Historians of mathematics whom I have consulted on the matter have been to a one convinced that Book III of the Elements is reliably traced back to writers very near Plato and the very young Aristotle — at the Early Academy near mid-4th century BC. There is some uncertainty and further discussion of this is ongoing still in 2020, about precisely where the work of Theaetetus belongs, within the litany of mathematicians set down originally by Aristotle’s student and colleague Eudemus of Rhodes. The date of Theaetetus’s death, early on in the scholarly discussion set to 369BC, remains a matter of lively presentday discussion amongst classical scholars. Strong arguments have been presented for pinpointing his death decades earlier. However this may be, this litany of early mathematicians is repeated and discussed in some detail by Proclus, his Commentary on Euclid Book I. This website will take it as one key point the dating of work within Books III , X and XIII of Euclid’s major work. This will require detailed attention to the Scholia to Euclid, and to his texts also. Here is a sample, from late in Book III, which will have relevance to both the logical and the sequences, both the sequence of history of mathematics, and that of the internal stemma of argument-structure. Here is a sample, which exhibits both:
Proclus, as officer within the succession of heads of the Academy (his standard descriptor is ‘ ‘, meaning just that: as Pope Pius XII was official successor Vatican, so Proclus in the Academy. One might expand the metaphor with help from both the ‘philosophenmosaik’ in Naples [a primary theme here at youngersocrates.com] and from Rafael’s famous mural. Men of the eminent stature of Michelangelo and Leonardo carry the special signifying weight on this side of depictions of Plato’s school. Here again we have standard descriptors in ‘divine’ Plato and ‘inspired’ Aristotle. It is a very Greek set of ideas, this of a Diadochos following on certain progenitors, higher up than mere Heroes or even Saints per se. A little etymological sidelight can help here, a comment on the name ‘Desdemona’ Her roots are a pair of Greek words, the first part being ‘fear of’. The second goes back to demonology, or above-mortal sources of inspiration.
Few would begrudge to him what E.R. Dodds claimed about his central ‘person’ (“Missing Persons” 1977, its final chapter) an indwelling ‘demon’, making use of Dodds’s name. Was he not patient and passive and experiencing some form of trance as he cried out words [in Greek, of course] — words from the chorus of Euripides Baccae ? He fairly exploded, and passively observed as there coursed out of his lips, there as he did a para-psychology exorcism. His report has him performing this exorcism, whereupon presto!, all the 4-footed earth-crawling vermin were tele-transported off the premisses. Clearly not Dodds’s own person at work; rather the indwelling demon of whom he writes, it was ‘using my name’. Odd, but likely true. Recall the desi-demonion that our colleague Socrates used, or that used Socrates. This Renaissance Man Dodds turned up historically somewhat too late to be a proper part of the Renaissance. But if we prescind from history, he stands as a spirit, capable of publishing an anti-Nazi screed warning in the London of war-torn 1939.
Europe has known wars aplenty. A certain jewish man in Amsterdam, when I spoke to him about ‘Armistice Day’, made a telling remark. This conversation took place place aboard his floating rental houseboat in a former jewish ghetto within Amsterdam. I had noted that in the US of A we commemorate Armistice Day, as near as possible to 11 AM local time. He posed a sharply worded question: ‘which war was that, settling of which is today commemorated on 11 November’? Probably 2010AD. Our family had invited a guest aboard, a man from Massachusetts and not so incidentally with a jewish ancestry. The landlord had named his houseboat ‘Te Koop’ which I later discovered means ‘For Rent’. In any case this houseboat owner was not entirely unlike Spinoza — about whom our landlord knew much more, as soon came out, than he did about the Versailles Treaty. The conversation stayed up at the trans-personal level, however, so it never came up that some of the inequities within The Versailles Treaty may have impacted a man his age (about 60). Spinoza and this man were rejected — by the ‘monde’ of the West of his day and by his own flock or school, or tribe. In those ways Spinoza and Dodds — and Oliver Sacks too, I may add, Sacks the grateful-for-life man — they all stand shoulder-to-shoulder ‘brother alongside fellow being’. They all went out, or were soon to go out, in a way our anglo-American mathematician Lord Bertrand Russell admired. Russell cordially disliked the hyper-Christian Leibniz, whilst he just as cordially revered Spinoza. I went ashore and fetched a photocopy of Russell’s popular book [from Free University’s copy] on Philosophy of the West. which my landlord read eagerly. Quasi Pythagorean, we may think these people, successors of Pythagoras. In Russell’s case, this traditional fealty was publicly avowed (preface to his Autobiography).
