scribe’s discretion, and no pattern was assumed at the outset. e two – full word and its abbreviation – acted as allographs . is may be seen as a consequence of the scribal culture within which Diophantus operated. e upshot of this chapter, then, is to situate Diophantus historically in terms of a precise deuteronomic, scribal culture, and within the context of practices available to him from elite Greek mathematics. 2. Notes on symbolism in D iophantus We recall Nesselmann’s observation: Diophantus belongs to the category of ‘syncopated algebra’, where the text is primarily arranged as discursive, natural language (if of course in the rigid style typical of so much Greek mathematics), with certain expressions systematically abbreviated. 7 In this, it is generally understood to constitute a stepping stone leading from the rhetorical algebra of, say (if we allow ourselves such heresy), Elements Book , to the fully symbolic algebra of the moderns. As a rst approximation, let us take a couple of sentences printed in Tannery’s edition (prop. I.10, T1893, . 28.13–15): (1) Τετάχθω ὁ προστιθέμενος καὶ ἀϕαιρούμενος ἑκατέρῳ ἀριθμῷ ςΑ. κἂν μὲν τῷ Κ προστεθῇ, γίνεται ςΑ Μ ο Κ. Let the <number> which is added and taken away from each number <sc. of the two other given numbers> be set down, <namely> ςΑ <:Number 1>. And if it is added to 20, result: ςΑ Μ ο Κ <:number 1 Monads 20>. We see here the most important element in Diophantus’ symbolism: a special symbol for ‘number’, ς. We also see a transparent abbreviation for ‘monads’, Μ ο . To these should be added especially: two transparent abbreviations, for ‘dunamis’ (e ectively, ‘square’), Δ υ , and for ‘cube’, Κ υ . Symbols for higher powers exist and are made by combining symbols for the low powers, e.g. ΔΚ υ , dunamis–cube, or the h power. An appended χ turns such a power into its related unit fraction: a dunamis, Δ υ , can become a dunamiston, Δ υχ , or the unit fraction correlated with a dunamis. ( e symbol itself is reminiscent in form especially of the standard scribal symbols for case endings.) Finally we should mention a special symbol for 7 For a previous, brief characterization of Diophantus’ symbolism in practice, see Rashed 1984 : lxxxi–lxxxii, whose position I follow here. Heath 1885 : 57–82 may still be read with pro t. In general, many of the claims made in this section were made by past scholars already, and my apology for going through this section in detail is that the point is worth repeating – and should be seen in detail as an introduction to the following and much more speculative discussion.
Reasoning and symbolism in Diophantus 331‘lacking’, roughly an upside-down Ψ (I shall indeed represent it in what follows by Ψ, for lack of better fonts. Note that this is to be understood as a ‘minus’ sign followed by the entirety of the remaining expression – as if it came equipped with a set of following parentheses.) Together with Greek alphabetic numerals (Α, Β, Ι, Κ, Ρ, Σ for 1, 2, 10, 20, 100, 200 . . .) one has the main system with which complex phrases can be formed of the type, e.g. (2) Κ υ ΒΔ υ Α ςΒ Μ ο Γ Ψ Κ υ Α Δ υ Γ ςΔ Μ ο Α Most of all, Diophantine reasoning has to do with manipulation of such phrases. Syntactically, note that such phrases have a xed order: one goes through the powers in a xed sequence (although in terms of Greek syntax, any order could be natural). e numeral, also, always follows the unit to which it refers (this, however, can be explained as natural Greek syntax). Finally, there is a xed order relative to the ‘lacking’ symbol: the subtrahend is always to the right of the symbol. is of course follows from the very meaning of ‘lacking’. Semantically, we may say that the ‘number’ functions rather like an ‘unknown’, on which the ‘dunamis’ or the ‘cube’ depend as well (a single ‘number’ multiplied by itself results in a single ‘dunamis’ which, once again multiplied by a ‘number’, yields a ‘cube’). e monads, on the other hand, are independent of the ‘number’. Let us consider the wider context. When we discuss symbolism in Diophantus, we need to describe it at three levels. First, there is the symbol-ism which Diophantus had explicitly introduced in the preface to his trea-tise. Second, Diophantus has a number of fairly specialized symbols which he did not explicitly set out. ird, we should have a sense of the entire symbolic regime of the Diophantine page, bringing everything together – the markedly Diophantine, and the standard symbolism of Greek scribal practice. e symbols explicitly introduced by Diophantus are those mentioned above (in the order in which Diophantus introduces them): Δ υ , Κ υ , Δ υ Δ, ΔΚ υ , Κ υ Κ, ς, Μ ο , χ , Ψ. ese then unmistakably belong to the phrases such as those of example (2), serving further to underline the importance of this type of expression. Beyond that, the manuscripts display a variety of further symbols. Tannery systematically represents symbolically in his edition such symbols as he feels, apparently, to be markedly Diophantine (on the other hand, he always resolves standard scribal abbreviations; more on this below). e following especially are noticeable among the markedly Diophantine:
e alphabetic numerals themselves. While Greek numbers are very o en written out by alphabetic numerals, they are more frequently spelled out in Greek writing as the appropriate number words – just as we have to decide between ‘5’ and ‘ ve’. e avoidance of number words and the use of alphabetical numerals, instead, is therefore a decision involving a numerical code. 䊐, for ‘square’ (used here in the meaning of ‘a square number’). 芯, for ‘the right sides’, in a right-angled triangle. Here they are studied as ful lling Pythagoras’ theorem and therefore o ering an arena for equalities for square numbers. Strangely, Tannery does not print this symbol. Α, Β, Γ, etc. for ‘ rst’, ‘second’, ‘third’, etc. is is used in the important context where several numbers are involved in the problem, e.g. what we represent by ‘ n 1 + n 2 =3 n 3 ’ which, for Diophantus, would be ‘the rst and the second are three times the third’, with ‘ rst’, ‘second’, etc. used later on systematically to refer to the same object. Of course, such symbols are not to be confused with their respective numerals and they are di erently written out. Β πλ , Γ πλ , for ‘two times’, ‘three times’, etc. is symbolism is based on the alphabetic numerals, tucking on to them a transparent abbreviation of the Greek form of ‘times’. Ε ΙΓ : this is an especially dramatic notation whereby Diophantus refrains from resolving the results of divisions into unit fractions, and instead writes out, like in the example above, ‘ ve thirteenths’ in a kind of superscript notation. Tannery further transforms this notation into a sort of upside-down modern notation. As long as we do not mean anything technical by the word, we may refer to this as Diophantus’ ‘fraction symbolism’. e last few mentioned symbols (with the possible exception of the frac-tion symbolism) are not unique to Diophantus, but for obvious reasons the text has much more recourse to such symbols than ordinary Greek texts so that, indeed, they can be said to be markedly Diophantine. One ought to mention immediately that many words, typical to Diophantus, are not abbreviated. ese fall into two types. First, several central relations and concepts – ‘multiply’, ‘add’, ‘given’, etc. – are written in fully spelled out forms. In other words, Diophantus’ abbreviations are located within the level of the noun-phrase, and do not touch the structure of the sentence interrelating the noun-phrases. ‘Lacking’ is the exception to the rule that relations are not abbreviated, but it serves to con rm the rule that abbreviations are located at the level of the noun-phrase. e ‘lacking’
