Jean Buridan

How to use this course

Jean Buridan is known to most people for a donkey he never wrote about. Historians of science know him for impetus, the quantity that a thrower gives a stone and that keeps it moving after it leaves the hand. Philosophers know him as one of the sharpest logicians of the Middle Ages. All three belong to a single career: forty years of teaching in the arts faculty of the University of Paris, in the form of commentaries and disputed questions on Aristotle.

Each technical lesson opens with a paragraph for the general reader, then states Buridan’s positions as closely as the texts allow and compares them with modern physics or logic. Buridan wrote in Latin prose, without equations. When we write formulas, they are our models of his claims, and we say so. The documented facts of his life are few: university records, papal letters, and remarks in his own works. Later stories about him are marked as legends.

Timeline

Picardy and Paris, c. 1300–1328

Almost nothing is known of Buridan’s family or childhood. He came from the diocese of Arras in Picardy, in northern France; a papal letter of about 1330 describes him as a cleric of that diocese. The town of Béthune is the traditional birthplace, but the tradition is late and some historians doubt it. His birth is usually placed around 1300.

The student

He came to Paris as a poor student. He held a bursary at the Collège du Cardinal Lemoine, one of the endowed houses that fed and lodged students of limited means. Some later accounts also associate him with the College of Navarre. The arts course of the University of Paris took six or seven years: grammar, and above all logic, followed by Aristotle’s natural philosophy, metaphysics and ethics. Teaching proceeded by reading a text with commentary and by disputation, in which a question was posed, arguments on each side were rehearsed, and the master determined the answer.

Buridan became a master of arts and received the licence to teach by the mid-1320s. Most masters of arts used the post as a stepping stone to the higher faculties of theology, law or medicine. Buridan never did. He stayed in the arts faculty for his whole career, a choice that was unusual for a man of his reputation and that shaped what he wrote: questions on Aristotle and textbooks of logic, rather than commentaries on theology.

Rector at about twenty-eight

A document dated 2 February 1328 names him rector of the university. The rector was elected for a short term by the arts faculty, which was organized into four “nations” (French, Picard, Norman and English-German). The office was administrative and ceremonial, but election to it marks him as a respected member of the faculty within a few years of his mastership.

A master of arts for life

Buridan was a secular cleric, not a member of a religious order. Like most masters he lived on church benefices, incomes attached to ecclesiastical offices that papal letters granted him. Records from 1329 to 1348 describe him as a celebrated philosopher who lectured on natural, metaphysical and moral philosophy. In 1349 he was listed among the masters able to support themselves without the university’s help, a sign of material security.

The question format

Almost all of Buridan’s philosophy survives as quaestiones. Each begins with a yes-or-no question (“Whether a projectile, after leaving the hand of the thrower, is moved by the air”), lists arguments for the answer he will reject, states the authority on the other side, and then gives his own determination and replies. He lectured on the same Aristotelian texts several times, and some works exist in more than one version. The last version of his questions on the Physics (the ultima lectura) and the questions on On the Heavens contain the physics of Lessons 04 to 08.

Travel and observation

Two passages in his works suggest that he visited the papal court at Avignon before 1334, and that he climbed Mont Ventoux in Provence to observe the clouds and the weather. Buridan’s natural philosophy is full of ordinary experiences of this kind: spinning tops, smiths’ grindstones, ships, bellows, bells. He used them as evidence, while never designing experiments in the modern sense.

Nominalism, Ockham and the statute of 1340

Buridan was a nominalist: he held that universal terms such as “man” or “whiteness” are signs in the mind and in language, and that nothing outside the mind is universal. Everything that exists is an individual. Gyula Klima has called his position “essentialist nominalism”, because Buridan still held that our concepts of substances capture real natures. He used nominalism mostly as a method of parsimony, refusing to multiply entities when an analysis of terms would do.

