"Constructivist mathematics" has a completely different meaning, which sometimes causes confusion at joint meetings of professional mathematicians and mathematics educators. The Dutch mathematician L.E.J. Brouwer developed a school of "intuitionist logic" that disallowed proof using the classical principle of "the excluded middle." Recall that in our logic section we included the rule of inference called "modus tollendo ponens" which says essentially that a statement must be either true or false -- we exclude any middle ground.
Modus Tollendo Ponens
Premise 1. P or Q
Premise 2. not P
Conclusion. Q
We make implicit use of this principle when we establish a proof by contradiction. Imagine trying to do mathematics without an occasional proof by contradiction. This is what the constructivist mathematics do. As far as I know, their use of the term predates the use by educators, and the two usages are completely independent.
"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."
http://www.math.fau.edu/Richman/html/construc.htm
Here is some further information on the excluded middle. (Compare this to our problem about the coins.)
"Non-Contradiction and Excluded Middle"
Peter Suber, Philosophy Department, Earlham College
* Three principles
* Denying one or more of these principles
* Rivals to these principles
* Kinds of logical opposition
"The Principle of Non-Contradiction (PNC) and Principle of Excluded Middle (PEM) are frequently mistaken for one another and for a third principle which asserts their conjunction.
Given a statement and its negation, p and ~p, the PNC asserts that at most one is true. The PEM asserts that at least one is true. The PNC says "not both" and the PEM "not neither". Together, and only together, they assert that exactly one is true.
Let us call the principle that asserts the conjunction of the PNC and PEM, the Principle of Exclusive Disjunction for Contradictories (PEDC). Surprisingly, this important principle has acquired no particular name in the history of logic.
PNC at most one is true; both can be false
PEM at least one is true; both can be true
PEDC exactly one is true, exactly one is false
Clearly the PEDC is not identical to either the PNC or the PEM, and the latter two are not identical to one another. Case 1. If the PNC were true of the world, and the PEM false, then there would be some pairs of contradictories for which neither member was true. The world would be underdetermined. The world would be thinner and more abstract than the PEDC would have it.
Case 2. If the PEM were true of the world and the PNC false, then there would be some pairs of contradictories for which both members were true. The world would be overdetermined; it would be richer and more concrete (in Hegel's sense, more articulated or differentiated and more dense and continuous) than the PEDC would allow.
Case 3. If both were false, the world would be underdetermined in some respects and overdetermined in others. These are the three ways in which the world may be said to be "inconsistent". Consistent logics can be developed that enable us to describe these inconsistent states of affairs; see e.g. Rescher and Brandom, Logic of Inconsistency, Rowman and Littlefield, 1979."
http://www.earlham.edu/~peters/courses/logsys/pnc-pem.htm
See also:
THE PRINCIPLE OF EXCLUDED MIDDLE THEN AND NOW: ARISTOTLE AND PRINCIPIA MATHEMATICA
© Floy E. Andrews
http://www.mun.ca/animus/1996vol1/andrews.htm