##
**Fragments and traces of non-Euclidean geometry in Aristotle.
(Fragmente und Spuren nichteuklidischer Geometrie bei Aristoteles.)**
*(German)*
Zbl 1219.01005

Beiträge zur Altertumskunde 280. Berlin: de Gruyter (ISBN 978-3-11-022415-3/hbk; 978-3-11-022416-0/ebook). xxiv, 425 p. (2010).

Ever since 1965, Imre Tóth (1921–2010) has published reports of varying length of his major discovery [I. Tóth, Mat. Lapok 16, 300–315 (1965; Zbl 0137.00201); Magyar Tud. Akad., Mat. Fiz. Tud. Oszt. Közl. 17, 1–49 (1967; Zbl 0189.00301); Arch. Hist. Exact Sci. 3, 249–422 (1967; Zbl 0154.24602); Aristotele e i fondamenti assiomatici della geometria. Prolegomeni alla comprensione dei frammenti non-euclidei nel “Corpus Aristotelicum” nel loro contesto matematico e filosofico. Temi Metafisici e Problemi del Pensiero Antico. Studi e Testi. Milano: Vita e Pensiero (1998; Zbl 0954.01002)], viz. that the Greek geometers of the time of Plato and Aristotle had looked at the consequences of the negation of what was to become the Euclidean parallel postulate and failed to find a logical contradiction among the resulting propositions, had realized that it was undecidable on the basis of the other (absolute) assumptions, had realized the circuitous nature of attempts to prove genuinely Euclidean propositions (i.e. those that do not belong to absolute geometry), such as I.29 of Euclid’s Elements, and were subsequently forced to admit the Euclidean parallel postulate among the archai.

Given the level of scrutiny the Aristotelian corpus has received over more than two millenia, but in particular during the 19th and the first half of the 20th century, the author was under the impression of having rediscovered something – living in communist Romania of the 1950s and 1960s with no access to more recent literature (such as [T. L. Heath, Mathematics in Aristotle. Oxford: Clarendon Press; London: Oxford University Press (1949; Zbl 0033.04903)]) – as he relates to his interviewer Gaspare Polizzi in [Iris 1, 29–80 (2009)]: “I did not have the slightest inkling that I had discovered something unknown and new, on the contrary, I was certain that I had read something that was common knowledge, that everyone had known for a long time, except for me who lived in this wretched closed space which Western literature did not penetrate. It took me a long time to realize that these non-Euclidean fragments had simply been ignored by the whole two-thousand-year exegesis of Aristotle.” Much like non-Euclidean geometry itself, which was stumbled upon by several mathematicians, none of whom could bring himself to embrace it, the discovery of the fragments in Aristotle pointing to awareness of the possibility of a non-Euclidean path was made, and rejected in the same breath, by Richard Walzer in 1929, Oskar Becker in 1935, and by Heath in the book cited above. There were some who did not reject the thesis of this early awareness of the possibility to negate the parallel postulate, such as Charles Mugler, in a book on Plato, who “came to the conjectural conclusion that there were already non-Euclidean reflections in the Academy community”, without making any “reference to the non-Euclidean fragments in Aristotle. His thesis was violently attacked by the famous classical scholar Harold Cherniss, who wondered how Mugler dared to conjecture about non-Euclidean geometry in Plato’s Academy without proof from any texts.” (cited from [Iris 1, 29–80 (2009)]). Much earlier, “by exclusively analyzing the first book of Euclid’s Elements, Charles S. Peirce [\(\dots\)] had already been led to the categorical conclusion that the Euclidean text implies manifestly non-Euclidean reflections underlying its preparation.” (cited from [Iris 1, 29–80 (2009)]).

