Sperduto - Explaining by Correction

7/27/2019 Sperduto - Explaining by Correction

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Explaining by correction: a guide to Poppers

suppressed derivations

Luke Sperduto

London School of Economics and Political Science

June 2013

Outline1. Context: what is Popper saying about the depth of scientic theories?2. Derivations: what maths does he suppress?3. Conclusion: what is left to say?

1 Context

In The Aim of Science, the fth chapter of his book Objective Knowledge , Karl

Popper discusses a set of standards for evaluating the satisfactoriness of a scienticexplanation. Included in this discussion are the following two conditions:

a) A satisfactory explanation must be independently testable. That is, it musthave empirical consequences other than those it is invoked to explain, otherwisethe explanation is unsatisfactorily ad hoc or perhaps circular.

b) An explanation is more satisfactory to the extent that it invokes structural orrelational properties of greater universality than both those constituting theexplicandum or invoked by competing, or previously held, explicans.

This universality of the explaining theory - along with such ephemeral qualities

as its simplicity, the wealth of its content, and a certain coherence or com-pactness (or organicity), - is part of what Popper refers to as its depth . Thoughdepth is, he says, hardly susceptible of logical analysis ... there does seem tobe something like a su cient condition for depth, or for degrees of depth, which

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can be logically analysed. To illustrate this su cient condition, Popper turns tothe Newtonian synthesis. The way Newtons law of universal gravitation entailsGalilean mechanics and Keplers third law of planetary motion paragons a kind of entailment relation that is his su cient condition for theoretic depth.

Compared to the gravitational theories of both Galileo and Kepler, Newtons the-ory of gravitation, Popper argues, exemplies the conditions above in the way it corrects them while explaining them. In other words, Newton explains why theearlier theories were correct, insofar as they were, while simultaneously indicatingexactly which phenomena the earlier theories got wrong, by how much they got itwrong and why. Because it makes predictions about projectile and planetary mo-tion that, strictly speaking, contradict the predictions made by Galileos and Ke-plers theories, Newtons theory meets condition (a); it has independently testableconsequences that are distinct from those of the earlier theories, so its explanationof them is not ad hoc . Newtonian theory meets condition (b) because:

i) It invokes a relational property of massive bodies that holds both within andbeyond the observational range (i.e. earthbound projectiles or the planets of our solar system) of the earlier theories, and which is therefore more universal.This more universal property of massive bodies, embodied in Newtons grav-itational constant, G , is the theoretical basis of the predictions that strictlycontradict those of earlier theories.

ii) Within the observational range addressed by Galileo and Kepler, its strictlycontradictory predictions are nonetheless observationally indistinguishable fromthose of the earlier theories, and thereby explain why the earlier theories func-tion so successfully despite being incorrect.

By satisfying enumerated conditions (a) and (b), Newtons theory meets Popperssu cient condition for greater depth. At the end of his presentation of the threetheories of gravitation, he suggests, that whenever in the empirical sciences a newtheory of a higher level of universality successfully explains some older theory by correcting it , then this is a sure sign that the new theory has penetrated deeperthan the older ones.

In his exposition of the correction made to Keplers third law by Newtonian theory,Popper compares the following two equations:

a 3

T 2 = m 0 + m 1 (1)

a 3

T 2= constant (2)

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Equation (2) is Keplers third law, where a is the mean distance between thetwo orbiting bodies and T is the orbital period (i.e. the time needed for a fullrevolution). Equation (1) can be derived, says Popper, from the application of Newtons theory to a two-body system, where m 0 and m 1 are the masses of thetwo bodies. That derivation is shown in the next section.

Before turning to the maths, note that Keplers constant is the same constant forall planets of the solar system, regardless of their mass. Thus, if equation (1) reallycan be derived from Newtons theory, it is clear that that theorys prediction for thevalue of a

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T 2 will strictly contradict Keplers prediction, since Newtons predictionwill vary with m 1, the mass of the planet. However, since in our solar system eachm 1 is negligible compared to m 0, the mass of the sun, the strictly contradictorypredictions are nonetheless observationally indistinguishable (given the state of observational technology available in the 18th century). Therefore, Popper claims,Newtons theory explains and corrects Keplers third law.

2 Derivations

The following diagram abstracts from several features of our actual solar system(e.g. elliptical orbits, additional massive bodies, relative sizes of planets and radii,etc.) in order to help pose the precise problem of the derivation, stated on thenext page.

Fig. 1

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Given: Newtons law of universal gravitation F =G m 0 m 1

R 2

Newtons second law of motion F = m

where F is the force acting between or upon the massive bodies,measured in Newtons (N = kg

ms2 );

G is Newtons gravitational constant, measured in units of m3

kg s2 ;m 0 and m 1 are the masses of the orbiting bodies, S and P,

measured in kilograms (kg);R is the distance between the two bodies, given by r 0 + r 1, where

r 0 and r 1 are the radii of the orbits of S and P, respectively,around their shared center of mass, C, and are measured in meters (m);

m is the mass of either orbiting body; and is the angular acceleration experienced by either orbiting body,

measured in rads2 .