Reasoning and symbolism in Diophantus 333abbreviation is used inside the noun-phrase of the speci c form of example (2) above, when a quantitative value is set out statically. e relation of subtraction holding dynamically between such noun-phrases – when one engages in the act of subtracting a value from a quantitative term – this operation is referred to by a di erent verb, ‘take away’ ( aphairein ), which is not abbreviated. Further, the logical signposts marking the very rigid form of the problem, such as ‘let it be set down’, ‘to the positions’, etc., are fully written out. In other words, just as symbolism does not reach the level of the sen-tence, so it does not reach the level of the paragraph. e rule is con rmed: abbreviations are con ned to the level of the noun-phrase. I shall return to discuss the signi cance of this limitation in Section 4 below. For the time being, I note the conclusion, that Diophantus’ marked use of symbolism is not co-extensive with Diophantus’ marked use of language. Over and above Diophantus’ marked use of symbolism, it should be mentioned that Greek manuscripts, certainly from late antiquity onwards, used many abbreviations for common words such as prepositions, con-nectors, etc.: our own ‘&’, for instance, ultimately derives from such scribal practices. ere are also many abbreviations of grammatical forms, espe-cially case markings, so that the Greek nominal root is written, followed by the abbreviation for ‘ον’, ‘ οις’, etc. as appropriate. Such abbreviations are of course in common use in the manuscripts of Diophantus. Most (but not all) of such symbols were transparent abbreviations and in general they could be considered as a mere aid to swi writing. eir use is as could be predicted: the more expensive a manuscript was, the less such abbreviations would be used; they are more common in technical treatises than in literary works; humanists, proud of their mastery of Greek forms, would tend to resolve abbreviations, while Byzantine scribes – o en scrambling to get as much into the page as possible – would also o en tend to abbreviate. We should mention one scribal abbreviation, which is not at all speci c to Diophantus, but which is especially valuable to him: the one for the sound-sequence /is/. It so happens that this common sound-sequence is the lexical root for ‘equal’ in Greek. Since it is a very common sound-sequence, it naturally has a standard abbreviation, so that Diophantus has ‘for free’ a symbol for this important relation. How are such symbols understood? at is, what is the relationship between Diophantus’ symbols, and the alphabetically written words that they replace? e rst thing to notice, as already suggested above, is that the symbols are most o en a transparent abbreviation of the alphabetical form. Diophantus’ own strategy of choice in the symbols he had himself coined was to clip the word into its rst syllable (especially when this is a simple,
consonant–vowel syllable), which he then turned into a symbol by placing the vowel as a superscript on the consonant: Κ υ , Δ υ , Μ ο . e symbols result from two reductions – a word into its initial syllable, a syllable into its con-sonant. All of this makes sense in terms of natural language phonology so that, in such cases, Diophantus’ symbolism may be tied to the heard sound and not just to the visible trace. (It may be relevant that in all three words – monas, dunamis, kubos – the stress falls indeed on the rst syllable.) With arithm- and leipsei this simple strategy fails. e symbols, in both cases, are more complex: perhaps some combination of the alpha and the rho of the arithmos (but this is a well-known palaeographic puzzle), certainly some reference to the psi of the leipsei. is is in line with the standard symbol-ism, e.g. for prepositions: these are o en rendered by a combination of their consonants (‘pros’, e.g., becoming a ligature of the pi and the rho). Note also that while alphabetical numerals do not directly represent the sounds of the number-words they stand for, the system as a whole is iso-morphic to spoken numerals (two-number words, ‘two and thirty’ become two number-symbols, ΛΒ). In this, the alphabetical numeral system di ers from its main alternative in Greek antiquity, the acrophonic system where each symbol had, directly, a sound meaning (Π for pente, ve, Δ for deka, ten, etc.: the only exception is the use of a stroke for the unit), but the acro-phonic number symbolism as a whole was equivalent to the Roman system with which we are familiar and was no longer isomorphic to spoken numer-als: not ΛΒ, but ΔΔΔΙΙ. e latter clearly is not meant to be pronounced as ‘deka-deka-deka-click-click’. In fact, it is no longer a pronounced symbol: the trace has become free of the sound. In the alphabetical system, every-thing can be understood as symbols standing for sounds in natural Greek: I believe this may be the reason why this system was nally preferred for most ordinary writing. With this in mind, we can see that Diophantus’ marked symbols are at least potentially spoken: the numbers, as explained above, as well as the symbols based upon them. A stroke turns a numeral into its depend-ent ordinal or unit-fraction (identical in sound, as in symbol: compare English ‘third’, ‘fourth’, etc.). Further, ordinals are sometimes rendered in an even more direct phonological system, e.g. Δ ευ , abbreviating δευτερος, for ‘second’. ( us the system for ordinals has three separate forms: the fully written-out word, the phonologically abbreviated form and the alphabetic numeral-based form. is is important, given the role of ordinals as a kind of unknown-mark in expressions such as ‘the rst number’.) e ×-times symbolism, too, merely adds the onset consonants of the abbreviated words: Β πλ for ‘double’.
