Abstract

This article follows on the introductory article “Direct Logic for Intelligent Applications” [Hewitt 2017a]. Strong Types enable new mathematical theorems to be proved including the Formal Consistency of Mathematics. Also, Strong Types are extremely important in Direct Logic because they block all known paradoxes[Cantini and Bruni 2017]. Blocking known paradoxes makes Direct Logic safer for use in Intelligent Applications by preventing security holes. Inconsistency Robustness is performance of information systems with pervasively inconsistent information. Inconsistency Robustness of the community of professional mathematicians is their performance repeatedly repairing contradictions over the centuries. In the Inconsistency Robustness paradigm, deriving contradictions has been a progressive development and not “game stoppers.” Contradictions can be helpful instead of being something to be “swept under the rug” by denying their existence, which has been repeatedly attempted by authoritarian theoreticians. Such denial has delayed mathematical development. This article reports how considerations of Inconsistency Robustness have recently influenced the foundations of mathematics for Computer Science continuing a tradition developing the sociological basis for foundations. Mathematics here means the common foundation of all classical mathematical theories from Euclid to the mathematics used to prove Fermat's Last [McLarty 2010]. Direct Logic provides categorical axiomatizations of the Natural Numbers, Real Numbers, Ordinal Numbers, Set Theory, and the Lambda Calculus meaning that up a unique isomorphism there is only one model that satisfies the respective axioms. Good evidence for the consistency Classical Direct Logic derives from how it blocks the known paradoxes of classical mathematics. Humans have spent millennia devising paradoxes for classical mathematics. Having a powerful system like Direct Logic is important in computer science because computers must be able to formalize all logical inferences without requiring recourse to human intervention. Any inconsistency in Classical Direct Logic would be a potential security hole because it could be used to cause computer systems to adopt invalid conclusions. After [Church 1934], logicians faced the following dilemma: • 1st order theories cannot be powerful lest they fall into inconsistency because of Church’s Paradox. • 2nd order theories contravene the philosophical doctrine that theorems must be computationally enumerable. The above issues can be addressed by requiring Mathematics to be strongly typed using so that: • Mathematics self proves that it is “open” in the sense that theorems are not computationally enumerable. • Mathematics self proves that it is formally consistent. • Strong mathematical theories for Natural Numbers, Ordinals, Set Theory, the Lambda Calculus, Actors, etc. are inferentially decidable, meaning that every true proposition is provable and every proposition is either provable or disprovable. Furthermore, theorems of these theories are not enumerable by a provably total procedure.