TURNING DATA INTO EVIDENCE Three Lectures on the Role of Theory in Science 1. CLOSING THE LOOP Testing Newtonian Gravity, Then and Now 2. GETTING STARTED Building Theories from Working Hypotheses 3. GAINING ACCESS Using Seismology to Probe

1
TURNING DATA INTO EVIDENCEThree Lectures on
the Role of Theory in Science1. CLOSING THE
LOOPTesting Newtonian Gravity, Then and Now2.
GETTING STARTEDBuilding Theories from Working
Hypotheses3. GAINING ACCESSUsing Seismology
to Probe the Earths Insides

• George E. Smith
• Tufts University

2
THE USUAL VIEW

• In science what turns a datum B into evidence for
  a claim A that reaches beyond it is a deduction
  from A of a sufficiently close counterpart of B.
• In particular, historically what made celestial
  observations evidence for Newtonian gravity were
  the increasingly accurate predictions derived
  from the theory of these observations
• The realization that Einsteinian gravity would
  all along have yielded no less accurate
  predictions tells us that scientists had all
  along over-valued the evidence for Newtonian
  gravity

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SO, WHY NOT SIMPLY HYPOTHESIS TESTING BY MEANS OF
DEDUCED PREDICTIONS?HEMPELS PROVISO PROBLEM

• Deduced predictions in celestial mechanics
  presuppose a proviso no other forces (of
  consequence) are at work.
• The only evidence for this proviso is close
  agreement between the predictions and
  observation.
• But then a primary purpose of comparing deduced
  predictions and observation is to answer the
  question, Are other forces at work?
• How then is the theory of gravity tested in the
  process?

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OUTLINE

• Introduction the issue
• The logic, as dictated by Newtons Principia
• How this logic played out after the Principia
• A. Then complications that obscure the
  logic
• B. Now in light of the perihelion of
  Mercury
• Concluding remarks

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GRAVITY RESEARCH THEN AND NOW

• IN CELESTIAL MECHANICS
• What are the true motions orbital and
  rotational of the planets, their satellites,
  and comets, and what forces govern these motions?
• IN PHYSICAL GEODESY
• What is the shape of the Earth, how does the
  gravitational field surrounding it vary, and what
  distribution of density within the Earth produces
  this field?

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CALCULATING PLANETARY ORBITS 1680
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NEWTONS EVIDENCE PROBLEM IN THE PRINCIPIA
By reason of the deviation of the Sun from the
center of gravity, the centripetal force does not
always tend to that immobile center, and hence
the planets neither move exactly in ellipses nor
revolve twice in the same orbit. Each time a
planet revolves it traces a fresh orbit, as in
the motion of the Moon, and each orbit depends on
the combined motions of all the planets, not to
mention the action of all these on each other.
But to consider simultaneously all these causes
of motion and to define these motions by exact
laws admitting of easy calculation exceeds, if I
am not mistaken, the force of any human mind.
Isaac Newton, ca. December 1684 (First
published by Rouse Ball in 1893)
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INFERRING LAWS OF FORCE FROMPHENOMENA OF
MOTION

• Phenomena Descriptions of regularities of motion
  that hold at least quam proxime over a finite
  body of observations from a limited period of
  time
• The planets swept out equal areas in equal times
  quam proxime with respect to the Sun over the
  period from the 1580s to the 1680s.
• Propositions, deduced from the laws of motion, of
  the form
• If _ _ _ quam proxime, then quam proxime.
• If a body sweeps out equal areas in equal times
  quam proxime with respect to some point, then the
  force governing its motion is directed quam
  proxime toward this point.
• Conclusions Specifications of forces (central
  accelerations) that hold at least quam proxime
  over the given finite body of observations
• Therefore, the force governing the orbital
  motion of the planets, at least from the 1580s to
  the 1680s, was directed quam proxime toward the
  Sun.