Ockham

The Oxford Franciscan William of Ockham had developed a nominalist logic in his Summa logicae in the 1320s, and his views divided the Paris arts faculty in the late 1330s. Buridan knew the Summa logicae but never names Ockham, and his logic owes little directly to it. In 1340, during his second term as rector, Buridan signed an arts faculty statute that censured masters who interpreted authoritative texts by their literal sense against the evident intention of their authors. The statute warned of errors harmful to both philosophy and Scripture. One article targeted a thesis of Nicholas of Autrecourt.

Hypothetical certainty

Nicholas of Autrecourt argued that from the existence of one thing we can never infer with certainty the existence of another, so that causal knowledge of nature is impossible. Buridan replied that the sciences of nature do not need absolute certainty. Their principles hold “on the supposition of the common course of nature”: God could interrupt that course by a miracle, but such cases do not concern natural philosophy. A principle of physics is accepted when it has been confirmed in many cases and found false in none. This view gave natural philosophy a standing of its own, separate from theology and from the mere exposition of Aristotle. It also let Buridan treat theories as hypotheses to be judged by how well they save the appearances, which is how he handles the rotation of the Earth in Lesson 08.

What keeps a projectile moving?

For the general reader. Aristotle taught that a body moves only while something pushes it. A thrown stone was a problem: after it leaves the hand, what pushes it? Aristotle said the air. Buridan collected everyday experiences showing that the air cannot be doing the work, and so cleared the ground for a new idea.

The problem

In Aristotle’s physics, a motion that is not natural requires a mover in contact with the moved body. In Physics VIII.10 Aristotle concluded that the thrower moves the air along with the stone, and gives the air a power of moving that passes from layer to layer and weakens until the stone falls. He also reported, without accepting it, the theory of antiperistasis (mutual replacement): air displaced in front of the stone flows round and pushes it from behind. Buridan opens question 12 of book VIII of his last questions on the Physics by asking which, if either, is right.

The experiences

Buridan concludes that the air resists projectiles and does not carry them, and that something inside the projectile must keep it moving. The next lesson gives his answer.

A modern assessment

The arguments are good. The top and grindstone, which rotate in place, are the strongest, since no theory of displaced air can explain them. The ship argument is correct in its observation: the apparent wind on deck blows from the bow. The feather argument rests on a real effect that Buridan explained with his own theory (Lesson 05). Modern fluid dynamics agrees that the air exerts a drag force opposite to the velocity of a projectile. A wake does form behind a moving body, but the pressure there is lower, not higher, than in front.

Impetus

For the general reader. Buridan’s answer was that the thrower puts something into the stone, a motive power he called impetus. It keeps the stone going until air resistance and the stone’s weight wear it down. More matter and more speed mean more impetus. This sounds close to the modern idea of momentum, and in several ways it is, though Buridan thought of impetus as a cause of motion rather than as a measure of it.

The definition

Combining the two theses, impetus increases with the quantity of matter and with the speed. Buridan gives no formula, and he gives no unit of impetus. Historians often write \(I\propto mv\), with \(m\) the quantity of matter, as a modern summary of what he says. Ernest Moody judged that impetus so defined corresponds to momentum.

A permanent quality

The key difference from earlier theories lies in what happens when nothing opposes the motion. The Alexandrian John Philoponus in the sixth century, and Franciscus de Marchia in Paris around 1320, had proposed an impressed power that spends itself and dies away of its own accord. Ibn Sina had held that the violent inclination (mayl) is not used up by itself. Buridan agreed with Ibn Sina on this point. In the related questions he argues that impetus is not the motion itself, since it produces the motion, and that it is a lasting quality of the body, “of a permanent nature”, comparable to the quality a magnet induces in iron. Moody quotes him saying that impetus would last indefinitely if it were not diminished by an opposed resistance or by a tendency to a contrary motion. How directly Buridan drew on Ibn Sina is debated.

