Logical completeness, truth, and proofs

Source abstract

Abstract When discussing such topics as ‘truth in mathematics’, it would be culpable negligence to ignore what logic has to say on the subject. What logic has to offer are theorems, not speculations. We teach these theorems in logic classes, out apparently to no avail, since the teaching does not seem to leave any trace in grown-up mathematicians. Perhaps this is due to the fact that when proving cheorems their significance is seldom discussed; only their mathematical usefulness, as opposed to general wisdom, is stressed. Theorems are admittedly not a detailed description of reality; they refer to idealized models; logic deals only with models of reasoning; but mathematicians, and scientists in general, should know what a tremendous amount of reliable and useful information is conveyed by properties of abstract models. We should sometimes pause to reflect on theprems, besides proving new ones, especially when, as is the case in logic, they concern our own activity.