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From Evidence that is Approximate to A
Law that is Taken to be Exact

• Rule 3 Those qualities of bodies that cannot be
  intended and remitted and that belong to all
  bodies on which experiments can be made should be
  regarded as qualities of all bodies universally.
  
• Rule 4 In experimental philosophy, propositions
  gathered from phenomena by induction should be
  regarded as either exactly or very, very nearly
  true notwithstanding any con-trary hypotheses,
  until yet other phenomena make such propositions
  either more exact or liable to exceptions.
• This rule should be followed so that
  arguments based on induction may not be nullified
  by hypotheses.

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PREREQUISITES FOR TAKING THE THEORY OF
GRAVITY AS EXACT

• The theory must identify specific conditions
  under which the phenomena from which it was
  inferred would hold exactly without restriction
  of time e.g.
• The area rule would hold exactly in the absence
  of forces from other orbiting bodies
• The orbits would be perfectly stationary were it
  not for perturbing forces from other orbiting
  bodies
• The theory must identify a specific configuration
  in which the macroscopic variation of gravity
  about a body would result from the microstructure
  of the body e.g.
• Gravity would vary exactly as the inverse-square
  around a body were it a sphere with a spherically
  symmetric distribution of density

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TAKING THE THEORY TO BE EXACTTHE PRIMARY
IMPLICATION

• Every systematic discrepancy between observation
  and any theoretically deduced result ought to
  stem from a physical source not taken into
  account in the theoretical deduction
• a further density variation
• a further celestial force

12
THE NEWTONIAN APPROACH CONTINUING EVIDENCE

• Taking the law of gravity to hold exactly was a
  research strategy, adopted in response to the
  complexity of the true planetary motions.
• Deductions of planetary motions etc. are
  Newtonian idealizations approximations that,
  according to theory, would hold exactly in
  certain specifiable circumstances -- in
  particular, in the absence of further forces or
  density variations.
• The upshot of comparing calculated and observed
  orbital motions is to shift the focus of ongoing
  research onto systematic discrepancies, asking in
  a sequence of successive approximations, what
  further forces or density variations are at work?
• Theory thus becomes, first and foremost, not an
  explanation (or even a representation) of known
  phenomena, but an instrument in ongoing research,
  revealing new second-order phenomena that can
  provide a basis for continuing testing of the
  theory.

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THE LOGIC OF THEORY TESTING

• The theory requires that every deviation from any
  Newtonian idealization be physically
  significant i.e. every deviation must result
  from some further force or density variation.
• Basic Testing pin down sources of the
  discrepancies and confirm they are robust and
  physically significant (within the context of the
  theory) while achieving progressively smaller
  discrepancies between (idealized) calculation and
  observation.
• Ramified Testing keep incorporating previously
  identified physical sources of second-order
  phenomena into the (idealized) calculation,
  thereby progressively constraining the freedom to
  pursue physical sources for new second-order
  phenomena that then emerge.
• The continuing evidence lies not merely in the
  aggregate of the individual comparisons with
  observation, but also in the history of the
  development of the sequence of successive
  approximations.

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NEPTUNE AS AN EXAMPLE OF PHYSICAL
SIGNIFICANCE
seconds of arc
15
THE GREAT INEQUALITY AS A MORE TYPICAL
EXAMPLE
minutes of arc
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OUTLINE

• Introduction the issue
• The logic, as dictated by Newtons Principia
• How this logic played out after the Principia
• A. Then complications obscuring the logic
• B. Now in light of the perihelion of
  Mercury
• Concluding remarks

17
Second-Order Phenomena Often Underdetermine
Their Physical Source

• Example

• Example
• Deviation of surface gravity from Newtons ideal
  variation implies the value of (C-A)/Ma2 and
  hence a correction to the difference (C-A) in the
  Earths moments of inertia, and the lunar-solar
  precession implies the value of (C-A)/C and
  hence a correction to the polar moment C these
  two corrected values constrain the variation ?(r)
  of density inside the Earth, but they do not
  suffice to determine ?(r) .

