There ain't no such thing as a proof. No formal argument can be convincing enough. All we can do is to support a newly coined statement by linking it to a number of earlier statements whose validity we tend to accept without further justification. The character of the link depends on the chosen logic, and no logic's adequacy is out of any suspicion at all. However rigorous, a proof implies a whole lot of implicit assumptions, while any attempt to clarify at least some of them inevitably throws us into the abyss of inexhaustible circularity.
That is why deduction cannot provide knowledge; at its best, it only may be of a heuristic value as a source of promising hypotheses. The same propositions could be formulated in an intuitive manner, with no reference to any deductive system. In fact most working mathematicians behave exactly this way, leaving the traditional derivation for academic reports. Moreover, stretching a clear idea to the frozen standards will often dilute the original thought in uncanny technicalities, and thus undermine the public trust instead of reinforcing it. People don't need to know why it works, as long as it works. No matter deductive or not, a formal theory is true when it is applicable in a range of practically important cases: science is a thing to heavily use rather than merely contemplate. Logic suggests, people decide.
To become truth, the outcome of a formal proof must be socially "digested", acknowledged and practically tried. Otherwise, any attempts to derive consequences of what has not yet been generally accepted as really proven would just pile up one doubt on top of another. That is why mathematical theories are often treated as whimsical toys, with their public value reduced to a couple of common recipes, or a pretentious claim. Their professional beauty cannot seduce those who needs friendliness rather than glam. In science, extensive development leads from obvious remarks to conventional generalizations, and then to an incomprehensible chaos to satisfy nobody. In every derivation, we proceed from direct observations to indirect, thus increasing the overall uncertainty.
Why should that happen? A well-developed theory is much like a thermodynamic system, with lots of assertions interacting in an almost chaotic manner. When such a science approaches self-contained existence, when in becomes conceptually closed, any inner motion would only increase the entropy of the system. The more fundamental is science, the less conclusive it is. A theory of everything is virtually a theory of nothing.
It is only in an open science that the full power of logic could be productively exploited. From one practical truth to another; from the well-established to the well-justified. Nobody prevents us from inventing a huge formalism, provided we never forget about the final goal of simplifying people's life rather than merely decorating it. What you may find quite logical may seem arbitrary to the others; it does not really matter, since we have to check the adequacy of the tool any time we need it. There is no final truth, as long as the world goes round. Things will change, and our ideas will follow.
We need deductive theories despite all their unreliability, and possibly just because of it. How can one fall in love with eternity, with death? We do not believe in logical rigor, but we want being persuaded to believe, just for a sign of compassionate esteem.