En grundläggande transhistorisk motsättning i västerländsk civilisation: greker mot hebréer. Historisk undersökning av de två perspektivens olika vikt.

Undviker skolastiska diskussioner: our business is with reason on the broadest possible social stage. [p7]

With our modern schooling we may imagine that a man only has to open his eyes to see that nature works by constant laws. But on the face of it is not obvious at all. Water normally flows downwards; yet in a thin glass tube it will rise. Hard substances burn or crack in intense heat; but not asbestos. Nature is full of such apparent anomalies. It takes considerable abstraction, standing on the shoulders of generations of speculation and research, to realize that these apparent breaches of law are only so because we have begun with too particular a law. [p10]

Tanken på en ordnad, regelbunden natur är något som måste skapas. Den skapades av de olika vetenskaperna (språk, teologi, sekulär och kanonisk rätt, från och med elvahundratalet även filosofi):  they opened the door to the influence of Greek and Arabic physics, which took little account of any god who intervened in nature. Equally important, they defined an area where human reason was autonomous. […] There is no clearer example of this process than the judicial ordeal. The growth of rational judicial procedures threw the ordeal into discredit. [p10]

[Om prövning, se Peter Landau The Development of Law i The New Cambridge Medieval History vol. 4.]

Den skapades också av matematiken, at its most practical level this merged with technology. [arkitektur – katedralerna] [p10]

What was new in the two scholastic centuries which culminated with Aquinas was not belief in miracle. It was the rationalistic context through which miracle was seen. Our purpose in this book is to search for the origins of this rationalistic context. We shall search for it not, now, in terms of its intellectual antecedents – Aristotle, Avicenna, etc. – but in terms of men and society. Our question is a social one. What distinct psychological aor social conditions, if any, urged men to think more rationally about nature, and hence generated that tension in their view of God and the world? [p13]

Murray överger de klassbaserade förklaringarna. Beliefs, credulities, enthusiasms, hostilities, and vices: all were often said to be common to different social classes – often enough, that is, to obstruct any general, neat equation between class and religious allegiance. [p17]

So no such neat equation will be attempted. Instead I shall suggest another equation. It will be based on another set of social categories, created from the first set by a simple operation. The operation is that which turns the concept of relative position into one of relative movement (for mathematicians dy/dt instead of y). Some three centuries before the Renaissance a drawing was in fashion of the Wheel of Fortune, surrounded by four human figures: one going up, one down, and two, at top and bottom, in precarious stability. These are the categories envisaged in this book. They are usually harder to identify in medieval flesh-and-blood than the stationary classes. But they are not so hard that knowledge need perish altogether; and there are reasons for thinking this social calculus the most promising basis for the following investigation. [p17]


  1. Beskriver den sociala mekanism som skapar den nya mentaliteten på elva- och tolvhudratalet. (acceleration of up-and-down social mobility: ekonomisk förändring 975-1125 sätter samhällsekonomin för resten av medeltiden)
  2. Kapitel 3, 4 och 5: hur ekonomin påverkar tanken. Kap 5: The chief instrument to power was the mind. Men recognized this, and cultivated their minds accordingly. Their recognition will be documented from twefth- and thirteenth-century sources.
  3. Kap 6-8: Matematiken
  4. Kap 9-12: De lärdas kultur
  5. Antirationalistiska motangrepp från de högre samhällskikten


Ekonomisk förändring

Pengar är ett medel för att utbyta varor. Pengarnas funktion måste sökas i utbytet, och därför i ägandet. Under medeltiden fördelades den mesta rikedomen inte genom utbyte, utan genom auktoritet, rätt och gåvor. Utbytet var perifert: det skedde i gränsområden som Venedig, eller bland marginaliserade grupper som judar. Utbyte skedde endast när rikedom inte kan fördelas på något annat sätt, i huvudsak när den skulle fördelas mellan sociala eller geografiska enheter. Murrays exempel är vin- och salthandeln i det karolingiska väldet, där produktiviteten steg kraftigt i Rhenområdet, vilket skapade ett överflöde som måste fördelas, och där den politiska enheten lät salt från kustområdena föras inåt landet. Sådana förändringar sker för snabbt för att de traditionella sätten att fördela rikedomen ska kunna förändras i takt, och utbytet fyller tomrummet. Pengar minskar transaktionskostnaderna i utbytet, vilket ökar den utbytta volymen. Den ökade volymen gör marknaden mer fluid, vilket minskar efterfrågan av naturabetalning, vilket ökar efterfrågan av pengar. En högre efterfrågan av pengar innebär att myntpräglingen ökar, vilket leder till att fler transaktioner, som hade varit olönsamma eller omöjliga utan pengar, kan genomföras. Processen är självaccelererande, och monetarisering kan ske väldigt snabbt. [p27-29]

