Nikolai Bugaev was a mathematician keenly interested in philosophy. He stressed the role of discontinuity in his mathematical research that he called arithmology. He also emphasized the importance of discontinuity in nature which he embodied in his version of monadology. The article discusses the viability of his philosophical investigations.

This essay considers the origins of the arithmological genre, the first specimen of which was an anonymous Neopythagorean treatise of the first century BCE. Arithmology as a special genre of philosophical writings dealing with the properties of the first ten numbers should be distinguished from number symbolism, which is a universal cultural phenomenon related to individual significant numbers. As our analysis shows, the philosophical foundations of arithmology were laid down in the treatise of Plato’s successor Speusippus On Pythagorean (...) Numbers, who relied on the Platonic doctrine of the ten ideal numbers, whereas in ancient Pythagoreanism arithmological notions, unlike number symbolism, are not attested. In the first century BCE, an epoch of revival of Platonism and Aristotelianism, Speusippus’ ideas received a second birth, thus marking the beginning of arithmology as a popular genre. (shrink)

In both ancient tradition and modern research Pythagoreanism has been understood as a religious sect or as a philosophical and scientific community. Numerous attempts have been made to reconcile these pictures as well as to analyze them separately. The most recent scholarship compartmentalizes different facets of Pythagorean knowledge, but this offers no context for exploring their origins, development, and interdependence. This collection aims to reverse this trend, addressing connections between the different fields of Pythagorean knowledge, such as eschatology, metempsychosis, metaphysics, (...) epistemology, arithmology and numerology, music, dietetics and medicine as well as politics. In particular, the contributions discuss how the Pythagorean way of life related to more doctrinal aspects of knowledge, such as Pythagorean religion and science. The volume explores the effects of this interdependence between different kinds of knowledge both within the Pythagorean corpus and in its later reception. Chapters cover historical periods from the Archaic Period (6th century BC) to Neoplatonism, Early Christianity, the European and Arabic Middle Ages, and the Renaissance through to the Early Modern Period (17th century AD). (shrink)

The reputation Theologoumena Arithmeticae has acquired is largely that of being an odd, and frequently opaque, compilation of arithmological lore. As a sourcebook for this aspect of the Pythagorean tradition it is, of course, invaluable. However, its poor reputation is increased, and its historical value lessened, by the depredations time has wrought on the text. ThA was never great prose: it is a compilation, largely from the lost Theologoumena Arithmeticae of Nicomachus of Gerasa and from Anatolius' Peri Dekados; and the (...) text which cannot be safely or conjecturally assigned to these two sources often reads like little more than the written-up notes of some student. However, the treatise contains passages of considerable lucidity, within the context of its peculiar framework. (shrink)

The reputation Theologoumena Arithmeticae has acquired is largely that of being an odd, and frequently opaque, compilation of arithmological lore. As a sourcebook for this aspect of the Pythagorean tradition it is, of course, invaluable. However, its poor reputation is increased, and its historical value lessened, by the depredations time has wrought on the text. ThA was never great prose: it is a compilation, largely from the lost Theologoumena Arithmeticae of Nicomachus of Gerasa and from Anatolius' Peri Dekados; and the (...) text which cannot be safely or conjecturally assigned to these two sources often reads like little more than the written-up notes of some student. However, the treatise contains passages of considerable lucidity, within the context of its peculiar framework. (shrink)

Pour parler de la chute ou de la déclinaison de l’Âme dans le cadre des récits cosmogoniques gnostiques, Plotin a recours, en sus du terme νεῦσις, au substantif σφάλμα, comme s’il s’agissait d’un synonyme de νεῦσις. L’enquête que nous avons menée montre le bien-fondé de cet usage, puisque le terme σφάλμα est, de fait, utilisé par les hérésiologues pour décrire, dans un langage hérité du néo-pythagorisme et donc rythmé par le nombre, la chute de Sophia, laquelle incarne le douzième éon (...) de la Dodécade en même temps que le trentième et dernier éon de l’ensemble du Plérôme. Rapporté à la métaphysique pythagoricienne dont il relève, le récit gnostique perd toute apparence d’arbitraire et révèle enfin sa véritable structure, commandée de bout en bout par le nombre. Il est raisonnable de penser que l’arithmologie rivale du traité 34 de Plotin est là pour y couper court. When speaking of the fall or decline of the Soul within the context of the Gnostic cosmogonic narratives, Plotinus, in addition to the term νεῦσις, resorts to the substantive σφάλμα, as if it were a synonym of νεῦσις. The following investigation demonstrates the validity of this usage, as the term σφάλμα is, in fact, used by the heresiologists to describe, in a language inherited from Neopythagoreanism, and therefore structured by number, the fall of Sophia, which embodies both the twelfth aeon of the Dodecade as well as the thirtieth and final aeon of the entire Pleroma. Related back to the Pythagorean metaphysics whence it is stems, the Gnostic narrative loses any appearance of arbitrariness and finally reveals its true structure, ordered from start to finish by number. It is reasonable to think that the rival arithmology of Plotinus’ Treatise 34 was created in order to undermine this very narrative. (shrink)