H. P. GRICE E J. L. SPERANZA: LA CONVERSAZIONE -- I VERBALI: RINALDINI

 G.: Rinaldini again, and there it sits in 1640, Opus algebricum, as if theology had merely been a customs checkpoint on the road to blue-collar mathematics. S.: You are very unfair to theology. G.: Only as unfair as mathematics deserves to its social betters. S.: You mean its social betters are the people who still count with Greek fingers and Roman dignity. G.: Precisely. Arithmos and numerus have manners. Algebra arrives with sleeves rolled up and a spanner in the pocket. S.: That is because algebra does work. G.: Exactly my complaint. It is blue-collar. One goes off to Macerata, takes a theology degree because the world insists on respectable Latin, and comes straight home to Ancona to print Arabic labour under a Latin title. S.: Opus algebricum is itself a compromise. G.: A very revealing one. Opus for the schoolmen, algebra for the workmen, and the whole thing pretending not to know it is imported trouble. S.: Austin would have liked that phrase. G.: He did, in effect. “Al-,” he said, “that’s the trouble.” Not even Latin trouble. Imported trouble with the definite article still attached. S.: A stowaway article. G.: Exactly. A little al sitting on the title-page like a Levantine port clerk demanding entry into Christendom. S.: And you object because Rinaldini, being at least nominally civilised, should have stuck to arithmetic. G.: To arithmos and numerus, yes. If one must count, count in Greek or Roman. Do not arrive in Oxford with algebraic trousers and expect sympathy. S.: Yet he was in Ancona, not Oxford. G.: Worse. Ancona is a port, which makes the whole thing more plausible and more morally suspect. S.: You are determined to class mathematics by income and wardrobe. G.: It is only a temporary aid to understanding. Arithmetic is upper middle. Geometry is landed. Algebra is industrious and morally earnest. S.: And analysis? G.: Analysis is the ambitious nephew. S.: Very good. G.: Keep it, but improve the insolence. S.: Gladly. Let us be exact for a moment. Why do you oppose arithmos and numerus to algebra? G.: Because arithmos names number in the Greek philosophical manner, and numerus in the Roman administrative one. Both have pedigree. Algebra enters later as a technique of transformation, operation, and unknowns, and therefore as a sort of social climber. S.: So arithmetic counts what one can point to, and algebra manipulates what one has not yet identified. G.: Exactly. Arithmetic says: here are three olives. Algebra says: let x be whatever survives the violence of the symbols. S.: Which is precisely why it frightened classicists. G.: And ought to. Unknowns are dangerous company. S.: Yet Rinaldini’s title says Opus algebricum, not Ars Arabica. G.: Because he wants the technique without the embarrassment. One Latinises the wrapper, leaves the labour inside. S.: Like many respectable societies. G.: Exactly. The degree is what one needs to be allowed to speak; the algebra is what one wants to say. S.: You have used that line before. G.: Because it remains useful. He goes to Macerata, collects theology as one collects a passport, returns to Ancona, and quietly says, if you do not mind, I shall now return to the subject. S.: Algebra. G.: Yes, blue-collar though it is. S.: But 1640 is late enough that algebra is hardly a novelty. G.: No, but novelty is not the issue. Social tone is. In a world of Latin titles, scholastic degrees, patrons, academies, fortifications, and bishop-adjacent expectations, algebra still sounds like trade entering the cloister by the side door. S.: Trade with very good symbols. G.: Quite. One must never underestimate the aesthetic power of labour. S.: Then where do the Arabs enter in your annoyance? G.: In the word itself, naturally. Algebra from al-jabr, carrying the article like contraband into Europe, then parading as if it had always belonged in a Latin sentence. S.: Which it does by 1640. G.: Institutionally yes, temperamentally no. S.: You do not really believe that. G.: Of course not. But one must tease civilization into self-recognition. The great irony is that the same Europe that prides itself on Rome and Athens quietly computes with Arabic inheritance and pretends the title-page has settled the matter. S.: So Opus algebricum is a diplomatic title for a mixed ancestry. G.: Precisely. It is a document of intellectual naturalisation. S.: There is your true interest, then, not the blue-collar sneer. G.: The sneer is a mode of affection. Mathematics after all is one of the few disciplines shameless enough to import useful things and only later discover etymology. S.: Philosophers do that too, but with worse conscience. G.: Indeed. S.: Now, tell me what Bostock would say. G.: Bostock would say that algebra is real rigour, by which he means not my sort of concern with what people mean, but the harder sort with what expressions allow, entail, transform, and preserve under rule. S.: And you