In analysis, constructive problems arise instantly because we start with the real numbers, and there is no finite procedure for deciding whether two given real numbers are equal (the real numbers are not discrete). The main thrust of constructive mathematics was in the direction of analysis, although several mathematicians, including Kronecker and van der Waerden, made important contributions to constructive algebra. Heyting, working in intuitionistic algebra, concentrated on issues raised by considering algebraic structures over the real numbers, and so developed a handmaiden of analysis rather than a theory of discrete algebraic structures. Paradoxically, it is in algebra where we are most likely to meet up with wildly nonconstructive arguments such as those that establish the existence of maximal ideals, and the existence of more than two automorphisms of the field of complex numbers.

It is important to realize that constructive algebra is algebra; in fact it is a generalization of classical algebra in that we do not assume the law of excluded middle, just as group theory is a generalization of abelian group theory in that the commutative law is not assumed. A constructive proof of a theorem is, in particular, a proof of that theorem. Every theorem in constructive mathematics can be understood as referring to the conventional universe of mathematical discourse, and the proofs are acceptable within that universe. We do not limit ourselves to a restricted class of "constructive objects," as recursive function theorists do, nor do we introduce classically false principles, as the intuitionists do.

[from the preface of * A course in constructive algebra *]

NMSU seminar in constructive mathematics

In the present context, the characteristic property is the rejection of the
law of excluded middle. It is somewhat remarkable how this one
metamathematical move embodies the essence of constructivism. Constructivism
suggests rejection of the law of excluded middle because there is generally
no computational basis for asserting "*p* or not *p*".
For example,
consider the statement, provable using the law of excluded middle, that some
digit appears infinitely often in the decimal expansion of *pi*. Here
the
existence of an integer is claimed to be proved, but no method for its
computation is indicated by the proof. It is less clear, but born out by
experience and theoretical constructions such as recursive realizability,
that theorems proved without the law of excluded middle automatically have a
computational interpretation.

Contemporary constructive mathematics can be regarded as arising from the
philosophical impact of the computer on pure mathematics. The computer is
changing the very way we regard mathematical objects. Our growing experience
with programming makes the idea of computation-in-principle quite real to
us, and this idea is the motivating force behind doing constructive
mathematics. We now have an immediate feeling for the distinction between a
very large but finite process, and an infinite process.

While there are a number of logical systems that do not incorporate the law
of excluded middle, so-called *intuitionistic logic* is the accepted
standard in all constructive schools. This logic, first made explicit by A.
Heyting, was based on the mathematical practice of L.E.J. Brouwer and his
intuitionistic school. Intuitionistic logic can be thought of as a fragment
of classical logic, so anything proved within it is simultaneously proved
classically. The full-blown intuitionism of Brouwer embraced principles
which are false under the classical interpretation: for example, that every
function defined on all the real numbers is continuous. These principles can
be thought of as flowing from a more detailed philosophical analysis of the
nature of mathematical objects and our relation to them.

The Russian school of constructive mathematics, initiated by A.A. Markov and
continued by N.A. Shanin, G.S. Tseitin, B.A. Kushner
and others, is probably best thought of as constructive recursive
mathematics. The underlying logic is intuitionistic, but the mathematical
objects are restricted to finite objects---including algorithms represented
by finite strings of symbols. Historically, but perhaps not necessarily,
this school adopted *Markov's principle*: to show that an algorithm
halts at some stage, it suffices to prove that it cannot possibly run
forever. Brouwer held Markov's principle to be false, and in certain
formalizations of his thinking it is refutable. However, as it is
classically true, it is not refutable in basic constructive mathematics.

I identify basic constructive mathematics with the constructive mathematical
practice of Errett Bishop, although there is a bit more to this than simply
adopting intuitionistic logic. The most salient addition is the axiom of
dependent choices, a strong countable choice axiom. This axiom conforms with
the practice of the Russian school and of the intuitionists. Bishop also
uses the notion of an *operation*, which is a completely informal
analogue of an algorithm in recursive mathematics. While this is probably
interpretable in intuitionism, it is confusing, at best, in classical
mathematics. With the possible exception of Tennenbaum's constructive
development of Noetherian rings, all of Bishop's mathematics can be
rephrased to avoid talking about operations.

There are a number of books on constructive mathematics available. Heyting's charming introduction to intuitionism is unique. The others divide roughly according to whether they emphasize mathematics or logic. Examples of the former are Aberth, Bishop, Bishop-Bridges, Bridges, Bridges-Richman, and Mines-Richman-Ruitenburg; examples of the latter are Dummett, Beeson, and Troelstra-van Dalen. I am inclined to classify Kushner and Shanin with the latter because, like texts that emphasize logic, they deal with conventional mathematics at one remove, studying a symbolism for algorithms rather than the objects of ordinary mathematical discourse themselves. Aberth, although he also is presenting Russian constructivism, has much more of the flavor of a conventional mathematical text---for the novice this is a good introduction to the Russian school.

Troelstra and van Dalen's book---actually a two-volume set---is the first comprehensive book on constructive mathematics written by mathematicians raised in the tradition of intuitionism---direct descendants of Brouwer, as it were. The introduction can be read with profit by anyone wishing to get an idea of what constructive mathematics is about.

**Aberth, O.**,*Computable analysis*, McGraw-Hill, 1980.**Beeson, M.J.**,*Foundations of constructive mathematics*, Springer-Verlag 1985.**Bishop, E.**,*Foundations of constructive analysis*, McGraw-Hill, 1967.**Bishop, E.**and**D.S. Bridges**,*Constructive analysis*, Springer-Verlag 1985.**Bridges, D.S.**,*Constructive functional analysis*, Pitman, 1979.**Bridges, D.S.**and**F. Richman**,*Varieties of constructive mathematics*, Cambridge Univ. Press 1987.**Dummett, M.**,*Elements of intuitionism*, Oxford Univ. Press 1977.**Heyting, A.**,*Intuitionism, an introduction*, North-Holland 1971.**Kushner, B.A.**,*Lectures on constructive mathematical analysis*, Amer. Math. Soc. 1985.**Mines, R., F. Richman**and**W. Ruitenburg**,*A course in constructive algebra*, Springer-Verlag 1988.**Shanin, N.A.**,*Constructive real numbers and constructive function spaces*, Amer. Math. Soc. 1968.**Tennenbaum, J.B.**,*A constructive version of Hilbert's basis theorem*, dissertation, University of California San Diego, 1973.**Troelstra, A.S**and**D. van Dalen**,*Constructivism in mathematics, an introduction*, two volumes, North-Holland, 1988.

[from a review of * Constructivism in mathematics, an introduction*,

by A.S. Troelstra and D. van Dalen. North-Holland, 1988]

Last modified May 23, 1997