Analog SFF, July-August 2007

by Dell Magazine Authors

  The medievals also introduced the arithmetic of fractions.Neither the Arabs nor the Latins understood Eudoxus's solution to irrational proportions. So the Latins created a “procedure of denomination” to convert geometric ratios into numerical ratios, and solved the geometric problem using arithmetic. Bradwardine devised algorithms for compound fractions to analyze Aristotle's theory of motion. (He found it contradictory.) Oresme developed a theory of commensurability of compound fractions that implicitly involved fractional exponents. (He used it to discredit astrology.)

  The continuum was the medievals’ greatest contribution to mathematical physics. It began, oddly enough, in a theological question about degrees of charity (Peter Lombard's Sentences, 17). Does a body participate in a constant form to varying degrees, or does the form itself vary? Arguing the latter, Scotus proposed that the addition of distinct similar parts to an existing form created a unified form of greater intensity. Thus, five parts of redness added to three parts of redness created eight parts of redness in exactly the same way that adding weights produced a new, increased weight. This seems obvious to us because we've had seven hundred years to get used to it; but until the “Calculators of Merton” developed this “intension and remission of forms,” Aristotelian contraries had been the rule (wet/dry, hot/cold, motion/rest, etc.) Now, “either/or” could be “more-or-less."

  If only they could measure them ... But the Calculators lacked instruments to actually measure color, temperature, et al. So their mathematical treatments remained thought experiments applied to abstract problems. However, “one must conceive of measuring heat and force before one sets about doing it. If the Calculators dreamed and did not act, that does not mean their dreams were irrelevant to the acts of others.” [Stanford Encyclopedia of Philosophy, on William of Heytesbury]

  Reply to Objection 1. Bradwardine and others consciously departed from Aristotle when they realized that natural processes had to be represented by mathematical functions that hold for all values, and are therefore continuous.

  Furthermore, to criticize the medievals both for clinging to Aristotle (as here) and for rejecting Aristotle (II.2, Obj. 2) makes it seem that the objection is really to the medievals themselves.

  Reply to Objection 2. Mathematics was “about unchanging things” because arithmetic and geometry comprised all of mathematics. But after Bradwardine's compound fractions revealed logical flaws in Aristotle's theory of motion, he defined a new concept: instantaneous motion. Heytesbury used intension and remission of forms to develop the limit concept and open and closed sets. This laid conceptual foundations for the infinitesimal calculus, but that was as close as the medievals got to “a mathematics of changeable things."

  * * * *

  Article 5. Whether medieval natural philosophers pursued natural philosophy as a research enterprise rather than as a body of knowledge.

  Objection 1. It would seem not, because the Middle Ages had no sense of the natural world as a thing to be explored. Aristotle had explained everything, so nothing fundamentally new could be found.

  On the contrary, John of Salisbury [Metalogicon] wrote, “Bernard of Chartres used to say that we are like dwarfs on the shoulders of giants, so that we can see more than they, and things at a greater distance, not by virtue of any sharpness of sight on our part, or any physical distinction, but because we are carried high and raised up by their giant size.” And Fra Giordano of Pisa wrote, “Not all the arts have been found; we shall never see an end to finding them.” Therefore, the medievals had a sense of research and progress.

  I answer that there can be no research enterprise without a defined body of knowledge in the first place. In the Latin West (and Islam), the natural sciences were organized into an Aristotelian schema that, in Peter Dear's words, “determined what was worth saying” about natural philosophy. At first, this meant understanding how particular facts “fit” into this structure; but by the 14th century, it had become clear that Aristotle's conclusions were sometimes wrong. Buridan at Paris, Bradwardine at Oxford, and others—working within the framework of Aristotelian principles and methods—had begun amending and correcting them. Meanwhile, outside the universities, engineering was being pursued using deliberate research.