would answer? G.: That rigour is admirable but not sovereign. Algebra does not become philosophy merely by being exact, any more than my navy memoranda became Euclid by being typed. S.: Yet Rinaldini is not merely a calculator. G.: No, and that is what complicates the sneer. He is friend of Galileo and Borelli, supervisor of fortresses, founder of the Cimento, wrangler with colleagues, proposer of a thermometric scale, and writer of Philosophia rationalis, naturalis, atque moralis. S.: Which is not blue-collar at all. G.: It is blue-collar with Latin gloves. S.: Better. Then perhaps the real contrast is not between arithmetic and algebra, but between inherited numerical dignity and operative symbolic labour. G.: That is very nearly right. Arithmos and numerus belong to counting, order, ratio, civic enumeration, even music and cosmos. Algebra belongs to manipulation, reduction, solution, procedure, and operational anonymity. S.: Unknowns again. G.: Yes. Unknowns are where the collars become blue. S.: You really ought to explain yourself. G.: Very well. With numerus and arithmos one still imagines objects, counts, measures, proportions, civic totals, perhaps celestial harmonies. With algebra one writes x+3=7x + 3 = 7x+3=7 and solves x=4x = 4x=4 without ever needing to know whether x was apples, ducats, or sinners. S.: Which is the whole advantage. G.: Precisely the advantage of labour. It gets on with the job regardless of pedigree. S.: Then Rinaldini’s blue-collar side is methodological. G.: Exactly. Algebra cares for rules of operation before it cares for the noble standing of the objects. It is practical abstraction. S.: And that made it useful for fortresses, scales, and all the rest. G.: Of course. Men who build, measure, defend, and calibrate naturally like symbols that work harder than social rank. S.: So the very “blue-collar” quality made it fit the Italy of patrons, engineers, academies, and patrons pretending not to be engineers. G.: Admirably put. S.: Thank you. G.: Do not become bourgeois about it. S.: Never beyond Bologna. Now, if one were truly classical, how would one resist algebra? G.: One would say that proper mathematical culture should remain tied to geometry, proportion, arithmos as intelligible multiplicity, numerus as counted order, not be surrendered to imported procedures whose very name begins with the foreign article. S.: A splendidly bad position. G.: Quite so. That is why I enjoy airing it. S.: And Austin? G.: Austin enjoyed the article. “Al-,” he said, “that’s the trouble.” He heard at once that the word carries its passport in the first syllable. S.: Mary Warnock laughed, I trust. G.: In the way moral philosophers laugh when something indecent turns out to be merely grammatical. S.: And the children? G.: They seized on the “al” and turned it into playground liturgy: AL, AL, AL. Which is what happens when Arabic philology meets English gravel. S.: That is almost too neat. G.: Childhood often is. S.: Let us return to Rinaldini’s route. Ancona to Macerata, theology degree, back to Ancona, then Opus algebricum. G.: Yes, and the route matters because it displays the old academic economy perfectly: take the respectable credential the world requires, then use it to say what you actually mean. S.: The degree is licence, the algebra is intention. G.: Exactly. The same pattern repeats more often than academic piety admits. S.: Then your punchline about “if you don’t mind” is serious. G.: Entirely serious. “If you don’t mind” is the whole philosophy. It is a politeness formula that means I shall do this regardless, but I should prefer not to force you to object aloud. S.: An implicature of survival. G.: Exactly. Seventeenth-century Italy, like Oxford, valued the art of getting on with the subject while appearing merely civil. S.: Then perhaps algebra is not blue-collar in opposition to theology, but in relation to social necessity. G.: Yes. It is the work one actually wants to do once the respectable forms have been satisfied. S.: You make theology sound like customs paperwork. G.: In this story it very nearly was. S.: Harsh on Macerata. G.: No harsher than Macerata was on young minds. S.: Fair. Now, could one not say that algebra itself had by then acquired dignity enough? G.: Certainly enough to be printed, taught, Latinised, and dedicated. But dignity acquired is not the same as dignity inherited. That difference is exactly what makes it amusing. S.: You are a snob of intellectual genealogy. G.: Only playfully. All real thought is mongrel sooner or later. S.: Then why cling to arithmos and numerus at all? G.: Because they remind us that there are older ways of conceiving number, as measure, ratio, ordered plurality, civic count, and cosmic relation, whereas algebra stresses operational transformability. S.: So the contrast is philosophical as well as social. G.: Yes. Arithmos belongs to ontology and proportion; algebra to procedure and solution. S.: That is too sharp, surely. G.: Of course. I am sharpening it for the sake of the joke, which is a respectable analytical instrument when used soberly. S.: Soberly. G.: In the Oxonian sense. S.: Then let us do some formalism, since you have asked for Arabic labour to appear. Suppose Rinaldini writes ax+b=cax + b = cax+b=c then x=c−bax = \frac{c-b}{a}x=ac−b​. That is not Greek numerus but symbolic operation on unknowns. G.: Precisely. One does not contemplate number; one rearranges relations. It is almost manual. S.: Manual in symbols. G.: The cleanest form of manual labour. S.: And if he moves to higher forms, systems, powers, perhaps even rhetorical equations in words, the same applies. G.: Yes. Algebra generalises procedure. It emancipates calculation from named particulars. S.: Which makes it useful to natural philosophy. G.: Immensely. Once one wishes to scale, compare, infer, calculate intervals, or handle unknown magnitudes, algebra is the servant with no concern for ancestry. S.: A useful servant then. G.: The most dangerous sort. S.: You really are enjoying the class language. G.: Because it is not entirely false. Arithmetic can sit with philosophers at dinner; algebra arrives later and solves the household accounts. S.: Which is why the philosophers despise it and borrow from it continuously. G.: Exactly. One must never trust a discipline that publicly sneers at what privately enables it. S.: That would disqualify philosophy. G.: In large part, yes. S.: Now tell me why Rinaldini, being also a founder of the Cimento, matters beyond the title. G.: Because the Cimento is proving and trying, which means mathematics under experimental pressure. Algebra in that context is no idle symbolic pastime. It is part of a culture of testing, measuring, resolving, composing, and resisting mere authority. S.: So blue-collar again, but scientifically so. G.: Exactly. Del Cimento is a society whose motto might as well be: if it will not work, do not ask us to admire it. S.: Which is almost your own view of many philosophical systems. G.: I prefer them at least to be incorrect elegantly. S.: Rinaldini’s termometric scale is another sign of the practical impulse. G.: Yes. Freezing and boiling water at ordinary atmospheric pressure, with the interval divided into twelve degrees. A man who thinks in calibrations rather than metaphors. S.: Though twelve is a very civilised number. G.: Quite. One must not make him too plebeian. S.: Then perhaps he is blue-collar only by your theatrical standard, not by his own. G.: Naturally. Theatrical standards are often the only honest ones in intellectual history. S.: That is a suspicious maxim. G.: Most accurate maxims are suspicious. S.: Then perhaps the real issue is that algebra, unlike arithmetic, exposed classicists to the possibility that thought can be exact without being noble in the ancient sense. G.: Splendid. That is exactly it. S.: Thank you. G.: Keep it, but make it a little less devastating. S.: Happily. So Opus algebricum is a title announcing that exactness no longer requires Greek pedigree. G.: Yes. It says: I can be exact with imported tools, and you may dislike the etymology but not the result. S.: Which is why even Austin was forced into philological admiration. G.: Indeed. He could sneer at the article and still know that nothing in Oxford would remove it. S.: The children understood all this better by chanting AL. G.: Children often reach the essence by barbarism. S.: Let us have one more pass at your social taxonomy. Arithmetic upper middle, geometry landed, algebra blue-collar, analysis ambitious nephew. What of logic? G.: Logic is the family solicitor. S.: And metaphysics? G.: The aunt with a title and no ready cash. S.: Ethics? G.: The clergyman cousin who knows too much family history. S.: Excellent. G.: Keep all of it and publish none. S.: Never intentionally. Now, if one were to rescue algebra from your class satire, what would one say? G.: One would say that algebra is the great instrument by which mathematics ceased to depend on immediate intuitive display of its subject matter and acquired a generality of operation that made later science possible. S.: Very sober. G.: Yes. And one would add that its linguistic foreignness is one of civilisation’s better lessons: Europe thinks with more borrowings than its pride allows. S.: There is your true point, then. G.: More or less. Opus algebricum is a title in which Latin respectability and Arabic labour coexist without peace and without divorce. S.: Which is why you like it. G.: Exactly. It is intellectually mixed and socially revealing. S.: And Rinaldini himself? G.: A mathematician natural philosopher and practical man who took the short road from theology to algebra because he knew which part was passport and which part subject. S.: The shorter route was not the road from Ancona to Macerata. G.: No. The shorter route was from respectability to work. S.: Dry enough? G.: Sufficiently Anconitan, with Arabic dust on Roman shoes.