Are all non-axiomatic truths (truths that cannot be proved as theorems) trivial, like the Cretan liar paradox or other self-referential statements? Only such statements are involved in Gödel’s proof of incompleteness. The same kind of statements would be excluded from formal logic by Bertrand Russell. But I suspect that Gödel left the door open to the existence of other, more significant, unprovable truths.
Unlike computers, which follow only algorithms, i.e. deductive logic, humans can “leap to conclusions”, which may be true or false. Their truth status cannot be determined deductively, but can sometimes be tested empirically, and at least partially confirmed by inductions, i.e. repetitions of the experiments (replication); and by approaching the question from different sides. If the results converge, the hypothesis is confirmed, at least until some other experiment falsifies it; then the theory can be either amended or abandoned for a better one. This is the essence of the scientific method.
Humans are very good at hypothesis-formation, sometimes too good. The left-brain interpreter (Michael Gazzanica, “The Mind’s Past”, University of California Press, Berkeley, 1998, 201 pp.) fills in gaps in sensory input to weave a complete scene or story; this sometimes leads to visual or auditory illusions, mental confabulations, or false memories, but sometimes gives rise to Eureka-type insights. It can be a step from data gathering and manipulation to understanding, which is quite a different mental state. Humans are inferior to computers in rapid accurate calculations; we cannot usually (except for idiot-savants) multiply two large numbers in our heads. But we are good at leaps of understanding.
The ability to seek patterns in data and to form hypotheses about why these patterns arose or why they exist is part of symbolic thinking, which distinguishes humans from animals. According to Terrence Deacon (“The Symbolic Species”, W.W. Norton, New York, 1997, 527 pp.), animals are capable only of iconic and indexic thought; the latter means associating inputs with results, antecedents with consequences, correlationally. Hence they are capable of learning by conditioned response, which can be reinforced or extinguished. Humans can do that too, but in addition they are also capable of symbolic thought, which asks WHY a certain patterns of antecedents and consequences exists — and then supplies an answer, from intuition or imagination. Possibly, the rewiring of the brain during evolution which made us human involved, among other features, the creation of the left-brain interpreter. This is a kind of confluence or “consilience” (E.O. Wilson’s word) of Gazzaniga and Deacon.
Thus humans can reach non-axiomatic (non-algorithmic) truths by intuition (pattern-seeking imagination) and induction (experimentation and confirmation); i.e. we can do science as well as mathematics and logic. But we are not perfect at it: hypotheses can be wrong, theories may be disconfirmed or included as special cases of broader theories. And intuition alone can lead us astray into elaborate magical or theological structures which do not even seek confirmation, except in direct revelation by a deity.
To indulge in one of these (and I leave its truth-status suspended): God knows all truths, axiomatic or not. He is quite beyond Gödel incompleteness, and beyond intractability as well (referring to problems which a computer could solve only in a time greater than the age of the universe), because he has eternity available for computing. (Actually, eternity is not just an infinite extension of time, but a different dimension, timelessness, the “eternal now”. So an intractable computation would in eternity be solved instantly.) This is what it means to say that He is all-knowing (omniscient). But He leaves a gap of uncertainty for human free will, voluntarily and graciously abandoning omniscience to leave room for the mind and life of His creatures. Hence the Bible Code contains predictions of disasters, but qualified by the (almost desperate) question “Can you change it?”
Human intuition is still hazy and imperfect; it can be wrong, false like false memories, like seeing through a haze or semi-darkness. We may grasp some unproveable truths, and accept them by faith, but doubt is ever on our heels. In science we try to remove doubt (or at least lessen it), but in metaphysics we lack the tools to do so. Humans are only imperfect images of God, who can reach ultimate truths unfailingly. Should we then accept His word in revelation? But then, how to guard against false prophets? And what if different prophets have different revelations? If they are combinable, well and good, we can take them as partial revelations. But if they are mutually inconsistent, we are in trouble.
Truth is like the Holy Grail, forever beyond our grasp.