PHILOSOPHY 21: HISTORY OF MODERN EUROPEAN PHILOSOPHY

PHILOSOPHY 21: HISTORY OF MODERN EUROPEAN PHILOSOPHY

FALL 2005

HANDOUT #3

MEDIEVAL “NATURAL PHILOSOPHY”

Natural Philosophy

The medieval study of natural philosophy took the form of commentaries on Aristotle's Physics. As we have seen, Arabic medieval philosophers like Avicenna noticed conflicts between Aristotelian philosophy of nature and doctrines of revealed religion – doctrines Islam, Judaism, and Christianity shared. Some of the conflicts over nature were:

- “The Eternity of the World” Whether the universe had existed for all eternity or had been created some finite amount of time before.

- In what way are the laws of nature necessary? Can God change them, or override them in particular instances (miracles)? Or does necessary mean nothing more than is willed by God?

One issue that is particularly relevant to early modern philosophy arose over the proper interpretation of Aristotle's accounts of

- projectile motion (Aristotle's 'violent' motion)

and

- the motion of a freely-falling body (Aristotle's 'natural' motion).^{^[1]}

As we saw in Handout #1, Aristotle's accounts of projectile motion and the motion of freely-falling bodies are special cases of his 'proportional' law of motion:

The velocity of a body is a function of the ratio between impelling force and resisting force:

V = I/M

where I is a measure of the ‘impelling force’ of the moving body, and M a measure of the resistance (or "density") of the medium.

There are two problems with Aristotle’s law of motion:

1. If the resistance of the medium is zero (a ‘void’), then 'velocity' is undefined (the “divide by zero” problem).

2. Aristotle’s law treats velocity as

the rate at which a body traverses a given distance.

Thus, we cannot use it to distinguish between a body's

- traversing a 100 meter course in 10 seconds at a constant velocity of 10 meters per second;

and

- accelerating from a state of rest at the beginning of the course and covering the 100 meter course in 10 seconds

Thus, it can give us no concept of 'velocity-at-a-time'.

The Law of Motion for Freely-Falling Bodies

Aristotle maintained that bodies have a "natural" motion, a direction in which they tend in accordance with their essence to go. A freely-falling body is one that is moving in its natural direction. It moves in that direction at a rate proportional to

(i) the motive power furnished by the body's natural tendency to move in that direction;

and

(ii) the power of the resistance to that motion furnished by the medium.

Aristotle's Law of Freely Falling Bodies:

The velocity of a freely-falling body is proportional to the motive power furnished by the body (the impelling force) and the resistive power of the medium.

Now, bodies of the same size can differ in their 'impelling force' depending on their essences. One body could have, say, a greater downward force than another body of the same size -- more 'weight'. To understand Aristotle’s physical intuition here, imagine two ball bearings, one steel, the other aluminum, dropping through motor oil. Since the proportion of the density of the steel to the density of the oil is greater, the steel ball will drop faster than the aluminum ball.

So, according to Aristotle’s law, then, if A is denser than B, A should fall faster than B.

Problem: This isn't true! All freely-falling bodies fall at the same rate. (To be fair, this is difficult to see experimentally: Drop a styrofoam cup and a lead weight from the roof and see which one hits the ground first.)

The Law of Projectile Motion

The application of Aristotle's general Proportional Law to the case of projectile motion (a baseball thrown into the air for example) takes the impelling force to be the ‘violence’ with which the stone is moved out of its natural downward movement. In this case the stone's natural downward movement – its impelling force, or weight – is the 'resistive medium'.

As we saw, Aristotle assumed that if a body is moving in a direction contrary to its natural motion, force must be continuously imparted on it in order to have that effect. And since a void is impossible, that means it must have some body in contact with it pushing it along.

Problem: As we saw, Aristotle had no very good explanation for the fact that the baseball clearly continues to rise for a good while after it leaves the centerfielder's hand. What causes it to keep rising?

Bradwardine's Criticism of the General Proportional Law

Thomas Bradwardine, a Franciscan on the Arts Faculty of Merton College, Oxford University, until 1335 (later Archbishop of Canterbury), saw that Aristotle's General Law made two assumptions:

1. Motion occurs only if the motive power is greater than the resistance.

2. Velocity is a function of the ratio of motive power to resistance.^{^[2]}

Bradwardine showed that if Aristotle intends the function in (2) to be simple proportionality, then these assumptions are inconsistent. If the velocity is determined by the simple ratio of motive power over resistance in (2), then (1) must be false.

Proof: A velocity divided is still a velocity. Something moving at “one-third speed” is still moving at a speed. Suppose we have a velocity produced by some ratio of I to M of, say, 3/1. If we divide that velocity by a number equal to the ratio – 3 in this case – we should still have a certain (lower) velocity, however low the original velocity was.

That is, the thing should still be moving, only slower.

But (3/1)/3 = 1.

Dividing the resulting velocity by one-third produces a situation in which I (the impelling power) and M (the resistive medium) are identical in value.

So there should be no motion at all.

But that contradicts assumption (1).

Bradwardine concluded that the function Aristotle had in mind must be something other than simple proportionality. He suggests that if a given velocity arises from a certain ratio I/M greater than 1/1, twice that velocity will be produced by a ratio (I/M)², three times the velocity by a ratio (I/M)³, one half the velocity by a ratio (I/M)^1/2....

Thus, if a certain velocity V arises from the ratio 3/1, then 2V arises not from the ratio 6/1, but from (3/1)² . The impelling force must be nine times as powerful as that of the resistive medium, not six times, to produce a speed twice as fast as the original speed. The velocity 3V arises from (3/1)³; the impelling force must be twenty-seven times as powerful as the resistive medium. Or to reduce the speed by half, ½V, you must have an impelling force of (3/1)^1/2, Ö3, or 1.732 times the force of the resistive medium.

Although this is not the correct law of motion, it does introduce two new ideas of great (and overlooked) historical significance:^{^[3]}

1. An exponential function is used in a law of nature for the first time in history.

2. The law is stated for 'instantaneous' velocities. It's not just intended to express the proportion of total distance traversed in the whole time of the motions compared.

Bradwardine maintained that two bodies might traverse equal distances in an equal time even though they are not moved at the same velocity during any finite part of the time.

Thomas of Hentisbury, a student of Bradwardine’s at Merton, asked this question:

If one body is uniformly accelerated from rest to a certain terminal velocity in a certain time, at what velocity would another body have to move if it were moved at a constant velocity for that same length of time?

Hentisbury's answer:

The moving body which is accelerated uniformly during some assigned period of time will traverse a distance exactly equal to what it would traverse in an equal period of time if it were moved uniformly at its mean degree of velocity.^{^[4]}

Which is the right answer. Where the acceleration is from rest, the mean degree will be one half the terminal velocity:

S_t = 1/2V × t

Hentisbury’s answer thus gives us, not just the distance S, but the “distance covered during some assigned period of time” – S_t. That is, Hentisbury’s formulation permits (in fact, it requires) us to think of the velocity as 'velocity at a time' -- that is, as instantaneous velocity.

Buridan's "Impetus" Theory

Jean Buridan, a Franciscan on the Arts Faculty at Paris from 1328 to 1358, offered an explanation of projectile motion that eliminated Aristotle's need for a continually impressed 'impelling force'.

The mover...gives to [the projectile] a certain impetus by which that projectile keeps moving. The more rapidly the mover moved it, the greater the impetus....The impetus is constantly diminished by air and the 'gravity' of the stone, and finally decreased to the point where the gravity of the stone exceeds it and causes the stone to fall downward.

....(T)he reception of all natural forms and conditions of matter is determined according to the matter. The greater the quantity of matter, the greater the impetus that can be received.^{^[5]}

Steel has a greater 'quantity of matter' per given volume than wood. That's why steel can receive more impetus; and so we can throw a piece of steel further than we can a piece of wood.

