Transcript for the lecture

Mathematics, motion, and truth:
the Earth goes round the Sun
Jeremy Gray
Open University and University of
Warwick
Osiander, preface to De revolutionibus
• it is the job of the astronomer -- since he cannot by
any line of reasoning reach the true causes of these
movements -• to think up or construct whatever causes or
hypotheses he pleases such that, by the assumption of
these causes, those same movements can be
calculated correctly from the principles of geometry
for the past and for the future too.
• It is not necessary that these hypotheses be true, or
even probably [so]
Wittenberg 1536
Opposition
• On the day when the
LORD gave the
Amorites over to the
Israelites, Joshua spoke
to the LORD; and he
said in the sight of
Israel, ``Sun, stand still
at Gibeon, and Moon, in
the valley of Aijalon.''
And the Sun stood still,
and the Moon stopped.
Galileo
HUYGENS ON “CENTRIFUGAL FORCE”
The tension in the string retaining
a body in uniform circular motion
varies as the product of
EG/δt2
which – by Euclid III,36 –
becomes
 (GC2/AG)/δt2
which, as G approaches C,
 v2/r
and the weight of the body.
NEWTON’S PROBLEM: TO INFER
FORCES FROM MOTIONS
The centripetal force retaining a
body in uniform circular motion
varies as the product of
EG/δt2
which – by Euclid 3,36 – becomes
 (GC2/AG)/δt2
which, as G approaches C,
 v2/r
and the mass of the body.
Problem: How to generalize from uniform circular to
arbitrary curvilinear motions – e.g. Kepler’s ellipse?
NEWTON ON CURVATURE – 1671
Newton -- Principia
• Given: Motion in an ellipse,
• force is directed to a focus of the ellipse,
• Deduce: force is inverse square in the distance
of the planet from the focus.
• But . . .
Problems
• the observations are necessarily approximate
and support a variety of conclusions about the
orbit;
• the Sun wobbles and so displaces the focus,
which means that the orbit cannot actually be
an ellipse.
• So: conclusions could only be approximate.
How robust?
Newton tested the inverse square law in a
variety of situations:
•
•
•
•
motion in an ellipse to an arbitrary point,
motion in eccentric circles,
motion in rotating ellipses,
motion in near circles.
The conclusion was remarkably robust
• He found that planetary precession was so
small that any departure from inverse square
could be ruled out,
• and that even the motion of the Moon
conformed to this hypothesis.
• The inverse square law even held up for orbits
that were markedly eccentric and for orbits
that were not even perfect ellipses.
Johann Bernoulli –
from the force law to the trajectory
acceleration / force
Fitting up a conic
• Inverse square force =>
• Best circle at each point
=>
• Trajectory is the unique
conic at that point
Elsewhere in the Principia
Newton had discussed this problem for central
forces of any kind;
His solution requires certain quadratures
(integrals) to be known in advance.
Bernoulli’s public letter of 1710 questioned the
extent to which Newton was able to turn such
problems into his calculus in 1687 or 1710 and
solve them there.
Newton
• Geometry captures
physics;
• Algebra is
useful/essential but
should disappear
vs Bernouilli
• Systematic mathematics
is better than ad hoc
techniques
Euler
Euler: the reality of space
• Derive mechanics from three fundamental
properties of bodies:
• position,
• impenetrability,
• Inertia (ad hoc -- Newton's laws of motion).
Euler remained throughout his life hostile to the
idea of force as a primitive notion.
1900
Poincaré
ICM Zurich 1897
Mathematics has three uses:
• it aids in the understanding of nature;
• it helps make precise notions of number, space
and time;
• it has an aesthetic purpose, by which
mathematics and physics advance inseparably
together.
Empiricist, not rationalist
laws of nature are drawn from experiment and
expressed in the language of mathematics. But:
Experiments are particular, laws are general.
Experiments are approximate, laws exact.
A law is a generalisation, but -- -- every truth can
be generalised in infinitely many ways.
Analogy is the only way forward
Kepler's laws and Newton's agree: a single
planet travels in an ellipse. But:
Newton's theory allows perturbed orbits though
no-one has written down their equations
Kepler's laws restricted to generalisations of an
ellipse.
Poincaré was not a realist.
• Favoured a plurality of possible theories.
• Poincaré’s geometric conventionalism – 1891.
• Geometry was to be understood in a physical
setting.
Is space Euclidean or non-Euclidean?
1890s, public discussion.
Poincaré’s surprising answer:
non-Euclidean geometry makes sense, but
there is no way of telling if Space is Euclidean or
non-Euclidean.
Dichotomy
• Either
Light rays are straight and the geometry of space
is non-Euclidean geometry
• Or
Light rays are curved and the geometry of space
is Euclidean
Choice by convention
No possibility of deciding on logical
grounds.
The only way forward is an arbitrary
choice based on human convenience.
Hypotheses
Natural and necessary – the influence of distant
bodies can be ignored.
Indifferent – lead to same conclusion
matter is continuous / matter is discrete.
