The mathematician and the wolf

Osvaldo Cunha Neto

Whenever I feel, I think[1]

Despite having different etymologies, the first one referring to the quest for knowledge and the second taking place by means of inspiration[2], Mathematics and Music harmonize with each other as curiously connected subjects[3]. While it is possible to find people who receive inspiration as well as musical compositions without knowing theories, how to read musical scores and let alone fractions and logarithms, it is widely known that almost every kind of music may be transcribed into codes which ultimately, as a rule, represent numbers. Moreover, there are even those who know the mathematics inherent to encoding and are able to “read” any piece of transcribed music, although not having any talent whatsoever for singing or playing instruments. Just like the legendary poet Orpheus (and his followers), Pythagoras has not only taught religious doctrines, such as metempsychosis, but has also been interested in music[4]. However, one major difference between them is that while Orpheus allegedly received music by means of inspiration and was famous for thrilling the audiences with his voice (Prot, 315 b), Pythagoras has only come up with theories in order to explain music, and we are not told whether he could sing or play any musical instrument (let alone if he thoroughly mastered the ability to well express himself orally). What we do know is that Pythagoras has also contributed in shaping the difference between two categories of knowledge that are supported by similar archaic foundations: Philosophy and Sophistry[5]. Thus, it is not accidental the fact that, even having similar etymologies, the ‘philosopher’ differs from the ‘sophist’ since the beginning, the former handling with a theoretical knowledge and the latter with a practical one[6]. Pythagoras was the first who called himself a ‘philosopher’ (Laercio D, Lives, I, 12): the theoretician built the wall with his bare hands in order to separate himself from the practical, but both the philosopher and the sophist are sage. A paradigmatic genius to rely on so as to understand the origins of scientific and academic thought, Pythagoras decisively contributes to the creation of important models of both systematic and intellectual, but also theoretical and proto-scientific practices. He not only assumed the need for social isolation; he has also become known for creating a segregate and secret community. Although being able to join the rational and the intuitive studies, Pythagoras was not a politically active citizen, something that reflects on his own way of thinking. Considered to be the first westerner who developed mathematical and musical theories, Pythagoras also transcends the idea that there are such things as divinations and human inspirations derived from polytheistic and pantheistic mythologies[7]. By identifying recurring forms in nature and understanding the functioning and the logic behind the sensitive phenomena based on rational conjectures (supposedly universal), Pythagoras believes that there are principles transcend those which are human. These principles are consistent in themselves and, therefore, are derived from a rational being who must be perfect and, hence, unique and eternal. Then it is from the one who forged the term ‘philosopher’ that the idea of the divine associated with a dhmiourgov"[8] emerges. Turning now our attention to the sophists, we know that the apologetic writers of philosophy[9] painted them as pseudo-sages, but we lack the sophists’ performances to know how they would define themselves. From the Classical Period onwards, there is predominance of written versions about the confrontation would have taken place between the sophists and the philosophers. Eight centuries later, in the context of Imperial Rome, the author of Lives of the Sophists forges a term to manage and explain what distinguishes a ‘sophist’ from a ‘philosopher’. It is from what Philostratus called ‘Second Sophistic’ that the hypothesis that a true ‘sophist’ is characterized by ‘divine inspiration’[10] also emerges. About these two kind of divinities (the dhmiourgov" proposed by Pythagoras and the ‘divine inspiration’ by Philostratus), one would ask if they are relevant in order to understand what is proper of Philosophy and Sophistry[11].  After all, was not Socrates, “the philosopher”, inspired by a divine entity?[12] Since the difference between ‘sophist’ and ‘philosopher’ has only been established among many writers after the Classical Period, it would not be unreasonable to ask whether, while alive, Pythagoras was called more as a ‘sophist’ rather than as a philosopher[13]. Being an educator averse to crowds and holder of an h[qo" e[rhmon, Pythagoras did not leave to posterity his theories and formulations because they were restricted to his initiated entourage. Indirect sources[14], however, show that, although alien to social and political practices, such thinker was proficuous in the development of theoretical reflections on astronomy, mathematics and music. Being the precursor of the “nerd profile” (which in the Clouds, 102, also applied to the ‘sophists’), Pythagoras was aware that his practice was quite different from all the other popular sages of his time. Note that his retirement to intellectual work, which until today inspires academics, has something that also for Philostratus does not fit that which well characterizes a sophist. In his “inspired” work, the Life of the Sophists, we see that the ability to well express ideas in public, the political engagement and, in particular, the inspired performance, are the characteristic features of the sophists. Answering the acidic portrayals of the sophists made by the literate philosophers, Philostratus is the only sophist who had a work of his own preserved to us, a work which offers a characterization of his own occupation. Still, it is impossible to deny that thinking about the meaning of ‘sophist’ is a really huge challenge due to the ambiguity, the polissemy[15] and the debate over literary genre and historiography, which are all connected to this task. To begin with, I believe that a methodology which focuses solely on scarce, fragmented and random doxography about those traditionally called sophists is misguided. Will not be completely found there, as shown by