Philosophy 226f: Philosophy of Science

Prof. Robert DiSalle (rdisalle@uwo.ca)Talbot College 408, 519-661-2111 x85763

Course Website: http://instruct.uwo.ca/philosophy/226f/

The nature of scientific explanation:

What is it that we understand when we have a scientific account of some phenomenon?

Does scientific explanation have a different character from explanation in other forms of thought, or in common-sense human reasoning?

Is the scientific way of explaining privileged in some way?

Or is it too severely restricted to give us the kind of deeper understanding that we expect in certain other ways of thinking?

A Kuhnian question: Did the adoption of the Copernican-Galilean viewpoint involve abandoning a degree of understanding for the sake of greater predictive power?

Did Galileo’s work reflect the replacement of “understanding why” with merely “describing how” as the ultimate aim of science?

René Descartes, 1638: Galileo only described particular cases of motion, without inquiring into their “first causes.”

In general, why do bodies in motion tend to persist in motion?

Aristotle: As a matter of fact, they don’t tend to persist; the little persistence that they do manage can be explained by the medium.

Galileo: Assume that they do persist in motion; then explain what forces are cause them to speed up, slow down, or change direction.

What keeps the planets in motion?

Aristotle: Planets are carried by spheres, each of which executes the rotational motion that is naturally suited to its spherical form.

Galileo: Suppose that the planets naturally persist in their rectilinear motion. Explain their curved orbits by the forces that cause them to deviate from the straight path.

A “deductive-nomological” theory of scientific explanation: (cf. Rudolph Carnap Carl Hempel, 1950's and 1960's).

Deductive: Implies that explanation is a kind of deduction from given premises.

Nomological: Implies that the given premises are universal or statistical laws of nature.

The point: that to explain an event, ideally, is to provide what suffices to deduce logically that the event must have occurred.

Explanation is symmetric with prediction.

Why did event E occur (the “explanandum”)?

Because it is logically deducible from the “explanans”:

Some set of laws of nature L1, L2,...LN , and a set of initial conditions C1, C2,…CN,

If laws of nature L1, L2,...LN hold And initial conditions C1, C2,...CN obtainThen event E necessarily follows.

If L1, L2,...LN are merely statistical laws rather than universal laws, then given L1, L2,...LN and C1, C2,...CN , event E follows with probability P.

The philosophical idea behind the D-N model: To give a conceptual analysis of what we mean by explanation, and to show that our commonsense idea of explanation is implicitly based on the D-N structure.

Explanations that don’t (at least implicitly) have these ingredients turn out to be pseudo-explanations.

Explanations that claim to provide more than what the D-N model provides are delusions.

Kepler’s early model of the solar system

Kepler’s model: the inner planets

Empirical laws of planetary motion discovered by Kepler

1. Planets orbit the sun in ellipses, with the sun at their common focus. This accounts for the inability of previous astronomers to find a circle, or combination of circles, that accounts for their irregular motions

2. A radius drawn from the sun to a planet sweeps out equal areas in equal times. This rule governs the way in which planets speed up as they approach the sun and slow down as they recede from it.

3. The square of the time it takes a planet to complete one orbit (the period t) is proportional to the cube of its average distance from the sun (the mean radius r). So: t2 r3 for each planet.

Kepler’s ellipse law: Planets orbit the sun in ellipses with the sun at their common focus.

Kepler’s area law: The radius drawn from the sun to a planet sweeps out equal areas in equal times.

Kepler’s “harmonic law”: The periodic time t and the mean radius r of any planetary orbit are related as t2 r3.

Or, t r3/2

Or, for any two planets a and b,

Ta2 / Tb

2 = Ra3 / Rb

3

Kepler’s physical astronomy

Newton’s laws of motion (The Mathematical Principles of Natural Philosophy, 1687)

Law 1. Every body, left to itself, maintains its state of uniform motion or rest until acted upon by a force.

Law 2. Acceleration is in the direction in which a forced is impressed, and is proportional to the magnitude of the force and the mass of the body.

Law 3. To every action there is an equal and opposite reaction.

