1.2

MARK MANDELKERN 2

1.2. The various approaches. Attempts to solve these problems fall

into four or more categories.

First were the attempts of Brouwer. Brouwer's critique of classical

mathematics forms the crucial motivation behind modern constructive

mathematics (see pp. vii-x and 1-10 in [B],and [S]). However, some of

his positive results are debatable. Brouwer [BJ proved that all

constructively defined functions on the closed unit interval are uniformly

continuous. Thus, while proving the (classically valid) uniform

continuity principle, at the same time he proves the (classically invalid)

continuity principle. An exposition of his results may be found in[H].

Brouwer's proofs, however, are not constructive in the modern sense. He

introduced methods of questionable constructivity which are still used in

what is now known as intuitionistic mathematics. It was sometime later

that it was shown by Bishop [B] how mathematics could be pursued in an

uncompromisingly constructive manner.

The second category consists of the analysis of intuitionistic

methods by means of symbolic logic. While illuminating certain aspects of

the problems, this approach not only shares certain of the

nonconstructivities of intuitionistic methods, but also suffers from the

limitations of any attempt to understand the continuum by means of formal

logic. There is good reason to suspect that the continuum will forever

resist all efforts to compress it within the confines of any formal

system.

Third is the attempt of recursive analysis, in which the concepts of

real number and of function are delimited in order to facilitate their

study. Recursive analysis offers a proof for the first principle [Mg] [C]

[KLS], and a counterexample to the second [L]. However, these results are

based on nonconstructive hypotheses, and on an interpretation of numbers

and functions as recursive, an interpretation more restrictive than is

appropriate for constructive analysis. Thus, while some success is

obtained in this way, the restrictions on the concepts of number and

function are so severe as to leave the original problems completely

unresolved — in the context of an unfettered constructive mathematics.