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Mathematics as the language of Nature – a
historical view
N. Mukunda
INSA Public Lecture – 29 September 2014
I. Introduction
II. The Galilean-Newtonian tradition
III. Two Phases in the Mathematics –Physics Relationship
IV. Mathematical formulation before physical understanding
V. Some significant lessons
VI. On the natures of mathematics and mathematical knowledge
VII. Some concluding thoughts
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"Philosophy (ie, physics) is written in this grand book – I
mean the universe – which stands continually open to our
gaze, but it cannot be understood unless one first learns
to comprehend the language and interpret the characters
in which it is written. It is written in the language of
mathematics, …, without which it is humanly impossible
to understand a single word of it; without these, one is
wandering around in a dark labyrinth".
— Galileo, 'Il Saggiatore' (1623)
Einstein on Galileo in 1933
‘ … the father of modern physics and in fact of the whole
of modern natural science’
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"Every one of our laws is a purely mathematical statement in
rather complex and abstruse mathematics…It gets more and
more abstruse and more and more difficult as we go on … it
is impossible to explain honestly the beauties of the laws of
nature in a way that people can feel, without their having
some deep understanding of mathematics".
— R.P. Feynman (1964)
Galileo on measurement
"Measure what can be measured, and make measurable what
cannot be measured".
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Galilean-Newtonian pattern for natural science
• Careful observation of natural phenomena, where
possible by controlled experiments, and description of
the results in mathematical form.
• Based on a proposed law or hypothesis, derivation of
predictions using mathematical analysis.
• Performing new observations or experiments to check
the predictions.
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Earlier phase in the relationship
Simultaneous progress in mathematics and physics –
Newton, Descartes, Fermat, Huyghens, Leibnitz, Euler,
Lagrange, Laplace, Poisson, Gauss, Hamilton, Jacobi
Exceptional case of Faraday – intensely intuitive, limited
mathematical powers.
Faraday to Maxwell on 25 March 1857:
"I was at first almost frightened when I saw such mathematical
force made to bear upon the subject, and then wondered to see
that the subject stood it so well".
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Einstein’s comparison of the two pairs
"… the pair Faraday – Maxwell has a most remarkable inner
similarity with the pair Galileo-Newton- the former of each
pair grasping the relations intuitively, and the second one
formulating those relations exactly and applying them
quantitatively.“
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Later phase – independent progress in mathematics
and physics, connections appear later. Examples:
(a) Non Euclidean Geometry – 1830’s – Gauss,
Lobachevsky, Bolyai.
1854 : Riemann’s Inaugural Lecture:
“On the hypotheses which lie at the foundations of
geometry”
Over few decades, led to Absolute
Calculus:
Differential
Christoffel, Ricci, Levi-Civita, Bianchi, Beltrami
1907 – 1915 : General Theory of Relativity created
by Einstein using Riemannian Geometry.
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(b) Non-Commutative algebra
Hamilton, Cayley, Sylvester, Grassmann — matrices, higher
dimensional spaces … Hilbert space and linear
transformations.
1925 – 26 : birth of quantum mechanics – Hilbert spaces and
operator theory most appropriate framework
"Non-Euclidean geometry and non-commutative algebra,
which were at one time considered to be purely fictions of
the mind and pastimes for logical thinkers, have now been
found to be very necessary for the description of general
facts of the physical world".
─ Dirac (1931)
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(c) Groups and Symmetry
Group concept – Lagrange, Galois, … , Lie –
First discrete, then continuous
Greatest gift of 19th century mathematics to 20th century physics.
Perfect language to describe symmetries of physical systems,
their consequences, especially within quantum mechanics Poincarè, Einstein, Noether’s theorem of 1918.
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Mathematical formulation before physical understanding – Examples
(a) Newton’s Law of Universal Gravitation
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Newton to Richard Bentley in 1692 -93
“That one body may act upon another at a distance through
a vacuum, without the mediation of anything else, by and
through which their action and force may be conveyed from
one to another, is to me so great an absurdity, that I believe
no man, who has in philosophical matters a competent
faculty of thinking, can ever fall into.”
Wigner even more forceful
“Philosophically, the law of gravitation as formulated by
Newton was repugnant to his time and to himself.”
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(b) Maxwell’s equations for electromagnetism - 1865
(a)
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Maxwell’s belief in need for the aether- given up only after Special
Relativity in 1905
Dirac in 1927 – application of principles of quantum
mechanics to Maxwell’s equations – structure retained,
meaning completely changed.
