Recursive Functions

Abstract

The point at issue in the study of recursive functions is to determine which functions are effectively computable. Church’s thesis asserts that the collection of effectively computable functions is precisely the collection of general recursive functions. One knows that a function is effectively computable only after one has effectively computed it. By contrast, the general recursive functions are generated from well-defined beginnings using very strict rules of construction, so the issue that is begged is one of explicit construction versus abstract existence.

We need education in the obvious more than investigation of the obscure. —OW. Holmes, II

To reverence superiority and accept a fact though it slay him are the final tests of an educated man. —Martin H. Fischer

Mathematics has not a foot to stand upon which is not purely metaphys ical. It begins in metaphysics; and their several orbits are continually intersecting. —Thomas de Quincey

A small inaccuracy can save hours of explanation. —H.H. Munro (Saki)

The validity of mathematical propositions is independent of the actual world—the world of existing subject-matters—, is logically prior to it, and would remain unaffected were it to vanish from being. —Cassius J. Keyser

Philosophy simply puts everything before us and neither explains nor deduces anything. Since everything lies open to view, there is nothing to explain. For what is hidden, for example, is of no interest to us. —Ludwig Wittgenstein

What men really want is not knowledge but certainty. —Bertrand Russell

Logic is a gamble, at terrible odds—if it was a bet you wouldn’t take it. —Tom Stoppard