Mathematics is beautifulMathematics is useful

Of these two aspects, which one is more important?

by V. Kannan

University of Hyderabad

on18th August 2012

atVBIT

Part I : Beauty of MathematicsPart II : Usefulness of MathematicsPart III : Place of Mathematics among other

subjectsPart IV : Srinivasa RamanujanPart V : Views of scientists Feynmann, Dirac, etc

Beauty of Mathematics

1. Mathematics is needed to understand beauty.

2. Mathematics exhibits beauty.

3. Mathematics can be used to create beauty.

4. Mathematics is tested by its beauty.

5. Mathematics enhances the beauty of life.

Beauty of Mathematics

Mathematics is needed to understand beauty“To those who do not know mathematics it is

difficult to get across a real feeling as to the beauty, the deepest beauty, of nature ... If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in”

- Richard Feynman

Richard Phillips Feynman (1918-1988)

American physicist Nobel prize in physics 1965

Beauty of MathematicsMathematics can show beauty

“Mathematics, rightly viewed, possesses not only truth, but supreme beauty -- a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.”

Bertrand Russell (1872-1970)

Bertrand Arthur William Russell British philosopher

Born in 1872 Ravenscroft,Wales

Nobel Prize for Literature In 1950

Died in 1970 February 02 at Penrhyndeudraeth, Wales

Bertrand Russell

Beauty of Mathematics

“The mathematician does not study puremathematics because it is useful; he studies itbecause he delights in it and he delights in itbecause it is beautiful.”

J.H.Poincare (1854-1912)(cited in H.E.Huntley, The Divine Proportion)

French Mathematician1854-1912The Last Universalist

Jules Henri Poincaré

Beauty of Mathematics

Mathematics can create beautyMathematics can be used to create beautiful

designs. Of the many ways to do so, here are three:

1. Iterated Function Systems to create fractals.2. Julia Sets of quadratic polynomials.3. Mandelbrot set.

Barnsley Fern

Michael Barnsley

Sierpinski Gasket

Sierpinski

Julia set

Julia

Mandelbrot Set

Mandelbrot

Beauty of Mathematics

Mathematics is tested by beauty

“The mathematician's patterns, like the painter‘s or the poet's must be beautiful; the ideas, like the colors or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics.”

G. H. Hardy (1877 - 1947) A Mathematician's Apology, Cambridge University Press, 1994.

An English Mathematician1877-1947Fellow of the Royal Society

Godfrey Harold Hardy

Beauty of Mathematics

Mathematics enhances the beauty of life

Russell's paradox :Call the set of all sets that are not members of

themselves "R.“If R is a member of itself, then by definition it must

not be a member of itself.Similarly, if R is not a member of itself, then by

definition it must be a member of itself.Discovered by Bertrand Russell In 1910

Beauty of Mathematics

Russel’s paradox :

A barber who shaves all those men who do not shave themselves.

When one thinks about whether the barber should shave himself or not, the paradox begins to emerge.

Usefulness of Mathematics

“The range of applications of matrix eigen values problems is incredibly large.”

- p.376, Kreyszig, Advanced Engineering Mathematics

• Stretching of an elastic membrane• Markov Processes• Population models• Vibrating systems of two masses on two springs

Usefulness of Mathematics

Application of triple integrals to mass distribution.

Application of triple integrals to moment of inertia.

Application of divergence theorem to heat flow.Application of Fourier analysis to mechanics and

electrostatics.

Place of Mathematics among other subjects

Antiquity:Some subjects are old but extinct

Example : ArcherySome subjects are new :

Example : Computer ScienceSome subjects are both old and new :

Example : Medicine and Mathematics

Place of Mathematics among other subjects

List of subjects studied 5000 years ago:Four upavedas : (a)Dhanurveda : Archery

(b)Ayurveda : Medicine(c)Gandharvaveda : Music(d)Sthapatya veda :

Archeology

Place of Mathematics among other subjects

Six ancilary subjects :a) Phoneticsb) Grammarc) Prosody : Rules of poetryd) Etymologye) Astronomyf) Rituals

Place of Mathematics among other subjects

From other Vidyasthanas :a) Logicb) Meemamsa : Interpretation and

Analysisc) Purana : Mythologyd) Ethics

Place of Mathematics among other subjects

64 Kalas : Many are extinct now.

Place of Mathematics among other subjects

Three categories :1. Subjects that are for beauty, charm, elegance

e.g. : Music, fine arts2. Subjects that are for application, uses, welfare

e.g. : Engineering, technology3. Subjects that are for search of truth

e.g. : Astronomy, PhysicsWhere does Mathematics belong to ?

In all the three.

Srinivasa Ramanujan(1887-1920)

Lived in Kumbakonam

Contributed to Number theory and combinatorics

Srinivasa Ramanujan, native to India, was an extraordinary mathematician. A child prodigy, he was self-educated.

-G.D.Anderson, M.Vuorinen

“Every positive Integer was one of his (Ramanujan’s) personal friends”

-J.E.Littlewood.

“The greatest mathematician of the century” -Prof.Julian Huxley.

“A gift from heaven” -E.T.Bell.

The Door – Number ProblemThere is a street in which the houses are numbered

serially from 1 onwards. The total number of houses is roughly between 50 and 500. A mathematician lives in a house whose number has this peculiar property :

The sum of all numbers in the left side = The sum of all numbers in the right side Find the number of this house.REFERENCE : A 2009 film “God, zero and infinity”.

Ramanujan’s Answer :204i.e., 1+2+…+203 = 205+206+…288.

The next pair of numbers is 1189 &1681.

“I have never met his equal and I can compare him only with Euler and Jacobi.”

“I learnt from him much more than he learnt from me.”

-G.H.Hardy

“A single look at them (Ramanujan’s formulae) is enough to show that they could only be written down by a mathematician of the highest class”.

-G.H.Hardy

“A single look at them (Ramanujan’s formulae) is enough to show that they could only be written down by a mathematician of the highest class”.

-G.H.Hardy

Views of scientists

As time goes on, it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those which Nature has chosen.

- Paul A M Dirac

Views of scientists

It is more important to have beauty in one's equations than to have them fit experiment... It seems that if one is working from the point of view of getting beauty in one's equations, and if one has really a sound insight, one is on a sure line of progress.

- Paul A M Dirac

Views of scientists

The mathematician plays a game in which he himself invents the rules while the physicist plays a game in which the rules are provided by nature, but as time goes on it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those which nature has chosen.

- Paul A M Dirac

Views of scientists

Theoretical physicists accept the need for mathematical beauty as an act of faith... For example, the main reason why the theory of relativity is so universally accepted is its mathematical beauty.

- Paul A M Dirac

Views of scientists

For many parts of Nature can neither be invented with sufficient subtlety, nor demonstrated with sufficient perspicuity, nor accommodated unto use with sufficient dexterity, without the aid and intervening of the mathematics, of which sort are perspective, music, astronomy, cosmography, architecture, engineery, and divers others.

-Sir Francis Bacon

Views of scientists

The burden of the lecture is just to emphasize the fact that it is impossible to explain honestly the beauty of the laws of nature in a way that people can feel, without their having some deep understanding of mathematics. I am sorry, but this seems to be the case.

-Richard Feynman