Mocking the Maths. a Preliminary Study on Three Centuries of Parodies of the Mathematical Language

Inter-studia humanitatis, 16, 2014, ISSN 1822-1114, p. 43-60 MOCKING THE MATHS. A PRELIMINARY STUDY ON THREE CENTURIES OF PARODIES OF THE MATHEMATICAL LANGUAGE

Moreno Bonda Vytautas Magnus University E-mail [email protected]

Summary While mathematics has always been considered a serious subject, the history of ideas reveals a number of satirical reinterpretations of the mathematical language, processes and principles. In Early Modern Europe, mathematics was the object of satires, critics and mockery. Rereading three famous satirical works – Alice in Wonderland, Gulliver’s Travels, and Rabelais’ Fifth Book – in which the maths are openly ridiculed, we investigate the reasons for these parodies and define them as reactions to reforms of this discipline. Alice is caricaturing negative and symbolical algebra; Swift lampoons Athanasius Kircher and his followers’ computational logic; Rabelais satirizes Ramon Llull’s thinking machines. However, we suggest that the debate was much broader and not limited to the reform of mathematics. As a matter of fact, a number of theologians understood mathematics as the language able to express in symbols the inexpressibility of the Creation’s infiniteness. The parodies of the mathematics are clearly a contestation of this epistemological system. Key words: Alice in Wonderland, Gulliver’s Travels, mathematical theology, parodies of mathematics, Rabelais, ramism, Ramon Llull, thinking machine. MOCKING THE MATHS. A PRELIMINARY STUDY ON THREE CENTURIES OF ON THREE CENTURIES STUDY A PRELIMINARY MOCKING THE MATHS. LANGUAGE OF THE MATHEMATICAL PARODIES

42 43 Introduction While mathematics is almost always considered a very serious subject and, possibly, the most respectable scientific discipline, the histories of literature and philosophy reveal a number of satirical reinterpretations of the mathematical language, its processes and its principles. At least for three centuries – from Rabelais’ desecrating critic of the Scholastic education (1564) to Lewis Carroll’s veiled parody (1865) of algebra – mathematics has been the subject of satires, critics and mockery. This paper is intended as a preliminary study on the reasons, methods, meanings and objectives of these parodies of the mathematics and the mathematical language. More precisely, we aim at defining the extent and charac- teristics of this phenomenon in Early Modern Europe. Specifically, it will firstly define the chronological limits of this attitude toward mathematics. Secondly, it provides a contextualization of these satires in the coeval epistemological debates. The ultimate ambition of the research is to understand whether these parodies were just elaborate jokes and, thus an end in itself, or rather coherent assertions of alternative definitions of knowledge meanings and processes. The paper portrays just the broader problem aiming at framing it. Herein, concrete circumstances and specific cases cannot be analysed in depth in these pages. Accordingly, it will touch on specific theories and personalities just in passing. Rather to provide references to specific studies or lay the foundations for fur- ther investigations.

1. The Parody of Mathematics: A Debate on Formal Logic The idea of parodies being a reaction to reforms or redefinitions of the processes of knowledge had been formulated in the past by Helena M. Pycior’s in her study on the intersection of mathematics and humour(Pycior, 1984). Focus- ing on the parody of symbolic algebra in Lewis Carroll’s Alice in Wonderland, the scholar investigates the reasons for such an irreverent attitude during the second half of the 19th century. This study has, first of all, the merit to contextu- alize the parody of algebra in the coeval intellectual debate about knowledge. Secondly, it makes evident 19th century scholars’ interest for the relation between worlds, the deductive processes of the human mind and the language onda

B used to express them. Similar considerations emerge in Melanie Bayley’s Alice’s Adventures in Al- gebra (2009), where the relation between the early 19th century reform of the oreno

M discipline and its parody is clearly expressed: ‘it becomes clear that Dodgson

44 45 [the real name of Lewis Carroll], a stubbornly conservative mathematician, used some scenes to satirise these radical new ideas’ (Bayley, 2009). Interest- ingly, Bayley indicates not only Carrol’s evident intent to satirize the mathematics, but illustrates also the technique used to achieve this goal: the recourse to the reductio ad absurdum – a very common logical process adopted by mathematicians devoted to Euclid. Moreover, in this article emerges, once again, the connection between the logical processes, the mathematical language and the meaning (or nonsense) of single terms. Every investigation of the comical-ironical use of the mathematical language seems a necessarily conduct to the study of the relation between formal logic and mathematics. This is what emerges in Ernest Nagel’s essay on the history of modern logic (Nagel, 1935). Dealing with imaginary numbers, the scholar indicates in William Fred one of the most tenacious critics of negative and symbolical algebra and an unexpected satirical writer able to comically use mathematical principles. Fred’s play, and consequently it analysis in Nagel, deals with the mathematical concept of “nothing” (Pycior, 1984. p155). Similarly, the same notion is ridiculed in another of Fred’s essay:Pantagruel’s Decision of the Question about Nothing (Fred, 1915. p214). In this imitation of Rabelais, Fred realizes a sketch centred on Pantagruel’s reflection on the concept of nothing in the universities. Fred indirectly refers to the mathematicians’ use of zero in operation as the division, which, while logically sustainable, has neither a practical function, nor a counterpart in nature – as in the example the character “De Morgan” offers his students to explain the meaning of half a horse and two men and three quarters. It is not by chance that Fred openly refers to Rabelais. In the French author’s works, and specifically in his Fifth Book, the mathematical speculations typical of medieval universities are criticised by the means of an irreverent parody. Rabelais’ interest for the mathematics is by far less relevant than his desire to comment on the empty logic of the nominalists. Nevertheless, the connection between logic and the parody of mathematics in his works has already been pointed out and investigated. Specifically, John Lewis tried to define Ramón Llull’s mathematical studies reception in Rabelais. Revealing, in this study, is -Ra belais use of the word mathematicusto refer to both astronomers and astrologers. Specifically, the idea of being able to predict the behaviour of a man based on the study of the stars is deemed ridiculous and contrary to the Christian concept of free will. Clearly, the serious intention to know the universe by the

