Someone asked me recently about constructivism in mathematics, which led me to the Douglas Bridges’ article in Stanford Encyclopedia of Philosophy on constructive mathematics. What is constructivism? It is a philsophy of mathematics that “asserts that it is necessary to find (or “construct”) a mathematical object to prove that it exists” (wikipedia), as opposed to classical mathematics where it is enough to assume that the object does not exist and derive a contradiction.
In the Bridges’ article, I found a priceless illustrative example, the Markov Principle (MP):
For each binary sequence (an), if it is contradictory that all the terms an equal 0, then there exists a term equal to 1.
Markov’s Principle represents an unbounded search: if you have a proof that all terms an being 0 leads to a contradiction, then, by testing the terms a1,a2,a3, in turn, you are guaranteed to come across a term equal to 1; but this guarantee does not extend to an assurance that you will find the desired term before the end of the universe. ”
[Photomedia Forum post by T.Neugebauer from May 6, 2006 ]