Tóth goes on to refer to the reaction to his prominently published [Arch. Hist. Exact Sci. 3, 249–422 (1967; Zbl 0154.24602)], which could no longer be ignored, and which incidentally received a very positive Zentralblatt review by Stamatis, in the following terms: “At best, the world of historians of science and philologists regarded the product of my Aristotelian archaeology with great skepticism, at times with accusations of ‘madness’. Only a minority – a negligible but perhaps homeopathic quantity – greeted my excavations with positive interest, some even with enthusiasm. Among the philologists, I recall Kurt von Fritz, the great historian of Greek thought, Evanghelos Stamatis, the publisher of Euclid, Lorenzo Minio Paluello, the great specialist in the texts of Aristotle, Giovanni Reale of course, and also Hellmut Flashar, who included my results in the last edition (1983) of his Aristoteles, the third volume of the classic and prestigious Geschichte der Philosophie, edited by Ueberweg; among the historians and philosophers I want to mention B. L. van der Waerden, Willy Hartner, Marshall Clagett, Thomas Kuhn, Sir Karl Popper, Ludovico Geymonat, Ferdinand Gonseth, Jules Vuillemin, Adolf Yuchkevitch, Izabella Bashmakova, Boris Rosenfeld and – last but not least – Hans Freudenthal, who discovered, at the same time as I did, the non-Euclidean nature of one of these fragments”. To this list one should add Theokritos Kouremenos, who expresses his support for Tóth on p. 441 of [Hermes 122, No. 4, 437–450 (1994)].

The present book, perhaps made necessary by the reception of his previous works, presents his arguments with the utmost care, based entirely on a meticulous textual analysis of not only the 19 passages (18 from Aristotle and one from Plato’s Cratylos 436a-e and 438c-e, discovered by V. Hösle [I fondamenti dell’aritmetica e della geometria di Platone. Milano: Vita e Pensiero (1994)] (where Plato is seen as presenting a very modern view of a logically consistent theory, a view which F. Ademollo [The Cratylus of Plato: A commentary. Cambridge: Cambridge University Press (2011)] finds “probably incorrect” (p. 436); his skepticism extends to the non-Euclidean interpretation of it (footnote 98 on p. 436))) on which the author’s conclusions are based, but also of Euclid’s Elements, and several other sources, such as Proclus.

Every reconstruction of ancient Greek mathematics runs the risk of being called speculative, as there are too few extant texts (and those extant ones usually do not provide uncontentious answers to the questions we ask of them, a case in point being the simple question on why Theodorus stopped at 17 (see M. Caveing [Centaurus 38, No. 2–3, 277–292 (1996; Zbl 0849.01001)] for a complete bibliography of the matter; some of the papers were omitted and should be added to the discussion on pages 39–44 of this book), and some, such as D. H. Fowler [The mathematics of Plato’s Academy. A new reconstruction. Oxford Science Publications. Oxford: Clarendon Press (1987; Zbl 0627.01002)], are admittedly highly speculative. Despite its conclusions – surprising or shocking to those who have forgotten that, as Littlewood once said to Hardy, the Greek mathematicians “are not clever schoolboys or ‘scholarship candidates’, but ‘fellows of another college’ ” – the present book belongs to the least specuative ones in the reconstruction of Greek mathematics. The author brings such an overwhelming collective evidence to the fore, with such a scrupulous avoidance of any extra-textual “insights”, that it is quite impossible to resist the argument. Indeed, the author has taken great care to answer the critics who provided some sort of argument against his thesis, and these make the dullest reading, for no fault of the author, but simply for the platitude of the arguments put forth by his critics (the most extensive being those of G. J. Kayas [Rev. Questions Sci. 147, 175–194, 281–301, 457–465 (1976; Zbl 0333.01005)]), whose main line of reasoning is that Aristotle wasn’t quite serious when he wrote what he wrote, or that his geometric examples amount to nothing more than banalities of singular triviality in the Aristotelian opus. The most striking case for Tóth’s argument comes in De Caelo I 12, 281b3, in which we are told that if the sum of the angles of a triangle is not two right angles, then it is no longer true that the diagonal of a square must be incommensurable with the side. This passage is being destroyed by his critics by taking away, through translational tricks, and by relying on a corrupted copy of the text, the if-then relationship between the two sentences.