the suggestions of Philostratus, what was proper to the actual sophists. Neither this material would help finding the meanings of ‘sophist’ or ‘Sophistry’ themselves; after all, it was not those sophists (supposedly canonical) who have concentrated themselves on defining their own profession. Some philosophers were much more focused on the semantic construction of these terms and they were the ones who have jointly hunted the “wolves” [16]. But, in the end, which of these versions should prevail? The one made by the philosophers, who were able to present their written account about was oral in nature? Or the version of the sophist Philostratus, who ventures himself in the literature, the same one which gave cause to the prevalence of the “enemy” tradition? We know that despite the meaning of ‘sophist’ presented by the Foreigner from Elea (Sophist, 216 d) is neither the only nor the more acidic, it is, beyond any doubt, that which resounded the most since its publication. But Protagoras’ “Truth” also had great repercussion; no one can deny that it has survived to us by means of a force that spoke and left behind its static state, representing the “the passage from enthusiasm to drafting”[17]! Despite having been burned on public square, the sophist’s Truth still reaches us today! Was Protagoras a specialist in oral discourse more than in written discourse? Just like almost all sophists of his time, probably yes! Notwithstanding, according to what Philostratus implies, Protagoras and many other sophists of the Classical Period mingled too much with other “philosophers” (Life of Sophists, 481). By accepting that, on one hand, writing is a more appropriate tool to the philosophers and that, on the other hand, the oral performance and inspiration are more easily found among the sophists, we must accept that in the Classical Period, the main exponents of these two different classes present themselves as inverted exceptions: Protagoras was a sophist who published copiously whereas Socrates was a philosopher who favored oral performance and unparalleled inspiration. In fact, exactly as the Foreigner from Elea indicates, the ‘sophist’ and ‘philosopher’ are similar just like the wolf and the dog. Let us think about hunting.... If, on one hand, the wolf is a wild animal which is even able to hunt the hunter, an independent, self-sufficient creature, having a sharp defense instinct; on the other hand, we know that the domestic breed of dogs is the result of human intervention[18]. Today, this breed is established among us due to its being docile and domesticated, but it is well known that if it is deprived of the human favoritism, it will not have the natural attributes necessary to remain free by itself. While the sophists were acknowledged for their own merits, since the dawn of Western Civilization, venturing in different human activities related to knowledge and the use of speech in public and private spheres, the philosophers, by their  turn, having a  more subtle and reserved nature, tried to separate themselves ideologically from sophists through the adoption of mechanisms that protected them from a possible direct confrontation, starting by creating “schools” with specific vocabularies and methods which thus favored the characteristic isolation of these sages. The following result was inevitable. While at the same time the sophists kept isolated from each other, plunged into some sort of inertia which made them emerge, and therefore continued interested in practices of spontaneous knowledge and speech exercises, especially in education and politics, the philosophers, by their turn, took “refuge” under the ever growing trend of written culture and sought to consolidate, within intellectual associations, the terminology, the behavior and the concepts which would later cause their triumph in history, which eventually took place. Observing the hunting we realize that even with their weaknesses the “dogs” had gained more space than the “wolves” because they had linked to men, who are those who defeat and hunt all the other species by means of their ingenious at the use tools that are not provided by nature. The “wolves”, that is, the “sophists”, having a more rustic essence and being connected to the spontaneity of the human faculties of thinking and speaking, kept isolated from each other, some eventually even associated themselves to writing, but as to what refers to the species as a whole, they were undoubtedly hunted by the “dogs”. Without establishing direct dialogue with the sophists (literally speaking), the philosophers entered their golden age that will remain uninterrupted until the appearance of the ‘Second Sophistic’. The literary production of the philosophers shows that the pejorative conceptualization of the terms ‘sophist’ and ‘Sophistry’ is completed by means of providing more details regarding the terms ‘philosopher’ and ‘Philosophy’, since it had established the basic paradigm of definition by means of pair of oppositions: “essence” versus “appearance”, “being” versus “not-being”, “knowledge” versusdoxa”, “good” versus “evil”. It is necessary to note that any definition process requires the establishment of some kind of alterity. Nor would it be possible to conjecture about the meaning of ‘Sophistry’ without looking for references to ‘Philosophy’. Part of the struggle for existence is about having the semantic domain in one’s  hand. But would it be allowed to explore new territory within the world of philosophy (through a scientific article, that is, a written speech) and thus present an apology for the sophists’ ancestral art of divination? As an interpreter of Philostratus, I hereby invite those interested in the subject to confront the classical legacy with the literary production of the philosophers in order to assess whether the issues put forth here so far deserve academic credibility or not. There will never be a consensus on what really characterizes the ‘sophist’ and the ‘philosopher’ not only because different authorities have taken distinct sides in this debate, but also because like the musician and the mathematician, both the philosopher and the sophist deal with common faculties that dialectically interrelate with each other; sometimes some of these faculties prevail over the others, also reciprocally defining and excluding each other.



