Looking From The Inside And From The Outside

Many times in mathematics there is a natural dichotomy between describing some object from the inside and from the outside. Imagine algebraic varieties for instance; they can be described from the outside as solution sets of polynomial equations, but one can also try to understand how it is for actual points to move around inside them, perhaps to parameterize them in some way. The concept of formal proofs has the interesting feature that it provides opportunities for both perspectives. The inner perspective has been largely overlooked, but in fact lengths of proofs lead to new ways to measure the information content of mathematical objects. The disparity between minimal lengths of proofs with and without \lemmas" provides an indication of internal symmetry of mathematical objects and their descriptions. A principal observation of this paper is that mathematical structures can be embedded into spaces of logical formulas and inherit additional structure from proofs. We shall look at nitely generated groups, rational numbers and SL(2; Z), and examples from topology and analysis. Why can a logical formula be \hard to prove"? Truth and provability are both connected to questions of algorithmic complexity. The existence of a polynomial time algorithm for deciding whether a propositional formula is a tautology or not is equivalent to the famous P = NP problem. The existence of polynomial-size proofs for all propositional tautologies is related to the complexity problem NP = co ?NP. In predicate logic the set of tautologies is not algorithmically decidable among all formulas, and this implies that minimal proofs can have non-recursive size as compared to the size of the tautology. It is not only the absolute size of proofs which matters. The minimal size of a proof of a given formula may be much shorter when one is permitted to use \lemmas" than when one is required to give a \direct" proof. These Both authors are grateful to IHES for its hospitality.