What makes a demonstration worthy of the name?

Abstract

This paper revisits the foundations of mathematical proof through the lens of Aristotle's threefold conception of truth: sensory evidence, axiomatic definition, and syllogistic deduction. I argue that modern mathematics has too often neglected the first of these - corroborated perception - in favour of axioms and theorems alone, thereby producing "proofs" that either disguise axioms as discoveries or repackage empirical observations in formal dress. Using Goedel's incompleteness theorem as a case study, I show that Goedel's reasoning covertly relies on an empirical notion of truth, smuggling physics into mathematics while claiming to remain purely formal. This reliance undermines the demonstration's claim to be "worthy of the name," since it oscillates between logical registers of differing validity - analytical, dialectical, and rhetorical. More broadly, the paper contends that mathematics often operates as a form of "virtual physics," with intuition and empirical models quietly shaping its axioms and results. By clarifying the requirements for a genuine demonstration, I aim to restore the Aristotelian distinction between logic proper and looser modes of reasoning, and to highlight the risks of conflating mathematics with physics.