Reasoning and symbolism in Diophantus 335 e symbols for square, and for sides in a right-angled triangle, are the exception, then. ere the trace, and not the sound, becomes the vehicle of meaning. e reason for this is clear, as the trace here has indeed such an obvious connotation. e sign and the signi ed are isomorphic. Even so, note that the understanding is that 䊐 stands not just for the concept ‘square’ but also and perhaps primarily for the sequence ‘tetragon’, as wit-nessed by the fact that the symbol is o en followed by case marking: 䊐 οις for ‘tetragonois’, ‘by the squares’. e most interesting exception is the form 䊐 䊐, sometimes used to represent ‘squares’, the plural marked not by the sound of the case ending, but by the tracing of duplication (compare our use of ‘pp.’, for instance, for ‘pages’; notice also that the same also happens occasionally with the ‘number’ symbol). Speaking generally for Greek writing in manuscripts, the phonological nature of abbreviation symbolism becomes most apparent through the rebus principle. To provide an example: there is a standard scribal abbrevia-tion for the Greek word ‘ara’, ‘therefore’. ere is also an important prepo-sition, ‘para’, meaning, roughly, ‘alongside’. e letter pi, followed by the symbol for ‘ara’, may be used to represent the preposition ‘para’. Such rebus writing is common in Greek manuscripts and shows that the symbol for ‘ara’ stands not merely for the concept ‘therefore’ but, perhaps more funda-mentally, for the sound-sequence ‘ara’. Obviously, Diophantus’ symbolism does not lend itself to such rebus combinations. One can mention, however, an important close analogue. We recall Diophantus’ symbol for ‘number’, meaning, e ectively, the ‘unknown’. is may be said to be the cornerstone of Diophantus’ symbol-ism: on it ride the higher powers; it is the starting point for investigation in each problem. It is thus, perhaps, not inappropriate that this symbol is the least transparently phonological. It is, so to speak, Diophantus’ cipher. Crucially, it is also clearly de ned by Diophantus in his introduc-tion: ‘ at which possesses none of these properties [such as dunamis, cube, etc.] and has in it an indeterminate number of monads, is called a number and its symbol is ς’ (Tannery 6.3–5). us the symbol is, strictly speaking, only to be used for the indeterminate, or unknown, goal of the problem. It should be used in such contexts as ‘Let the <number> which is added and taken away from each number <sc. of the two other given numbers> be set down, <namely> ςΑ <:Number 1>.’ Notice the two occurrences of ‘number’ in this phrase. e rst is ‘number’ in its standard Greek meaning (which therefore, one would think, should not be abbreviable into the symbol ς). In the phrase ‘from each number’, the word ‘number’ does not stand for an unknown number, but just for
‘number’. It is only the second number – the one counted as ‘1’ - which serves as an unknown in this problem. Only this, then, by Diophantus’ explicit de nition, counts as a ς; appropriately, then, Tannery prints the rst ‘number’ as a fully spelled-out word and the second as a symbol. But as the reader may guess by now, there are many cases in the manuscripts where ‘number’ of the rst type is abbreviated, as well, using Diophantus’ symbol ς. 8 us the symbol is understood, at least by Diophantus’ scribes, to range not across a semantic range (the unknown number), but across a phonological or orthographic range (the representation of the sound, or trace, ‘arithm-’). It would indeed be surprising if it were otherwise, given that scribal symbolism, as a system, was understood in such phonological or orthographic terms. e text in example (1) above followed closely (with some variation of orthography) Tannery’s edition. It is clearly punctuated and spaced (as it is not in the manuscripts, not even the Renaissance ones). It has accents and aspiration marks (like the Renaissance manuscripts, but most probably unlike Diophantus’ text in late antiquity). It also sharply demarcates the two kinds of writing: explicit and markedly Diophantine symbols, which, in the proof itself, Tannery systematically presents in abbreviated form, on the one hand; and standard scribal abbreviations, which Tannery systemati-cally resolves (as, indeed, philologers invariably do). As Tannery himself recognized, his systematization of the symbolism was not based on manuscript evidence. I shall not say anything more on the unmarked symbolism, such as the case markings, whose usage indeed di ers (as one expects) from one manuscript to another. ey should be mentioned, so that we keep in mind the full context of Diophantus’ symbols. But even more important is that Diophantus’ own marked symbolism is not systematically used in the manuscripts. e symbols described above are o en interchanged with fully written words. is is as much as can be expected. Both Δ υ and Δυναμις stand for exactly the same thing – the sound pattern or trace /dunamis/ – and so there is no essential reason to use one and not the other. us a free interchangeability is predicted. Notice rst the form of example (1) in all the Paris manuscripts, comparing the (translated) form of Tannery’s text to that of the manuscripts: 8 is was pointed out already by Nesselmann 1842 : 300–1. Indeed, my impression is that awareness of such quirks of Diophantus’ text was more widespread prior to Tannery: following the acceptance of his edition, knowledge of the manuscripts (as well as of the early printed editions – whose practices, I note in passing, are comparable to those of the manuscripts) became less common among scholars of Diophantus’ mathematics.
