Mathematics, motion, and truth: the Earth goes round the Sun

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Mathematics, motion, and truth the Earth goes
round the Sun

• Jeremy Gray
• Open University and University of Warwick

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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

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Wittenberg 1536
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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.

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Galileo
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HUYGENS ON CENTRIFUGAL FORCE

• The tension in the string retaining a body in
  uniform circular motion varies as the product of
• EG/dt2
• which by Euclid III,36 becomes
• ? (GC2/AG)/dt2
• which, as G approaches C,
• ? v2/r
• and the weight of the body.

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NEWTONS PROBLEM TO INFER FORCES FROM
MOTIONS

• The centripetal force retaining a body in
  uniform circular motion varies as the product of
• EG/dt2
• which by Euclid 3,36 becomes
• ? (GC2/AG)/dt2
• 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. Keplers
ellipse?
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NEWTON ON CURVATURE 1671
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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 . . .

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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.

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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.

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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.

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Johann Bernoulli from the force law to the
trajectory
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acceleration / force

• Fitting up a conic

• Inverse square force gt
• Best circle at each point gt
• Trajectory is the unique conic at that point

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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.
• Bernoullis 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.

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Newton vs Bernouilli

• Geometry captures physics
• Algebra is useful/essential but should disappear

• Systematic mathematics is better than ad hoc
  techniques

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Euler
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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.

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1900
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Poincaré
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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.

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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.

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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.

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Poincaré was not a realist.

• Favoured a plurality of possible theories.
• Poincarés geometric conventionalism 1891.
• Geometry was to be understood in a physical
  setting.

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Is space Euclidean or non-Euclidean?

• 1890s, public discussion.
• Poincarés surprising answer
• non-Euclidean geometry makes sense, but
• there is no way of telling if Space is Euclidean
  or non-Euclidean.

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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

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Choice by convention
No possibility of deciding on logical grounds.
The only way forward is an arbitrary choice
based on human convenience.
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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.
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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.

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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.

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. . . .

• 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.

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. . . .

• 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.

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Geometry is different
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.
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Edouard Le Roy Pierre Duhem
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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.

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. . . .

• 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 neo-Kantianism,
  as he
• admitted, but the article went further in
  advocating a

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. . . .

• Radical conventionalism
• there are no facts in science, only inventions
• which are entirely arbitrary even though they may
  be necessary on pragmatic grounds.

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scientific facts' that are onlyinventions

• Le Roy cited
• the atom,
• the phenomenon of eclipses, and
• the rotation of the Earth.

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Catholic Church did nothing wrong

• The Earths 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.

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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

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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.

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Poincaré a succession of gradations

• ignorance
• astronomical predictions,
• Newton's laws,
• the deduction of the rotation of the Earth (and a
  defence of Galileo).

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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.

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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.

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. . . .

• 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.

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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.

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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

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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.

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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.

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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.

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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?