  In the early twentieth century, L.E.J. Brouwer pioneered a new philosophy of
mathematics, called intuitionism. Intuitionism was revolutionary in many
respects but stands out -mathematically speaking- for its challenge of
Hilbert's formalist philosophy of mathematics and rejection of the law of
excluded middle from the 'classical' logic used in mainstream mathematics. Out
of intuitionism grew intuitionistic logic and the associated
Brouwer-Heyting-Kolmogorov interpretation by which 'there exists x' intuitively
means 'an algorithm to compute x is given'. A number of schools of constructive
mathematics were developed, inspired by Brouwer's intuitionism and invariably
based on intuitionistic logic, but with varying interpretations of what
constitutes an algorithm. This paper deals with the dichotomy between
constructive and non-constructive mathematics, or rather the absence of such an
'excluded middle'. In particular, we challenge the 'binary' view that
mathematics is either constructive or not. To this end, we identify a part of
classical mathematics, namely classical Nonstandard Analysis, and show it
inhabits the twilight-zone between the constructive and non-constructive.
Intuitively, the predicate 'x is standard' typical of Nonstandard Analysis can
be interpreted as 'x is computable', giving rise to computable (and sometimes
constructive) mathematics obtained directly from classical Nonstandard
Analysis. Our results formalise Osswald's longstanding conjecture that
classical Nonstandard Analysis is locally constructive. Finally, an alternative
explanation of our results is provided by Brouwer's thesis that logic depends
upon mathematics.

Logic