Reasoning and symbolism in Diophantus 337 Tannery: Let the <number> which is added and taken away from each number <sc. of the two other given numbers> be set down, <namely> ςΑ <:Number 1>. And if it is added to 20, result: ςΑ Μ ο Κ <:number 1 Monads 20>. Manuscripts: Let the <number> which is added and taken away from each number <sc. of the two other given numbers> be set down, <namely> One number. And if it is added to 20, result: One number, 20 Monads. Here we see Tannery’s most typical treatment of the manuscripts: abbreviat-ing expressions which, in the manuscripts, are resolved, within the problem itself. Note the opposite, inside enunciations. For example, the enunciation to .10 which, in Tannery’s form, may be translated: Tannery: To nd three numbers so that the <multiplication> by any two, taken with a given number, makes a square. Compare this with, e.g., Par. Gr. 2379: Manuscript: To nd three numbers so that the <multiplication> by any two, taken with a given ς, makes a 䊐. Tannery, we recall, followed a rational system: inside the proof, all mark-edly Diophantine symbols were presented in abbreviated form, while in the enunciation no symbolism was used. We nd that the manuscripts some-times have abbreviated forms where Tannery has fully written words, and sometimes have fully written words where Tannery has abbreviations. In other words, Tannery’s rational system does not work. I had systematically studied the marked Diophantine symbols through the propositions whose number divide by ten, in Books to , in all the Paris manuscripts. ese are only eight propositions, but the labour, even so, was considerable: essentially, I was busy recording noise. As a consequence of this, I gave up on further systematic studies, merely con rming the overall picture described here, with other manuscripts. One notices perhaps a gradual tendency to introduce more and more abbreviated forms as the treatise progresses (do the scribes become tired, in time?): Par. Gr. 2378, for instance, has no symbolism in my Book speci-mens at all, while they are frequent in Book . e ordinal numbers, with their three separate forms (fully spelled out, phonologically abbreviated, alphabetical numeral based), are especially bewildering. Consider once again .10, once again in Par. Gr. 2378. I plot the sequence of ordinals, using N for the alphabetic numeral, P for phonological abbreviation and F for the full version: NNNNPPFFFFFFFNFFFFPFNNFFFPF.
Tannery has all as alphabetical numerals. e most we can say is that, in the manuscripts, there is an overall tendency to prefer using the same form within a single phrase, though exceptions to this are found as well. Here we see Tannery homogenizing, turning numbers into numerals. But we may also nd the opposite, e.g. in .20, an expression we may translate as Tannery: Let the two <numbers> be set down as ς3 τετάχθωσαν οἱ δύο ς3 Par. Gr. 2485: Let the 2 <numbers> be set down as Numbers, ree. τετάχθωσαν οἱ Β ἀριθμοὶ τρεῖς Tannery has spelled out the word ‘two’, to signal that it functions here in a syntactic, not an arithmetical way. But it is neither syntactic nor arith-metical, it is phonological/orthographic. In the manuscripts, we have the phonological/orthographic object /duo/ which may be represented, as far as the scribes are concerned, by either B or δύο: both would do equally well. Signi cantly, it is di cult to discern a system even in the symbols intro-duced by Diophantus himself. Consider Par. Gr. 2380, inside .10: ς ενος μοναδων Γ, that is ‘ς one, monads 3’ (I quote this as an elegant example where both Diophantus’ special symbols, as well as numerals, are inter-changed with fully spelled out words). Very typical are expressions such as Par. Gr. 2378, .20: Δ υ Δ αριθμους Ε Μ ο , that is ‘Δ υ 4, numbers 5, Μ ο 1’. e ‘numbers’ – alone in the phrase – are spelled out. In general, one can say that monads appear to be abbreviated more o en than anything else in Diophantus’ symbolism: this may be because they are so common there. But the main fact is not quantitative, but qualitative: one nds, in all manu-scripts, the full range from Diophantine phrases fully spelled out in natural Greek, through all kinds of combinations of symbols and full words, to fully abbreviated phrases. My conclusion is that symbols in Diophantus are allographs : ways of expressing precisely the same things as their fully spelled out equivalents. And once this allography is understood, the chaos of the manuscripts becomes natural. For why should you decide in advance when to use this or that, when the two are fully equivalent? One should now understand Tannery’s plight. at he systematized his printed edition is natural: what else should he have done? I am not even sure we should criticize him for failing to provide a critical apparatus on the symbols. e task is immense and its fruits dubious. In particular, given Tannery’s goal – of reconstructing, to the best of his ability, Diophantus’ original text – a critical study of the abbreviations seems indeed hopeless. e interrelationships between manuscripts, in terms of their choice of
Reasoning and symbolism in Diophantus 339abbreviation as against a fully spelled out word, are tenuous. Sometimes one discerns a nities: the same sequence of symbols is sometimes used in a group of manuscripts, suggesting a common origin (and why shouldn’t a scribe be in uenced by what he has in his source?). But such cases are rare while, on the whole, patterns are more o en found inside a single manu-script: a tendency to avoid abbreviations for a stretch of writing, then a tendency to put them in . . . However, Tannery did not make appeal to this argument – which would have put his edition in the uncomfortable position of being, in a central way, Tannery’s rather than Diophantus’. So he made appeal to another argument. When criticized by Hultsch ( 1894 ) for his failure to note scribal variation for symbolism in his apparatus, Tannery replied that he had found that tedious, 9 because – so he had implied – Diophantus had purely abbreviated forms, that is in line with Tannery’s edition – which then were corrupted by the manuscript tradition. is question merits consideration. In the handful of thirteenth-century manuscripts we possess (the earli-est), symbolism is more frequent. us the tendency of scribes, during the historical stretch for which we have direct evidence , was to resolve abbrevia-tions into words. e simplest hypothesis, then, would be that of a simple extrapolation: throughout, scribes tend to resolve abbreviations – hence, Diophantus himself must have produced a strict abbreviated text. is is false, I think, for the following reasons. First, the relevant con-sideration is not that of Diophantus’ manuscript tradition alone, but that of scribal practice in general. We may then witness a peak in the use of abbreviations in Byzantine technical manuscripts of the relevant period of the twel h and thirteenth centuries – which are in general characterized by minute writing aiming to pack as much as possible into the page. Early minuscule manuscripts, and of course majuscule texts, o en are more of luxury objects and have fewer abbreviations; humanist manuscripts, again, for similar reasons, tend to have fewer abbreviations. us the evidence of the process of resolution of abbreviations, from the thirteenth to the six-teenth centuries, may not be extended into the past, as an hypothetical series of resolution stretching all the way from as far back as the fourth century . Second, I nd it striking that the Arabic tradition knows nothing of Diophantus’ symbols. ere are of course good linguistic reasons why Arabic (as well as Syriac and Hebrew) would not rely as much on the kind of abbreviation typical to the Greek and Latin tradition. Indeed, to continue with the linguistic typology, symbolism is also independently 9 T1893/5: xxxiv–xlii.
