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 vs Bernouilli

• Geometry captures physics;

• Algebra is useful/essential but should disappear

• 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

• EitherLight 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 differentGeometry 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 neo-Kantianism, 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 onlyinventions

• 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?