Augustus De Morgan affirms the play was published in 1803 inThe Gentlemen’s Monthly Miscel-

lany. OF ON THREE CENTURIES STUDY A PRELIMINARY MOCKING THE MATHS. LANGUAGE OF THE MATHEMATICAL PARODIES

44 45 means of the study of stars and, generally, through a logic-symbolic language as that of mathematics was, at the time of Rabelais, were already the perceived remains of the empty logic of the nominalists or, even worse, as astrology. The chronological limits of the research is impose by themselves. While there are plenty of examples of recourse to the mathematical language in literature and philosophy before the 16th century, it is precisely in this period – that is, at the transition from the symbolic and transcendent universe of the Middle Ages to the concrete and “useful” nature of the humanists’ world – that the most abstract branches of the mathematics were perceived as a ridiculous rel- ict of the useless science of the nominalists. It is with Rabelais that the parody of the mathematics became an instrument to repel the empty logic to the “darkness” of Middle Ages. This aspect of the quarrel between the apologists of the ancients and the advocates of the Scholastic developed for three centuries and became very relevant in connection to the Cartesian reform of knowledge during the 17th century, and once again, with the birth of the symbolic algebra in the 19th century.

2. Why to Parody? Logic and Mathematics in Alice in Wonderland As pointed out by H. M. Pycior (1984, p149), it is evident that certain passages of Alice’s Adventures are at least echoing the language of well-known mathematical treatises published at the time when Dodgson was still teaching maths. As an example, in the famous scene of the trial of the Knave, Alice com- ments on a White Rabbit’s sentence stating: ‘I don’t believe there is an atom of meaning in it’ (Pycior, 1984. p149 and Gardner, 1960. p159). The linguistic coincidence of this expression with a declaration of Augustus De Morgan is evident. In his Trigonometry and Double Algebra De Morgan had affirmed: ‘With one exception, no word nor sign of arithmetic or algebra has one atom of meaning throughout this chapter’ (De Morgan, 1849). The reference to mathematical problems and specifically to the nonsense of negative and impossible numbers, and to the incapability of mathematics to provide any sort of result constitutes an evident thread in Carroll’s two books. In Through the Looking-Glass, Alice reiterates the absurdity of subtracting a greater from a lesser number; points out the nonsense of the mathematical onda

B terminology trying to divide a loaf by a knife and subtracting a bone from a dog; and finally concludes that mathematics cannot bring to any certitude:

oreno I’ll try if I know all the things I used to know. Let me see: four times five is

M twelve, and four times six is thirteen, and four times seven is – oh dear! I

46 47 shall never get to twenty at that rate! However, the Multiplication Table doesn’t signify: let’s try Geography (Gardner, 1960. p38). In other Alice’s passages, the reference to the debate on symbolic algebra is more indirect, but equally irreverent. As an example, it is known that Alice has fallen down a rabbit hole, eaten a cake that has shrunk her and, later, a mushroom that can restore her to her proper size. As cleverly pointed out by M. Bay- ley, the use of terms of Arabic origins in the episode of the mushroom – hookah and algebra – suggest that the Alice “restoration and reduction” is precisely a rendition of el jebr e al mokabala, that is the full Arabic name of algebra. In- terestingly, De Morgan, in his treaty on double algebra, had used precisely this extended Arabic expression to refer to algebra. The continuous reference to De Morgan apology of mathematics (as the best instrument for mental training) is evident in Carroll’s works. In mathematics ‘every term is distinctly explained, and has but one meaning, and rarely two words are employed to mean the same thing’ (De Morgan, 1898. p3 and Pycior, 1984. p151), argued the 19th century mathematician, but some Cam- bridge scholars objected that hardly can meaning be found in the recently formulated definition of negative (quantity less than nothing) and imaginary numbers (numbers which, when squared, produce a negative numbers). De Morgan himself criticized, in his Trigonometry and Double Algebra (De Morgan, 1849), George Peacock’s symbolic algebra. Other mathematicians saw in the symbolic approach and the ambiguous use of certain symbols a betrayal of mathematics distinctive traits – non-ambiguity and concreteness. Condemned was, as an example, Peacock’s introduction of “x” as a meaningless symbol, while usually stood for the arithmetical multiplication. During the early 19th century mathematics was undergoing a revision that distanced it from reality and usefulness. For this reason, Carroll’s writings, as well as the above-mentioned Fred’s play, have to be understood as a reaction to De Morgan’s double algebra and George Peacock’s symbolic algebra. Clearly the 19th century’s parodies of the mathematics were connected with the spread of new theories. Such an action-reaction process seems logical when speaking about this century: it was a century when mathematics was undergoing profound reinterpretations, and symbolic algebra was being defined. Nevertheless, as mentioned above, the parodies of the mathematical language were very common even before the introduction of, for example, impossible numbers. It is therefore necessary to understand whether it is likely to interpret earlier parodies, as those in Gulliver’s Travels or Rabelais’ oeuvres, as reactions

to concrete problems posed by the academic debates too. OF ON THREE CENTURIES STUDY A PRELIMINARY MOCKING THE MATHS. LANGUAGE OF THE MATHEMATICAL PARODIES