used in Sanskrit mathematics. 10 Indo-European words are a concatena-tion of pre xes, roots and su xes. Each component is phonologically autonomous, so that it is always possible to substitute some by alternative symbols. A written word can thus naturally become a sequence concatenat-ing symbols, or alphabetic representations, for pre xes, roots and su xes. Semitic words, on the other hand, are consonantal roots inside which are inserted patterns of vocalic in xes. e components cannot be taken apart in the stream of speech, so that it is no longer feasible to substitute a word by a concatenation of symbols, each standing for a root or a grammatical element. Quite simply, the language does not function in terms of such con-catenations. Arab translators, then, had naturally resolved standard Greek abbreviations into their fully spelled out forms. But they did respect some symbols: for instance, magical symbols, similar in character to those known from Greek-era Papyri (though not derived from the Greek), are attested in the Arabic tradition; 11 most famously, the Arabs had gradually appropri-ated Indian numeral symbols. In such cases, the symbols were understood primarily not as phonological units, but as written traces. I suggest that, had Diophantus’ use of symbolism been as consistent as Tannery makes it, an astute mathematical reader would recognize in it the use of symbolism which goes beyond scribal expediency, and which is based on the written trace – especially, given Diophantus’ own, explicit introduction of the symbols. e Arab suppression of the symbolism in Diophantus suggests, then, that they saw in it no more than the standard scribal abbreviation they were familiar with from elsewhere in Greek writing. I conclude with two comments, one historical, and the other cognitive. Historically, we see that Diophantus’ symbols are rooted in a certain scribal practice. is should be seen in the context of the long duration of Greek writing. In antiquity, Greek writing was among the simplest systems in use anywhere in human history: a single set of characters (roughly speaking, our upper case), used with few abbreviations. rough late antiquity to the early Middle Ages, the system becomes much more complex: the use of abbreviations becomes much more common, and a new set of characters (roughly speaking, our lower case) is introduced while the old set remains in use in many contexts. In other words, the period is characterized by an explosion in allography. 12 is may be related to the introduction of the 10 See the lucid discussion in H1995: 87–90. 11 Canaan 1937 –8/2004, especially 2004: 167–75. 12 I t i s d i cult to nd precise references for such claims that are rather the common stock of knowledge acquired by palaeographers in their practice. e best introduction to the practices of Greek manuscripts probably remains Groningen 1955 . For abbreviations in early Greek script, see McNamee 1982.
Reasoning and symbolism in Diophantus 341codex, and with the overall tenor of the culture with which it is associated: a culture where writing as such becomes the centre of cultural life, with much greater attention to its material setting. It is in this context that Diophantus introduces his symbols: they are the product of the same culture that gave rise to the codex. Cognitively, we see that those symbols introduced by Diophantus are indeed allographs. at is: they do not suppress the verbal reading of the sign, but refer to it in a di erent, visual way. It was impossible for a Greek reader to come across the symbol Μ ο and not to have suggested to his mind the verbal sound-shape ‘monad’. But at the same time, the symbol itself would be striking: it would be a very common shape seen over and over again in the text of Diophantus and nowhere else. It would also be a very simple shape, immediately read o the page as a single visual object. us, alongside the verbal reading of the object, there would also be a visual rec-ognition of it, both obligatory and instantaneous. I thus suggest that what is involved here is a systematic bimodality . One systematically reads the sign both verbally and visually. One reads out the word; but is also aware of the sign. To sum up, then, Diophantus’ symbolism gives rise to a bimodal (verbal and visual) parsing of the text (at the level of the noun-phrase). I shall return to analyse the signi cance of this in Section 4 below, where I shall argue that this bimodality explains the function of Diophantus’ symbols within his reasoning. Before that, then, let us acquaint ourselves with this mode of reasoning. 3. Notes on reasoning in Diophantus A sample of Diophantus e following is a literal translation of Diophantus’ .10. I follow Tannery’s text, with the di erence that, for each case where a symbol is available (including alphabetical numerals which, when symbolic, I render by our own Arabic numerals), I toss a couple of coins to decide whether I print it as symbol or as resolved word. (25% I make to be full words, which is what I postulate, for the sake of the exercise, might have been the original ratio.) e translation follows my conventions from the translation of Greek geometry, 13 including the introduction of Latin numerals to count steps of construction and Arabic numerals to count steps of reasoning. 13 See Netz 2004.