Den monetära ekonomins sammansmältning med byteshandeln – the intermediate character of late dark-age exchange – förklarar den de snabba samhällsförändringarna kring millenieskiftet. [p33?]

Under the ever-present ghost of money as a measure of value, then, the real thing – in this also like a ghost – hovered on the edge of the payments-system, in both quantity and quality half-in and half-out. [p33. Eftersom det är ont om pengar används de endast i undantagsfall eller i kombination med natura som betalningsmedel: en häst och 13 solidi, eller pepparkorn istället för mynt. Pengarnas främsta funktion var som värdemått.]

This partial, and infinitiely variable, role of money had a third aspect. Different social groups and institutions differed in their attitudes to what little money there was. As with other useful devices, a minority had discovered the profitable uses to which this one could be put. But this minority formed only a faint silver or gold fleck in a scene dominated by other colours. This transpires from those records which speak of money in the contect of other elements in wealth. The paradox could occur here that the richer you were, the less money you had. [p33]

Pengarnas fortlevnad som money of account innebär att systemet i princip är en penningekonomi: det kan snabbt absorbera en ökad reell mängd pengar: [p35]

[Otto] Kontakten med Islam – ökad handel, ökad införsel av silver genom både öst och syd (Donau och Po). land reclamation, growing population, active public authority, towns, specialization. local exchange in food, distant trade in luxury. [p52]

Otto’s empire was only the centre of an economic system stretching beyond it. Its main appendix was England. [p52]

Ytterligare inflöde på tusentalet: erövringarna. Korsika, Sicilien, Jerusalem, Spanien. Räderna mot Sydfrankrike upphör: tillväxt i Languedoc. Pengar (faktiska pengar – mynt) flödar från normandiska erövringar: Spanien, Neapel, Palermo, Antiokeia, England. Veckomarknad i Laon 1071; årlig marknad i Paris 1074. [p55-57]

In 1100 and later, the main Latin word for money was pecunia. The word had a secondary connotation as ‘treasure’. This double meaning was classical. But it had not applied in the dark age. Back in the 750s documents could speak of merchants as a class that ‘has no pecunia’; or of a man obliged to live ‘in the pecunia‘ of another. This usage recalled a Latin of long before Cicero, when the word pecunia grew up as a brother to pecus (flock). In the dark age pecunia meant lands, buildings, animals, etc. with occasionally some treasure thrown in: roughly, in fact, what the rich man in Bishop Ratherius’ polemic listed as his property. As late as the eleventh century, and even later, vestiges of this usage remained in the expression pecunia viva for livestock. But that was by then a specialized meaning. From just before the year 1000 the strictly monetary meaning of the word gained ground, to become virtually exclusive by c. 1100. A similar shift is traceable in other related words. [p58 – se också The Poor in the Middle Ages, och Money and the Middle Ages]


The qualities of money have been enumerated: it moves freely from hand to hand; it travels; it divides almost anyhow; a lot fits in a small space; it can be left to pile up without suffering natural vicissitudes. These qualities are reflected in societies with money in them. Men’s mutual relations shift, as if liquefied by their medium of exchange; men travel; social blocks split, like sums of cash, into changeable groupings of individuals; people herd in towns, like coins in a chest; and power, finally, like value, is increasingly abstracted from the perishable to the imperishable, from individuals to institutions. [p60]

Den sociala rörligheten och institutionernas ökade makt skapar nya förutsättningar, nya möjligheter, nya handlingsmönster – nya tankemönster. (i princip North). Två nya tankemönster: girighet och ambition. [p60] The logic by which money could give rise to the first of these [mental] habits, money-mindedness, is simple. Money makes wealth mobile. The mind, apprehending that, can start hoping for wealth. [p60-61]