One should not imagine that Tóth makes outlandish claims about these ancient discoveries: quite the contrary. He phrases all of his conclusions with particular care, and dismisses apparently similar ones, such as those made by A. E. Busurina [Istor. Metodol. Estestv. Nauk 11, 161–171 (1971; Zbl 0251.01002)], who believed the Greeks had discovered the existence of triangles with angle sum greater than two rights by means of spherical models (by showing that it is very unlikely that mathematicians of those times had enough knowledge of spherical geometry, or would consider a spherical triangle to be a “triangle” at all). For Tóth, the Greek forays into non-Euclidean geometry were made by following deductively a non-Euclidean hypothesis (such as the one regarding the angle sum in a triangle), much like Bolyai or Lobachevsky did, without any recourse to “models”. He also notes, after pointing out that we find the very modern sounding view in Aristotle that there is freedom in choosing the axioms for geometry, with parallels in ethics, in the Magna Moralia I 10-1, 1187a29-b14 and in the Ethica Eudemia II 6, 1222b15-42, that this stands in stark contrast with the image we get of Aristotle’s views on the axiomatization of any science, and of geometry in particular, from Analytica posteriora 1 1-10 (and II 19, where we are told that the axioms are apprehended by intuition). Far from being a reason to doubt Tóth’s interpretation, this is, in view of the comment (a subjective confession of perplexity, very rarely found in the Aristotelian corpus) in Ethica Eudemia II 6, 1222b38-39, “here we can say precisely neither that it is nor that it is not so, except this much”, a very strong element of support. If the triangles with angle sum three and four right angles that Aristotle refers to a bit earlier were there just for numbers-play sake, then there wouldn’t have been any reason for the embarrassment encapsulated in the previous admission. Independent confirmation of the unusual honesty characterizing Aristotle, who could have omitted the insertion of the effects of changing first principles on the ensuing theories, he most likely was aware of from the research of contemporary geometers, to avoid conflict with the views he had presented in Analytica posteriora 1 1-10, comes from B. Cassin [Aristote et le logos. Paris: Presses Universitaires France (1997)]: “Aristote m’a toujours paru un philosophe d’une honnêteté terrible”. Confirmation of the existence of contradictions in Aristotle’s corpus comes from M. Mignucci [L’argomentazione dimostrativa in Aristotele. (Commento agli Analitici secondi, vol. I). Padova: Antenore (1975)]: “Aristotele non può esere classificato come un pensatore sistematico e coerente, il cui discorso sia raffigurabile come il sviluppo ordinato di una serie di idee primitive.”

That Tóth could have come up with his reading of Aristotle is due to a great extent to his willingness to grant the ancient Greek geometers the status of “fellows of another college”. Taking this Littlewoodian idea seriously will necessarily lead to incredulity from the firm believers in the myth of progress. Another scholar of Greek mathematics, Aram Frenkian, who in his [Le postulat chez Euclide et chez les modernes. Paris: Librairie Philosophique J. Vrin (1940)] explictly states that he would like to, for a change, start from the opposite myth, that of perfection in the beginning, of the perfect Adam, suffered the same fate of not being cited in any book on Greek mathematics written in English (Árpád Szabó cites him, but he wrote his [Anfänge der griechischen Mathematik. München-Wien: R. Oldenbourg (1969; Zbl 0192.31603)] in German) with the exception of D. Rappaport Lachterman [The ethics of geometry. A genealogy of modernity. New York: Routledge (1989)], an outsider in the field of history of Greek mathematics. Frenkian’s book is cited by Tóth, who also dedicated [Arch. Hist. Exact Sci. 3, 249–422 (1967; Zbl 0154.24602)] to Frenkian’s memory. In his interview printed in [Iris 1, 29–80 (2009)], Tóth mentions another reason for the rejection of his interpretation (which likely happens without a careful reading of his arguments), which will perhaps be read, by those schooled in the cover-your-back-first tradition of analytic philosophy (for which Tóth has no respect, and which hardly deserves any in this context, given the amount of patent nonsense with regard to the foundations of geometry solemnly declared by its founding fathers Russell and Frege), as self-incrimination:

