Mathematics as the language of Mathematics as the language of Nature – a historical viewNature – a historical view

N. MukundaN. Mukunda

INSA Public Lecture – 29 September 2014INSA Public Lecture – 29 September 2014

I. IntroductionII. The Galilean-Newtonian traditionIII. Two Phases in the Mathematics –Physics RelationshipIV. Mathematical formulation before physical understandingV. Some significant lessonsVI. On the natures of mathematics and mathematical knowledgeVII. 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 Differential Calculus:

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

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."

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."

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