Corollaries to Newton’s Laws:

Corollary 1. A body moved by two forces will follow the diagonal of the parallelogram formed by the motions that those bodies would separately produce.Corollary 2. Every motion can therefore be described as the composition of motions according to Cor. 1.Corollary 3. The total quantity of motion in a system of bodies is not changed by the mutual interactions of those bodies, because every action will be balanced by an equal and opposite reaction.Corollary 4. The center of gravity of a system of bodies is not changed by the mutual interactions of those bodies; therefore it will always remain at rest or moving uniformly in a straight line (unless the entire system is acted upon by some force originating outside the system).

Corollary 1 at work:

The projectile’s motion combines inertial motion with free fall according to the parallelogram of forces.

Corollaries 5 and 6: The Newtonian theory of relativity

Corollary 5. The motions of the bodies in a given space are the same among themselves whether that space is at rest or moving uniformly in a straight line (Newton’s precise formulation of the “Galilean” principle of relativity). I.e., because force is determined by acceleration, no experiment can measure the velocity of the system in which it takes place. Acceleration is “absolute,” but velocity is relative.

Corollary 6. The motions of the bodies in a given space are the same among themselves whether that space is at rest, moving uniformly in a straight line, or accelerated uniformly in parallel directions, by forces that act equally and in parallel directions on all the bodies in the system.

Examples of Corollary 6:

The system of Jupiter and its moons behaves (almost) as if it is at rest or moving uniformly in a straight line, because the attractive force of the sun acts (almost) equally on every part of the system.

In an orbiting spacecraft, bodies behave as if no forces were acting on any of them (as if they were “weightless”) because the attraction of the earth acts equally on all of them.

The system of Jupiter and its moons, considered as an isolated system

Descartes on the origin of centrifugal forces:

Descartes’ vortex theory of planetary motion:

The universe is completely filled with vortices, each surrounding a rotating star

Descartes: What motion is, taking the term in its common use.

But motion…in the ordinary sense of the term, is nothing more than the action by which a body passes from one place to another. And just as we have remarked above that the same thing may be said to change and not to change place at the same time, so also we may say that the same thing is at the same time moved and not moved. Thus, for example, a person seated in a vessel which is setting sail, thinks he is in motion if he look to the shore that he has left, and consider it as fixed; but not if he regard the ship itself, among the parts of which he preserves always the same situation. Moreover, because we are accustomed to suppose that there is no motion without action, and that in rest there is the cessation of action, the person thus seated is more properly said to be at rest than in motion, seeing he is not conscious of being in action.

Descartes: What motion is properly so called (motion “in the philosophical sense”)

But if, instead of occupying ourselves with that which has no foundation, unless in ordinary usage, we desire to know what ought to be understood by motion according to the truth of the thing, we may say, in order to give it a determinate nature, that it is THE TRANSPORTING OF ONE PART OF MATTER OR OF ONE BODY FROM THE VICINITY OF THOSE BODIES THAT ARE IN IMMEDIATE CONTACT WITH IT, OR WHICH WE REGARD AS AT REST, to the vicinity of other bodies. By a body as a part of matter, I understand all that which is transferred together, although it be perhaps composed of several parts, which in themselves have other motions….

Newton:

It is indeed a matter of great difficulty to discover, and effectually to distinguish, the true motion of particular bodies from the apparent; because the parts of that immovable space, in which those motions are performed, do by no means come under the observation of our senses. Yet the thing is not altogether desperate; for we have some arguments to guide us, partly from the apparent motions, which are the differences of the true motions; partly from the forces, which are the causes and effects of the true motion. For instance, if two globes, kept at a given distance one from the other by means of a cord that connects them, were revolved about their common center of gravity, we might, from the tension of the cord, discover the endeavor of the globes to recede from the axis of their motion, and from thence we might compute the quantity of their circular motions.