Impetus and momentum

Model Buridan’s claims by a quantity \(I=mv\) that obeys

where \(R\) is the resistance of the medium and \(G\) the opposing component of the body’s gravity. With \(R=G=0\), \(I\) is constant forever. This is formally Newton’s second law with the impetus playing the role of momentum. Two differences remain. For Buridan, impetus is a cause that the motion requires at every moment, a power in the body; for Newton, momentum needs no cause to persist. And Buridan kept the Aristotelian rule that in a resisting medium the speed is set by the ratio of motive power to resistance, which sits uneasily with his own theory.

Buridan’s other examples

• The long jump. A jumper runs before jumping in order to acquire impetus, and in flight feels the air resisting from in front.
• The large grindstone. A large smith’s wheel is harder to stop than a small one, because it has more matter and so more impetus.
• Rebound. A ball dropped on hard ground is compressed and springs back up with new impetus; a bent bow, a plucked string and a swinging bell continue their motion in the same way.

Falling bodies and accidental gravity

For the general reader. A falling stone speeds up. Why? Aristotle had little to say. Buridan’s answer was that weight keeps adding impetus: in the first moment gravity alone moves the stone, slowly, and gives it some impetus; in the next moment gravity and impetus together move it faster, and add more impetus; and so on. This is the right qualitative picture of uniform acceleration, three centuries before Galileo stated the law.

Buridan’s account

In modern terms: if gravity adds equal increments of impetus in equal times, then \(dI/dt=W\), the weight, and \(v=gt\). Buridan did not say clearly whether the increments come in equal times or equal distances. Moody concluded that he probably treated the two as equivalent. They are not.

Time or distance?

Proof. Suppose \(\dot x=kx\) with \(x(0)=0\). The function \(x\equiv0\) is a solution, and since the right side is Lipschitz, it is the only one. A body starting from rest never moves. With \(\dot x=gt\) instead, integrating gives \(x=\tfrac12gt^2\), and eliminating \(t\) gives \(v^2=2gx\). Galileo first assumed the distance law, in a letter to Paolo Sarpi in 1604, and later replaced it with the time law in the Two New Sciences (1638).

The mean speed theorem

Buridan’s Paris contemporaries and the Oxford Calculators at Merton College studied uniformly “difform” motion, that is, motion with uniformly increasing speed, as an exercise in the mathematics of intensities. Nicole Oresme gave a geometric proof by representing speed as the height of a figure over a base line of time.

In Oresme’s figure the distance is the area under the speed line. The trapezoid with parallel sides \(v_0,v_1\) and width \(T\) has the same area as the rectangle of height \(\tfrac12(v_0+v_1)\). Neither Buridan nor Oresme applied the theorem to free fall; Galileo did, and with it derived the law of fall.

Impetus in the heavens

For the general reader. Medieval philosophers taught that the heavenly spheres are moved by intelligences, spiritual beings who keep them turning. Buridan suggested something simpler: God gave each sphere an impetus at the creation, and since nothing in the heavens resists, the spheres have kept turning ever since. This brought the heavens and the Earth under one mechanics.

Buridan offers this tentatively and says he would like to learn from the theologians how these things happen. It is a hypothesis about how the heavens might work, offered on the principle of economy.

Circular impetus

Buridan allowed impetus in any direction, including circular. His grindstone and his top are bodies that keep a rotational impetus. In modern terms, the quantity that persists in a freely rotating rigid body is the angular momentum \(L=I\omega\), where \(I=\sum m_ir_i^2\) is the moment of inertia. A torque \(\tau\) changes it according to \(dL/dt=\tau\). Buridan’s claim that a frictionless sphere keeps turning is the case \(\tau=0\). His remark that a large wheel is harder to stop than a small one is the dependence of \(I\) on mass and radius.

Newtonian mechanics differs in one important respect. Linear and angular momentum are separate conserved quantities: a body moving in a circle in a plane needs a force toward the centre, and without it the motion becomes straight. Buridan’s circular impetus in the spheres works because each sphere is a rigid body rotating about its axis, where the inner cohesion supplies the centripetal forces. For a single planet moving freely, circular impetus would not be enough.

Does the Earth turn?