46 47 3. Who is Mocking Who? Doing the Maths in Gulliver’s Travels In Jonathan Swift’s writings, and specifically in the third book of Gulliver’s Travels, a number of scholars have perceived a harsh critic of the man as a rational being. Some, basing on an often-quoted letter to Alexander Pope (dated September 9th, 1725), says Swift was a rationalist with no faith in reason (Flohr, 1997). Others, contextualizing the Travels in the Querelle des Anciens et des Modernes, saw in them a pessimistic attitude toward modernity, particularly in regard to modern science (Hamou, 2007). Undeniable is Swift’s interest for the coeval scientific debate and, generally, for the development of sciences. The author’s comparison of ancient and modern thinking and, specifically, his implicit references to both John Locke’s Essay concerning human understanding and George Berkeley’s Essay toward a new theory of vision seem to confirm his intention to inquire into modern epistemology. Correspondingly, in Gulliver’s description of the Grand Academy of Lagado, the satirical allegory of the 17th and 18th centuries’ scientific community clearly emerges. In the space of two chapters, Swift indulges himself in representing every sort of absurd research: extract sunbeams out of cucumbers, calcine ice into gunpowder, write treatises on the malleability of fire, contrive a new method to build houses by commencing at the roof, etc. In some passages Swift certainly refers to real theories and experiments conducted in the early 18th century. As an example, he is commenting on some learned men’s opinions, who maintain that it is not impossible for the blind to distinguish colours by touch, or on Francois-Xavier Bon success in spinning spider silk in 1709. These and similar references induced scholars to define Swift a critic of the rational method and sciences tout court. Nevertheless, a more careful analysis of these and other passages and their sources makes evident that Swift’s criticism was not directed, generically, against the modern science or, specifically, in opposition to speculative learning. Rather, the aim of the whole third book of the Travels is to ridicule specifically scientists’ manner of posing problems and finding solutions, and their inappropriate linguistic formulation of arguments. That is to say that criticized are the dialectical principles on which modern sciences are founded upon. Re- vealing, in this sense, is the comparison with Rabelais’ description of the oc- onda

B cupations of the courtiers of Quintessence. As an example, patent are the parallels between the (ridiculed) medieval education of Gargantua (based on a particularly complex and articulated rheto- oreno

M The Grand Academy of Lagado is described in the 5th and 6th chapters of the 3rd book.

48 49 ric), the mathematical formulations of the Laputians and the absurd reasoning of the scholars of Lagado. Derided are in particular Laputa’s astronomers, icons of speculative thinking, who ‘spend the greatest part of their lives in observing the celestial bodies’ (Swift, 1865. p204). However, Swift is not inveighing exclu- sively against the uselessness of their researches, but mainly against the abuse of the mathematical language in the expositions of illogically formulated problems. The source of these passages is Cornelius Agrippa’s De Vanitate Artium et Scientiarum (Swift, 1865. p204). In this treatise, astronomers, astrologers and their disputes are deemed vain ‘being the works of neither God nor Nature, but the fiddle-faddle and trifles of mathematicians rooted in a corrupted philosophy and on the fables of the poets’ (Swift, 1865. p204 and Provvidera, 2004). The German philosopher denounces astronomers’ perverted understanding of natural processes and language in their attempt to deduce from the abstract rhetoric of the mathematic the causes of “inferior” accidents. The connection between astronomers and mathematicians is evident in the 16th century meaning of the word mathematicus – a practitioner of both arts (Lewis, 2009. p261). And, as a matter of fact, the irony exhibited by Swift writ- ing about Laputa targets indistinctly both astronomers and mathematicians as it becomes clear since Guliver’s first description of the chamber of presence where he saw the king Seated on his throne, attended on each side by persons of prime quality. Before the throne there was a large table filled with globes and spheres, and mathematical instruments of all kinds. (Swift, 1865. p192) While this introduction to the courtesan of Laputa might seem absolutely neuter, it immediately becomes clear that the register of the whole passage is the comical-ironic one – the king is so engaged in scientific enquires that needs to be “flapped” into the exercise of his hearing and speech. The need for the ‘assistance of flappers is a happy stroke of ridicule’ (Swift, 1865. p192). If it is still possible to doubt about Swift’s intention to criticize specifically mathematicians, unambiguous is the meaning of the portrayal of the banquet offered by the same king of Laputa. We had two courses, of three dishes each. In the first course there was a shoulder of mutton cut into an equilateral triangle, a piece of beef into a rhomboid, and a pudding into a cycloid. The servants cut our bread into cones, cylinders, parallelograms and other mathematical figures. (Swift, 1865. p192) MOCKING THE MATHS. A PRELIMINARY STUDY ON THREE CENTURIES OF ON THREE CENTURIES STUDY A PRELIMINARY MOCKING THE MATHS. LANGUAGE OF THE MATHEMATICAL PARODIES