To two given ςς: to add to the smaller of them, and to take away from the greater, and to make the resulting <number> have a given ratio to the remainder. Let it be set forth to add to 20, and to take away from 100 the same ς, and to make the greater 4-times the smaller. (a) Let the <number> which is added and taken away from each ς <sc. of the two given numbers> be set down, <namely> number, one. (1) And if it is added to twenty, results: ς1 Μ ο 20. (2) And if it is taken away from 100, results: Μ ο 100 lacking number 1. (3) And it shall be required that the greater be 4- tms the smaller. (4) erefore four- tms the smaller is equal to the greater; (5) but four- tms the smaller results: Μ ο 400 Ψ ς4; (6) these equal ς1 Μ ο 20 (7) Let the subtraction be added <as> common, (8) and let similar <terms> be taken away from similar <terms>. (9) Remaining: numbers, 5, equal Μ ο 380. (10) And the ς results: monads, 76. To the positions. I put the added and the taken away on each ς, ς 1; it shall be Μ ο 76. And if Μ ο 76 is added to 20, result: monads, 96; and if it is taken away from 100, remaining: monads, 24. And the greater shall stand being 4- tms the smaller. Diophantus the deuteronomist: systematization and the general To understand the function of the text above, I move on to compare it withthree other, hypothetical texts. I argue that all were possible in the late ancient Mediterranean. However, only the rst two had existed, whilethe third remained as a mere logical possibility, never actualized in writing. Text 1: A: I have a hundred and a twenty. I take away a number from the greater and add it to the smaller. Now the smaller has become four times that which was greater. How much did I take away and add? B: ? A: Seventy six! Check for yourself. Text 2: Hundred and twenty. I took away from the greater and added the same to the smaller, and the smaller became four times that which had been greater. Take the greater, a hundred. Its four times is four hundred. Take away the smaller,twenty. Le is three hundred eighty. Four plus one is ve. Divide three hundred eighty by ve: seventy six. Seventy six is the number taken away and added.
Reasoning and symbolism in Diophantus 343 Text 3: Given two numbers, the rst greater than the second, and given the ratio of a third number to unity, to nd a fourth number so that, added to the second and removed from the rst, it makes the ratio of the second to the rst equal to the given ratio of the third number to unity . Let the fourth have been found. Since the second number together with the fourth has to the rst lacking the fourth the ratio of the third to unity, make a h number which is the third multiplied by the rst lacking the third multiplied by the fourth. is h number is equal to the second together with the fourth. So the third multiplied by the rst lacking the third multiplied by the fourth is equal to the second with the fourth.So the third multiplied by the rst is equal to the second with the fourth with the third multiplied by the fourth, or to the second with the fourth taken the third and one times. at is, the third multiplied by the rst, lacking the second, is equal to the fourth taken the third and one times. Multiply all by the third and one fraction. us the third multiplied by the rst, multiplied by the third and one fraction, lacking the second multiplied by the third and one fraction, is equal to the fourth taken the third and one times, multiplied by the third and one fraction, which is the fourth. So the third multiplied by the rst, multiplied by the third and one fraction, lacking the second multiplied by the third and one fraction, is equal to the fourth. So it shall be constructed as follows. Let one be added to the third to make the sixth. Let the seventh be made to be the fraction of the sixth. Let the third be multiplied by the rst and by the seventh to make the eighth. Again, let the second be multiplied by the seventh to make the ninth. Now let the ninth be taken away from the eighth, to make the fourth. I say that the fourth produces the task. [Here it is straightforward to add an explicit synthesis, showing that the ratio obtains; for brevity’s sake, I omit this part.] I suggest that we see Diophantus’ text with reference to texts 1 and 2 – of which it must have been aware – and with reference to text 3 – which it deliberately avoided. 14 Based on Høyrup’s work, 15 I assume that texts such as text 1 were widespread in Mediterranean cultures from as far back as 14 Text 3 is my invention; perhaps not the most elegant one possible. All I did was to try to write, in an idiom as close as possible to that of Diophantus, a general analysis of the problem, following a line of reasoning hewing closely to the steps of the solution in Diophantus’ own solution. ( is is not a mechanical translation: obviously, a particular solution such as Diophantus’ underdetermines the general analysis from which it may be derived, since any particular term may be understood as the result of more than one kind of general con guration.) 15 See, for instance, H2002: 362–7. It is fair to say that my summary is based not so much on this reference from the book, as on numerous discussions, conference papers and preprints from
the third millennium (if not earlier), surviving, arguably, into our own time. ey persisted almost exclusively as an oral tradition (sometimes, perhaps, taking a ride for a couple of centuries on the back of written tra-ditions of the type of text 2, and then proceeding along in the oral mode). Such texts are called by Høyrup ‘lay algebra’. Occasionally, lay algebra gets written and systematized (to a certain extent) in an educational context. It then typically gets transformed into texts such as text 2: the mere question-and-answer format of text 1 is trans-formed into a set of indicative and imperative sentences put forward in the rigid, authoritarian style typical of most written education prior to the twentieth century. is is school algebra which has appeared several times in Mediterranean cultures. One can mention especially its Babylonian (early second millennium ), Greek (around the year zero) and Italian (early second millennium ) forms. e Babylonian layer is important as the rst school algebra of which we are aware; the Greek layer is important, for our purposes, as providing, possibly, a context for Diophantus’ work; the Italian layer is important, for our purposes, as providing a context for the interest in Diophantus in the Renaissance. e historical relationship between various school algebras is not clear and it may be that they depend on the persistence of lay algebra no less than on previous school algebras. It should be said that, while essentially based on the written mode, this is a use of writing fundamentally di erent from that of elite literary culture. Writing is understood as a local, ad-hoc a air. e di erence between the literacy of school algebra and the oralcy of lay algebra is huge, in terms of their archaeology : clay tablets, papyri and libri d’abbaco o en survive, spoken words never do. But the clay tablets, papyri and libri d’abbaco of school algebra do not belong to the world of Gilgamesh, Homer or Dante. ey are not faithfully copied and maintained, and the assump-tions we have for the stability of written culture need not hold for them. What would happen when such materials become part of elite literate culture itself? One hypothetical example is text 3: a reworking of the same material, keeping as closely as possible to the features of elite literate Greek mathematics (which was developed especially for the treatment of geom-etry). is may be called, then – just so that we have a term – Euclidean algebra. 16 When transforming the materials of lay and school algebra into the author, and that as such summaries go it is likely to deviate in some ways from the way in which Høyrup himself would have summed up his own position. 16 I use the term ‘Euclidean’ to refer to elite, literate mathematical practices. It is true that Euclid – especially Books and – could have been occasionally part of ancient education (the three papyrus as fragments P. Mich. 3. 143, P. Berol. Inv 17469 and P. Oxy. 1.29, with de nitions