ABDOUNUR, O. J. Matemática e música: pensamento analógico na construção de significados. Escrituras Editora: São Paulo, 1999.

ARISTOFANES . Clouds. Translated by Kenneth James Dover .Claredon: Oxford, 1989.

CASSIN, B. Jacques le Sophiste. Lacan logos et psycanalyse. Epel: Paris, 2012

L'effet sophistique. Gallimard: Paris, 1995.

GUTHRIE, W. K. The Sophists. Cambridge University Press: Cambridge, 1971.

LAERCIO, D. Lives of eminent philosophers. With an English translation by R .D. Hicks. Harvard University Press: London, 1991.

LIDDELL, H. G. and SCOTT, R. Greek English Lexicon. With a revised supplement. Clarendon Press: Oxford, 1996.

ONIONS, C. T. The Oxford Dictionary of English Etymology. Oxford: Clarendon, 1996.

PHILOSTRATUS The Life of Apollonius of Tyana. Translation of F. C. Conybeare. The Loeb Classical Library. Harvard University Press. London, 1989.

Lives of Sophists. Translation of Wilmer Cave Wright. The Loeb Classical Library. Harvard University Press. London, 1921.

PLATO  Protagoras. Translation of W. R. M. Lamb. The Loeb Classical Library. Harvard University Press. London, 1999.

Timaeus. Translation of R. G. Bury. The Loeb Classical Library. Harvard University Press. London, 1999.

Sophist. Translation of Harold North Fowler. The Loeb Classical Library. Harvard University Press. London, 1996.

ROUGIER, L. A Religião Astral dos Pitagóricos. Tradução de Aydano Arruda. Ibrasa: São Paulo, 1990.

VAZ PINTO, M. J. SofistasTestemunhhos e fragmentos. Introdução e notas: Maria José Vaz Pinto e Ana Alexandre Alves de Sousa. Imprensa Nacional Casa da Moeda: Lisboa, 2005.

XENOPHON Memorabilia, Oeconomicus, Symposium, Apology. The Loeb Classical Library. Translated by  E. C. Marchant and O. J. Todd. Harvard University Press: London, 1997.

On hunting. The Loeb Classical Library. Translated by  G. W. Bowersock. Harvard University Press: London, 1925.


[1] Title of poem of Fernando Pessoa, a Portuguese poet, quoted by Abdounur in his Book Matemática e Música: pensamento analógico na construção de significados, 1999, p. XVII.