48 49 Very interesting is the certitude that mathematicians themselves feltin- voked and directly involved in the debate. The already mentioned British mathematician De Morgan read Swift and vehemently reacted to his parody, exhibit- ing a good dose of sarcasm himself when commenting Swift’s technical knowledge is of a poor kind. According to him beef and mutton were served up in the shape of equilateral triangles, rhomboids, and cycloids. These plane figures have no thickness, and I defy all your readers to produce a mathematician who would be content with mutton of two dimensions. As to the bread, which appeared in cones, cylinders, and parallelograms, the mathematicians would take the cones and cylinders, and leave the parallelograms for Swift. (Swift, 1865. p204) There are no doubts: the targets of Swift parody are specifically the mathematicians and the mathematics. Accordingly, Swift’s mathematical description of the functioning of the stone that allows Laputa to fly has to be considered a parody of the mathematical language of scientists – a complex, and empty, rhetoric as that of the medieval nominalists of Rabelais’ oeuvres: The explanation which Gulliver gives of its [the loadstone] system of pro- gression, by the assistance of a diagram and letters is exactly the language and style of mathematical demonstrations, which it is evidently intended to parody. (Swift, 1865. p204) However, it seems Swift is condemning, echoing Petrus Ramus (who recog- nized in the dialectic the foundation of a scientia universalis), expressly certain dialectic aspects of the scientific debate because of the deemed contrary to the natural processes of the human thought (Lara, 2008). In Rabelais the rhetoric of the nominalists is ridiculed because it is unable to transmit knowledge; Swift mocked the mathematical language of the scientists since it is a language unable to translate the perception into applied knowledge. While medieval rhetoric and logic were not fitted for narrating the Truth of God, mathematics is inappropriate for representing the language of nature. Unmistakably, the fulcrum of Swift’s critical argumentation is the rhetorical language of sciences. It is precisely a figure of speech – that of the contrast – to reveal in full the debate in which Swift critics have to be framed; while the scholars of Lagado onda

B are speculating about ill-formulated questions, ‘the people in the street walked fast, and were generally in rags’ (Swift, 1865. p212). Gulliver’s account is evidently aimed at pointing out the Laputians’ (representation of coeval scientists) oreno inability to translate the acquired awareness into knowledge because of a dis- M

50 51 torted perception of the world and an inappropriate translation of nature’s language into that of men. Most of the third book of Gulliver’s Travels is focusing on language, from the very long (and unexpected) philological excursus about the name Laputa (Swift, 1865. p194-195), to the needs for flappers: I observed many in the habit of servants, with blown bladders, fastened like a flail to the end of a stick, which they carried in their hands. In each bladder was a small quantity of dried peas, or little pebbles. With these bladders they now and then flapped the mouth and ears of those who stood near them. It seems the minds of these peoples are so taken up with intense speculations that they can neither speak, nor attend to the discourses of others, without being roused by some external action upon their organs of speech and hearing. (Swift, 1865. p190-191) One last passage confirms that the centres of Swift’s critic are both the logical and linguistic aspects of ‘the great invention of demonstration’ (Swift, 1865. p222), which he clearly considered decayed into the dialectic syllogism since it has lost his capability to produce concrete knowledge. The first professor met by Gulliver in the Academy of Lagado was involved in an experiment to improve ‘speculative knowledge by practical and mechanical operations’ (Swift, 1865. p221). By means of a contrivance, developed by the professor, even ‘the most ignorant person might write books in philosophy, poetry, politics, law, mathematics and theology’ without genius or study (Swift, 1865. p221). This machine consisted of bits of woods with all the words (in all their forms and moods) written on them. By activating the engine, the pieces of wood were randomly shifted in a new position creating a new sequence of words. When a new suc- cession of words was obtained, some students had to write down the pieces of sentences, which made some sense. In this way new scientific treatises would have been produced. Clearly, the lack of concreteness and the inability to understand and use nature is connected, in Swift’s critic, to the incapability to organize speech according to more natural rhetorical principles. Scholars, based mainly on Sir Walter Scott’s commentaries (Scott, 1884), saw in this passage an intention to ridicule Ramon Llull’s (1232-1315)- me chanical (computistic) logic. Llull had actually invented a “thinking machine” consisting of movable circles with inscribed words and topics. However, more probably, Swift intends to criticize Llull’s followers: Athanasius Kircher (1602- 1680) who invented several machines of a somewhat similar nature; Quirinus Kahlmann (1651-1689) who announced that he had invented a mechanical instrument able to master all sciences, languages and knowledge; and certainly

the Jesuit Kaspar (Gaspard) Knittel’s Via Regia ad omnes Artes et Scientias and OF ON THREE CENTURIES STUDY A PRELIMINARY MOCKING THE MATHS. LANGUAGE OF THE MATHEMATICAL PARODIES

