The acid test of concepts

A striking passage from Frege’s unpublished essay “Boole’s logical calculus and the concept-script,” which he submitted in 1881 first to Zeitschrift für Mathematik und Physik, then the Mathematischen Annalen, and finally the Zeitschrift für Philosophie und philophische Kritik, but was rejected each time:

I now return once more to the examples mentioned earlier, so as to point out the sort of concept formation that is to be seen in those accounts. The fourth example gives us the concept of a multiple of 4 […]. The eighth example gives us the concept of the congruence of two numbers with respect to a modulus, the 13th that of the continuity of a function at a point etc. All these concepts have been developed in science and have proved their fruitfulness. For this reason what we may discover in them has a far higher claim on our attention than anything that our everyday trains of thought might offer. For fruitfulness is the acid test of concepts, and scientific workshops the true field of study for logic.

This is pp. 32-33 in the translation by Peter Long and Roger White, with the assistance of Raymond Hargreaves, in Posthumous Writings, edited by Hans Hermes et al (Blackwell, 1979).

This remains one of the lesser appreciated themes in Frege’s work. The standard fare in a philosophy major mentions sense and reference, concept and anti-psychologism, but I’d bet even very few working philosophers are familiar with these suggestive passages on fruitfulness. Of course, every time I read the word I can only think of Keats, on autumn: “Season of mists and mellow fruitfulness…”

Some helpful discussions appear in:

  • Jamie Tappenden, “Fruitfulness as a Theme in the Philosophy of Mathematics,” The Journal of Philosophy (2012)
  • Jamie Tappenden, “The Mathematical and Logical Background to Analytic Philosophy,” The Oxford Handbook of the History of Analytic Philosophy
  • Jamie Tappenden, “Extending knowledge and ‘fruitful concepts’: Fregean themes in the philosophy of mathematics,” in Gottlob Frege: Critical Assessments of Leading Philosophers, Vol. III, Frege’s philosophy of mathematics, edited by Michael Beany and Erich Reck

That unto logik hadde longe y-go

Two weeks ago I read Charles Homer Haskins’s slim volume The Rise of Universities (1923), a charming collection of three lectures—”The Earliest Universities,” “The Mediaeval Professor,” “The Mediaeval Student”—on the birth of universities, especially at Bologna and Paris.

I came to Haskins to get my bearings after the disorientation of discovering, while skimming David Bressoud’s new book Calculus Reordered, that the history of science took an important step forward as early as the early 1300s—centuries before Galileo, et al.—when William Heytesbury and colleagues at Merton College in Oxford clarified the relationship between kinematics and dynamics, giving the first purely mathematical treatment of motion. (Heytesbury’s most important work, the Regulae solvendi sophismata—Rules for Solving Sophisms—seems not to have been translated in full into English.) The dark ages were not quite so dark, after all. Clifford Truesdell sums up the contributions of these so-called Oxford Calculators in his Essays in the History of Mechanics:

The now published sources prove to us, beyond contention, that the main kinematical properties of uniformly accelerated motions, still attributed to Galileo by the physics texts, were discovered and proved by scholars of Merton college. […] In principle, the qualities of Greek physics were replaced, at least for motions, by the numerical quantities that have ruled Western science ever since. The work was quickly diffused into France, Italy, and other parts of Europe. Almost immediately, Giovanni di Casale and Nicole Oresme found how to represent the results by geometrical graphs, introducing the connection between geometry and the physical world that became a second characteristic habit of Western thought.

Contrary to the received image of abortive medieval scholasticism, Haskins paints a portrait of rich intellectual ferment, drawing a great deal more continuity with the present than we usually assume [cf. the dispute over the so-called “continuity thesis” in the history of science]:

The occasion for the rise of universities was a great revival of learning, not that revival of the fourteenth and fifteenth centuries to which the term is usually applied, but an earlier revival, less known though in its way quite as significant, which historians now call the renaissance of the twelfth century. So long as knowledge was limited to the seven liberal arts of the early Middle Ages, there could be no universities, for there was nothing to teach beyond the bare elements of grammar, rhetoric, logic, and the still barer notions of arithmetic, astronomy, geometry, and music, which did duty for an academic curriculum. Between 1100 and 1200, however, there came a great influx of new knowledge into western Europe, partly through Italy and Sicily, but chiefly through the Arab scholars of Spain—the works of Aristotle, Euclid, Ptolemy, and the Greek physicians, the new arithmetic, and those texts of the Roman law which had lain hidden through the Dark Ages. In addition to the elementary propositions of triangle and circle, Europe now had those books of plane and solid geometry which have done duty in schools and colleges ever since; instead of the painful operations with Roman numerals—how painful one can readily see by trying a simple problem of multiplication or division with these characters—it was now possible to work readily with Arabic figures; in the place of Boethius, the “Master of them that know” became the teacher of Europe in logic, metaphysics, and ethics. In law and medicine men now possessed the fullness of ancient learning. This new knowledge burst the bonds of the cathedral and monastery schools and created the learned professions; it drew over mountains and across the narrow seas eager youths who, like Chaucer’s Oxford clerk of a later day, “would gladly learn and gladly teach,” to form in Paris and Bologna those academic gilds which have given us our first and our best definition of a university, a society of masters and scholars.

Later in the book, Haskins notes that this renaissance

added to the store of western knowledge the astronomy of Ptolemy, the complete works of Euclid, and the Aristotelian logic, while at the same time under the head of grammar great stimulus was given to the study and reading of the Latin classics. This classical revival, which is noteworthy and comparatively little known, centered in such cathedral schools as Chartres and Orleans, where the spirit of a real humanism showed itself in an enthusiastic study of ancient authors and in the production of Latin verse of a really remarkable quality. Certain writings of one of these poets, Bishop Hildebert of Le Mans, were even mistaken for “real antiques” by later humanists. Nevertheless, though brilliant, this classical movement was short-lived, crushed in its early youth by the triumph of logic and the more practical studies of law and rhetoric. In the later twelfth century John of Salisbury inveighs against the logicians of his day, with their superficial knowledge of literature; in the university curriculum of the thirteenth century, literary studies have quite disappeared. Toward 1250, when a French poet, Henri d’Andeli, wrote his Battle of the Seven Arts, the classics are already the ancients, fighting a losing battle against the moderns:

Logic has the students,
Whereas Grammar is reduced in numbers.
Civil Law rode gorgeously
And Canon Law rode haughtily
Ahead of all the other arts.

If the absence of the ancient classics and of vernacular literature is a striking feature of the university curriculum in arts, an equally striking fact is the amount of emphasis placed on logic or dialectic. The earliest university statutes, those of Paris in 1215, require the whole of Aristotle’s logical works, and throughout the Middle Ages these remain the backbone of the arts course, so that Chaucer can speak of the study of logic as synonymous with attendance at a university—

That un-to logik hadde longe y-go.

In a sense this is perfectly just, for logic was not only a major subject of study itself, it pervaded every other subject as a method and gave tone and character to the mediaeval mind. Syllogism, disputation, the orderly marshalling of arguments for and against specific theses, these became the intellectual habit of the age in law and medicine as well as in philosophy and theology. The logic, of course, was Aristotle’s, and the other works of the philosopher soon followed, so that in the Paris course of 1254 we find also the Ethics, the Metaphysics, and the various treatises on natural science which had at first been forbidden to students. To Dante Aristotle had become “the Master of them that know,” by virtue of the universality of his method no less than of his all-embracing learning. “The father of book knowledge and the grandfather of the commentator,” no other writer appealed so strongly as Aristotle to the mediaeval reverence for the text-book and the mediaeval habit of formal thought. Doctrines like the eternity of matter which seemed dangerous to faith were explained away, and great and authoritative systems of theology were built up by the methods of the pagan philosopher. And all idea of literary form disappeared when everything depended on argument alone.

The teat of interpretation

Andrea Long Chu in the Chronicle of Higher Education:

Structural problems are problems because real people hurt real people. You cannot have a cycle of abuse without actually existing abusers. That sounds simple, which is why so many academics hate it. When scholars defend Avital—or “complicate the narrative,” as we like to say—in part this is because we cannot stand believing what most people believe. The need to feel smarter is deep. Intelligence is a hungry god.

In this way, Avital’s case has become a strange referendum on literary study. Generations of scholars have been suckled at the teat of interpretation: We spend our days parsing commas and decoding metaphors. We get high on finding meaning others can’t. We hoard it, like dragons. We would be intellectually humiliated to learn that the truth was plain: that Avital quite simply sexually harassed her student, just as described. Sometimes analysis is simply denial with more words. Sometimes, as a frustrated student in a first-year literature course always mutters, the text just means what it says it means.

The cleansing and deepening of the dispute

Erich Heller, The Disinherited Mind, 1952:

Anyone engaged in the study, teaching, and criticism of literature as a university discipline is likely to become at some time or other aware of one fundamental problem raised by his own pursuits, of a difficulty that is all his own and is not, or at least not to the same extent, shared by his colleagues in other subjects. For however sincerely he may struggle against bias and prejudice in his own approach and appreciation, his work will still be very intimately related to his experiences in wider fields. […] and this, he will see, is no shortcoming of his own discipline, to be conquered in scientific campaigns or disguised by scientific masquerades, but is in fact its distinctive virtue. […] thus he would be ill-advised to concentrate exclusively on those aspects of his discipline which allow the calm neutrality of what is indisputably factual and “objective”. His business, I think, is not the avoidance of subjectivity, but its purification; not the shunning of what is disputable, but the cleansing and deepening of the dispute.

Too comfortable in his own skin

Arthur Krystal on Jacques Barzun and Lionel Trilling:

As students and instructors at Columbia they had only a nodding acquaintance. Barzun, tall, fair-haired, Gallically handsome, was self-assured and interested in history, theater, music, and detective stories. Trilling, shy, intense, on the short side, was keen on Freud, Marx, and American fiction. To a budding and brooding intellectual like Trilling, the young Barzun seemed too comfortable in his own skin; there was no angst, no alienation. “Such awareness as we first had of each other,” Trilling recalled, “was across a barrier which had about it something of a barricade.” Meanwhile, in Barzun’s eyes, Trilling seemed “content to do well, with little exertion, in what he liked and to stumble through the rest.” Upon learning they would be paired up, neither one jumped at the prospect.