The word 'innumerable' entered English in the fourteenth century from Old French 'innumerable,' which descends from Latin 'innumerābilis' (that cannot be counted, countless). The Latin adjective is transparently constructed: the negative prefix 'in-' (not) combined with 'numerābilis' (that can be counted), from 'numerāre' (to count), from 'numerus' (number), from Proto-Indo-European *nem- (to assign, to allot). 'Innumerable' thus means, with etymological precision, 'not number-able' — exceeding the capacity of counting.
The word belongs to a productive class of Latin adjectives formed with 'in-' plus '-ābilis': 'intolerabilis' (not bearable — intolerable), 'insuperabilis' (not surmountable — insuperable), 'inexorabilis' (not movable by entreaty — inexorable), 'inexpugnabilis' (not takeable by storm — impregnable). These adjectives express impossibility — the quality named by the root word cannot be applied. Something innumerable is not merely difficult to count; it is, by implication, impossible to count.
In practice, however, English speakers use 'innumerable' as a hyperbolic intensifier rather than a literal claim of impossibility. 'Innumerable stars,' 'innumerable difficulties,' 'innumerable occasions' — in each case, the speaker means 'very many,' not 'literally impossible to count.' This gap between etymological precision and everyday usage is characteristic of many superlative adjectives: 'incredible' (not credible) means merely 'very impressive'; 'indescribable' (not describable) precedes descriptions; 'innumerable' (not countable) applies to quantities that could, in principle, be counted if anyone cared to try.
The concept of the innumerable — quantities beyond counting — has a rich philosophical and theological history. In ancient Greek thought, the question of whether the cosmos was finite or infinite occupied the pre-Socratic philosophers. Anaximander posited an 'ápeiron' (the boundless, the unlimited) as the fundamental principle of reality. Archimedes, in 'The Sand-Reckoner,' famously set out to prove that even so apparently innumerable a quantity as the number of grains of sand that could fill the universe was, in fact, finite and expressible — his point being that nothing physical is truly innumerable if the right notation is available.
In Christian theology, the innumerable became an attribute of divine abundance. God's mercies are innumerable (Psalm 40:12); the host of heaven is innumerable (Hebrews 12:22); the sand of the sea and the stars of the sky are invoked as figures of innumerable quantity in the divine promise to Abraham (Genesis 22:17). These biblical usages shaped the word's English career, lending it a sense of overwhelming abundance that persists in literary and poetic usage.
The mathematical treatment of the innumerable became rigorous in the nineteenth century with Georg Cantor's development of set theory. Cantor proved that there are different sizes of infinity — that the set of real numbers is 'larger' (more 'innumerable') than the set of natural numbers, even though both are infinite. The natural numbers (1, 2, 3...) are 'countably infinite' — they can be put in one-to-one correspondence with a listing. The real numbers are 'uncountably infinite' — no listing can include
This mathematical distinction between countable and uncountable infinity gives the everyday word 'innumerable' an unexpected precision test. Are the natural numbers innumerable? In everyday English, yes — there are too many to count in a human lifetime. In mathematics, no — they are countable (enumerable), just infinite. Are the real numbers innumerable? In both everyday English and mathematics, yes — they are both too many to count and provably impossible to enumerate
The related noun 'innumeracy' — the mathematical equivalent of illiteracy — was popularized by the mathematician John Allen Paulos in his 1988 book 'Innumeracy: Mathematical Illiteracy and Its Consequences.' Paulos used the word to describe a widespread inability to reason with numbers, understand statistical arguments, or assess quantitative claims. The word was modeled on 'illiteracy' and has since entered standard English.
Cognates across European languages are consistent: French 'innombrable' (which uses the native French form 'nombre' rather than the Latinate 'numéro'), Spanish 'innumerable,' Italian 'innumerabile,' Portuguese 'inumerável.' German, characteristically, prefers a native compound: 'unzählbar' (un-count-able), which achieves the same meaning with Germanic elements.