50 51 his other philosophical-mathematical work Cosmographia elementaris propo- sitionibus physico-mathematicis proposita (Swift, 1865. p222). Similarly but in a somewhat indirect manner, the Academy of Lagado is mocking Knittel’s pre- cursors: the Jesuits Christopher Clavius, Athanasius Kircher, Caspar Schott and Sebastian Izquierdo. Finally, Swift indicates in the natural rhetoric of the spirits of the ancient philosophers (met by Gulliver in Glubbdubdrib) an alternative to the ill-formulated speculations of the moderns and their inability to master the language (Swift, 1865. p225-249). On the one hand, the instruments and the language of the scientists of Lagado deform the perception and representation of reality: their reasoning is always structurally (formally) absolutely correct because it is independent from the meaning of the words, but, being for this reason a pure syllogism, it is unable to produce knowledge. On the other hand, the plain (natural) rhetoric of the ancients allows to directly experience nature (being an active part of it) and, thus, make use of it. These and similar contrasts, together with the references to Agrippa and Rabelais and the critics of Llull, Kircher, Kahlmann and Knitter makes the third book of Gulliver’s Travels a sort of compendium of the centuries-long dispute (Llull and Rabelais defining its beginning, and Francois-Xavier Bon and Swift himself being the sign of the contemporary relevance of the debate) between humanists’ practical attitude founded on a natural dialectic and the speculations of medieval nominalists formulated by the means of an unnatural language. The “cure” offered to Gargantua in order to get a proper education consisted in reading classical treatises where the nature is described with a natural language. Likewise, in Swift the balancing for the abstractness of the specula- tion of the moderns has to be found in the natural dialectic of the spirits of the classical philosophers. Clearly, the third book of the Travels is a direct and manifest answer to Wil- liam Wotton’s defence of the Modern (Wotton, 1694). Interesting is that both Wotton and Swift individuate a dichotomy between the natural language (the rhetoric of the ancients) and the formal logic of the moderns. Both focus on the relation between language, sciences, learning processes and the capability to represent the universe by the means of an intelligible language. The debate about language, logic and knowledge seems particularly rele- onda

B vant among ecclesiastics. Swift was a cleric (and dean of St. Patrick) and most of the other personalities directly or indirectly criticized in hisTravels were Jesuits or, generally, clergymen. This aspect of the dispute delineated by Swift has not oreno been satisfactorily investigated yet. Scholars often turned their attention either M

52 53 to Ramus’ dialectic reform, or to the persistence of a medieval logic. On the contrary, this study reveals that for almost three centuries the fulcra of the scientific debate were the language of scholars and its capability to translate the nature into knowledge. In Rabelais’s oeuvres, the reasons for such an interest in the relation between maths and the language of the universe, especially among clergymen, are revealed.

4. The Mathematical language of God in Rabelais’ Parody An in-depth analysis of the use and understanding of the mathematical language among clergymen, from Ramon Llull’s inventions and speculations to the late 18th centuries, delineates unexpected sceneries. Mathematics constituted, for a large number of clerics, an integral part of theology and a privileged language to understand and transcribe that of God. Based on John N. Crossley’s study on Llull and computer science, it could be stated that the Majorcan philosopher contributed innovatively to define: the idea of a formal language; the idea of a logical rule; the computation of combinations; the use of binary and ternary relations; the use of symbols for variables; the idea of substitution for a variable; and finally, and most revolu- tionary, the use of a machine for logic (Crossley, 2010). Revealing is the reason, which guided him in the development of the Great Art that produced all these contributes to modern informatics: the conversion of non-Christians based on the definition of an objective world attained by the means of a perfect language (Eco, 1993). Clearly Llull’s aim in developing his system was not to improve logic, but rather to provide means for the conversion of non-Christians that is, to use Pous’s words, el mecanisme l`ogic de l’Art esta al servei d’una finalitat religiosa (Pous, 1979. p120 and Crossley, 2010. p3). Language and religion are relevant topics in Athanasius Kircher’s Ars magna sciendi sive combinatorica (1669) too. The probability, its calculation and the search for a perfect language are interconnected topics in the third chapter of this book, where a new version of the Llullistic method of combination of notions is presented. While Llull used Latin words, Kircher began filling the tables of his reasoning machine with signs and symbols of a different kind: symbols able to represent abstract concepts and repetitive structures and, consequently, non-finite and non-definite notions. Furthermore, Kircher tried to calculate the possible combinations of all signs of all linguistic systems including the mathematical one. It is precisely in the use of meaningless symbols that Kircher in-

dividuated the direction in which a perfect language had to be developed. The OF ON THREE CENTURIES STUDY A PRELIMINARY MOCKING THE MATHS. LANGUAGE OF THE MATHEMATICAL PARODIES