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RESPONDING TO UNDERDETERMINATION20TH CENTURY
DETERMINATION OF ?(r)
density
core-mantle boundary
density
19
ROBUSTNESS OF PHYSICAL SOURCES

• Examples
• Mass of Moon inferred from lunar nutation
  supported by calculated tides and lunar-solar
  precession
• Mass of Venus inferred from a particular
  inequality in the motion of Mars supported by
  calculated perturbations of Mercury, Earth, and
  Mars
• The far reach of the gravity fields of Jupiter
  and Saturn supported by variations in period of
  Halleys comet

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PROBLEMS IN ISOLATING DISCORDANCES

• The motion of the lunar perigee can be got
  from observation to within about 500,000th of
  the whole. None of the values hitherto computed
  from theory agrees as closely as this with the
  value derived from observation. The question
  then arises whether the discrepancy should be
  attributed to the fault of not having carried the
  approximation far enough, or is indicative of
  forces acting on the moon which have not yet been
  considered.
• G. W. Hill, 1875

• Newcombs Discordances, 1895
• Mercurys perihelion
• was 29 times probable error
• Venuss nodes
• was 5 times probable error
• Marss perihelion
• was 3 times probable error
• Mercurys eccentricity
• ? was 2 times probable error

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ANOTHER EXAMPLE OF DIFFICULTY
Many professional lives have been dedicated to
the long series of meridian circle (transit)
observations of the stars and planets throughout
the past three centuries. These observations
represent some of the most accurate scientific
measurements in existence before the advent of
electronics. The numerous successes arising from
these instruments are certainly most impressive.
However, as with all measurements, there is a
limit to the accuracy beyond which one cannot
expect to extract valid information. There are
many cases where that limit has been exceeded
Planet X has surely been such a case.
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THE MANY SOURCES OF DISCREPANCIES

• In observations
• Simple error bad data
• Limits of precision
• Systematic bias in instruments
• Inadequate corrections for known sources of
  systematic error, incl.
• Imprecise fundamental constants
• Not yet identified sources of systematic error

• In theoretical calculations
• Undetected calculation errors
• Imprecise orbital elements
• Imprecise planetary masses
• Insufficiently converged infinite-series
  calculations
• Need for higher-order terms
• Forces not taken into account
• Gravitation theory wrong

The ultimate goal of celestial mechanics is to
resolve the great question whether Newtons law
by itself accounts for all astronomical
phenomena the sole means of doing so is to make
observations as precise as possible and then to
compare them with the results of calculation.
The calculation can only be approximate.
Henri Poincaré, 1892

23
SECULAR MOTION OF THE MOON

• 18th Century
• Acceleration in motion of Moon announced by
  Halley (1693)
• A physical source identified by Laplace (1787)

• 19th Century
• Adams finds that Laplace has accounted for only
  half of the secular motion (1854)
• A further physical source earth is slowing from
  tidal friction

Owing to perturbations from gravity toward the
planets, eccentricity of Earths orbit changing.
24
EXAMPLE OF SPECTACULAR SUCCESS SPENCER JONES
(1939)

• Residual discrepancies in the motions of Mercury,
  Venus, and Earth correlate with unaccounted-for
  discrepancy in lunar motion
• Common cause gt Earths rotation irregular (in
  more ways than one)
• Expose a still further systematic observation
  error, requiring correction
• 1950 replace sidereal time with ephemeris time

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This form of evidence can be very strong

• It is evidence aimed at the question of the
  physical exactness of the theory, as well as the
  question of its projectibility
• The sequence of successive approximations leads
  to new second-order phenomena of progressively
  smaller magnitude
• New second-order phenomena presuppose not only
  the theory of gravity, but also previously
  identified physical sources of earlier
  second-order phenomena, thereby constraining the
  freedom to respond to these new phenomena
• Theory becomes entrenched from its sustained
  success in exposing increasingly subtle details
  of the physical world without having to backtrack
  and reject earlier discoveries