Evidence for a growth in money-mindedness

  1. Stölder. Vikingar, magyarer, saracener. Avarerna på sjuhundratalet plundrade Bysans; magyarerna gick västerut. Chronicle-refences to cases of robbery and theft suggest a pattern of crime roughly corresponding to the growth in the currency of gold and silver. […] The main motif [in hagiography] is that a saint miraculously saves a thief from hanging […]. This motif was dormant before the millennium and increased sharply in Lives written just afterwards, as if there were more thieves then. [p62-63]
  2. Gregorianska reformerna 1049 och framåt. Penningekonomin tillåter spekulation i kyrkliga ämbeten. Mer simoni, mer fördömande av simoni.
  3. At the start of the [eleventh] century we hear next to nothing of the persecution of Jews. […] By 1100 this friendly coexistence had largely gone. [p68] Första korståget; Rhenlandet. Två huvudsakliga förklaringar: kristen väckelse, eller det inhemska (kristna) borgerskapet som gör sig av med sina föregångare och konkurrenter. Den ekonomiska förklaringen bevisad felaktig. Ny förklaring – ideologisk: The Jews, [Dr. Lea Dasberg] argues, were victims of the Investiture Contest, the great struggle between Pope and Emperor. The Jews were royal protéges both in law and fact. The papalists’ quarrel with the king, above all when linked to their Cluniac friends’ old, ascetic mistrust of usury, was bound to put their subject laity at loggerheads with the most conspicuous group of royal satellites. [p68-69]. [Skulle progromerna ha något med ett asketiskt ockerhat att göra? Dubbelkolla med Rubin.] Murrays egen förklaring: Dasberg demolerade fel ekonomiska teori; judarna var faktiskt ekonomiska pionjärer; judehatet är de fattigas och den etablerade rikedomens hat mot nyrika uppkomlingar. [p69]
    1. Jewish social structure [i.e. international cohesion] and traditions […] helped them adapt with exceptional speed to the finer skills of money-making, not least of that of finance at interest. The growing circulation of money thus enabled Jews to become Europe’s first nouveaux riches. The reaction against them came, not from the ranks of their counterparts and successors, but from the usual enemies of nouveaux riches: the old-rich, and the poor. To the blemishes usually attached to the swaggering upstart, Jews added their status as a caste with its own religion. Their new wealth enabled them to enlarge Jewish religious institutions, and consequently self-confidence, to a point where Christians felt threatened. So there was violence, abetted in single instances by the motives of individual debtors. [p69]
    2. Peter [the Venerable]’s appointment as abbot had coincided with the first aches and pains felt by Cluny’s agrarian empire as it passed into the new conditions of the early twelfth century. These conditions included inflation; for which and other reasons Cluny became short of cash. So she borrowed: during Peter’s time in office, the equivalent of some 10,000 silver marks, five times her annual revenue. Peter had apparently done what he could to steer the abbey’s debts into Christian waters. But the tide was too strong. From an early stage, gold objects from Cluny’s sacristy were in the hands of Jews in nearby Mâcon; and other obligations and gages probably followed them. [p70-71]
    3. Germany’s invidious prominence in twelfth-century persecution, for instane, was largely due to the shortness of her commercial past – the nouveaux riches were all the more so there. [p71]
  4. Vi ser samma utveckling – samma upptagenhet vid pengar och de problem det medför – i predikningar som i satiren. Det kan därför inte vara fråga om bara en litterär trend – att skrivkunnigheten breder ut sig, att fler texter finns bevarade. Predikningen är en kontinuerlig genre.


[I vilken utsträckning ökade institutionernas makt – alltså kyrkans och statens? makt – institutioner became more comprehensive. p60]



Pengar à social rörlighet à ambition.

[…] the existence and activities of centralized political authority depend on money. [p83] […] because liquid wealth can be concentrated, it facilitates the concentration of public authority. [p84] Men staten är en abstraktion som kräver individer för att upprätthållas: de tar del dess prakt och härlighet och kan, genom staten, avancera. It is true that service often presupposed wealth. But the wealth could be granted by the same authority that called for the service. [p86]

Vilka var det som befodrades?