52 53 mathematical codification offered the instrument to have both meaningless symbols and a language able to represent non-finite objects and concepts. Nevertheless, while more and more theologians were exploiting the epistemological possibilities of the mathematics, a number of scholars reacted -ve hemently to this unexpected development. The most clarifying critic, and an explicit reference to those who are condemned, is in Rabelais’ Pantagruel and in his Fifth Book. Particularly explicit is a letter Gargantua writes his son Pan- tagruel J’entens et veulx que tu aprenes les langues parfaictement. Premiere- ment la grecque comme le veult Quintilian, secondement, la latine, et puis l’hébraïcque pour les sainctes lettres, et la chaldaïcque et arabicque pareillement; et que tu formes ton style quant à la grecque, à l’imitation de Platon, quant à la latine, à Cicéron. Des ars libéraux, géométrie, arisméticque et musicque, je t’en donnay quelque goust quand tu estoit encores petit, en l’eage de cinq à six ans; poursuys la reste, et de astro- nomie saiche-en tous les canons; laisse-moy l’astrologie divinatrice et l’art de Llullius, comme abuz et vanitéz. (Saulnier, 1946. p132) While it is known that the whole work of Rabelais is an allegory of the 16th century society, this is one of the few passages in which a renowned scholar is openly mentioned. And he is even mentioned a second time in the 1542 edi- tion of Pantagruel where ‘R. Llullius, De batisfolagiis principium’ is included in the comically expanded version of the catalogue of the Library de Saint Victor (Lewis, 2010. p260). The reference to Llull in the context of an educational pro- gramme urging to learn languages, investigate both the micro and macrocosms and, generally, imitate the ancients rather than the medieval scholars, confirms that the debate on language, mathematics and knowledge was central in Ra- belais too. Unmistakably, Gargantua is suggesting there is only one way to know God – the reading of the Writs. Consequently, Rabelais is evidently contesting Llull’s claim that his logical-mathematical Art could explain, by various combinatory processes, the actual workings of the universe, and therefore the workings of God’s mind. Such an ambition is very close to those of the astrologers, both in real history and on the flying island of Laputa (Lewis, 2010. p261).

onda [I intend and want you to learn the languages perfectly. Firstly the Greek as recommended by

B Quintilian, secondly, the Latin, and then the Hebraic for the holy Writs, and the Chaldean and Ara- bic too; and that you form your style imitating Plato for the Greek, and Cicero for the Latin. […] Of the liberal arts, geometry, arithmetic and music, I gave you a taste when you were still young, at oreno the age of five; pursue the rest and the dry astronomy in every canon; leave to me the divinatory

M astrology and the art of Llull, as abuse and vanity.]

54 55 The reason for Rabelais to openly refer to a scholar and warn against his theories is the 16th century revival in the fortune of Llull’s epistemological system. As an example, the neo-Platonic Pico della Mirandola opposed astrology, but, paradoxically, acted as one of the greater supporters of Llullism (Garin, 1952 and 1976). This fact is revealing because there might be a clue on the connection between the recourse to mathematics to describe the universe and God, and a general revitalization of Platonism. The idea of a consistent recourse to mathematics in connection with a growing consciousness of the infinity of the Creation has already been proposed. Remarkable, in this context, are the dictionaries of philosophy’s expressions used to write about the concept of infinite: Plato sees in the infinite and in the finite two components of all the things, in his Filebo, he distinguishes these two concepts noting that in a way the infinite is multiple since it is what is devoid of measure and limit, it is therefore indeterminate. Consequently, the infinite is subject to a “plus” and a “minus”, to infinite reduction and restoration. (Lamanna et al. 1991. p170) Reduction and restoration were precisely the two terms used to refer to Alice and algebra in the first part of this article. A relation between the attempt to use mathematics to understand the language of the universe, theology’s ambition to understand that of God and a revival of the Platonism should be investigated. It should noted that these debates were not just theoretical speculations. A brief analysis of the history of epistemology in the ecclesiastical world reveals a great number of concrete uses of mathematics and probability to inquire into theological matters. The history of Christianity was radically refor- mulated, based on complex calculation, by the French Jesuit Jean Hardouin. Building on mathematical estimates and the Holy Writings, he contested the traditional chronology and drew a new one. In a similar manner, and based on methodological processes very close to Descartes’, Pierre-Daniel Houet wrote (in 1679) a Demonstratio evangelica in which the truth of the Writs is demon- strated. A glaring example of mathematical theology is John Craig’s Theologiae Christianae Principia Mathematica(1698). In this work, the calculation of probability is used to determine the reliability of historical sources. The aim of the author is to determine when the historical narration of Jesus’ life and actions will be unsustainable because of “a too small probability” it actually occurred. In turn, this calculation would allow the estimation of the date of Christ’s -sec ond advent, based on a passage of the Bible in which it is told that Christ will MOCKING THE MATHS. A PRELIMINARY STUDY ON THREE CENTURIES OF ON THREE CENTURIES STUDY A PRELIMINARY MOCKING THE MATHS. LANGUAGE OF THE MATHEMATICAL PARODIES

54 55 come again when not a single believer will be left on Earth. The whole book is filled with equations and calculations even though the subject matter is strictly theological. It is evident that the opportunity to use the mathematical language to understand God’s actions and plans was not just a hypothesis. Rather, it was already a common practice for many clergymen.

5. Again on the Meaning and Function of the Parody in Gulliver’s Travels. As mentioned in the introduction, language and logic seem to be the fulcra of several passages of Gulliver’s Travels. However, at a macro level it looks like the whole book is built around different applications of these two intellectual exercises. Swift defines an opposition between the natural rhetoric of the ancients and the medieval formal logic. Evident, as an example, is the opposition of rhetoric and formal logic in the comparison of the 1st and the 3rd books of the Travels. The results of the sophisticated rhetoric of the Lilliputians and those of the formal logic of the inhabitants of Laputa are manifest in their way to dress: the elegance of the former contrasts with the rags of the latter. In this comparison not only Swift’s predilection for the ancient in the tail of the Querelle des Anciens et des Modern is evident, but two centuries of debate are summarized with ingenious metaphors, contrasts and similitudes. While the Lilliputians are great orators (Swift, 1865. p8), the philosophers and astronomers of Laputa are barely able to listen to or maintain a conversa- tion. Analogously, the civilization of the formers is largely based on the rhetorical construction of their debates, while the latters prefer to use the abstract language of mathematics. The consequences of these opposite attitudes are striking: Lilliput’s harmony of proportions, the elegance of its courtesans’ clothes and the perfect exactitude of the costume Lilliputians tailor for Gulliver, contrast with the rags of the people of Lagado and the ‘very ill-made, and quite out of shape,’ clothes sewn by the mathematicians of Laputa – the result of a ‘mistake of a figure in the calculation’ (Swift, 1865. p195) Evidently the clothes of Lilliput have to be compared with the badly sew attire that the tailors-mathematicians of Laputa prepare for Gulliver, in the terms of absurdly complicated mathematical data. The contrast reveals that the modern mathematicians are on the side of the pure relativity onda

