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
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.
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