  • De vars färdigheter ökar effektiviteten (ett ekonomiskt och politiskt centrums förmåga att leva upp till sina åtaganden) steg i graderna.
    • Lekmän. Framgångsrika bönder i Mâconnais blev biskops- och greveprévôt. In Languedoc, the few rural land-exploiters who made the leap into the military class were those who used or misused the comital offices of vigueier or bayle. A similar story can be read in Normandy. There, rather than among merchants, it is in the duke’s entourage or under the shadow of churches’ that an eleventh-century aristocracy of money has been found. [p87]
    • [läs klart]


[hoppar till p101]

The observation of social movement, I have suggested, gave rise to the philosophical allegory of Fortune’s wheel. Andras framgångar ger en möjlighet att drömma om egna. [p101]

Är ambition en moral disorder? It was certainly questioned quite keenly in the thirteenth century, when books by down-to-earth Greeks were persuading ascetic Christians to come to terms with the hard realities of social life. But in the eleventh century, and even for most of the twelfth, the ascetic Christians were innocent of such worldly wisdom. Ambition was a sin. [p102]

Thirteenth-century preachers who lashed love of money spoke also of social competitiveness as if it were a twin vice of the same milieu. [p104] Avarice, after all, could serve ambition; as vice versa. But the connection went deeper. Both tendencies, expressing social competitiveness in different ways, were ultimately facilitated by the same economic and political conditions. [p104-5]

Vad var det för sorts ambition att stiga i rikedom och aktning? Ambition, firstly, made a man busy, in both senses: externally active, and pressed for time. Peter Damianus, John of Salisbury, biskop Anselm II av Lucca talar om kyrkans karriärister, om courtier-clergy, om hur hårt simonisterna sliter för att nå gynnsamma ställningar. De avskyr lättja, men även streberns slitsamma fjäskande. [p105]

Servants of government were busy in a second sense: they tended to be short of time. […] Humbert has been heard speaking of the ‘vigils’ of his ambitious careerists, laboring ‘day and night’ for their masters. Peter Damian, always the hermit, warns of time-consuming business that attends the abbot of a cenobitic monastery, as he ‘prolongs his day of negotiations and general business half-way into the night … misses compline with the others, and has to say his matins before sunrise [i.e. and snatch what sleep he can afterwards]’. […] Bracton records the assize of Novel Disseisin was only produced after ‘many thoughtful vigils’. [p106]

Tidspressen var inte något nytt för trettonhundratalet. For in the economic field, signs of pressure on time occur early. Peasants and artisans near Fleury about the year 1000 ignored feast-days through a need to work in their fields (a complaint which would be endemic in sources about 1200). […] And there were merchants and usurers in Lyons in the 1160s ‘working night and day’ to accumulate money. [p106-107]

[Dock, Murray sager ju självt att det blev värre under det långa tolvhundratalet – a complaint which would be endemic etc. Vad som var nytt var arbetsklockor.]


Reason and Power

Det uppstår under den här tiden en idé om att människan med förnuftet kan påverka sin omgivning – bortom å ena sidan feudalismens nakna våld, och å andra sidan det övernaturliga, en tredje samhällelig kraft: vad vi kan kalla humankapitalförädling.[p110-111]

Människans tankar (förnuft) kan påverka den naturliga omgivningen. Att de medeltida var medvetna om detta (dvs blev medvetna om detta) under the central middle ages. Det framgår av texter om teknologi, magi och astrologi. När Theophilus (en pseudonym) skrev sin De diversis artibus kring 1100 var han uppenbart medveten om den psykologiska effekten av de tekniker för glassfärgning, färgblandningar för illustrationer, klockor etc han diskuterar. It is the mind that does the work. […] Theophilus is one of the first writers of the medieval renaissance to mention that noticeably intellectualist concept, The Sevenfold Gifts of the Spirit. Four these Gifts are Wisdom, Understanding, Counsel, and Knowledge. [p112]

Hugo av Sankt Viktor could scarcely bring himself to speak of so base a branch of knowledge as ’mechanics’. But he did so for long enough to betray an appreciation that artefacts are a triumph specifically of man’s reason. […] We hear an early Italian humanist in 1238 declaring that ‘by art heavy weights are lifted; by art fish are caught; by art men are carried dryfoot over sea’. If the liberal arts merged at one side with technology, at the other they were coming to merge, at the same date, with the bulky Greek and Arabic newcomer, philosophy. [p112-113]

[är detta ett ömsesidigt mergande, eller snarare en avspaltning?]