For the general reader. Is the sky turning around us once a day, or is the Earth turning under the sky? Buridan asked this question two centuries before Copernicus. He agreed that the turning Earth explains the appearances equally well, and is simpler. But he chose a resting Earth, because an arrow shot straight up falls back where it started. The argument fails, and the reason it fails is a consequence of impetus itself.

The question

In question 22 of book II of his questions on On the Heavens, Buridan asks whether the Earth always rests at the centre of the world. One of his doubts is whether a daily rotation of the Earth from west to east, with the sphere of fixed stars at rest, could save all the appearances. He reports that some people of his time considered it probable.

Buridan also reports replies to physical objections: the air turns with the Earth, so we feel no wind; heating by motion comes from friction, and there is none if earth, water and air move together. He grants that the hypothesis saves the astronomical appearances.

He concludes that the Earth rests at the centre, while granting that the hypothesis cannot be refuted by astronomy alone. He adds a geological refinement: since rivers carry earth from the land into the sea, the Earth’s centre of gravity, not its geometric centre, is the one that lies at the centre of the world, and the Earth shifts slightly as material is moved.

The answer from impetus

The flaw lies in the premise that the arrow has no eastward motion. Before it is released, the arrow moves east with the ground. By Buridan’s own principle, a body keeps the impetus it has unless something opposes it. So the arrow keeps its eastward motion during the flight, and the ground and arrow move east together. Nicole Oresme in his Livre du ciel et du monde (1377) answered in this spirit that the arrow shares the motion of the Earth, and concluded that neither rotation nor rest can be demonstrated. He nonetheless accepted a resting Earth, mainly because Scripture seems to assume it. Albert of Saxony took the opposite line, arguing that the arrow’s lateral impetus would be too weak to carry it so far. Galileo made the shared-motion argument the basis of his relativity principle in the Dialogue of 1632.

The real effect

Derivation. In the rotating frame the acceleration is \(\mathbf a=\mathbf g-2\boldsymbol\Omega\times\mathbf v\) (neglecting the small centrifugal correction, absorbed into \(\mathbf g\)). Take \(x\) east and \(z\) up. To first order in \(\Omega\), the vertical motion is \(v_z=v_0-gt\), and the east component obeys \(\ddot x=-2\Omega\cos\lambda\,v_z\). Integrating twice, \(x(t)=-\Omega\cos\lambda\,(v_0t^2-\tfrac13gt^3)\). At landing \(t=2v_0/g\), so \(x=-\Omega\cos\lambda\,\big(\tfrac{4v_0^3}{g^2}-\tfrac{8v_0^3}{3g^2}\big)=-\tfrac43\Omega\cos\lambda\,v_0^3/g^2\).

Logic: supposition and consequence

For the general reader. Buridan’s most widely used book was a logic textbook, the Summulae de dialectica. It began as a commentary on the standard textbook of Peter of Spain and became an independent work, read in universities from Paris to Prague and Vienna. Two of its central ideas are supposition, a theory of what the words in a sentence stand for, and consequence, a theory of when one sentence follows from another. Together they amount to a theory of quantifiers and a theory of valid inference.

Supposition

A term in a proposition supposits for the things it stands for there. Buridan recognized two main kinds. In material supposition a term stands for itself or for another word, as in “‘man’ is a noun”. In personal supposition it stands for what it signifies, as in “a man runs”. He rejected the “simple” supposition for universals that realists used. Common terms in personal supposition are divided further according to how one may descend to singulars.

In modern notation the three modes correspond to an existential quantifier with wide scope, a universal quantifier, and an existential quantifier in the scope of a universal: \(\forall x\,(Mx\to\exists y\,(Ay\wedge x=y))\). Supposition theory was a way of handling quantifier scope in natural language without variables.

Consequences

Buridan’s Treatise on Consequences sets out the theory of inference in an axiomatic style. A consequence is a conditional of the form “\(A\), therefore \(B\)”. The natural definition is that a consequence is valid when it is impossible for the antecedent to be true while the consequent is false. Buridan saw that this fails, because for him propositions are sentence-tokens that exist only when someone forms them.

Among his theorems: from a true antecedent a false consequent never follows; from the impossible anything follows; the necessary follows from anything; if \(A\) implies \(B\) and \(B\) implies \(C\), then \(A\) implies \(C\). Material consequences are valid “as enthymemes”: “a man runs, so an animal runs” becomes formal when one adds the premise “every man is an animal”.

Self-reference and the Liar

For the general reader. “This sentence is false.” If it is true, it is false; if it is false, it is true. Medieval logicians called such sentences insolubilia, insolubles, and wrote treatises on them. Buridan’s Sophismata, the last treatise of his Summulae, contains a whole chapter of self-referential puzzles and one of the most influential medieval solutions.

Virtual implication

Earlier, Buridan had held that every proposition signifies or asserts its own truth. In the Sophismata he rejected this and adopted a weaker principle.

Apply this to a Liar sentence \(L\): “\(L\) is false.” It signifies that \(L\) is false, and it virtually implies “\(L\) is true”. Both cannot hold together, so \(L\) is not true; hence \(L\) is false. That things are as \(L\) signifies (\(L\) is false) does not make it true, because the second condition fails. The paradox is blocked: \(L\) is simply false.

The solution is close in spirit to modern treatments that refuse to let a sentence’s truth be fixed by its content alone. It was influential through Albert of Saxony and later Paris and Central European logicians, and it remains a subject of research, notably by Stephen Read and Gyula Klima.

Buridan’s bridge

Among the sophisms is a puzzle often called Buridan’s bridge. Plato guards a bridge and tells Socrates: if the first proposition you utter is true, I will let you cross; if it is false, I will throw you into the water. Socrates replies: “You will throw me into the water.” Buridan used the case to examine what a conditional promise obliges and when it can be kept.

Last years and legends

Buridan kept teaching through the Black Death of 1348–49 and into the 1350s. The last record of him alive dates from 1358, when he served as adjudicator in a dispute over territory between the English and Picard nations of the university. In 1361 one of his benefices was conferred on someone else, which suggests that he had died. Neither the date nor the cause of his death is known.

The legends

• The sack and the Seine. A story told after his death says that a king of France had him tied in a sack and thrown into the Seine for an affair with the queen. François Villon alludes to it in his Ballade des dames du temps jadis. There is no evidence for it.
• Vienna. Another story says he was expelled from Paris for his nominalism and founded the University of Vienna. It was his follower Albert of Saxony who became Vienna’s first rector in 1365.
• The shoe. A third says he struck the future Pope Clement VI with a shoe in a quarrel over a woman. It too lacks support.
• Buridan’s ass. A donkey placed exactly between two equal bales of hay starves because it has no reason to choose. The example does not occur in Buridan’s surviving works, and Pierre Duhem found no trace of it there. Aristotle had used a similar case, a man equally hungry and thirsty between food and drink (On the Heavens II.13, 295b32). The ass may have begun as a parody of Buridan’s theory of the will, in which the will can suspend its choice between alternatives that the intellect judges equal.

The age of Buridan

Paris and the new universities

Buridan’s questions and his Summulae were copied and taught across Europe. Nicole Oresme, often described as his pupil, developed the geometry of qualities, the graphical proof of the mean speed theorem and the most careful medieval discussion of the Earth’s rotation. Albert of Saxony carried impetus and Buridan’s logic to Vienna, and Marsilius of Inghen, another Paris master in the Buridanian tradition, became the first rector of the University of Heidelberg in 1386. Buridan’s works were taught at Prague, Cracow, Vienna, Heidelberg, Cologne and in northern Italy. A Cologne document of 1425 refers to the previous century as the age of Buridan, and in 1516 a Paris master could still say that Buridan ruled in physics there.

The ban of 1474

In March 1474 King Louis XI issued an ordinance against the nominalists at Paris that forbade the teaching of Ockham, Buridan, Albert of Saxony, Marsilius of Inghen and others; some of their books were chained in libraries. The ban was lifted in 1481.