[2] Bellow I quote the etymology of mathematic and music, in English, as well as, the meaning of máthēma and mousikḗ, in Greek. As shown in the ‘References’, the dictionaries consulted were Dictionary of English Etymology and Lidell & Scott Greek English Lexicon.

>mathematic, -ical. XVI or further, of Gr. mathématique, máthēma.

>μάθ-ημα , ατοςτό, (μαθεῖν) A. that which is learnt, lesson, 2. learning, knowledge, 3. esp. the mathematical sciences,  4. astrology, 5. creed.

>music - of musique – Latin mūsica – Greek mousikḗ pert. To a Muse or the Muses, concerning the arts, poetry, literature. >μουσι^κή (sc. τέχνη), , A. any art over which the Muses presided, esp. poetry sung to music, II. generally, art or letters.

[3] Since the publication of article “Neanderthal Notes” in Scientifc American of 1997 (Volume 276, Number 5310, Issue of April), it is discussed whether it would be possible to affirm that diatonic scale (with proportional numerical distance) has existed for at least 40.000 years, and then, the notion about the relationship between music and mathematic is older than previously thought, Cf. Abdounur, 1999, XVII.

[4] Rougier, 1989, p. 71.

[5] The occurrences here of the word ‘Sophistry’ actually refer, in different moments, to a neologism, that is, a meaning not found in all dictionaries. After explained that I would use the word ‘Sophistic’ in future papers, here I prefer to maintain the traditional word ‘Sophistry’ in order to avoid of misunderstanding.

[6] The “practice” dissociated from “theory” is just the semantical feature of word sofisthv" (with the suffix - thv", Cf. Guthrie, 1971, p. 27-34). As well as the opposite, that is, the “theory” without commitment to what is effective and “practical” is what defines the perpetual feeling (never actually realized) of filosovfo", Cf. Cassin, 1995, 447.

>σοφισ-τής , οῦ, master of one's craft, adept, expert, of diviners, Hdt.2.49

>φιλόσοφος , , A.lover of wisdom; Pythagoras called himself  φιλόσοφος, not σοφός,  Cic  Tusc.5.3.9D.L.Prooem.12.

[7] Or even by recognized entities present in nature and in cultural and social life.

[8] Literally speaking “who does”, “the God Creator”, Cf. Timaeus, 30 a, 30 b, 32 c.

[9] That is, Xenophon, Isocrates (who, despite being called as a ‘sophist’, also created his own ‘Philosophy’) , Plato and Aristotle.

[10]  ἡ δὲ τῇ θεσπιῳδῷ τε καὶ χρηστηριώδει, Life of Sophists, 481.

[11] The question which deserves be searched is related to the influence of the idea of a dhmiourgov" in all monotheist religious and orthodox thought, whereas the idea of qeoivand daivmone" are related to the heterodox and general approach of Greek thinkers’ thought. It is interesting that, exactly when Philostratus writes about Apollonius, a kind of holy man (pythagoric philosopher or sophist?), he uses these terms (qevo", daivmwn) more frequently than he had used them in his Life of Sophists. Cf. Life of Apollonius, book I, cap. XVIII, similar meaning between qevo" and daivmwn.

[12]  Memorabilia, book I, cap. I, § 2.

[13] It is noteworthy to remember that Aeschylus called Prometheus a sophist in his homonymous piece, 459, as well as Aristophanes thus called Socrates in Clouds, 331, 360, Herodotus thus called Solon in his History, I 29, and Plato thus called Homer, Simonides, and Hesiod in Protagoras, 316 d. Cf. Guthrie, 1971, p. 28.

[14]  Such as Aristotle, Plato, Iambicus, Porphyry, Diogenes Laertius, Sextus Empiricus, and so on.

[15] Vaz Pinto, 2005, p. 16.

[16] I am referring to Plato’s jargon created in the Sophist, 231 a. There the character of the Foreign from Eleia says it is difficult to differentiate the philosopher from the sophist, as well as the dog from the wolf. Xenophon interacts with such language in his book on Hunting.

[17] Cassin, 2012, p. 17.

[18] Selecting and breeding only the tamest wolves.


Published 26.9.2016

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