Newton’s bucket experiment: The Cartesian definition of motion vs. the dynamical measure of motion

The bucket and water are at rest: No motion in Descartes’ sense, and no dynamical effect

The bucket spins: Now the water moves in Descartes’ sense, but no dynamical effect

The water spins along with the bucket: No motion in Descartes’ sense, but an evident dynamical effect

The bucket stops and the water continues: The water moves in Descartes’ sense, with the same dynamical effect

Newton’s thought-experiment on rotation: Even if there is nothing else in the universe-- therefore no relative motion-- the rotation of these spheres about their common centre of gravity can be known from the tension on the cord joining them.

Newton’s argument for universal gravitation

Theoretical premises of the argument:

1. The laws of motion and their corollaries.

2. Propositions on centripetal forces, mathematically derived from the laws of motion

Empirical premises of the argument:

• That all the planets, with respect to the Sun, obey Kepler’s 2nd and 3rd laws

2. That the satellites of Jupiter, Saturn, and Earth obey Kepler’s 2nd and 3rd laws with respect to their central planets

Methodological premises: Rules of Reasoning in Philosophy

I. We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances.

II. Therefore to the same natural effects we must, as far as possible, assign the same causes.

III. The qualities of bodies, which admit neither intensification nor remission of degrees, and which are found to belong to all bodies within the reach of our experiments, are to be esteemed the universal qualities of all bodies whatsoever.

IV. In experimental philosophy we are to look upon propositions inferred by general induction from phenomena as accurately or very nearly true, notwithstanding any contrary hypotheses that may be imagined, till such time as other phenomena occur, by which they may either be made more accurate, or liable to exceptions. In experimental philosophy we are to look upon propositions inferred by general induction from phenomena as accurately or very nearly true, notwithstanding any contrary hypotheses that may be imagined, till such time as other phenomena occur, by which they may either be made more accurate, or liable to exceptions.

I. “The areas which revolving bodies describe by radii drawn to an immovable centre of force do lie in the same immovable planes, and are proportional to the times in which they are described.”

II. “Every body that moves in any curve line described in a plane, and by a radius, drawn to a point either immovable, or moving forward in a uniform rectilinear motion, describes about that point areas proportional to the times, is urged by a centripetal force directed to that point.” [This is the converse of Proposition I.]

III. Proposition II can be generalized to the case of a centre of force in any sort of motion.]

Propositions from Principia, book I:

IV. “The centripetal forces of bodies, which by equable motions describe different circles, tend to the centres of the same circles; and are one to the other as the squares of the arcs described in equal times applied to the radii of the circles.” [In other words, F 2/R, where F = centripetal force, = arc of the circle, and R = radius of the circle.]

Corollary 1. Since the arcs described are proportional to angular velocity (V), i.e. V, then FV2/R.

Corollary 2. Since the periodic time around the circle (T) is proportional to R/V (and V R/T), then F R/T2.

Corollary 6. If T R3/2, then F 1/R2. This is the case of the planets orbiting the sun.

Corollary 7. In general, if T Rn, then F 1/R2n-1.

Proposition 2, Book I

The argument for universal gravitation (Principia, Book III)

I. The force that draws Jupiter's moons out of rectilinear motion and into their orbits (a) tends to Jupiter's centre, and (b) is an inverse-square force [i.e., it varies as 1/D2, where D is the distance from the centre of Jupiter].Proof: (a) Phenomenon I and Proposition II and III; (b) Corollary 6, Proposition IV. The same proposition holds for Saturn and its moons.

II. The forces that draw the planets into their orbits (a) tend to the centre of the sun and (b) are inverse-square forces.Proof: (a) Phenomenon V, and Propositions II and III; (b) Corollary VI, Proposition IV.

III. The force that holds the moon in orbit around the earth (a) tends to the centre of the earth and (b) is an inverse-square force.Proof: (a) Phenomenon VI, and Propositions II and III; (b) observations of the moon’s distance and motion.

IV. “The moon gravitates toward the earth, and by the force of gravity is continually drawn off from a rectilinear motion and retained in its orbit.”

Proof: Since the force on the moon is inversely proportional to the square of the distance from the earth, and since its distance from the earth’s centre is 60 times the radius of the earth, then the force acting on the moon is 1/602, i.e. 1/3600, as powerful as the same force acting at the earth’s surface (i.e., at one radius from the centre). And it turns out that the acceleration of the moon is exactly 1/3600 of the acceleration of a falling apple at the earth’s surface. Hence the force acting on the moon is just gravity, attenuated by distance.

