FOM: foundational problems

[Martin Davis]

Something I miss in many of the postings here is the realization that
foundational work tends to be in response to specific foundational problems.

There are various things in mathematical practice that lead mathematicians
beyond what they are able to justify in terms of their contemporary
understanding. When this happens, a foundational problem arises. In the
happiest instances, foundational work will not only provide a satisfactory
explanation, but also will provide new directions for mathematical research.

Historically, one of the forces leading to such over extension, and thus to
foundational problems, has been the sheer power of formalism. Rules of
algebraic manipulation led to expressions involving sqrt(-1) even though
everyone knew there couldn't be such a thing. Working with these
impossibilities even led to the correct roots for cubic equations with three
*real* roots. The Gauss-Argand-Hamilton solution to this foundational
problem opened up vast new vistas: complex analysis and geometry.

Analogy is another force leading mathematicians beyond what they are able to
justify. (This has great contemporary relevance for set theory: the infinite
by analogy with the finite, the huge infinite by analogy with the countably
infinite.) Euler's masterful method for summing \Sigma_1^infty (1/n^2) by
factoring the Maclaurin series  for the sine as though it were a polynomial
and looking at the symmetric functions of its roots (see Polya "Induction &
Analogy in Mathematics") is a wonderful example. On the other hand, without
a proper theory of  convergence and of summation methods for divergent
series, Euler felt free to propose setting x=2 in the expansion of 1/(1-x)
as an infinite geometric series to obtain:
              -1 = 1 + 2 + 4 + 8 + 


The obvious success of infinitesimal methods, used freely by engineers and
physicists long after mathematicians had insisted they were illegitimate,
was an example of a foundational problem not resolved until Abraham Robinson
explained it. Robinson and Roquette were convinced that there is a
foundational explanation for the analogy between algebraic number theory and
algebraic function theory, but their results in this direction remain
inconclusive. 

There are many other examples: the free use of algebraic manipulation of
operators in solving ordinary differential equations (see George Boole's
classical text for masterful use of these techniques without a shred of
justification), the Heaviside operator calculus, the Dirac delta function.

I would like to urge that thinking about fom in terms of foundational
problems needing to be addressed, rather than in terms of which concepts or
theorems are truly "foundational", is  likely to be a more fruitful approach.

Martin Davis