The law of the excluded muddle - useful books
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Mathematical HistoryWorries about nonconstructive proofs began to surface in the 19th century, but the possibility of having a separate branch of mathematics restricted to constructive proofs really started in the 1920's with L. E. J. Brouwer's Intuitionism. If you want a non-technical look at the history of the subject then Mathematics, The loss of certainty by Morris Kline has a fair amount on constructivism. If you are more interested in original papers then you should look at From Frege to Gödel : a source book in mathematical logic edited by Jean Heijenoort (although Brouwer's paper is very short). The recent book Gnomes in the fog by Dennis Hesseling is a look at the Brouwer's development of Intuitionism, and its reception by the mathematics community.
Constructive mathematics for the more advanced readerIntuitionism as devised by Brouwer can be somewhat obscure, and if you want to study the subject of constructive mathematics it is as well to look for later developments, such as that by Errett Bishop as found in Constructive analysis (co-authored with Douglas Bridges). Alternatively you could consider Elements of intuitionism by Michael Dummett. If you are looking for a discussion of different aspects of the subject rather than wanting to study it in detail then you should take a look at Truth in Mathematics by H G Dales and and G Oliveri which has a chapter by Douglas Bridges on Constructive truth in practice. The chapter by A.S. Troelstra in the Handbook of Mathematical logic(edited by Jon Barwise) gives a useful overview of the different schools of constructive mathematics - of course the whole of this book is relevant to the ideas of constructivism .
A course in constructive algebra by Ray Mines deals with a specific area of constructive mathematics, but would be useful for those wanting to know more about the constructive version of the Hilbert basis theorem.
Online resourcesAn overview of the subject can be found in the Stanford Encyclopedia article on Constructive Mathematics. Several current experts in the field have websites on the subject, for instance Douglas S. Bridges has some useful pages at Constructive mathematics and Eric Schechter has written an article on the Axiom of choice and a piece explaining why Constructivism is Difficult . There are also interesting articles on A.S. Troelstra's website including an Encyclopedia article on Constuctivism and proof theory (pdf file) and a Survey of the history of constructivism (ps file).
If you are interested in constructive algebra, or want to get the flavour of a constructive proof, then you should look at A constructive version of the Hilbert Basis theorem (pdf file)