Douglas J. Dailey | Searcharxiv

arXiv2026

The Purpose of Mathematics according to Plato and Augustine

In 1973, Russian mathematician I.R. Shafarevitch delivered a lecture to the G\"ottingen Academy of Sciences on the purpose of mathematics. The conclusion he reached in his address is that the ultimate purpose of mathematics must be religious. In this talk, we will explore a possible way in which this claim can be justified by understanding the purpose that mathematics served within a person's intellectual formation according to Plato. To place Plato's view into a Christian perspective, we will then investigate the thought of St. Augustine of Hippo, the great fifth century theologian and bishop. Augustine's insight on the role that number plays in the development of reason sheds light on how knowledge of mathematics conduces to knowledge of God.

math.HO↗

arXiv2017

Detecting finite flat dimension of modules via iterates of the Frobenius endomorphism

It is proved that a module $M$ over a Noetherian ring $R$ of positive characteristic $p$ has finite flat dimension if there exists an integer $t\ge 0$ such that $\operatorname{Tor}_i^R(M, {}^{f^{e}}\!R)=0$ for $t\le i\le t+\dim R$ and infinitely many $e$. This extends a result of Herzog, who proved it when $M$ is finitely generated, and strengthens a result of the third author and Webb in the case $M$ is arbitrary. It is also proved that when $R$ is a Cohen-Macaulay local ring, it suffices that the Tor vanishing holds for one $e\ge \log_{p}e(R)$, where $e(R)$ is the multiplicity of $R$.

math.AC↗