Show that: a3

T 2 = m 0 + m 1

where a = ( r 0 + r 1), the average distance between the orbiting bodies, andT is the orbital period of a body in a two-body system, measured in seconds (s).

The derivation involves seven steps. The rst 6 of which solve for T , and the lastsimply inserts that expression into the left-hand side of (1) to yield the New-

tonian correction of Keplers third law. Actually, the derivation yieldsa 3

T 2 =(m 0 + m 1) (

G

4 2 ), but the scaling term at the end is left out of Poppers pre-sentation because it is constant.

To begin , use the second law of motion to express the force acting on P due toits angular acceleration around its orbit.

angular velocity: ! 2T rads

angular acceleration: d !dT = (2 r 1 )(2 )

T 2ms (since 2 rad = 2 r m)

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= v2

r 1ms , where velocity is v

2 rT

ms

so F = m yields F = m 1 v2

r 1 N

Second , use the law of universal gravitation to express the gravitational forceoperating between the two bodies.

F =G m 0 m 1(r 0 + r 1)2

N

Third , since there are assumed to be only two bodies in the system and we abstract

from all other extenuating forces, equate the expressions from the previous twosteps and solve for the velocity of P.

m 1v2

r 1=

G m 0 m 1(r 0 + r 1)2

v 2 =G m 0 r 1(r 0 + r 1)2

Fourth , nd an expression for r 1 by identifying the center of mass of the system.For a visual aid to interpreting the denition of the systems center of mass, consultthe following diagram, taken from [6].

Fig. 2

By subtracting 1 from each of the mass subscripts for notational continuity, wesee that r 1 from Fig. 1 is equal to x 2 x cm , and x 2 x 1 = r 0 + r 1 = R .

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Consequently,

r 1 = x 2 x cm

=m 0(x 2 x 1)

m 0 + m 1

=m 0R

m 0 + m 1

Fifth , instantiate one of the r 1s in the expression from step three with the ex-pression from step four.

v 2 =G m 0 r 1

(r 0 + r 1)2

=G m 0 (

m 0 Rm 0 + m 1 )

(r 0 + r 1)2

=G m 20

(m 0 + m 1)R

Sixth , we know that, in general, the velocity of an orbiting body is given by:

v 02 =4 2r 02

T 2

so equate v for our system (from step 5) with v 0 and plug in R for r 0 to write

4 2R 2m 20T 2(m 0 + m 1)

=G m 20

(m 0 + m 1)R

and conclude that T = 2

q ( r 0 + r 1 )3

G(

m 0+

m 1)s .

Finally , a3

T 2 = ( m 0 + m 1)( r 0 + r 1)3 (G

4 2 ( r 0 + r 1 )3 ) = ( m 0 + m 1) (G

4 2 )

Q.E.D.

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A quick check for the appropriate units conrms the conclusion.

a 3

T 2has units

m3

s2

and

(m 0 + m 1) (G

4 2) also has units kg N (

mkg

)2 =m3

s2.

3 Conclusion

We now see more fully how Newtons theory of gravitation explains Keplers theoryby correcting its predictions about the relation between the size of a planets orbitand its orbital period. Whether or not Keplers third law contradicts Newtonstheory in precisely the same way as does Galileos, as Popper maintains, is an openquestion for further inquiry. There may very well be a wide variety of di erent waysa later theory can explain an earlier one by correcting it. It is also worth notingthat explaining-by-correction is not the only relation that successful descendanttheories can bear to their forbears, or at least Popper has not argued as much.

In any case, this distinctive inter-theoretical correspondence does more than dene

and illustrate the elusive notion of theoretical depth - a notion discussed as greaterconceptual economy by Thomas Kuhn (1957, pg. 37), as a theorys tendencyto sooth the imagination by Adam Smith (1980, pg. 46) and referred to by ImreLakatos variously as the heuristic superiority of a research programme or as theunity or beauty of science, (Lakatos 1978, pg. 52, 88 and 183).

What is depth? In addition to getting a bit of purchase on that question, explaining-by-correction as a type of entailment helps characterize more clearly those aspectsand posits of scientic theories we should expect to survive revolutionary paradigmshifts. In this capacity it informs an endless debate between the Pessimistic In-duction of the anti-realists and the No Miracles Arguments of their opponents, as

well as a variety of intermediating positions on the truth of scientic claims andthe nature of scientic inquiry itself.

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Sources

1. Kuhn, Thomas. The Copernican Revolution . Cambridge: Harvard UP, 1957.

2. Lakatos, Imre. The methodology of scientic research programmes . Ed. JohnWorrall and Gregory Currie. Cambridge: Cambridge UP, 1978.

3. Popper, Karl. Objective Knowledge . Oxford: Clarendon Press, 1966.

4. Smith, Adam. Essays on Philosophical Subjects . Ed. W.P.D. Wightman andJ.C. Bryce. Oxford: Oxford UP, 1980.

5. Center of Mass. http://hyperphysics.phy-astr.gsu.edu/hbase/cm.html.

6. The Physics of Binary Stars. http://www.egglescli e.org.uk/physics/ gravita-tion/binary/binary.html.

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Sperduto - Explaining by Correction