26
OVERALL HISTORICAL PATTERNA FEEDBACK LOOP

• Idealized calculated orbits presupposing
  theory and various physical details
• Comparison with astronomical observations
• Discrepancy with clear signature!
• Physical source of discrepancy still further
  physical details that make a difference!
• New idealized calculation incorporating the new
  details and their further implications

• Ever smaller discrepancies
• Ever many more details that turn out to make a
  difference

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OUTLINE

• Introduction the issue
• The logic, as dictated by Newtons Principia
• How this logic played out after the Principia
• A. Then complications obscuring the logic
• B. Now in light of the perihelion of
  Mercury
• Concluding remarks

28
INEXACTNESS EXPOSEDTHE PERIHELION OF MERCURY
The secular variations already given are derived
from these same values of the masses, the
centennial motion of the perihelion being
increased by the quantity ?Dt? 43.?37 In order
to represent the observed motion. This quantity
is the product of the centennial mean motion by
the factor 0.000 000 0806
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PERIHELION OF MERCURY CURRENT
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FROM NEWTONIAN TO EINSTEINIAN GRAVITY

• Discrepancy between Newtonian calculation and
  observation
• 43.37 2.1 gt 43.11 0.45
• Increment from the Einsteinian calculation
• 43. gt 42.98
• Newtonian gravity is the static, weak-field
  limit of Einsteinian!
• A limit-case idealization
• The orbital equation becomes, where µ
  G(Mm), u 1/r

31
CONTINUITY OF EVIDENCE ACROSS THE CONCEPTUAL
DIVIDE

• 43 per century was a Newtonian second-order
  phenomenon
• From limit-case reasoning, evidence for Newtonian
  gravity carried over, with minor qualifications,
  to Einsteinian
• Earlier evidential reasoning for Newtonian
  gravity, even though requiring some
  qualifications, was not nullified
• Previously identified physical sources of
  Newtonian second-order phenomena remained intact
  in Einsteinian

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• though the world does not change with a
  change of paradigm, the scientist afterward works
  in a different world. I am convinced that we
  must learn to make sense of statements that at
  least resemble these.
• Thomas S. Kuhn, SSR, p. 121
• The continuity of evidence across the conceptual
  divide between Newtonian and Einsteinian gravity
  highlights an extremely important sense in which
  the scientist afterward works in the same world.

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PRIMARY CONCLUSIONS

• The most important evidence in classical
  gravitational research came from the complexities
  of the actual motions and of the gravitational
  fields surrounding bodies.
• This evidence consisted of success in pinning
  down physical sources of deviations from
  Newtonian idealizations, in a sequence of
  increasingly precise successive approximations.
• This evidence carried forward, continuously,
  across the tran-sition from Newtonian to
  Einsteinian gravity and remains an important
  source of continuing evidence today.

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CLOSING THE LOOP

• Thrust of the Evidence
• Not merely numerical agreement, a curve-fit
• Increasingly strong, still continuing evidence
  that certain physical details make specific
  differences

• Idealized calculated orbits presupposing
  theory and various physical details
• Comparison with astronomical observations
• Discrepancy with clear signature!
• (Revised theory when deemed necessary)
• Physical source of discrepancy still further
  physical details that make a difference!
• New idealized calculation incorporating the new
  details and their further implications

35
THE KNOWLEDGE ACHIEVED IN GRAVITY SCIENCE

• Interpenetration of theory and an ever growing
  multiplicity of details that make a difference
• Details evidence for theory and values for
  parameters
• Theory lawlike generalizations supporting
  counterfactual conditionals that license
  conclusions about differences a detail makes
• Two requirements for generalizations to do this
• They must hold to high approximation over a
  restricted domain
• They must be lawlike i.e. they must be
  projectible over this domain
• Just what Einstein showed about Newtonian
  gravity, and Newton took the trouble to show
  about Galilean gravity