Following upon my previous posts, where I began to explore the possibility of framing mathematical understanding within the Thomistic modes of beauty proportio and claritas, I now consider the third mode of beauty, that of integritas. This mode brings beauty in mathematical understanding to its fulfillment and serves to complete and unite the other two modes. For it is in the crafting, rendering, and communicating mathematical truths that integritas is to be sought and developed. In being a measure of beauty, integritas measures the extent to which a work, rendering, or communication expresses a subject-matter of mathematics in such a way as to uncover ever deeper proportios and orders, enable ever more profound claritas to be revealed, and so allow for ever greater works to be developed which, in turn, are themselves measured for their individual integritas. Furthermore, it is in the acts of verification through defining, axiomatizing, and producing proofs that we move from claritas to integritas. For, it is by the movement toward integritas that the proportios form order, as seen with claritas, and become formalized in order to be communicable to others. In the end, the aim is the advancing of mathematical knowledge to those within the community of mathematicians.
(Here is the link to part 3: