Against the Mathematicians is the larger of the two surviving works of Sextus Empiricus, a Pyrrhonist sceptic and physician of the Empiric school who wrote, probably, in the late second century CE. Its title addresses not mathematicians in the modern sense but all who profess a learned discipline, and its eleven books join two works: six against the professors of the arts, and five against the dogmatic philosophers. The whole survives in medieval manuscripts, one part to each book.
Book I is against the grammarians, II the rhetoricians, III the geometers, IV the arithmeticians, V the astrologers, meaning the Chaldaeans who cast birth horoscopes, and VI the musicians. Books VII and VIII are against the logicians, on the criterion of truth and then on truth, the sign and proof; IX and X against the physicists, on the gods, cause, body, place, motion, time and number; XI against the ethicists, on good, bad and indifferent, and on whether there is an art of living. Throughout, Sextus sets the claims of the schools, the Stoics and Epicureans above all, against one another and against counter-argument, to bring the reader to suspend judgement and so to live undisturbed. His reviews of earlier doctrine, such as the account of every school's criterion in Book VII, keep much that is otherwise lost.
Sextus the Empiricist Against the Mathematicians: summary and contents (PDF)
Sextus the Empiricist Against the Mathematicians: summary and contents (Markdown)
Contents
-
Ζʹ Book VII
The first of two books against the logicians takes up the criterion of truth. Sextus begins with the ancient divisions of philosophy into one, two or three parts, then reviews what each school, from the early natural philosophers to the Stoics, held the criterion to be, including the dispute over Zeno's definition of the impression, which Cleanthes took as a literal imprint, like a seal in wax, and Chrysippus as an alteration. He closes by arguing that no criterion can be established.
-
Ηʹ Book VIII
Having disposed of the criterion, Sextus examines in turn truth, the sign, things not evident, proof, and the material and existence of proof. The middle of the book answers the Stoics, for whom a sign is the antecedent proposition of a sound conditional. It ends with the sceptic's argument against proof cancelling itself, like fire consuming its fuel or a man kicking away the ladder he has climbed, before turning to the physicists.
-
Θʹ Book IX
Physics, the first of two books against the natural philosophers, covers the physical principles, the gods, cause and what is acted upon, whole and part, and body. Sextus promises to press only the chief points, unlike Cleitomachus and the Academics, whose rebuttals ran on at length. He argues that neither body nor the incorporeal can be a cause of either, and closes by concluding that body is neither perceptible nor intelligible.
-
Ιʹ Book X
Continuing the case against the physicists, Sextus treats place, motion, time, number, and coming-to-be and perishing. He starts from Epicurus, for whom the same intangible thing is called void, place or room according to whether a body occupies it, and tests motion against a distance divided into lettered parts. The book ends with a wall that can neither perish while its stones touch nor once they are apart, and announces the ethicists.
-
ΙΑʹ Book XI
The closing book of the series turns to the ethicists and asks how the things of life are divided, what good, bad and indifferent are, and whether anything is good or bad in itself. Sextus argues that belief in such goods brings disturbance, while the one who suspends judgement lives undisturbed, as Timon promised. He then denies that there is an art of living or that it can be taught, and with this completes his account of the sceptical way.
-
Αʹ Book I
Grammar is the first of the disciplines Sextus attacks. He sets the Pyrrhonist objection apart from Epicurus's, which he puts down to Epicurus's own lack of learning and his quarrel with Nausiphanes, a pupil of Pyrrho. The arguments test the grammarians' parts of speech, the division of a Homeric verse, and their spelling and etymology, and close by finding the interpretation of poets useless, since clear poems need none and grammarians cannot agree on an author's meaning.
-
Βʹ Book II
Rhetoric is approached first through the definitions the philosophers gave of it, beginning with the Gorgias, where Plato calls it the craftsman of persuasion through speech. Sextus argues that rhetoric is no art, that fine diction is not its property, since the same words offend from a grave speaker and amuse from a mime, and that its parts and its proofs cannot stand. He ends by turning to the geometers and arithmeticians.
-
Γʹ Book III
Sextus begins his case against the geometers with their habit of taking their archai by hypothesis, following Timon, who had made the same question his starting point against the physicists. He then attacks the point, the line, the surface and the body as geometry defines them, arguing for instance that two lines placed side by side cannot become one, and closes by denying that anything can be cut or taken away from a line.
-
Δʹ Book IV
The book against the arithmeticians starts from the Pythagoreans, who declared that all things are like number and swore by Pythagoras, who gave them the tetractys. Sextus argues that the one cannot be conceived by itself or with the things counted, nor can one idea of the one, by sharing in which a thing is called one, be shared by many. He ends by showing that number can arise neither by addition nor by subtraction, so that there is no number.
-
Εʹ Book V
Sextus here turns on the astrologers, and makes clear that he means neither the science built from arithmetic and geometry nor the forecasting of droughts, rains and earthquakes practised by Eudoxus and Hipparchus, but the Chaldaeans' casting of birth horoscopes. He sets out their zodiac and its divisions, then argues that the moment of birth or conception cannot be fixed, since women conceive faster or slower, and that observation could never span the Great Year the astrologers' claims would require.
-
Ϛʹ Book VI
Music, the last of the disciplines, is taken in three senses: the science of melody and rhythm practised by Aristoxenus, skill on instruments, and success in any work. Sextus rejects the claims that music orders the cosmos by harmony or prepares the young for wisdom, quoting verses on its pull towards indulgence, and then denies the existence of sound and of time. With no time there are no feet or rhythms, and the case against the disciplines is complete.