I define mathematical theories as stable self-contained (autonomous?) systems of reasoning, and formal theories – as mathematical models of such systems. Working with stable self-contained models mathematicians have developed their ability to draw a maximum of conclusions from a minimum of premises. This is why mathematical modeling is so efficient (my solution to the problem of “The Incomprehensible Effectiveness of Mathematics in the Natural Sciences” – as put by Eugene Wigner).
For me, Goedel’s results are the crucial evidence that stable self-contained systems of reasoning cannot be perfect (just because they are stable and self-contained). Such systems are either non-universal (i.e. they cannot express the notion of natural numbers: 0, 1, 2, 3, 4, …), or they are universal, yet then they run inevitably either into contradictions, or into unsolvable problems.
For humans, Platonist thinking is the best way of working with imagined structures. (Another version of this thesis was proposed in 1991 by Keith Devlin on p. 67 of his Logic and Information.) Thus, a correct philosophical position of a mathematician should be: a) Platonism – on working days – when I’m doing mathematics (otherwise, my “doing” will be inefficient), b) Formalism – on weekends – when I’m thinking “about” mathematics (otherwise, I will end up in mysticism). (The initial version of this aphorism was proposed in 1979 by Reuben Hersh (picture) on p. 32 of his Some proposals for reviving the philosophy of mathematics.)