“One reason for my split from the community of science historians is the fact that my conception of history is very different from theirs. My exergue lies in Marx’s phrase stating that human anatomy is the key to understanding the anatomy of the ape. If you don’t know what non-Euclidean geometry is today, you can’t understand the non-Euclidean factors in past works. This is my conception of history: if you don’t know the current or 19th-century situation in mathematical analysis, you cannot understand Archimedes or Leibniz. If you aren’t at least a bit familiar with modern mathematics, you cannot decode the mathematical passages in the writings of Plato or the non-Euclidean passages in Aristotle.”

Given the level of scrutiny the Aristotelian corpus has received over more than two millenia, but in particular during the 19th and the first half of the 20th century, the author was under the impression of having rediscovered something – living in communist Romania of the 1950s and 1960s with no access to more recent literature (such as [T. L. Heath, Mathematics in Aristotle. Oxford: Clarendon Press; London: Oxford University Press (1949; Zbl 0033.04903)]) – as he relates to his interviewer Gaspare Polizzi in [Iris 1, 29–80 (2009)]: “I did not have the slightest inkling that I had discovered something unknown and new, on the contrary, I was certain that I had read something that was common knowledge, that everyone had known for a long time, except for me who lived in this wretched closed space which Western literature did not penetrate. It took me a long time to realize that these non-Euclidean fragments had simply been ignored by the whole two-thousand-year exegesis of Aristotle.” Much like non-Euclidean geometry itself, which was stumbled upon by several mathematicians, none of whom could bring himself to embrace it, the discovery of the fragments in Aristotle pointing to awareness of the possibility of a non-Euclidean path was made, and rejected in the same breath, by Richard Walzer in 1929, Oskar Becker in 1935, and by Heath in the book cited above. There were some who did not reject the thesis of this early awareness of the possibility to negate the parallel postulate, such as Charles Mugler, in a book on Plato, who “came to the conjectural conclusion that there were already non-Euclidean reflections in the Academy community”, without making any “reference to the non-Euclidean fragments in Aristotle. His thesis was violently attacked by the famous classical scholar Harold Cherniss, who wondered how Mugler dared to conjecture about non-Euclidean geometry in Plato’s Academy without proof from any texts.” (cited from [Iris 1, 29–80 (2009)]). Much earlier, “by exclusively analyzing the first book of Euclid’s Elements, Charles S. Peirce [\(\dots\)] had already been led to the categorical conclusion that the Euclidean text implies manifestly non-Euclidean reflections underlying its preparation.” (cited from [Iris 1, 29–80 (2009)]).

Tóth goes on to refer to the reaction to his prominently published [Arch. Hist. Exact Sci. 3, 249–422 (1967; Zbl 0154.24602)], which could no longer be ignored, and which incidentally received a very positive Zentralblatt review by Stamatis, in the following terms: “At best, the world of historians of science and philologists regarded the product of my Aristotelian archaeology with great skepticism, at times with accusations of ‘madness’. Only a minority – a negligible but perhaps homeopathic quantity – greeted my excavations with positive interest, some even with enthusiasm. Among the philologists, I recall Kurt von Fritz, the great historian of Greek thought, Evanghelos Stamatis, the publisher of Euclid, Lorenzo Minio Paluello, the great specialist in the texts of Aristotle, Giovanni Reale of course, and also Hellmut Flashar, who included my results in the last edition (1983) of his Aristoteles, the third volume of the classic and prestigious Geschichte der Philosophie, edited by Ueberweg; among the historians and philosophers I want to mention B. L. van der Waerden, Willy Hartner, Marshall Clagett, Thomas Kuhn, Sir Karl Popper, Ludovico Geymonat, Ferdinand Gonseth, Jules Vuillemin, Adolf Yuchkevitch, Izabella Bashmakova, Boris Rosenfeld and – last but not least – Hans Freudenthal, who discovered, at the same time as I did, the non-Euclidean nature of one of these fragments”. To this list one should add Theokritos Kouremenos, who expresses his support for Tóth on p. 441 of [Hermes 122, No. 4, 437–450 (1994)].