Books of magic enjoyed a growing vogue in the thirteenth century. The vogue was partly just an unexpected result of the growth of literacy. But it also witnessed the widespread popular penetration of the concept under discussion. Magic promised power through knowledge. […] ‘so ends the book of the life of the rational soul … the book by which every creature can be subjected except the nine orders of angels … the book by which all science can be learned’. […] But the ideas in magic were not all bombast. One in particular was shared with more pregnant regions of medieval science. This was the idea of ‘properties of things’. Common to natural philosophy as a whole, this idea got its main impetus from medicine. […] You only had to learn the ‘properties’ (or ‘natures’) of things, and you could exploit them, for example to heal disease. [p113-114]

… astrology was well-fitted to rehearse the principle of applied science. It occupied good minds. Its astral laws prefigured the mathematics of post-Renaissance physics. [p114] Astrologin rättfärdigade sig gentemot tvivlarna genom att sitt praktiska värde: om man kan förutsäga en katastrof är man bättre rustad att utstå den. Denna logik skulle komma att rättfärdiga obduktioner: in the last instance at least, the plague, the astrologers’ principle certainly contributed to a science destined to have tangible results. [p116]

Reason in the face of other men

[…] the two fields [makt över naturen och makt över människor] certainly shared one feature: the key to power in both was the mind. [p116]

De intellektuella [vilka?] var medvetna om det. They evinced this consciousness in three main ways: in theories of historical development; in views of government, civil and military; and in the evaluations they made of knowledge and intelligence when discussing, on one hand, the value of historical study, and on the other, the virtue of prudence. [p116]

Among the most striking aspects of the twelfth-century renaissance was its interest in the rise and fall of empires. [p116] Mellan astrologisk determinism och teologisk ersättningslära (judar till kristna romare) fanns the apologists of learning [p117]. Det är de fria konsterna som garanterar imperiers livskraft. Konsten knyts till kungamakten. På så sätt görs exempelvis julevangeliets vise män till kungar, med början under det sena niohundratalet; vid mitten av elvahundratalet blir de heliga tre konungar Kölns skyddshelgon, där deras reliker också förvaras. Man talar om the literary accomplishments of Julius Caesar, och om Salomo. [p118] Secretum Secretorum, Alexander den stores lärobok när han studerade under Aristoteles, säger the head of policy and judgement is Reason. It is the health of the soul and the mirror of faults. … It is the chief of all praiseworthy things, and the fountain-head of all glories. [p120]

In the the Latin Secretum entered, the political field was still divided, in many people’s minds, between human and superhuman force. In such a field, the Secretum’s bold apotheosis of intellect had the attraction of novelty – just the novelty its readers were ready to discover.


The intellect in government (a) Kings and ministers

Makten under elva- och tolvhundratalen utövades i allt högre grad av ett centraliserat styre. Ämbetsmän behövde instrueras: det skrevs handböcker. Nämnda Secretum säger att en härskare skall göra till sina ministrar de (bland andra kvalifikationer) som är snabba i tanken, har gott minne, och show[s] skill in ’all sciences’, and know[s] the sources of expenditure. [p121-122]

Coutumes de Beauvausis från 1290. […] detailed working advice for those mobile viceroys, the baillis […]. The first chapter of the Coutumes lists ten qualities needed by the bailli. In the list, neither loving God nor loving the king comes higher than second place. The first is held by that virtue “which is and ought to be lady and mistress of all the others, and without which the other virtues cannot be employed. This virtue is called wisdom […]” [p122]

Li Tresors, från 1268. One section of Li Tresors treats the office of podestà elected chief executive of a commune; and Latini includes here a list of twelve virtues a candidate should have. Faith in God […] now comes at the very end of the list. And what comes at the head? Once more, it is wisdom […] [p123]

(b) warriors



The inherent usefulness of learning and reason

Prudentia görs till kristen dygd, utan att helt frigöras från konnotationen ’list’. Från 1000 och framåt. [p132-135] […] prudence was the virtue which distinguished aids and obstacles to the love of God. [p134] But we also find – and there is no getting away from the fact that the texts where we find it were the ones most widely copied – a tendency to merge the two meanings. The merging had the effect of usurping, for a non-Christian virtue, the Christian virtue’s place in the ethical scheme. [p135]


Matematiken kap 6

Under den tidiga medeltiden var aritmetiken i allmänhet förbehållen prästerskapet, de skriftlärda, och då i allmänhet som ett sätt att studera Bibeln och beräkna påskhögtidens rätta plats i kalendern. Dess tekniska utveckling stagnerade efter den karolingiska renässansen: tabeller och andra räknehjälpmedel användes, men inga större framsteg gjordes. Den ottonska renässansen innebar en utbredning av matematisk färdighet: “it is clear that in both England and Germany practical reckoning was taught to school-children.” [p144-157] [citat: p157]

Alcuinus slutade på sjuhundratalet räkna vid 9000 – större tal än så var alltför otympliga att representera; med abakusens hjälp når Gerbert bara tvåhundra år senare 1010. [p164]

Från slutet av niohundratalet evidence thickens, if slowly at first, about society generally och mathematics as a science takes wing, [p162] Från den ottonska renässansen och framåt kan man skriva en matematikens samhällshistoria.

In the four centuries after Gerbert there emerged, in a significant proportion of society, that familiarity with numbers which was a precondition of Renaissance mathematics. [p162]

Abakusen (åter)upptäcktes möjligen så tidigt som 970; den var känd i England senast 1079. Mellan tusentalet och trettonhundratalet skrevs minst arton avhandlingar om abakusen och dess användning. Firstly, the spread of a device for accurate calculation presupposed a preparedness, in the people it spread among, to explore in this direction. […] Secondly, once the device was known, it did not merely, like a mechanism, do the sums put to it. It had to operated; and it shaped its operators’ minds. […] From about the millennium, but above all in the thirteenth and fourteenth centuries, there radiated from this device, wherever it was used, a stimulus towards an accurate concept of numbers. [p165-166]

Men abakusen innebär också att matematiken begränsas: det behövs ett sätt att enkelt notera stegen i en uträkning, men också ett sätt att snabbare genomföra beräkningar. Arabiska siffror ersatte abakusen. De var kända i Europa 1200, men började inte bli allmänt använda förrän fjortonhundratalet. Inom bokföringen däremot användes de så tidigt som 1299 då Florens bankirers och penningväxlares gille förbjöd sina medlemmar att använda dem. Gillet upprepade förbudet minst tre gånger mellan 1299 och 1316, förmodligen, skriver Murray, för att undvika anklagelser om ocker och fusk av det lokala prästerskapet, för vilka de arabiska siffrorna lär ha varit otydbara. [p166-170] [dock, se kritiken]

Jag kan inte källmaterialet tillräckligt väl för att säga vem av dem som har rätt. Det spelar hur som helst ingen större roll: rationaliseringen under den senare hälften av medeltiden är tillräckligt väl belagd av Crosby, Murray och Le Goff oavsett den specifika frågan om abakusen och de arabiska siffrorna.

Handeln med läroböcker i matematik och astrologi antyder att tusentals människor räknade vid trettonhundratalets början. Bara i Florens studer över tusen barn abaco och algorismo 1345. [p172] Men what were the thousands of practical reckoners, in the teeming countryside of Europe before the Black Death? [p174]

We have learned where to find non-arithmeticians: in literature. Because literature normally took a different path from mathematics it provides a measure for the influence of mathematics on non-specialist thought. The more deliberately the paths diverged (e.g. because of literary prejudice against numbers), the more telling are the signs of mathematical influence. Literary conventions might resist the infiltration of precise number, but they could not hold out for ever. [p174-175]

[…] a shift can be shown to have occurred in the mental habits of the literate. There emerged among them what can be called an ‘arithmetical mentality’. [p175]