  Reply to Objection 1. Unlike ingeniators and magicians, university scholastics were indeed more focused on understanding known facts than on discovering new facts. Kuhn called this “normal” science (vs. “paradigm-shifting” science.) But if Aristotelian natural philosophy was essentially a taxonomic scheme, the same is true of any of its contradictory successors—positivism, instrumentalism, etc. The sense of revolution that animated 17th century science came partly from “unpacking” the consequences of these new philosophies. There was a sense of “doing something new” that Europe had not felt since ... “unpacking” Aristotelianism in the 11th and 12th centuries.

  That's why Columbus’ voyages symbolize the start of the Modern Age. It wasn't the discovery of the New World that mattered. It was “the discovery of discovery."

  * * * *

  Article 6. Whether medieval natural philosophers reconstructed the social basis of knowledge around a positive evaluation of cooperative research.

  Objection 1. It would seem otherwise, because the positive evaluation of cooperative research begins with The Royal Society and similar bodies.

  On the contrary, Stock writes of the Middle Ages that “the same commerce that re-monetized the economy established, for the first time since antiquity, a self-conscious community of intellectuals whose uninhibited communication with each other was the necessary condition for the advancement of learning."

  I answer that this new class of university scholars moved with ease from university to university. They adopted the trappings of knighthood, with titles, robes, ceremonies of initiation, and the like. People even called them “the new chivalry.” But while they were in correspondence with one another, their work was essentially independent. The peer review used in theology to ensure orthodoxy had no counterpart in natural philosophy. Consequently, there was seldom a consensus that a Question had been “solved."

  Furthermore, in the “impetus of thought,” the velocity of knowledge counts for as much as its mass. In the manuscript culture, documents reproduced slowly and accumulated copyist errors. (Two transposed ratios in Jordanus’ Elementa super demonstrationem ponderum muddled his explanation of the work principle.) Arguably, the greatest medieval invention was the printing press, which replaced one-off manuscripts with proofread, mass-produced typescripts, allowing ideas to spread more quickly.

  Reply to Objection 1. In 1025, Ragimbold of Cologne and Radolf of Liege engaged in a “mathematical tournament,” with other scholars participating as judges. Their letters provide the first example of the simultaneous investigation of a scientific question by different parties in contact with one another. They got it screwed up—Greek geometry was still undigested—but the remarkable thing is that it happened.

  Informal groups like the Calculators of Merton prefigured organizations like the Royal Society.

  * * * *

  Question IV.

  The medieval foundations of the Scientific Revolution.

  * * * *

  Article 1. Whether the 17th century revolution could have occurred without the work of the late medieval natural philosophers.

  Objection 1. It would seem it could have, because the 17th century was revolutionary to the very extent that it rejected medieval categories of thought. The Scientific Revolution occurred in spite of, not because of medieval scholasticism.

  On the contrary, Duhem says, “the physicists of the Paris school posited the foundations of the mechanics that Galileo, his contemporaries, and his disciples developed.” Copernicus repeated arguments made by Buridan and Oresme. Galileo reproduced Oresme's proof of the mean speed theorem and de Soto's law for falling weights using Bradwardine's definition of instantaneous motion. Consequently, writes Grant, “What the medieval scholastics started, their successo
rs in the Age of Reason completed."

  I answer that , Duhem's Continuity Thesis disturbed several centuries of assumptions about the Middle Ages. Not until recent times was the medieval era studied with professional dispassion; and if the continuity is not as great as Duhem contended, it is now acknowledged to be much greater than formerly supposed.

  But the medieval “pre-discoveries” of modern theories noted by Duhem demonstrate continuity only in “the facts and lore” of the sciences. More crucially, the scholastics accomplished certain preconditions enabling the methodology of Science to develop. Following Grant, these preconditions include:

  1. The independence of church and state. Charlemagne had modeled his empire on Rome, including imperial control of the priesthood. It took the papacy two centuries to secure the right to appoint bishops, preside over church councils, etc.; but by thus stripping princes of their spiritual roles, the medievals created something new: the secular state. Consequently, in the Middle Ages, there was always another authority to appeal to. In the social space between them, independent, freestanding institutions like guilds and universities could grow, which were elsewhere subordinate to emirs or bureaucrats.

  2. The cathedral schools maintained Roman learning and passed it on. The Latins were “prepped” well before Aristotle's works were translated.

  3. The translations furnished the Latin West with a ready-made curriculum that gave coherence to the study of nature. Had the medieval Church rejected pagan and Islamic learning, or had these translations been of literature and poetry, European history would have been very different—and Science perhaps never born.

  4. The universities gave Science, for the first time in history, a home base with freedom of inquiry where the “ready-made curriculum” could be taught.

  5. The theologian-natural philosophers. Since natural philosophy was a prerequisite for a theology degree, theologians were trained in natural philosophy and regarded it favorably.

  6. Freedom of inquiry into nature. Parens scientiarum and the administrative struggles of the arts faculties at the universities helped establish the principle. The disputatio and the Questions genre, which required arguments for both sides of a question, encouraged a “culture of poking into things.” William of Ockham declared, “Assertions ... concerning natural philosophy, which do not pertain to theology, should not be solemnly condemned or forbidden to anyone, since in such matters everyone should be free to say freely whatever he pleases."

  7. The maintenance and improvement of the exact sciences. A few examples from the medieval West are: the explanation of the rainbow (Theodoric of Fribourg), the work principle in physics (anon., Aliud commentum), motion on an inclined plane (Jordanus de Nemours), image formation on the retina (Witelo), laws of magnetism (Pierre Maricourt), and “Gresham's” law of money (Nicole d'Oresme).

  8. The problems of science. The Middle Ages established the subject matter of modern science: the nature of space and time, the existence of a vacuum and the possibility of motion through it, the kinematics and dynamics of local motion, etc. The scholastics posed hundreds of different questions and cited a massive amount of empirical data (bellows, siphons, etc.) Galileo and the others did not work on new questions; they found new answers. As Grant says, “Without the natural philosophy of the universities, the 17th century would have had little to discuss."

  9. The language of science. The medievals passed along Aristotelian terms like potential, cause, matter, substance, analogy, relation, quantity and quality, genus and species, and created new terms like numerator and denominator, uniform motion, acceleration, gravity, momentum, impetus, inertia, kinematics and dynamics, intensity and quantity. Without this language, we could not talk about the problems of science.

  Dear's “six innovations” of the Scientific Revolution did not pop out of nowhere. Without these medieval preconditions, a tipping point could not have occurred in the 17th century, if indeed ever. As R. R. Palmer wrote in A History of the Modern World, “scholastic philosophy laid foundations on which later European thought was to be reared. It habituated Europeans to great exactness, to careful distinctions, even to the splitting of hairs. It called for disciplined thinking. And it made the world safe for reason."

  Reply to Objection 1. Etienne Gilson [La Philosophie au Moyen Age] wrote, “It is necessary ... to relegate to the domain of legend the history of a renaissance of thought succeeding to centuries of sleep, obscurity, and error. Modern philosophy did not have to undertake the struggle to establish the rights of reason against the Middle Ages; it was, on the contrary, the Middle Ages that established them for it, and the very manner in which the seventeenth century imagined it was abolishing the work of the preceding centuries did nothing more than continue it.” Galileo, Harvey, and even Newton used methods and principles that were recognizably Aristotelian.

  * * * *

  Article 2. Whether a Scientific Revolution could have preceded the 17th century.

  Objection 1. It would seem that a scientific revolution could have occurred in the 14th century because all of the preconditions for the revolution were in place by then, and most of Dear's six transformations had at least started. By the 14th century the medievals had rethought the science of motion and introduced crucial kinematical and dynamical concepts—and the 17th century revolution occurred primarily in the science of motion.

  On the contrary, medieval natural philosophy was “holistic.” It tried to explain the world as perceived by human senses. This was trying to explain too much, too soon. The 17th century succeeded because they restricted themselves to the simple, well-behaved domain of primary qualities, contrived experiments, and efficient causes.

  I answer that the 14th century “Paris school” and “Oxford Calculators” did not between them ignite a scientific revolution. But might they have done so? Counterfactuals are easily imagined, less easily substantiated. History happened as it did for reasons often centuries in the making, and can seldom be altered by changing whimsically this event or that.

  What were the drivers? The medievals were devoted to reason and had separated philosophy from theology. They developed the very concept of natural laws governing a tangible universe accessible to reason. They gave science an independent home base in the universities. They conceived of qualities as measurable, applied mathematical models, and required that philosophical conclusions be “saved by the appearances” of sensory experience. Outside the universities, alchemists conducted experiments; and ingeniators engaged in deliberate research and innovation. A continent-wide network of scholars accustomed to asking probing questions about nature had begun researching non-Aristotelian theories of motion. Letter symbolism was in use, and the arithmetical operators had appeared. Subject/object duality was genuinely lacking—but an Aristotelian alternative to atomism existed that did not require it.

  What were the inhibitors? The effort to explain the world as perceived by humans was biting off more than they could chew. (Imagine teaching physics and psychology as a single, combined science!) Their reliance on empiricism did not overcome their suspicion of deliberate experimentation. Their conviction that qualities were measurable awaited instruments to measure them. They applied mathematical thinking, but, except in the exact sciences, little mathematical calculation. Letter symbolism and operators were still used as shorthand in otherwise verbal discussions. Lastly, the “velocity of knowledge” was too slow: ideas didn't circulate fast enough to start a “chain reaction."

  But if the abortive 14th century revolution came so close, why did three centuries pass before the successful 17th century version? Three factors suggest themselves. During the Renaissance, Neoplatonic idealism grew at the expense of Aristotelian empiricism, and intellectual emphasis shifted to art and literature. In the Reformation, reasoned theology retreated before personal piety. Like al-Ghazali, Luther dismissed “the whole of Aristotle.” Scientifically, these eras marked time. But one additional possibility is that there simply weren't enough people.

&
nbsp; At the height of the 14th century, the Black Death wiped out a third of Europe. Ockham and Bradwardine were two who perished. Populations did not recover 14th century levels until ... the 17th century. And if the velocity of ideas is as important as the neutrons in an atomic pile, so is the critical mass of minds to emit and be excited by them. Only by Galileo's time would there again be as many natural philosophers as in Buridan's day.

  Reply to Objection 1. That a “true science” is impossible within Aristotelianism stems from its conscious rejection by 16th/17th century Neoplatonists. Arguably, the Scientific Revolution was more an increased understanding of how to apply mathematics than it was a rejection of the principles of Aristotelian physics. The conclusions of Aristotelian physics are another matter—many of them were factually wrong. Yet, we do not reject Galileo's principles merely because he insisted on circular orbits, claimed comets were atmospheric phenomena, or cited the ocean tides as proof of the earth's rotation. In fact, Aristotle had sound empirical reasons for rejecting heliocentrism [On physics]—perhaps even for saying that men had more teeth than women [On animals]. By the 14th century, Aristotelian conclusions were being questioned and corrected within an Aristotelian framework. Even the conceptual hurdle of deliberate experimentation might have been overcome. The “artful vexation of nature” certainly had important advocates.

  The medievals did not draw the objective/subjective line in quite the same way, and their theory of minima naturalia was not that of atoms; yet modern “atoms” seem much like “minimae.” Heisenberg's uncertainty principle and “spooky action at a distance” have battered 17th century positivism—and resurrected 14th century ghosts. Form and telos have been making a quiet comeback under new names. “The philosophy of nature produced by authors such as Aristotle or Thomas Aquinas is perhaps less out-of-date than expected,” writes Tanzella-Nitti.