Reasoning and symbolism in Diophantus 345elite-educated, literate form, Diophantus chose to produce not Euclidean algebra, but Diophantine algebra. I note in passing that the character of Diophantus – as intended for elite literate culture – is in my view not in serious doubt. e material does not conform to elementary school procedures; it is ultimately of great complex-ity, suitable only for a specialized readership. It had survived only inside elite literate tradition; and, as is well known, it quickly obtained the primary mark of elite literate work – having a commentary dedicated to it (that of Hypatia). 17 In other words, I suggest that Diophantus is engaged primarily in the rearrangement of previously available material into a certain given format, of course then massively extending it to cover new grounds that were not surveyed by school algebra itself. is is very much the standard view of Diophantus, and I merely wish to point out here what seem to me to be its consequences. Let us agree that Diophantus is engaged in the re tting of previous traditions into the formats of elite writing sanctioned by tradition. en it becomes open to suggest that he belongs to the overall practice of late antiquity and the Middle Ages which I have elsewhere called deutero-nomic: the production of texts which are primarily dependent upon some previous texts . 18 Typically, deuteronomic texts emphasize consistency, sys-tematicity and completion. ere is an attention to the manner of writing of the text. is means that they bring together various elements that might have been originally disparate. e act of trying to bring disparate 17 e evidence is the imsiest imaginable – a mere statement in the Suidas (Adler IV:644.1–4: Yπατια . . . εγραψεν υπομνημα εις Διοφαντον) which, however, if not proving beyond doubt that Hypatia wrote a commentary on Diophantus, makes it at least very likely that someone did. 18 Virtually everyone, from Tannery to Neugebauer onwards, has agreed that Diophantus was acquainted with many arithmetical problems deriving from earlier Mediterranean traditions and was therefore at least to some extent a systematizer. Some, such as Heath, had thought that Diophantus’ systematization of earlier problems may not have been the rst in the Greek world, making comparison with Euclid as the culmination of a tradition of writing Elements (I doubt this for Euclid and nd it very unlikely for Diophantus). e dates are xed, based on internal evidence, as –150 to +350. What else is argued concerning Diophantus’ dates is based on scattered, late Byzantine comments which are best ignored. e e silentio , together with Diophantus’ very survival, suggest – no more – a late date. ( e silence is not meaningless, as it encompasses authors from Hero to the neo-Platonist authors writing on number.) A late date was always the favourite among scholars (not surprisingly, then, the thesis of an early of Book I, Propositions .8–10 and .5, respectively – most likely derive from a classroom context). However, the bulk of papyri nds with mathematical educational contents are di erent in character, involving basic numeracy and measuring skills or, in more sophisticated examples, coming closer to Hero’s version of geometry. e impression is that, in antiquity itself, Euclid was fundamentally a cultural icon, which occasionally got inducted into the educational process.
components into some kind of coherent unity then would lead to a certain transformation. e way this applies to Diophantus is obvious. He brings together pre-viously available problems. He arranges them in a relatively clear order, ranging from the simple to the complex. He classi es, creating clear units of text, for instance the Greek Book , all dedicated to right-angled triangle problems. In the introduction he discusses his way of writing down the problems, and introduces a special manner of writing for the purpose. e structuring involves large-scale and small-scale transformations. e large-scale transformation is a product of the arrangement of the disparate problems in a rational structure. e problems o en become combinatorial variations on each other, e.g. .11–13: 11. To add the same number to two given numbers, and to make each a square. 12. To take away the same number from two given numbers, and to make each of the remainders a square. 13. To take away from the same number two given numbers, and to make each of the remainders a square. In such cases, it seems clear that Diophantus had used the rational structure as a guide, actively searching for more problems, bringing completion to his much more fragmentary sources. e huge structure – thirteen books, of which, in some form or another, ten survive, with perhaps four hundred problems solved – was built on the basis of such rational, combinatorial completion. e small-scale transformation involves each and every problem, which is presented, always, in the form above. It is immediately obvious that, in this respect, Diophantus consciously strove to imitate elite literate Greek mathematics though (as suggested by the examples above) this in itself would not determine the form of his text. Quite simply, there was more than a single way of producing numerical problems in elite literate Greek date was defended by Knorr 1993 ). I shall assume such a late date, while realizing of course the hypothetical nature of the argument: the dating of Diophantus is the rst brick of speculation in the following, speculative edi ce. I would like to question, though, the very habit of treating the post quem and the ante quem as de ning a homogeneous chronological segment. One’s attitude ought to be much more probabilistic – and should appreciate the fact that not all centuries are alike. Here are two probabilistic claims: (1) the rst century , and the rst century , both saw less in activity in the exact sciences; the second century , as well as the rst half of the fourth century , saw more. (2) e e silentio is more and more powerful, the further back in time we go. I think it is therefore correct to say that Diophantus most likely was active either in the second century or the rst half of the fourth century .