Buridan’s impetus is a new factor not found in Aristotle's theory. First of all, unlike Aristotle's 'impelling force', which can not only be 'corrupted' and disappear over time but requires continued action by the 'impressing' agent in order to continue, impetus is an enduring reality, a property of the matter itself. This property remains constant except as forces operate to increase or decrease it. But, Buridan says,

The impetus would endure for an infinite time, if were not diminished...by an opposed resistance.^{^[6]}

A moment's reflection shows that

Buridan's 'quantity of matter' functions in the definition of impetus exactly as 'mass' does in Newton's definition of 'momentum'.

Since impetus is defined by the product of mass and velocity, it is equivalent to 'momentum' in Newtonian mechanics. Indeed, the passage just quoted suggests Newton's first law of motion:

I. Every body continues in its state of...uniform motion...unless it is compelled to change that state by forces...upon it.

But Buridan went even further. He introduced the concept of impetus into the analysis of the uniformly accelerated movement of freely falling bodies: the force of gravity is defined as that which continuously increases the impetus of the body on which it acts. Thus, the effect of gravity is to change the momentum of a body continuously with time.

So Buridan is very close to the modern conception of 'force', as measured by the rate of acceleration of a body and by its mass -- or by the rate at which it changes the momentum of a body. Just as Newton's second law

II. The change of motion is proportional to the...force.

follows from his first, so Buridan's conception of gravity as a force whose effect is to change the momentum of the body on which it acts follows as a consequence of his definition of impetus as an enduring condition measured by velocity and quantity of matter.

To illustrate how advanced this 14th century physical speculation was, compare these medieval contributions with Galileo's early dynamics two hundred fifty years later.

Galileo's "Pisan Dynamics"

In his "Pisan Period" (1589) Galileo studied the movements of freely falling bodies (originating the legend that he dropped weights off the Leaning Tower of Pisa.) Galileo's assumptions during this period were:

1. Gravity is the essential and universal physical property of material bodies.

2. Space is empty -- a void -- weightless and immaterial. Nevertheless, it is real and endowed with mathematical properties: an extended emptiness filled with or occupied by bodies having weight.

3. The world has a center which determines absolute position for bodies in space and which determines the direction of their 'natural' motion, arising from their intrinsic gravity.

Heavier objects are more dense: they have more matter in a given volume of space. (Sound familiar?)

Galileo avoided the problems that arise when one combines belief in a void -- Galileo's assumption (2) -- with a proportional law of motion like Aristotle's. Galileo gave a law of motion that was not proportional but arithmetical:

V = I - M

Thus movement through a void becomes coherent: A body moving through a void (where the value for resistance M = 0) will move at a velocity determined only by the motive power (I) impressed upon it.

Applying this arithmetical law to a freely-falling body, however, will have the result that a body will fall through 'space' at a velocity that is determined entirely by the motive power furnished by its density -- its 'intrinsic gravity'. This has one inconvenient consequence: If the velocity of bodies in a void is determined entirely by their intrinsic gravity, then a denser body should fall faster, thus agreeing with Aristotle's false prediction.

Galileo's concept of 'intrinsic gravity' is thus no advance over Buridan's impetus. Galileo's Pisan law of motion is not even as advanced as Bradwardine's.

Conclusion

The 14th century precursors of the scientific revolution of the 17th century had a different conception of what they were doing. Bradwardine, Hentisbury, and Buridan did not think of themselves as uncovering fundamentally new knowledge; they thought of themselves as interpreting Aristotle, or at most as deriving explicitly conclusions that Aristotle had left implicit.

Again, their assumption was that the Ancients had it mostly right; all that was needed was clarification of obscurities within the texts, which continued to function as their intellectual authorities.

It was only when later philosophers came to realize that the Aristotle and the other Ancients didn't know everything that they could see themselves as trying to develop new knowledge.

But how? What was the proper way to find out what was true if you couldn't just look it up in a book? A new way of finding things out was needed.

NOTES