PATHWAYS OF DEDUCTION

Abstract

Cyclic structures underlie formal mathematical reasoning, and replication and folding play a crucial role in the complexity of proofs. These two aspects of the geometry of proofs are discussed. 1 Deductions, foldings and the brain Different models of various regions of the brain have been proposed and they stimulated the discussion on the way our mind works. The essential feature of most of these models is the hierarchical structure which is underlying the organization. What we “see” is nevertheless not necessarily the basic mechanism. Recent studies in computational complexity and proof theory reveal that hierarchical organizations, even though structurally appealing, are computationally inefficient. In fact, our brain seems to be “fast” in performing certain tasks (such as perceiving the presence of an animal in the landscape, or intuitively grasping a complicated mathematical idea) and extremely “slow” in performing others (as the construction of a mathematical proof). No hierarchical structure conceived by man displays similar features. In this paper we would like to show how the usual hierarchical approach to the construction of formal mathematical proofs (introduced with the work of Frege and established through the work in logic until this last decade) is inappropriate to reveal the intricate structures underlying proofs.