It seems that no one has claimed that an explicit statement of
the "inductive step" P(n) => P(n + 1) occurs in Greek mathematics.
Remarkably, however (to my mind), Euclid (Elements, VII.31) *does*
explicitly use the principle which we now call Fermat's principle of
infinite descent.
At any rate, Unguru, not content with arguing the *historical*
question of whether the Greeks *did* use mathematical induction, goes
on to make the *metaphysical* claim that they *could not* have used
it. I will not summarize his whole argument; but here is one strand
of it ("Greek math & induction", p. 276): "What concerns us here,
however, is to grasp that the mathematical thought process displayed
by proposition IX.8 is *not* an inductive mathematical process.
First, it is clear that Euclid did not, and could not, rely on the
axiom of mathematical induction which was not yet invented and
formulated."
The point seems to be that you can't use something which
hasn't been invented, just as you might argue that Archimedes couldn't
have driven a car, since they didn't exist then, or that Hilbert
couldn't have used the Internet, since it wasn't invented in his time.
Applied to the history of mathematics, however, I think that
Unguru's principle is wrong. I would almost go so far, in fact, as to
say that he has it exactly backward: actually, a mathematical
principle can't be invented and formulated until *after* it has been
used.
A good example is Zermelo's Axiom of Choice. In the paper in
which Zermelo gave his first proof of the Well-ordering Theorem
(1904), he stated the Axiom explicitly, though he had used it without
explicit mention in an earlier paper of 1901. His proof led to a
controversy about the validity of the Axiom. With hindsight it was
seen that others had already used the Axiom implicitly ---including
some of those (such as Borel) who objected to Zermelo's use of it.
(For the story, see Gregory H. Moore, _Zermelo's Axiom of Choice_,
Springer, 1982.)
Another example ---this one from ancient Greek mathematics---
is the "Archimedean ordering principle" (given a and b, we have na > b
for some whole number n). Archimedes states this as the fifth
postulate in his "On the sphere and cylinder", Book I, and reiterates
it in several other works. And in the "Quadrature of the parabola" he
remarks, "The earlier geometers have also used this lemma; [list of
results omitted] they proved by assuming a certain lemma similar to
the aforesaid. And, in the result, each of the aforesaid theorems has
been accepted no less than those proved without the lemma" (Heath's
edition, p. 234).
Unfortunately, we don't have as much information about
Archimedes's predecessors as we do about Zermelo's. It would appear that
some of the earlier mathematicians had used the principle without
formulating it explicitly. Indeed, we find it "half-formed", as it were,
in Euclid, not in the form of a postulate, but of a definition (V.4).
Note that in X.1, Euclid seems to apply this *definition* to any pair of
"unequal magnitudes". (Heath, Euclid's Elements, vol. 3, p. 16, finds a
similar principle stated in Aristotle's Physics.)
I think it would not be difficult to find many other examples
of mathematical principles which are used before they are explicitly
formulated.
Stacy Langton
University of San Diego
langton@acusd.edu