Litterära konventioner innan tolvhundratalet (levde kvar efteråt men gre less intense and less universal p175): för det första att undvika exakta mängder och tal, especially dates. Tal i hagiografisk litteratur mellan 1000 och 1200 handlar i allmänhet om små mängder eller små tidsperioder – tjugo solidi, tjugo år – and very occasionally the achievements of repetitive prayer (’he said the Paternoster three hundred times each night’: numbers are exceptionally high in this context). Hagiografin lyder under vissa genrekrav – den var sifferlös i större utsträckning än annan litteratur, men: Between the millenium and the fourteenth century, hagiography as a whole lost its dominant position in literature. That is to say, the characteristaclly innumerate genre commanded less of the literary scene. Trenden stämmer dessutom för annan litteratur: De diversis artibus till exempel anger endast undantagsvis exakta mått. [p175-176] I norra Frankrike – the heartland of the intellectual renaissance [in the twelfth century] – skrev abbot Suger av St. Denis om Ludvigs VI liv och gärningar: alla slag och härtåg, alla uppbåd, listor över de döda och sårade uppmäts till quamplures, multi eller rätt och slätt ’likt gräshoppor’. Två gånger anges ett antal över ett hundra: ett rent symboliskt tiotusen. (For such numbers were commonly used by chronichlers as if, rather than numbers, they were names, denoting broad orders of magnitude. [p177] denna second mental habit illustrative of the innumerancy of early medieval literature [p179]) Årtal anges aldrig. Abelardus (friend of the Chartres mathematicians) räknar i sin självbiografi med ‘många’, ‘en mängd’, ‘somliga’: inga antal över fem specificeras; datum används inte. [p177]

Shyness of figures; exaggeration of large numbers: in medieval Europe these figures marked most literature in most places at least down to the late thirteenth century. More specimens of both could be assembled from this and even later periods. Psychologically revealing because uncontrived, they show how the antipathy to arithmetic inherited by medieval literature from its forebears struck root deep in medieval soil. The limit of clear numerical consciousness for the bulk of the literate people remained, in effect, roughly that of arithmetic in Roman numerals. Above that, exact quantity was a stranger. [p180]

Första försöket att räkna – Domesday Book. I slutet av tusentalet kunde många fler än royal commissioners grundläggande aritmetik; mot slutet av elvahundratalet var förmågan att räkna pengar och mäta tyg test of legal majority för en borgarson.  Det gick långsammare i Frankrike: den gotiska arkitekturen (som kräver ett visst matematiskt kunnande) uppstod circa 1140; kontrakt och juridiska dokument pick up their taste for numbers after [1125]. By the early thirteenth centiry France had certainly made up for any ground it had lost earlier. In 1223 a French royal official will be found tossing off from his head the annual revenues of the last two kings, to the tune of hundres of thousands of livres (not in round numbers, either) – while a visiting cleric from Lausanne, to whom the boast was made, will be found remembering the figure. [p181]

Så det fanns en viss nummerkunnighet bland vissa grupper under elvahundratalet. Men det fanns också bredare kunnighet, som blir tydligast i Italien under tolvhundratalet. Fra Salimbene av Parma skrev krönikor under sena 1280-talet: han använder årtal, månader, datum, sidhänvisningar, han anger antal med stor noggrannhet; han anger matvarupriser under missväxtår och under år med god skörd. Bonvesino della Riva skrev 1288 en (för tiden) enorm statistisk studie av Milano: han räknar med exakthet dess invånare; dess kyrkor, marknader, fontäner, hospital, helgedomar, harnesksmeder, jurister, trumpetare, kloster; han beräknar den dagliga åtgången av säd, kött och fisk utslaget på festdagar och fastedagar. Bonvesino utgick dels från redan beräknade siffror (government accounts), dels från de siffror han och hans medhjälpare kunde få från slaktare, tullindrivare, etc. Giovanni Villani skrev i Florens mot slutet av 1340-talet sin Cronica, the biggest thesaurus of figures in medieval historiography. [p184] [M]ore widely copied than any other chronicle of the late middle ages. [p184] Där mäts brospann, hyror och avgifter, sädeskornpriser, tullavgifter, löner; han räknar sjukhus, nattvardsvinsmängder och allt mellan himmel och jord. When he could not discover a number […] Villani left a blank. That is to say, the numbers he put in were not wild guesses. Vi ser liknande utveckling i annan litteratur från 1200-talet: kanske framför allt den apokalyptiska talmystik som utgick från Joakim av Floris studier av Uppenbarelseboken. Och det var inte bara litteratur: bankirer listar anställda med löner, skulder och nettotillgångar; Dante förväntar sig att läsaren förstår uttryck som ’a man worth a thousand florins.’ Alltihop [p181-184].


