mid-14c., "incurable (of diseases, venom, etc.); extravagant (of expense); implacable (of hearts)," from Old French incorrigible "perfect, beyond rebuke or discipline" (14c.) or directly from Latin incorrigibilis "not to be corrected," from in- "not, opposite of" (see in- (1)) + corrigibilis, from corrigere "to correct," from com-, intensive prefix (see com-), + regere "to lead straight, rule" (from PIE root *reg- "move in a straight line," with derivatives meaning "to direct in a straight line," thus "to lead, rule"). From mid-15c. as "incapable of improvement" (of persons). Related: Incorrigibly. As a noun, from 1746.
word-forming element making abstract nouns from adjectives and meaning "condition or quality of being ______," from Middle English -ite, from Old French -ete (Modern French -ité) and directly from Latin -itatem (nominative -itas), suffix denoting state or condition, composed of -i- (from the stem or else a connective) + the common abstract suffix -tas (see -ty (2)).
Roughly, the word in -ity usually means the quality of being what the adjective describes, or concretely an instance of the quality, or collectively all the instances; & the word in -ism means the disposition, or collectively all those who feel it. [Fowler]
<a href="https://www.etymonline.com/word/incorrigibility">Etymology of incorrigibility by etymonline</a>
Harper, D. (n.d.). Etymology of incorrigibility. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/incorrigibility