h₂eḱ-mn̥-₄
“principle, self-evident truth”axiom, axiomatic
Nominal giving Greek axiōma, English axiom, axiomatic.
Discussion
The PIE form *h₂eḱ-mn̥- (worthy thing, thing of value) derives from the root *h₂eḱ- (sharp, pointed), with the result noun suffix *-mn̥-. The development from "sharp" to "worthy/valued" to "axiom" traces a remarkable conceptual path: sharpness → keenness → worthiness → self-evident truth.
Greek áxiōma (ἀξίωμα, "that which is deemed worthy, a self-evident proposition") is the derivative that entered philosophical and mathematical discourse. Aristotle used áxiōma for the basic principles that need no proof — propositions so "worthy" (worthy of acceptance, worthy of the name "truth") that they serve as foundations for all subsequent reasoning. English axiom, axiomatic, and the combining form axio- (as in axiology, the study of values) all descend from this single Greek formation.
The intermediate step is crucial: Greek áxios (ἄξιος, "worthy, deserving, of equal value") bridges the gap between the physical "sharp" and the abstract "self-evident." Something áxios is weighty, substantial, deserving of attention — and an áxiōma is a proposition worthy of acceptance without argument. The verb axioûn (to deem worthy, to consider deserving) preserves the evaluative sense.
The base root *h₂eḱ- (sharp) is independently one of PIE's most productive roots. It gave Latin ācer (sharp, keen), acidus (sour, acid), acūtus (sharpened, acute), and acuere (to sharpen). The English derivatives include: acid, acrid, acute, acumen (sharpness of mind), acupuncture (needle-sharpening), acme (the sharp peak, the highest point), and edge (from PGmc *agjō, from *h₂eḱ-).
The semantic chain sharp → keen → worthy → axiomatic encodes a recurring IE metaphorical pattern: physical qualities become intellectual qualities. Sharpness of blade becomes sharpness of mind (acumen), becomes worthiness of proposition (axiom). The metaphor is so deeply embedded that we no longer notice it: to call an argument "sharp" or "pointed" is to invoke the PIE root *h₂eḱ- without knowing it.