Some would certainly think it presumptuous to construct parallels between ecclesiastical and secular this way. and of the divine to the merely super-heroic amongst or mortals, stand-out from the flock or herd though they surely are. The Renaissance Human was more-than=human, even where of impiety in the very in his official successor. No unclarityissues concerning quadratic and cubic irrationality, to include specifically what Plato has his young Theaetetus point to in the classical locus of ‘dunamis’ theory in the form we now have elegantly preserved by recent editors Heiberg [late 19th cent.] and Stamatis [late 20th]
Please give it the necessary close look, this snapshot of the TLG text of Euthydemus. In particular please look closely at that dialogue’s cluster of the key term aei/aiei,
Now in Chapt. 22, which runs to only 32 lines, it is given a striking emphasis by being put inside quotation marks by the always-careful Oxford editor John Burnet.
Such ‘encrustation’ is of course open to a reasonable reader’s scepticism. John Cooper inaugurated this point of criticism in his Complete Plato, (Hackett, Indianapolis 1997). On the other hand this is Plato’s way of writing our ‘eternal’ or ‘always’.
This is clearly a term Plato puts much emphasis on. Further, it is a concept close to the heart of Amphinomus, who is well known to Proclus and other commentators on Euclid. It is likely he to whom we owe Scholion #18 to Euclid I, which complains about the opening proposition. What complaint, exactly ? Well it is not a Theorem proper, but rather a construction. Thus its product appears to be one of those hitherto-non-existent items, just ‘at this moment’ built by our geometer. He is scornful toward the ‘tote=trigwnon’, i.e. the ‘then-triangle’. It is as if Triangle weren’t an eternal object !
Plato had used the term in many of the centrally platonic contexts — well over 700 specimens of this occur corpus-wide — either in its 3-letter or its 4-letter variant.
This seemingly small variation of spelling has a very direct bearing on the report in Dionysius of Halicarnassus — the remark that Plato was fond of and preferential towards a diphthongal pair of letters, ahead of using the simpler monophongal variant. This latter variant, == which I have echoed here just a dozen or so lines below this — harks back to an earlier, Pre-Plato Attic.
Quite possibly young Theaetetus grew up writing this older style of Attic in Sunium, as witness the 70% preference shown at the beginning of Euclid X, where it very likely to be his authorship. This is exactly where potentially infinite ongoing processes come over the mathematical horizon in early Greek mathematics. the Theaetetus
On the literary side a chief model of that centenary=earlier writing is Thucydides, who favors it by a 128:0 ratio over the simpler form ἀεὶ .
Here is the snapshot of our ‘flock’ of αἰεὶ
Prof. Joseph Arel of No. Arizona State has published an insightful piece elaborating on recollection in Meno and bringing in connections between self-awareness and blindness averted or inverted. He does this with much vigor and clarity and makes ample references to Derrida on the ‘aveugle’
Here, in this pair of links, you will see a set of curious signs of A. De Morgan’s achieving deeply witty results with subjects in ancient physics, especially Thales. He plays artfully on a word like ‘niemand‘ and/or Odysseus’s adopted proper name : ‘ou-tis‘ Odysseus is not often credited with a Sisyphus kind of wit there in Polyphemus;s epic auditorium. The giant’s infinite appetite (alas, anthropophagous !) is heard to reach its end with this very mortal — with Outis. Thales’s publications are the front of the atomists’s physics, h.e. the ou- part of their ou-den.
A major point Slings was fond of in his interpreting of Rep. was his bringing out the elaborate wordplay on ‘ourano’ and soundalike words, related to seeing. Here Slings can rely on the strength of the witty byplay, in that Plato’s text confesses to the peccadillo in point. That is the little prank on the reader which is to shift about amongst the cognate sounds and near-cognate meanings. Here in our Anglo-America and its customary ways of speaking, we can find speakers offering their apologies for seeming to perpetrate a pun or trick of words. And clearly in Plato’s Athens one could find those offended by exhibits of philological theatrics. Plato locates this wicked word-play in the penultimate chapter of the key middle Book 5 of his masterwork, Republic. Notice Plato’s signalling his own self-consciousness and conscientiousness over this remark near this seeing-or-heaven passage:
εἰπὼν σοι σοφίζεσθαι περὶ τὸ ὄνομα [509d3] [= ‘I seem to you to be speaking a sophism over the word.’ ]
In the myth at the end of Phaedo, recall, the soul is meant to see with newfound power after being freed from its erstwhile body; it is calculated to be advantaged with a new heaven-specific clarity. One will see well soon enough (I might put it into idiomatic French.)