B and do not see utility as an aim in itself, while the harmonic principles of the courtesans of Lilliput make them exponents of the [school of the] useful sciences, which is based on the understanding of the harmonic oreno relations existing in nature. (Hamou, 207. p30-31) M

56 57 It is clear that the different perception of the reality in Lilliput and Laputa (but generally in the whole Travels) is the result of the adoption of different instruments to understand and describe the nature. Specifically, telescopes and language are the main tools used to acquire and transmit knowledge. However, the distortion of proportions caused by the observation through lenses and the unnatural language of mathematics and formal logic are the causes respective- ly of a distorted perception of nature and of the impossibility to benefit from the acquired knowledge (Laputa). On the contrary, the immediate perception of natural proportion and their expression by the means of an elegant rhetoric provide immediate practical benefits (Lilliput). The mistake of those trying to learn through formal logic analysis is, precisely, to focus on the enunciates’ pure form. Hence, their rhetoric is empty not being based on a process of translation founded primarily on the natural reality. Their tools and processes cannot lead to the knowledge of the Truth: telescopes and microscopes (instruments of external analysis) offer, in the best cases, a partial (distorted) image of the reality; the mathematical language cannot represent the super-sensible awareness. Nicolas Malebranche himself sug- gested, in his Recherce de la vérité (1674), that the mathematical-experimental approach could lead only to the definition of the infiniteness of the knowable, not to the knowledge: Speaking about these small animals, we could suppose that there are other smaller animals that devour them, and that are unperceivable to them because of their terrible littleness, exactly like these are invisible to us. [...] and it is possible that there are in nature smaller and smaller animals to infinity. (Malebranche, 1976. p55) It is clear that the three-centuries debate about ancient and modern epistemological systems or, specifically, about the possibility to use the mathematical language to describe the Creation, dealt first of all with the dialectic aspects of the process of acquisition and transmission of knowledge, rather than on a generic critic or apology of modern sciences.

Conclusions A number of theologians and scholars thought to have found in the mathematical language an instrument to make intelligible the super-sensible essence of the Creation: they understood mathematics as the language able to express in symbols the inexpressibility of the infiniteness. MOCKING THE MATHS. A PRELIMINARY STUDY ON THREE CENTURIES OF ON THREE CENTURIES STUDY A PRELIMINARY MOCKING THE MATHS. LANGUAGE OF THE MATHEMATICAL PARODIES

56 57 However, according to those who opposed this attitude, the absolute certitude derived from the logically perfect mathematical processes cannot coin- cide with, or substitute, the absolute Truth of transcendental knowledge. The parodies of the mathematics are clearly a contestation of the epistemological system represented by this discipline. This broader meaning clearly emerges in Alice’s reflections: The Alices refer not only to the problem of the negatives but also, directly and indirectly, to the breakdown of mathematics as a science of absolute truths following a development of the symbolical approach to algebra. In this epistemological crisis, reminiscent of Descartes’ in the Discourse on Method, Alice imitates the French philosopher by trying to establish what she knows with certainty. It is not surprising that Alice turns first to mathematics. But at this point, mathematics fails to prove a bedrock of knowledge. (Pycior, 1984. p165) Specifically, Dodgson’s reflections on algebra express a very common idea shared by those who refused the epistemological possibilities of the mathematics: the mathematical one is a purely symbolic language, which is apparently able to reproduce the Creation. However, the radical incompatibility between the two systems lies precisely in algebraic emphasis on structures over meaning. According to the detractors of mathematics, algebra is an empty language unable to represent the meaning of the Creation and describe its concrete manifestations as it was for the language of the medieval nominalists. In conclusion, it could be affirmed that the investigation points out a close connection between parodies of the mathematics and coeval attempts to reform not just this subject, but the whole epistemological system construed on it. In turn, both these new theories and their critics seem to relate to the con- ception of the finiteness or infiniteness of the Creation and, correspondingly to the necessity to adopt or renounce to a language able to describe, transcribe, and make intelligible the infiniteness. Consequently, it seems possible to frame the intellectual mocking of the mathematics in the centuries-long conflict between Platonism and Aristotelianism and their modern reinterpretations.

