Title: 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
1TURNING 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
2THE 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
3SO, 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?
4OUTLINE
- 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
5GRAVITY 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?
6CALCULATING PLANETARY ORBITS 1680
7NEWTONS 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)
8INFERRING 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.
9 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.
10PREREQUISITES 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
11TAKING 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
12THE 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.
13THE 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.
14NEPTUNE AS AN EXAMPLE OF PHYSICAL
SIGNIFICANCE
seconds of arc
15THE GREAT INEQUALITY AS A MORE TYPICAL
EXAMPLE
minutes of arc
16OUTLINE
- 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
17Second-Order Phenomena Often Underdetermine
Their Physical Source
- 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) .
18RESPONDING TO UNDERDETERMINATION20TH CENTURY
DETERMINATION OF ?(r)
density
core-mantle boundary
density
19ROBUSTNESS 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
20PROBLEMS 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
21ANOTHER 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.
22THE 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
23SECULAR 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.
24EXAMPLE 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
25This 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
26OVERALL 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
27OUTLINE
- 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
28INEXACTNESS 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
29PERIHELION OF MERCURY CURRENT
30FROM 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
31CONTINUITY 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
32- 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.
33PRIMARY 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.
34CLOSING 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
35THE 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