Plato’s own wit, especially when challenged by his two students Aristotle and Philip of Opus, gave rise to published writing. To include, I judge the little work that makes efforts at etymological wit — the seventh and final chapter, about the names of gods. I refer to Chapt.. 7 of the π. κοσμου (=P. Kosmou). A little anacoluthon of our best texts of Timaeus 28b seems to be executing a play of wit: call your book what you may, they both look to me to be about The All. This little aside continues ‘. . . or whatever we may opt to call it’. The text uses the optative of urbanity here, a usage later called the optativus urbanitatis. This usage is urbane in the sense of one often issued by an Athenian addressing another Athenian. But it is just as fitting if a Londoner such as Jonathan Barnes is addressing a fellow Oxonian J. Annas. Something we might imagine Plato to have added here might read this way : ‘. . .my good and early editors have wanted to add an alternate title p, Physews ) . Were Plato to emulate Philodemus and his tradition he might opt for a Latin version, De Natura. Each of you two, my early academic disciples may opt differently, Aristotle using Π. οὐρανοῦ and Philip Π. κόσμου A still further aside might refer ahead to still later editors. I mean the option of the p. kosmou of pseudo-Aristotle.
An additional follower of Plato, Holger Thesleff, has for many decades now admired Plato’s wit. There are many of us Plato-admirers, both “Quick and Dead” as A.E. Taylor wrote. He exercised a specialist wit in thus including himself as one amongst the dedicatees, even today. Some of us are still quick today. Plato, the man and his work. One can reasonably say even that Holger is a major participant in the Ideal of Platonic Wit. Thesleff’s admiration of Plato is very durable indeed, and he exhibits many of the virtues of Plato-lovers all. A notable virtue, which Thesleff is witty in his naming of it. I feel it is likely that he will plead ‘no contest’, before St. Peter if necessary, of the seeming vice, ‘The Over-Attentive Reader’.
Where do we find a more worthy target for such (truly virtuous) Thesleffian reading than our writer Plato, who can match a reader’s over-attentiveness with intricacies of composition we might want to call virtuous over-attention, on the side of the writing or re=writing ? Dionysius of Halicarnassus says he might hesitate between a pair of phrasings such as ‘yesterday I went down’ and ‘I went down yesterday’, a nuanced difference certainly, but not lost on a practitioner of many tropes, one such being Anaphora. An author of very distinctly lower value is the writer of Epinomis and De Kosmou. They are both ‘pseudo-‘ in their little presumptuous appendicatings, the first an appendix to Plato, the second to Aristotle. I believe the author of both these little appended works to be the very same man, Philip of Opus, a.k.a. Younger Socrates. Likely also, for reasons I spell out elsewhere, this same expert astronomer, Philip may well have made appendages to Books IV and XIIA of Euclid’s Elements. In the case of XIIA he added a little more than an entire book. He contrived to make it over-attentively Platonistic. I spell this out elsewhere here also. In this last case, Philip has earned the descriptive name ‘pseudo-Eudoxus’, for his hyperplatonic ‘exhaustion’ arguments.
in OCT 1995, D. Robinson emends the final lines, to insert Socrates-Simpliciter as final speaker. He must thus remove Socrates-Younger. This maneouvre wrongly diminishes Younger Socrates. But this is ,recall please — the man whom Ephraim’s text — fol. 67r, above — had named SOKRATES ALLOS [Socrates-alternate]
Prof. Joseph Arel of No. Arizona State has published an insightful piece elaborating on recollection in Meno and bringing in connections between self-awareness and blindness averted or inverted. He does this with much vigor and clarity and references to Derrida on the ‘aveugle’ Here you will see many signs of A. De Morgan’s achieving insights and deeply witty results with antiquity, playing on a word like ‘niemand‘ and/or ‘ou-tis‘ Thales and ou-den are sometimes coupled. What has come to have the name ‘Socratic Paradox’ is brought closer to de Morgan if formulated, the One thing I do indeed know is about my knowing not-even-One. [Less vision than Polyphemus, after the son of Sisyphus was done with him.]