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although attempts to provide a proof all turned out to be circular. Yet the
controversial character of many primary premisses in no way deterred
investigators from claiming their soundness. Th e demand for arguments
that are unshakeable or immovable, unerring or infallible, infl exible in the
sense of not open to persuasion, indisputable, irrefutable or incontrovert-
ible is expressed by diff erent authors with an extraordinary variety of terms.
Among the most common are akinēton (immovable), used for example by
Plato at Timaeus 51e, ametapeiston or ametapiston (not subject to persua-
sion), in Aristotle’s Posterior Analytics 72b3 and Ptolemy’s Syntaxis 1 1
6.17–21, anamartēton (unerring), in Plato’s Republic 339c, ametaptōton
(unchanging) and ametaptaiston (infallible), the fi rst in Plato’s Timaeus 29b
and Aristotle’s To p i c s 139b33, and the second in Galen, K 17(1) 863.3, and
especially the terms anamphisbētēton , incontestable (already in Diogenes of
Apollonia Fr. 1 and subsequently in prominent passages in Hero, Metrica
3 142.1, and in Ptolemy, Syntaxis 1 1 6.20 among many others) and anel-
egkton , irrefutable (Plato, Apology 22a, Timaeus 29b, all the way down to
Proclus in his Commentary on Euclid’s Elements 68.10). 18
Th e pluralism of Greek mathematics thus itself has many facets. Th e
actual practices of those who in diff erent disciplines laid claim to the title of
mathēmatikos varied appreciably. Th ey range from the astrologer working
out planetary positions for a horoscope, to the arithmetical proofs and use
of symbolism discussed by Mueller and Netz in their chapters, to the proof
of the infi nity of primes in Euclid or that of the area of a parabolic segment
in Archimedes. Th ere was as much disagreement on the nature of the
claims that ‘mathematics’ could make as on their justifi cation. One group
asserted the pre-eminence of mathematics on the grounds that it achieved
certainty, that its arguments were incontrovertible. Many philosophers and
quite a few mathematicians themselves joined together in seeing this as the
great pride of mathematics and the source of its prestige. But the disput-
able nature of the claims to indisputability kept breaking surface, either
in general or in relation to particular results. Moreover while there was
much deadly serious searching aft er certainty, there was also much playful-
ness, the ‘ludic’ quality that Netz has associated with other aspects of the18 It is striking that the term anamphisbētēton may mean indisputable or undisputed, just as
in Th ucydides (1 21) the term anexelegkton means beyond refutation (and so also beyond
verifi cation). In neither case is there any doubt, in context, as to how the word is to be
understood. Th at is less clear in the case of the chief term for ‘indemonstrable’, anapodeikton ,
which Galen has been seen as using of what has not been demonstrated (though capable
of demonstration) although in Aristotle it applies purely to what is incapable of being
demonstrated (see Hankinson 1991 ).