Alonzo ChurchMost of my time studying philosophy has concentrated on Alfred North Whitehead, Bertrand Russell, Ludwig Wittgenstein, and Willard van Orman Quine. I have, in my opinion solved some problems dealing mainly with opacity of reference. I have written on this on another web site. It is technical, trying to make the points by showing consequences of certain definitions and assumptions. The points are made in the computer languages Prolog And WildLIFE. Anyway this is very technical and takes a lot of time and effort to evaluate. My theory is of a language less complete that usually imagined in logic. Russell says in 1923 in “Vagueness” (in CPBR, Vol 9, p. 151), “We are capable of imagining what a precise symbolism would be, though, though we cannot actually construct such a symbolism. Hence we are able to imagine a precise meaning for such words as “or” or “not”. We can, in fact, see precisely what they would mean if our symbolism were precise. All traditional logic habitually assumes that precise symbols are being employed. It is therefore not applicable to this terrestrial life but only to an imagined celestial existence.”
In 1931 Kurt Godel brought the celestial down to earth. Whitehead and Russell had attempted to show in 1910-1913 in Principia Mathematica that mathematics could in theory all be derived from a small number of axioma and inference rules. Godel demonstrated that there would always be mathematical truths impossible to prove from any such system of axioms and rules of inference.
In my discussion of Alonzo Church, I try (at least), to show that there are facts of mathematics that cannot be even represented in any known system of logic, let alone proven.
This does not mean logic or mathematics – or the part we can know – is not useful. It is known to be very useful in science. But there are limits to it.
I am satisfied with this – no one has shown me errors in my reasoning – Although Church’s thesis is not proven, no counterexample has been found. If one were, it would be disproven.