Reasoning and symbolism in Diophantus 347format - not a single monolith to start with. In tting his text into the established elite Greek mathematical format, Diophantus had a certain freedom. e rst decision made by Diophantus was to keep the basic dichotomy of presentation from standard Greek mathematics, with an arrangement of a general statement followed by a particular proof. is indeed would appear as one of the most striking features of the Greek mathematical style. But most important, this arrangement is essential to the large-scale transformation introduced by Diophantus. To produce a structure based on rational completion, Diophantus needed to have something to complete rationally: a set of general statements referring to each problem in terms transcending the particular parameters of the problem at hand. I therefore argue that Diophantus’ general statements can be under-stood, at two levels, as a function of his deuteronomic project. He needs the general statements so as to conform to the elite form of presentation he sets out to emulate. Even more important, he needs them to provide building blocks for his main project of systematization. e upshot of this is that Diophantus does not need the general statements for the logical ow of the individual problem. is is indeed obvious from an inspection of the problems, where the general statements play no role at all. is observation may shed some light on the major mathematical ques-tion regarding Diophantus, that is, did he see his project in terms of provid-ing general solutions ? In some ways he clearly did. e clearest evidence is in the course of the propositions (extant in Arabic only) .13–14. We are given a square number N which is to be divided into any three numbers (i.e. N= a + b + c ) so that either N+ a , N+ b , N+ c are all squares ( .13), or N− a , N− b , N− c are all squares ( .14). It is not surprising that, in both cases, we reach a point in the argument where we are asked to take a given square number and divide it into two square numbers 19 – the famous Fermatian problem .8. Now, Diophantus (or his Arabic text) explicitly says that this is possible for ‘It has been seen earlier in this treatise of ours how to divide any square number into square parts.’ 20 ere, of course, the divided square is a particular number, 16. ( e particular number chosen as example in .13–14 is 25.) is reference is hardly a late gloss, as the very approach taken to the problem is predicated upon the reduction into .8. Indeed, the natural assumption on the part of any reader familiar with elite Greek 19 By iteration, this allows us to divide a square number into any number of square numbers; Diophantus, in fact, requires a division into three parts. Note however that even the basic operation of iteration itself calls for a generalization of the operation of .8. 20 Sesiano 1982 : 166.
geometry would be that results should be transferable from one set of numerical values to any other soluble set, on the analogue of the transfer-ability of geometrical results from one diagram to another: this would be the implication of picking a mode of presentation which is so suggestive of that of elite geometry. It is also likely that the very exposure to certain quasi-algebraic prac-tices (basically those of additions or subtractions of terms until one gets a simple equation of species) as well as the choice of simple parameters would instil the skills required for the nding of solutions with di erent numerical values from those found by Diophantus himself, so that the text of Diophantus, taken as a whole, does teach one how to nd solutions in terms more general than those of the particular numerical terms chosen for an individual Diophantine solution. 21 Having said that, however, the fundamental point remains that Diophantus allows his generality, such as it is, to emerge implicitly and from the totality of his practice . ere is no e ort made to make the generality of an individual claim explicit and visible locally . He does not solve the problem of dividing a square number into two square numbers in terms that are in and of themselves general – which he could have done by pursuing such problems in general terms . Why doesn’t he do that? ere are three ways of approaching this. First, readers’ expectations on how generality is to be sustained would have been informed with their experience in elite Greek geometry. ere, generality is not so much explicitly asserted, as it is implicitly suggested. 22 It is true that the nature of Greek geometrical practice – based on the survey of a nite range of diagrammatic con gurations – does not map precisely into Diophantus’ practice. Greek geometry allows a rigorous, even if an implicit, form of generality, which Diophantus’ technique does not support. is mismatch, in fact, may serve as partial explanation for the emerging gap in Diophantus’ generality. Second, if indeed I am right and Diophantus’ goals were primarily completion and homogeneity, and that the general statement may have been introduced in the service of such goals, than our problem is to a large extent di used. Diophantus did not provide explicit grounds for his gen-erality, but this is because he was not exactly looking for them. He did not introduce general statements for the reason that he was looking for general solutions. Rather, he introduced general statements because he perceived such statements to be an obligatory feature of a systematic arrangement 22 As argued in N1999: ch. 6 (a comparison also made by omaidis 2005 ). 21 is, if I understand him correctly, is the claim of omaidis 2005 .
Reasoning and symbolism in Diophantus 349of mathematical contents. Of course, I imagine that he would still prefer a general proof to a particular one – but only as long as other, no less impor-tant characteristics of the proof were respected as well. But this, I suggest, was not the case. I will try to show why in the next section. Even before that, let us mention the third and most obvious account for why Diophantus did not present a more general approach. An argument that comes to mind immediately is that Diophantus did not produce more general arguments because he did not possess the required symbolism. Fundamentally, what we then do is to put side by side our symbolism and that of Diophantus so as to observe the di erences and then to pronounce those di erences as essential for a full- edged argument producing a general algebraical conclusion. Of course, the di erences are there. In par-ticular, Diophantus has explicit symbols for a single value in each power: a single ‘number’ (a single x ), a single ‘dunamis’ (a single x 2 ), a single ‘cube’ (a single x 3 ), etc. ere is thus no obvious way of referring even to, say, two unknowns such as x and y . is is a major limitation, and of course it does curtail Diophantus’ expressive power. Some scholars come close to suggesting that this, nally, is why Diophantus does not produce explicit general arguments. 23 But by now we can see how weak this argument is, and this for two reasons. First, it is perfectly possible to express a general argument without the typographic symbolism expressing several unknowns, by the simple method of using natural language (over whose expressive power, a er all, typographic symbols have no advantage). is is the upshot of text 3 above. Of course, even though a text such as text 3 does prove a general claim, it does so in an opaque form that does not display the rationality of the argu-ment. But this helps to locate the problem more precisely: it is not that, with Diophantus’ symbolism, it was impossible to prove general claims; rather, it was impossible to prove general claims in a manner that makes the rational-ity of the argument transparent . Second, and crucially, note that it was perfectly possible for Diophantus to make the rather minimal extensions to his system so as to encompass multiple variables. Indeed, since the most natural way for him of speaking of several unknowns was to speak of ‘the rst number’, ‘the second number’, etc., he e ectively had the symbolism required – all he needed was to make the choice to put together the less common symbol for ‘number’ together with the standard abbreviation for numerals: αʹ ʹς would be ‘the rst number’, β´ ς would be ‘the second number’, etc. A bit more cumbersome than 23 See e.g. Heath 1885 : 80–2.