"One cannot escape the feeling that these mathematical
formulae have an independent existence and an intelligence
of their own, that they are wiser than we are, wiser even than
their discoverers, that we get more out of them than was
originally put into them.“
— Heinrich Hertz
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(c) The Case of the Dirac Equation 1928
Result of combining quantum mechanics and special
relativity. Introduced Spinors into relativistic physics.
Unexpected Successes – electron spin, anomalous magnetic
moment, hydrogen fine structure, positrons and antimatter.
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"… the difficulties…concerning the problem of
'physical content versus mathematical form'…the
greatest difficulties did not lie in the mathematics, but
at the point where the mathematics had to be linked to
nature. In the end, after all, we wanted to describe
nature, and not just do mathematics.“
— Werner Heisenberg
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Some Significant Lessons
(a) Newton’s ‘hypotheses non fingo’
‘Newton… still believed that the basic concepts and laws of his
system could be derived from experience; his phrase
‘hypotheses non fingo’ can only be interpreted in this sense’
— Einstein (1933)
The effort needed to go from experience to new concepts for
physics has increased enormously.
Einstein (1933)
“… the axiomatic basis of theoretical physics cannot be
abstracted from experience but must be freely invented.”
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(b) Mathematical formulation of physical laws ‘touches’
physical reality.
Wigner : "… the mathematical formulation of the physicist's
often crude experience leads in an uncanny number of cases
to an amazingly accurate description of a large class of
phenomena. This shows that the mathematical language has
more to commend it than being the only language which we
can speak: it shows that it is, in a very real sense, the correct
language".
Heisenberg : "If nature leads us to mathematical forms of
great simplicity and beauty that no one has previously
encountered , we cannot help thinking that they are 'true',
that they reveal a genuine feature of nature.“
(c) Physical interpretation always evolving, never final.
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Views on the nature of mathematics and mathematical knowledge
Platonic realist school versus constructivist school: Descartes,
Newton, Leibnitz, Hermite, Cantor, Godel, Hardy…
Versus
Kronecker, Poincare’, Brouwer…
Mathematics as a part of human language with precision,
conciseness, efficient manipulative power.
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End of 18th Century, attempt by Immanuel Kant to
explain the successes
of Galilean – Newtonian
approach. Profound theory of nature of human
knowledge, bringing together rationalist and empiricist
schools of philosophy.
Distinction between a priori and a posteriori forms of
knowledge.
Included among the (synthetic) a priori – Euclidean
geometry of space, simultaneity of events in time,
determinism, permanence of objects, law of mass
conservation, Newton's Third Law of Motion.
Later progress in physics and mathematics showed
need to revise many of the Kantian a priori's.
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David Hilbert in 1930
" . . .the most general basic thought of Kant's theory of knowledge
retains its importance . . . But the line between that which we
possess a priori and that for which experience is necessary must be
drawn differently . . . Kant has greatly overestimated the role and
the extent of the a priori . . . Kant's a priori theory contains
anthropomorphic dross from which it must be freed. After we
remove that, only that a priori will remain which also is the
foundation of pure mathematical knowledge."
Einstein
" I am convinced that the philosophers have had a harmful effect
upon the progress of scientific thinking in removing certain
fundamental concepts from the domain of empiricism where they
are under our control, to the intangible heights of the a priori.”
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Konrad Lorenz in 1940’s, Max Delbruck in 1970’s —
explanation of the origin of the Kantian a priori. Based on
Darwinian evolution guided by natural selection, role of
“World of Middle Dimensions.”
Phylogenetic Learning – by the species
Ontogenetic Learning – by each individual
Delbruck
" . . . two kinds of learning are involved in our dealing with
the world. One is phylogenetic learning . . . during evolution
we have evolved very sophisticated machinery for perceiving
and making inferences about a real world . . . what is a priori
for the individual is a posteriori for the species. The second
kind of learning involved in dealing with the world is
ontogenetic learning, namely the life long acquisition of
cultural, linguistic and scientific knowledge.“
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Can the capacity for mathematical thinking be a result of
biological evolution , of phylogenetic learning ?
" This must be ascribed to some mathematical quality in Nature, a
quality which the casual observer of Nature would not suspect,
but which nevertheless plays an important role in Nature's
scheme.“
— Dirac (1939)
C. N. Yang in 1979 " At the fundamental conceptual level they
amazingly share some concepts, but . . . the life force of each
discipline runs along its own veins."
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Wigner : "The miracle of the appropriateness of the language
of mathematics for the formulation of the laws of physics is a
wonderful gift which we neither understand nor deserve".
Einstein : "The eternal mystery of the world is its
comprehensibility… The fact that it is comprehensible is a
miracle."