Orsaker till matematisering:

  • Uppstår på samma ställen vid samma tidpunkt: Italien under tolvhundratalet, bland icke-latintalande. De utgår båda från vernakulären: vernakulära vikt- och mängdmått tränger ofta in i latinsk litteratur både vad gäller kommersiella ämnen och matematik. [W]riters about reckoning will sa or suggest that merchants were good reckoners [på 1100-talet]. Och ju mer monetariserad ekonomin blir, ju mer beroende av penningväxlare och penninglånare allmänheten blir, desto mer allmän blir matematiken. De problem Fibonaxxi diskuterar i Liber Abaci tar ofta formen av ekonomiska transaktion: ’The man who went to Constantinople to sell three pearls” etc. [p190]
    • Money divides and multiplies easily and precisely. Goods can be reduced by it to a common measure; and its use encourages their circulation. The spread of money through a society is a direct invitation to it to calculate with number. [p191]
  • Å andra sidan: ingen matematiker var köpman. Den specialiserade matematiken överlever i väldigt få manuskript: de välspridda, välanvända läroböckerna är långt mindre avancerade. Det fanns gränser för kommersialiseringens matematisering. Det var en framträdande faktor, men inte den enda. Pengar genomsyrade samhället (se ovan om snabb monetarisering). Och some otherwise dynamic merchants […] can even appear puzzlingly conservative in the field […] [Se kritiken nedan]. Det var alltså inte främst köpmännen specifikt, utan en mer allmän monetarisering. [p192-194]
  • Och det är ena halvan av sanningen. Den andra: styret. Under the umathematical dark ages var matematikens två främsta entusiaster Karl den store och Otto den store. De fem främsta företrädarna för medeltida matematik – Bede, Alcuin, Gerbert, Adelard av Bath, Leonardo av Pisa – lived the latter part of their lives as friends of monarchs. Tolvhundratalets Italien för födelseplatsen för matematiken, handeln, och den mogna stadsstaten, byggd på statssuveränitet och romersk rätt. Rätten blomstrade under tolvhundratalet. [p194] Kopplingen mellan centraliserad statsmakt och aritmetik är partly a mere exention of the ’countinghouse’ theory. […] Trade was often the ultimate stimulus to government arithmetic. Nearly all the great governmental numbering enterprises of the central middle ages, from Domesday on, owed at least part of their origin to some ruler’s desire to sponge up liquidity, created by trade. The same went for the complex taxation arrangements of twelfth- and thirteenth-century Italian communes […]. […] internal accounting between ruler and agent resembled in principle that of a big business. [p195]
  • Igen, å andra sidan: kronan räknade så mycket mer än pengar (människor och soldater bl. a.), och den räknade i så mycket större skala och över långt större områden. (och Murray talar om styrande i allmänhet: inklusive biskopsdömen och ordnar.) Statskonst är att räkna: behov av matematiker. [p197-198]
  • Arithmetic, then was a skill in pure reason. […] If reason led to power, so also and especially – must arithmetic. […] The two areas where arithmetic most throve, commerce and government, were, among many other things, society’s two main staircases. In a word: numerical skill could raise a man. [p206] Många matematiker var courtiers och professional astrologers, “for the most part men who had begun life with little substance of their own.” [p208]


[his discussion of arithmetic ignores the classical and medieval distinction between arithmetic (the science of numbers) and logistic (the art of reckoning). Failing to appreciate how systems of numeration fit techniques of calculation, he inverts the historical relation of abacus and algorism. Convinced of the inherent superiority of decimal place-value reckoning, he sets up as a paradox the retention of Roman numerals and the abacus by otherwise “progressive” men. But there is no mystery. The abacus is faster than written calculation and takes up no paper; Roman numbers betray tampering. The paradox is specious. Mahoney i The American Historical Review vol 84 no 5 dec 1979]

[Makt fås genom intellektuellt arbete – folk blir intellektuella.] But this is as far as Murray takes the influence of commerce on rationality. He misses the opportunity to show at this point how commerce and business themselves stimulated the rational process by requiring calculation, records, accounting, and the methodical pursuit of profit. Baldwin i Speculum vol 5 no 3 jul. 1980] [Men varför gör handel det? – varför kan man inte, som en gång trobrianderna, utveckla ett förhållandevis avancerat cirkulationssystem utan något av detta? Här knyter vi äntligen ihop säcken tiden/beräkning –  värdet är unikt som diskret kvantifierbart mått på en abstrakt rikedom; beräkningen ligger inbyggd. Postone & Gurevitj.] [är inte kapitalismen just beräknande? – se Om judefrågan]