Real generalisations, confirmed or refuted by
experiment.
Established
experimental
results
A
N
Y
T
H
I
N
G
Solid
mathematical
theorems
Poincaré on Fresnel and Maxwell
• The differential equations are always true, they
may be always integrated by the same
methods, and the results of this integration still
preserve their value. They express relations,
and if the equations remain true, it is because
the relations preserve their reality.
The reality of relations
• They teach us now, as they did then, that there
is such and such a relation between this thing
and that; only, the something which we then
called motion, we now call electric current.
....
But these are merely names of the images we
substituted for the real objects which Nature
will hide for ever from our eyes. The true
relations between these real objects are the
only reality we can attain, and the sole
condition is that the same relations shall exist
between these objects as between the images
we are forced to put in their place.
....
If the relations are known to us, what does it
matter if we think it convenient to replace one
image by another?
• That a given periodic is really due to the
vibration of a given atom, which, behaving
like a pendulum, is really displaced in this
manner or that -all this is neither certain nor essential.
Geometry is different
Our knowledge of the external world
– derived from our senses
– organised and ‘made sense of’ by our
brains
Arithmetic is synthetic a priori knowledge -- the
principle of induction.
Edouard Le Roy – Pierre Duhem
Eduoard Le Roy
• adapted Bergsonian vitalism to a modernist
philosophy of Catholicism:
• dogma a source of moral values without being either
inscrutable or in contradiction to rational knowledge.
• Attacked by Pope Pius X in his encyclical of 1907,
when the Pope moved to shut down the Catholic
Modernist movement.
....
• True knowledge -- an authentic and immediate
relationship with one's surroundings, and all
• Theoretical knowledge is a matter of
invention. This is not far from Boutroux's neoKantianism, as he
• admitted, but the article went further in
advocating a
....
• Radical conventionalism:
• there are no facts in science, only inventions
• which are entirely arbitrary even though they
may be necessary on pragmatic grounds.
scientific `facts' that are only
inventions
•
•
•
•
Le Roy cited:
the atom,
the phenomenon of eclipses, and
the rotation of the Earth.
Catholic Church did nothing wrong
• The Earth’s rotation is only an invention
• So Protestant and anti-clerical criticisms of the
Church seeking to accuse it of bigotry and
hostility to science were profoundly misplaced.
Poincaré had said:
• . . . the Earth turns round, has no meaning,
since it cannot be verified by experiment, . . .
• in other words, the earth turns round, and
• it is more convenient to suppose that the Earth
turns round,''
• have one and the same meaning.
• Science et Hypoth\`ese, p. 117
Poincaré's replies
• 'La science est-elle artificielle?'
• `La Science et la Réalité‘.
• Poincaré 1905b, La Valeur de la Science,
213--247 and 248--276.
Poincaré: a succession of gradations
•
•
•
•
ignorance
astronomical predictions,
Newton's laws,
the deduction of the rotation of the Earth (and
a defence of Galileo).
The role of convention
• was restricted to:
• the choice of units of length and time in
physics
• and definitions and postulates in mathematics.
• Thereafter, scientific facts were merely the
translation of brute facts into the language of
science.
The rotation of the Earth
• the two claims: that the Earth and that it does not
rotate
• cannot be told apart kinematically -- there is no
absolute space.
• But the claim of rotation has a much richer dynamical
theory -• the apparent motion of the stars, Foucault's
pendulum, and
• much else that would be disparate phenomena on a
Ptolemaic theory.
....
• the rotation of the Earth is not on the same
footing as the parallel postulate.
• Rather, it belongs with claims about the
existence of the external world.
The role of theory
• scientific facts are brute facts translated into
the language of science by being incorporated
in a theory.
• The choice of theory is arbitrary,
• the facts are inter-translatable.
Duhem in Bordeaux, 1894 and 1906
• Philosophy of physics in neo-Thomist journals
• Revue de philosophie and the Revue des
questions scientifiques,
• Société scientifique de Bruxelles.
• Neo-Thomist in 1890 obeying Pope Leo XIII's
instructions.
• La théorie physique. Son objet et sa structure.
1908
Duhemian holism
• Physicist's language may be translated
into facts in an infinity of different ways.
• A network of physical permits different
interpretations of any given result.
• No `crucial experiment' in physics.
Duhem opposed
• English models (Maxwell! and Kelvin!)
• `French or German physicists‘ would never have
done [this] `of their own free will'.
• But Hertz had reduced mathematical physics to
algebraic models.
• Poincar\'e: spread on a fashion for all things English
-- piles of faulty reasoning, false calculation, a
confusion of science and industry, and the rejection of
abstract and deductive theories.
Duhemian science
• Science is an exercise in classification,
• independent of any metaphysics.
• Scientific laws are incapable of being true
because they were only representations
• And so science was not capable of conflicting
with religion.
The Earth goes round the Sun
• A question in dynamics – the theory of motion.
• It is a matter of theory – one, or many –
• Is it True? Or just the right thing to say?