished 1945 essay, Carl G. Hempel stated that the method employed in mathematics "is the method of mathematical demonstration, which consists in the logical deduction of the proposition to be proved from other propositions, previously established." Hempel added the qualification that mathematical systems rest ultimately on axioms and postulates, which cannot themselves be secured by deduction. Hempel's claim concerning the method of mathematics is widely shared .

I accepted it as a young enthusiast of mathematics, but was uneasy with two aspects of it. First, it seemed to make mathematicians unnecessary by implying that a machine programmed with appropriate rules of inference and, say, Euclid's definitions, axioms, and postulates could deduce all 465 propositions presented in his Elements. Second, it reduced the role of historians of mathematics to reconstructing the deductive chains attained in the development of mathematics.

I came to realize that Hempel's claim could not be correct by reading a later publication, also by Hempel; in his Philosophy of Natural Science (1966), he presented an elementary proof that leads to the conclusion that

deduction cannot be the sole method of mathematics. In particular, he demonstrated that from even a single true statement, an infinity of other true statements can be validly deduced. If we take "or" in the nonexclusive

sense and are given a true proposition p, Hempel asserted that we can deduce an infinity of statements of the form "/? or q" where q is any proposition whatsoever. Note that all these propositions are true because with

the nonexclusive meaning of "or," all propositions of the form "p or q" are true if p is true. As Hempel stated, this example shows that the rules of logical inference provide only tests of the validity of arguments, not methods of discovery.2 Nor, it is important to note, do they provide guidance as to whether the deduced propositions are in any way significant.

Thus we see that an entity, be it man or machine, possessing the deductive rules of inference and a set of axioms from which to start, could generate an infinite number of true conclusions, none of which would be significant. We would not call such results mathematics. Consequently, mathematics as we know it cannot arise solely from deductive methods. A machine given Euclid's definitions, axioms, and postulates might deduce thousands of valid propositions without deriving any Euclidean theorems. Moreover, Hempel's analysis shows that even if definitions, axioms, and postulates could be produced deductively, still mathematics cannot rely solely on deduction. Furthermore, we see from this that historians of mathematics must not confine their efforts to reconstructing deductive chains from the past of mathematics. This is not to deny that deduction plays a major role in mathematical methodology; all I have attempted to show is that it cannot be the sole method of mathematics.