From Paris to Galileo

When Copernicus studied at Cracow in the 1490s, Buridan’s physics was part of the curriculum. Impetus theory was still taught in Italy in the sixteenth century, often in the mixed form that combined Buridan’s permanent impetus with the self-spending impressed force of Franciscus de Marchia. Galileo’s early De motu (c. 1590) uses such a self-spending impressed force; in his mature work he dropped the self-decay and treated uniform motion as something that persists without a cause. How much Galileo owed to the Paris school is still debated.

Rediscovery

Pierre Duhem, physicist and historian, found Buridan’s impetus theory in manuscripts while working on Leonardo da Vinci, and in his Études sur Léonard de Vinci (1906–1913) presented the Paris masters as the founders of modern mechanics. Anneliese Maier, Marshall Clagett and Ernest Moody later corrected and refined the picture, showing both the real anticipations and the deep differences. Modern philosophers have recovered Buridan as a logician of the first rank; his Summulae appeared complete in English, translated by Gyula Klima, in 2001.

Connections

People linked to Jean Buridan across this collection. A name without a link has no page here yet. See Jean Buridan in the network of all 337 lives →

Students

• Albert of Saxony
• Nicole Oresme

Influenced by

• Aristotle
• Ibn Sina
• John Philoponus
• Franciscus de Marchia
• Peter of Spain
• William of Ockham

Influenced

• Galileo Galilei
• Nicolaus Copernicus
• Marsilius of Inghen
• Pierre Duhem

Rivals

• Nicholas of Autrecourt

Also mentioned in this life

• Gaspard-Gustave Coriolis
• Isaac Newton
• Emmy Noether
• Plato
• Bernhard Riemann

Fields

• natural philosophy 12 lives
• mechanics 27 lives
• logic 8 lives
• cosmology 7 lives

On CoursesHub

• Classical Mechanics
• Cosmology

Sources and further reading

Biography

1. J. Zupko, John Buridan, Stanford Encyclopedia of Philosophy.
2. E. A. Moody, Buridan, Jean, Dictionary of Scientific Biography, vol. 2, Scribner, 1970.
3. J. Zupko, John Buridan: Portrait of a Fourteenth-Century Arts Master, University of Notre Dame Press, 2003.
4. G. Klima, John Buridan, Oxford University Press, 2009.
5. Jean Buridan, Wikipedia.

Physics

1. M. Clagett, The Science of Mechanics in the Middle Ages, University of Wisconsin Press, 1959 (translations of Buridan on impetus).
2. John Buridan, Questions on the Eight Books of the Physics, VIII, q. 12, translated in G. Klima et al., Medieval Philosophy: Essential Readings, Blackwell, 2007.
3. John Buridan, Questions on De caelo II, q. 22, translation from H. Shapiro (ed.), Medieval Philosophy, Modern Library, 1964.
4. E. A. Moody (ed.), Iohannis Buridani Quaestiones super libris quattuor De caelo et mundo, Medieval Academy of America, 1942.
5. Theory of impetus, Wikipedia.
6. A. Maier, On the Threshold of Exact Science, ed. and trans. S. D. Sargent, University of Pennsylvania Press, 1982.
7. P. Duhem, Études sur Léonard de Vinci, 3 vols., Hermann, 1906–1913.
8. N. Oresme, Le livre du ciel et du monde, ed. A. D. Menut and A. J. Denomy, University of Wisconsin Press, 1968.

Logic

1. John Buridan, Summulae de Dialectica, trans. G. Klima, Yale University Press, 2001.
2. John Buridan, Treatise on Consequences, trans. S. Read, Fordham University Press, 2015.
3. G. E. Hughes, John Buridan on Self-Reference: Chapter Eight of Buridan’s Sophismata, Cambridge University Press, 1982.
4. P. V. Spade and S. Read, Insolubles, Stanford Encyclopedia of Philosophy.