LITERATURE: • Bayley, M., 2009. Alice’s Adventures in Algebra. Newscientist, 2739. http://www.newscientist. onda com/article/mg2042739 B • 1.600-alices-adventures-in-algebra-wonderland-solved.html?full=true#.UlVKQiiQna0 [Ac- cessed on October 12th, 2013].

oreno • De Morgan, A., 1849. Trigonometry and Double Algebra. London: Taylor, Walton, and Maberly. M

58 59 • De Morgan, A., 1898. On the Study and Difficulties of Mathematics. 2nd ed. Chicago: Court Publishing Company. • Flohr, B., 1997. Swift’s Attitude to Reason in Book IV of Gulliver’s Travels, Essay no. 1 for the course Literature of the Eighteenth Century at the King’s College of London. http://www.itp. uni-hannover.de/~flohr/papers/m-lit-18-century1.pdf [Accessed on September 9th, 2013]. • Fred, W., 1915. Pantagruel’s Decision of the Question about Nothing. In: De Morgan, A. A Budget of Paradoxes. 2nd ed., Chicago: Open Court. • Gardner, M., ed., 1960. The Annotated Alice. New York: Clarkson Potter. • Garin, E., ed., 1952. Pico della Mirandola. Disputationes adversus astrologiam divinatricem. Firenze: Vallecchi. • Garin, E., 1976. Lo zodiaco della vita: la polemica sull’astrologia dal Trecento al Cinquecento. Roma-Bari: Laterza. • Grafton, A., 1999. Jean Hardouin: The Antiquary as Pariah. Journal of the Warburg and Cour- tauld Institutes, 62, 241–267. • Hamou, P., 2007. L’optique des Voyages de Gulliver. Revue d’historire des sciences, 60, 25–45. • Lamanna, E. P., et al., eds., 1991. Dizionario dei termini filosofici. Firenze: Le Monnier. • Lara, L. A., 2008. Petrus Ramus y el ocaso de la retórica cívica. Utopía y Praxis Latinoameri- cana, 43, 11–31. • Lewis, J., 2010. Rabelais and the reception of the ‘art’ of Ramón Llull in early sixteenth-century France. Renaissance Studies, 24 (2), 260–280. • Malebranche, N., 1976. La Recherche de la vérité. In: de Geneviève, L. R., ed. Oeuvres de Nicolas Malebranche. Paris: Vrin. • Nagel, E., 1935. Impossible Numbers: A Chapter in the History of Modern Logic. Studies in the History of Ideas, 3, 429–474. • Provvidera, T., 2004. Dell’incertitudine e della vanità delle scienze di Heinrich Cornelius Agrippa von Nettesheim. Torino: N. Aragno. • Pous, E. C., 1979. De Ramon Llull a la moderna informatica. Estudios Lulianos, 23, 113–135. • Pycior, H., 1982. Historical Roots of Confusion among Beginning Algebra Students: A Newly Discovered Manuscript. Mathematics Magazine, 55, 150–156. • Pycior, H., 1984. At the intersection of mathematics and humor: Lewis Carroll’s Alices and symbolical algebra. Victorian Studies, 28, 149–170. • Rabelais, F., 1946. Pantagruel. In: Saulnier, V., ed. Textes Littéraires Français. Geneva: Droz. • Scott, W., ed., 1884. The Works of Jonathan Swift, Dean of St. Patrick’s, Dublin. Containing Ad- ditional Letters, Tracts, and Poems Not Hitherto Published. With Notes and a Life of the Author by Sir Walter Scott. London: Bickers & Son. • Swift, J., 1865. Gulliver’s Travels into Several Remote Regions of the World. London: Cassell, Petter, and Galpin. • Wotton, W., 1694. Reflections Upon Ancient and Modern Learning. London: L. Fleake. MOCKING THE MATHS. A PRELIMINARY STUDY ON THREE CENTURIES OF ON THREE CENTURIES STUDY A PRELIMINARY MOCKING THE MATHS. LANGUAGE OF THE MATHEMATICAL PARODIES

58 59 MOCKING THE MATHS. A PRELIMINARY STUDY ON THREE CENTURIES OF PARODIES OF THE MATHEMATICAL LANGUAGE Moreno Bonda

Santrauka Matematika visą laiką laikyta rimta disciplina, o idėjų istorija atskleidžia daugelį satyrinių matematikos kalbos, procesų ir principų interpretacijų. XVI–XIX a. Europoje matematika buvo tapusi satyros, kritikos ir pajuokos objektu. Naujai skaitydami tris garsius darbus – Alisa Stebuklų šalyje, Guliverio kelionės ir Rablė Penktoji knyga, kuriuose atvirai išjuokiama matematika, nagrinėjame šio parodijavi- mo priežastis, jas įvardydami kaip reakciją į šioje disciplinoje vykdytas reformas. Al- isa karikatūrina simbolinę algebrą; Sviftas ironizuoja Athanasius Kircherio ir jo sekėjų skaičiavimo logiką; Rablė pajuokia Ramono Llullo mąstančias mašinas. Vis dėlto manome, kad vyko daug platesnis disputas, kuris neapsiribojo vien matematikos reforma, mat daugelis teologų matematiką suvokė kaip kalbą, galinčią simboliais išreikšti neišreiškiamą Kūrybos neaprėpiamumą. Matematikos parodijomis aiškiai- kri tikuojama ši epistemologinė sistema. Pagrindiniai žodžiai: Alisa Stebuklų šalyje, Guliverio kelionės, teologijos matematika, matematikos satyra, Rablė, ramismas, Ramonas Llullas, mąstanti mašina. onda B oreno M

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