The present book, perhaps made necessary by the reception of his previous works, presents his arguments with the utmost care, based entirely on a meticulous textual analysis of not only the 19 passages (18 from Aristotle and one from Plato’s Cratylos 436a-e and 438c-e, discovered by V. Hösle [I fondamenti dell’aritmetica e della geometria di Platone. Milano: Vita e Pensiero (1994)] (where Plato is seen as presenting a very modern view of a logically consistent theory, a view which F. Ademollo [The Cratylus of Plato: A commentary. Cambridge: Cambridge University Press (2011)] finds “probably incorrect” (p. 436); his skepticism extends to the non-Euclidean interpretation of it (footnote 98 on p. 436))) on which the author’s conclusions are based, but also of Euclid’s Elements, and several other sources, such as Proclus.

Every reconstruction of ancient Greek mathematics runs the risk of being called speculative, as there are too few extant texts (and those extant ones usually do not provide uncontentious answers to the questions we ask of them, a case in point being the simple question on why Theodorus stopped at 17 (see M. Caveing [Centaurus 38, No. 2–3, 277–292 (1996; Zbl 0849.01001)] for a complete bibliography of the matter; some of the papers were omitted and should be added to the discussion on pages 39–44 of this book), and some, such as D. H. Fowler [The mathematics of Plato’s Academy. A new reconstruction. Oxford Science Publications. Oxford: Clarendon Press (1987; Zbl 0627.01002)], are admittedly highly speculative. Despite its conclusions – surprising or shocking to those who have forgotten that, as Littlewood once said to Hardy, the Greek mathematicians “are not clever schoolboys or ‘scholarship candidates’, but ‘fellows of another college’ ” – the present book belongs to the least specuative ones in the reconstruction of Greek mathematics. The author brings such an overwhelming collective evidence to the fore, with such a scrupulous avoidance of any extra-textual “insights”, that it is quite impossible to resist the argument. Indeed, the author has taken great care to answer the critics who provided some sort of argument against his thesis, and these make the dullest reading, for no fault of the author, but simply for the platitude of the arguments put forth by his critics (the most extensive being those of G. J. Kayas [Rev. Questions Sci. 147, 175–194, 281–301, 457–465 (1976; Zbl 0333.01005)]), whose main line of reasoning is that Aristotle wasn’t quite serious when he wrote what he wrote, or that his geometric examples amount to nothing more than banalities of singular triviality in the Aristotelian opus. The most striking case for Tóth’s argument comes in De Caelo I 12, 281b3, in which we are told that if the sum of the angles of a triangle is not two right angles, then it is no longer true that the diagonal of a square must be incommensurable with the side. This passage is being destroyed by his critics by taking away, through translational tricks, and by relying on a corrupted copy of the text, the if-then relationship between the two sentences.

One should not imagine that Tóth makes outlandish claims about these ancient discoveries: quite the contrary. He phrases all of his conclusions with particular care, and dismisses apparently similar ones, such as those made by A. E. Busurina [Istor. Metodol. Estestv. Nauk 11, 161–171 (1971; Zbl 0251.01002)], who believed the Greeks had discovered the existence of triangles with angle sum greater than two rights by means of spherical models (by showing that it is very unlikely that mathematicians of those times had enough knowledge of spherical geometry, or would consider a spherical triangle to be a “triangle” at all). For Tóth, the Greek forays into non-Euclidean geometry were made by following deductively a non-Euclidean hypothesis (such as the one regarding the angle sum in a triangle), much like Bolyai or Lobachevsky did, without any recourse to “models”. He also notes, after pointing out that we find the very modern sounding view in Aristotle that there is freedom in choosing the axioms for geometry, with parallels in ethics, in the Magna Moralia I 10-1, 1187a29-b14 and in the Ethica Eudemia II 6, 1222b15-42, that this stands in stark contrast with the image we get of Aristotle’s views on the axiomatization of any science, and of geometry in particular, from Analytica posteriora 1 1-10 (and II 19, where we are told that the axioms are apprehended by intuition). Far from being a reason to doubt Tóth’s interpretation, this is, in view of the comment (a subjective confession of perplexity, very rarely found in the Aristotelian corpus) in Ethica Eudemia II 6, 1222b38-39, “here we can say precisely neither that it is nor that it is not so, except this much”, a very strong element of support. If the triangles with angle sum three and four right angles that Aristotle refers to a bit earlier were there just for numbers-play sake, then there wouldn’t have been any reason for the embarrassment encapsulated in the previous admission. Independent confirmation of the unusual honesty characterizing Aristotle, who could have omitted the insertion of the effects of changing first principles on the ensuing theories, he most likely was aware of from the research of contemporary geometers, to avoid conflict with the views he had presented in Analytica posteriora 1 1-10, comes from B. Cassin [Aristote et le logos. Paris: Presses Universitaires France (1997)]: “Aristote m’a toujours paru un philosophe d’une honnêteté terrible”. Confirmation of the existence of contradictions in Aristotle’s corpus comes from M. Mignucci [L’argomentazione dimostrativa in Aristotele. (Commento agli Analitici secondi, vol. I). Padova: Antenore (1975)]: “Aristotele non può esere classificato come un pensatore sistematico e coerente, il cui discorso sia raffigurabile come il sviluppo ordinato di una serie di idee primitive.”

That Tóth could have come up with his reading of Aristotle is due to a great extent to his willingness to grant the ancient Greek geometers the status of “fellows of another college”. Taking this Littlewoodian idea seriously will necessarily lead to incredulity from the firm believers in the myth of progress. Another scholar of Greek mathematics, Aram Frenkian, who in his [Le postulat chez Euclide et chez les modernes. Paris: Librairie Philosophique J. Vrin (1940)] explictly states that he would like to, for a change, start from the opposite myth, that of perfection in the beginning, of the perfect Adam, suffered the same fate of not being cited in any book on Greek mathematics written in English (Árpád Szabó cites him, but he wrote his [Anfänge der griechischen Mathematik. München-Wien: R. Oldenbourg (1969; Zbl 0192.31603)] in German) with the exception of D. Rappaport Lachterman [The ethics of geometry. A genealogy of modernity. New York: Routledge (1989)], an outsider in the field of history of Greek mathematics. Frenkian’s book is cited by Tóth, who also dedicated [Arch. Hist. Exact Sci. 3, 249–422 (1967; Zbl 0154.24602)] to Frenkian’s memory. In his interview printed in [Iris 1, 29–80 (2009)], Tóth mentions another reason for the rejection of his interpretation (which likely happens without a careful reading of his arguments), which will perhaps be read, by those schooled in the cover-your-back-first tradition of analytic philosophy (for which Tóth has no respect, and which hardly deserves any in this context, given the amount of patent nonsense with regard to the foundations of geometry solemnly declared by its founding fathers Russell and Frege), as self-incrimination:

“One reason for my split from the community of science historians is the fact that my conception of history is very different from theirs. My exergue lies in Marx’s phrase stating that human anatomy is the key to understanding the anatomy of the ape. If you don’t know what non-Euclidean geometry is today, you can’t understand the non-Euclidean factors in past works. This is my conception of history: if you don’t know the current or 19th-century situation in mathematical analysis, you cannot understand Archimedes or Leibniz. If you aren’t at least a bit familiar with modern mathematics, you cannot decode the mathematical passages in the writings of Plato or the non-Euclidean passages in Aristotle.”

Reviewer: Victor V. Pambuccian (Phoenix)

### MSC:

01A20 | History of Greek and Roman mathematics |

51-03 | History of geometry |

03-03 | History of mathematical logic and foundations |

03A05 | Philosophical and critical aspects of logic and foundations |

01-02 | Research exposition (monographs, survey articles) pertaining to history and biography |