Stacy Langton wrote:
[[I think it would not be difficult to find many other examples
of mathematical principles which are used before they are explicitly
formulated.]]
This is an extremely interesting point. It is maybe interesting
to observe that an analogous point was made by John Locke in his _Essay_
in connection with formal logical principles, and raises the question
about how a formulated principle differs from an unformulated one. it
would be interesting to see if other principles, such as "mathematical
induction" occupy the same sort of position:
Let me illustrate the issue by reference to the conflict between
Locke and Leibniz on the matter of the Principle of NonContradiction.
Locke held that if we think carefully about "triangular circle" we would
come to see that such a thing was impossible, without appealing the
Principle of Noncontradiction. Think about triangularity and
circularity and the evidently exclude one another. But for Leibniz (in
_New Essays_) they exclude on another by virtue of the Principle of
Noncontradiction. And Locke made the point that by consider A and B we
come to see that B follows logically from A, we historically, and in
principle, do not need at all to appeal to formal syllogisms
("otherwise no one made sound inferences before Aristotle").
Hilbert-Ackermann in their opening clearly distinguish between informal
and formal logic, the latter as a tool for investigating the former.
Hilbert's _Foundations of Geometry_ though logical in its concerns
through and through never deploys formal logic. But is it even correct
to say that it relies on logical principles, even unformulated logical
principles? Even here might it not be Locke rather than Leibniz who is
right?
What about now carefully articulated and demonstrated
"principles" such as "the" Fundamental Theorem of the Calculus? At one
point did "it" become a formulated principle and did those who first
exploited the relationship between derivative and integral somehow or
other rely on it? (A near maddening question, since of course those
who exploited that relationship did not have the post-Cauchy et al
concept of derivative/integral.) How do we characterize what was
floating around out there being exploited through such basic changes in
the conceptual frame of "the calculus" and of which e.g. the fundamental
theorem of line integrals, Green's Theorem, Stokes' Theorem...are
regarded as extensions or analogues?
That's the characterizational problem: to characterize what is
unformulated without appealing to subsequent formulations (and therefore
distortions of the underlying mathematical experience).
Robert Tragesser
[Dry]West[running]brook, Connecticut