V. Jupiter’s moons gravitate toward Jupiter; Saturn’s moons gravitate toward Saturn; the planets gravitate toward the sun, and are held in their orbits by gravity.

Proof: Rule of Reasoning II states that to like effects, we should assign like causes. The force that urges Jupiter’s moons toward Jupiter, and Saturn’s moons toward Saturn, obeys the same law as the force that urges the moon toward the earth, so they must be the same force; but by Proposition 4, this force is gravity.

Corollary 1. All planets have gravity (i.e. a gravitational force directed toward their centres); “for, doubtless, Venus, Mercury, and the rest are bodies of the same sort with Jupiter and Saturn.” Furthermore, by Law III, all attraction must be mutual, so these planets must in turn gravitate towards their satellites.

Corollary 2. The power of gravity towards any planet varies inversely as the square of the distance from that planet.

Corollary 3. It follows from Corollaries 1 and 2 that all planets gravitate towards each other. “And hence it is that Jupiter and Saturn, when near their conjunction, by their mutual attractions sensibly disturb each other’s motions. So the sun disturbs the motions of the moon; and both the sun and the moon disturb our seas....”

VI. All bodies gravitate toward every planet, and the weights of bodies toward any given planet, at equal distances from the centre of that planet, are proportional to their respective masses.

Proof: This follows from Galileo's demonstration that bodies fall at the same rate regardless of composition, which Newton proved to great accuracy using pendulums of different kinds. Further demonstration is the fact that the accelerations of Jupiter's moons (or Saturn's, or of the planets around the sun) depend only on their distances from Jupiter (or Saturn or the Sun).

VI. That there is a power of gravity tending to all bodies, proportional to the several quantities of matter which they severally contain." [i.e.,gravitation is universal, and the force of gravity toward any body is proportional to its mass.

Proof: The previous propositions and proposition LXIX and its corollaries (book I) jointly show that all planets have a power of gravity and that this power is proportional to their respective quantities of matter. “Moreover, since all the parts of any planet A gravitate towards any other planet B; and the gravity of the part is to the gravity of the whole as the matter of the part to the matter of to the matter of the whole; and (by Law III) to every action corresponds an equal reaction; therefore the planet B will, on the other hand, gravitate towards all the parts of the planet A, and its gravity towards any one part will be to the gravity towards the whole as the matter of the part to the matter of the whole.”

The system of the world: what is really moving, and what is really at rest?

Hypothesis I. “That the centre of the system of the world is immovable. This is acknowledged by all, while some contend that earth, others that the sun, is fixed in that centre. Let us see what from hence may follow.”

XI. That the common centre of gravity of the earth, the sun, and all the planets is immovable.

Proof: By Corollary 4 of the Laws of Motion, “that centre is either at rest or moving uniformly in a right line; but if that centre moved, the centre of the world would move also, against the Hypothesis.” [i.e., if that centre moved, all bodies in the system would share its motion, and nothing would be at rest.]

XII. “That the Sun is agitated by a perpetual motion, but never recedes far from the common centre of gravity of all the planets.”Proof: The ratio of the mass of the sun to the mass of Jupiter is 1067 to 1; of the sun to Saturn is 3021 to 1; therefore, the centre of gravity of the sun and Jupiter is just outside the surface of the sun, while the centre of gravity of the sun and Saturn is just inside the surface of the sun. “And, pursuing the principles of this computation, we should find that though the earth and all the planets were placed on one side of the sun, the distance of the common centre of gravity of all from the centre of the sun would scarcely amount to one diameter of the sun….the sun, according to the various positions of the planets, must perpetually be moved every way, but will never recede far from that centre.”

Corollary. “Hence the common centre of gravity of the earth, the sun, and all the planets, is to be esteemed the centre of the world....”

To put the Earth in the centre makes as much sense as thinking that the Earth’s mass can balance all the other masses together.

That makes as much physical sense as this picture: