Mathematical Pathways in Mystical

Expedition

Presented byMayamma Joseph,

Faculty, Department of MathematicsChrist University, Bangalore

MwB-2011, Bangalore, January 08, 2011

Introduction

“God is an infinite circle whose centre is everywhere and whose circumference is nowhere”

“The book of nature is written in the language of mathematics”

– Galileo

"God does not play dice" – Albert Einstein

Introduction………contd.• None of the ‘definitions’ of mysticism seem to indicate any

explicit connection between mathematics and mysticism

• Then, how can we explore a theme like ‘mysticism without bounds’ with Mathematics as the back drop?

• Mystical experience- the immediate experience of the divine is not the outcome of any human effort

• Mysticism viewed as an expedition calls for the preparation of the explorer and it demands the disciplining of his/her body, mind, will , heart etc.

• Mathematics plays an important role in the disciplining of the mind.

Introduction………contd.

• Mathematics is a universal language• The present culture is to a great extent caught up in the truths of science,

which consistently uses mathematics as the primary language for both exploration and communication; which means mathematics is also an exquisite candidate for contemporary spiritual exploration.

• Hence , identifying mathematical pathways for mystical journey is relevant and significant for the ‘knowledge societies’ of today’s world.

• The Mathematical experience will help the individual to sense the mystical spark in his or her quest for truth beyond the realm of intellect.

• Yes; even a mathematical idea can be instrumental for a ‘mystical illumination’ or a channel for ‘mystical experience’ for the seeker.

To the roots of Mathematics

• Mathematics is one of the oldest intellectual instruments • Root word- ‘Mathemata’(Greek) which indicated any subject of

instruction or study. • The Pythagorians used the term Mathematics specifically for

arithmetic and geometry ( Burton,1991). • Originated from the practical problems that involved the use of

numbers- counting, recording of business transactions, calendar preparation etc.

• Beginning of learning Mathematics for its own sake – by Greeks (6th century B.C.). They theorized Mathematics rather than using it as a practical tool.

• Plato was instrumental in letting Mathematics reach the place of higher education.

Mathematics and the divine- a historical review

• A careful observation of the growth and development of Mathematics indicate that the ‘priestly class’ played a major role.

• “ When all the inventions had been discovered, the science which are not concerned with the pleasures and necessities of life were developed first in the lands where men began to have leisure. This is the reason why mathematics originated in Egypt, for there the priestly class was able to enjoy leisure”– Aristotle ( Metaphysics)

• Indian Mathematics originated in service of the religion- Sulbhasutras,(gives the measurement and construction of sacrificial altars). These had to be done with precision to ensure the effectiveness of the sacrifice. ( Joseph, 2000)

• The authors of Sulbhasutras - priests- craftsmen responsible for the construction / maintenance of the altars.

Construction of Sacrificial Altar- Example

"He who desires heaven is to construct a fire-altar in the form of a falcon…”

Mathematics and the Divine- Historical Review contd…

• Temples as institutions for acquiring and disseminating scientific knowledge

• Parallels between Church in Medieval Europe and Temple in Medieval Kerala.

• Younger sons of Namboodiri’s –members of Kerala school freed of all economic and family responsibilities because of the social system ( Makkathayam&Marumakkathayam) were a leisured class with few religious duties . They consistently pursued their interest in Mathematics and astronomy.

• Study of infinite series by Kerala school- indicator to the study of mathematics for non-utilitarian purposes.

Mathematics and the Divine- Perspectives of

Mathematicians

‘Mathematics and the Divine’Perspectives of Mathematicians

Pythagoreans(5th-4th century BC)

What set the Pythagoreans apart from the other sects was the philosophy that “knowledge is the greatest purification”, and to them knowledge meant mathematics. Never before or since has mathematics had such an essential part in life and religion as it did with the Pythagoreans. At the heart of their scheme of things was the belief that some sort of an operative reality existed behind the phenomena of nature and that through the volition of this supreme architect , the universe was created- that beneath the apparent multiplicity and confusion, of the world around us there was a fundamental simplicity and stability that reason might discover. ( Burton,1991)

Plato ( 429-347 BC)

Plato, who was basically a philosopher could see the constantly guiding force of mathematical reason in shapes, forms, planets, music etc. In fact the prominence of the theme of Mathematics is obvious from the inscription he is supposed to have given at the entrance of the door to his academy:

‘‘Let no one ignorant of mathematics enter here.’’

Plato related Mathematics to the divine in two different ways.

1. Mathematics as used by the ‘divine’ in fashioning the

world

2.knowledge of Mathematics is an important step on the pathway to knowing forms- eternal intelligible objects.

Platonic solids

Nicholas of Cusa (1401-1464)

Mathematics played an important role in His thoughts.

“ Infinity may be unknowable directly for our reason, but there are means to get to know it and it is Mathematics that gives us these means. Consideration of Mathematical properties can open up a way to infinite” and hence to a mystical illumination.

Make the radius of the circle is made larger and larger, then difference between the circle and the line also disappear. The infinite line and the infinite circle thus becomes identical.

Cusanus’s vew

• Everything that is possible or potential in the domain of finite figures becomes actual in the case of corresponding infinite figures.

Eg.

Circle can be traced from a single line segment and circle will itself generate a sphere

Therefore, the finite line contains the sphere potentially and what is potentially true for a finite segment becomes actual reality for an infinite line; so the infinite line is triangle, circle and sphere and is also identifiable with all other geometrical figures.

In a general way, an infinite geometrical figure is identical with all of space because the infinite can lack nothing and hence all infinite figures coincide with one another. By going to the limit of theological figures, the human mind can get a feeling of the genuine coincidence of opposites as it exists in God.

….a mystical journey!!!

Georg Cantor (1845–1918)

• Mathematics serves metaphysics and religion.

• Mathematical research corresponds to considerations on creation, and its results are therefore steps towards God.

• “Every extension of our insight into the origin of the creatively-possible therefore must lead to an extension of our knowledge of God”.

• “Reason alone cannot decide the foundation of science, we need some grain of metaphysics which is supplied by God’s existence.”

( eg. What is a point? We do not know. A point is that which has no parts-

(Euclid)… then, what do we mean by part? )

• There are sets containing infinite objects.

Cantor…contd.

“Far be it from me to take credit personally for my discoveries.

I am merely the tool of a higher power, that will pursue its course when I am gone, even as it revealed itself thousands of years ago in Euclid and Archimedes.” (Cantor)

Isn’t the theme similar to the following lines by Mechthild of Magdeburg?

“Unless thou lead me, Lord, I cannot dance;

Would’st thou have me leap and spring,

Thou thyself, dear Lord must sing,

So shall I spring into thy love,

From thy love to understanding,

From understanding to delight,

Then, soaring human thought far, far above,

There circling will I dwell, and taste encircling love”

…an experience of union with God, the truth!

Srinivasa Ramanujan (1887-1920)

• He was intensely religious.

• He often united mathematics and spirituality together.

• Zero represented Absolute Reality, and that infinity represented the many manifestations of that Reality.

• Each mathematical discovery was a step closer to understanding the spiritual universe.

• "An equation for me has no meaning unless it expresses a thought of God."

• He could never explain the way he arrived at his deep insights in mathematical terms.

• According to him, many of his discoveries came through dreams, from the goddess Namakkal

Math blended with mysticism- Goethic arches

Mathematical concepts for vision

beyond comprehension

Mobius Transformations- Truly mystical!

Apparantly finite region having 1-1 correspondence with an infinite region!!!

Isn’t this mystical?

Infinite curve with finite length!

Logarithmic spiral (r = aeθ cot b)

Already…not yet !!! Curve appears to meet (0,0); but not really

The topologist’s sine curvef(x)=sin(1/x)

A to A’; B to B’…N to infinity???

Stereo Graphic Projection

Meditate Mathematically!!!

Sriyantra

Mystical creativity

“ To see the world in a grain of sand

And heaven in a wild flower;

Hold infinity in the palm of your hand

And eternity in an hour”

( William Blake)

Journey from Simplicity to complexity made simple!

.

Fractals generated using Mathematica 4.0 (http://www.bugman123.com/Fractals/index.html#DLA)

Newton-Raphson Fractal

f(z) = z5-1, zn+1 = zn - f(zn)/f'(zn)

Golden Ratio Spiral Orbit Trap

Diffusion Limited Aggregation (DLA) Fractal

Apollonian Gasket

Fractals in nature

We all stand on one ground- The Holy Ground

That holds us together and That cannot be separated

References

1. Brown, James R. Philosophy of Mathematics. London: Routledge, 2005.

2. Burton, David M. History of Mathematics. Chicago: WCB, 1995.

3. Byers, William . How Mathematicians think. Princeton: Princeton UP, 2010.

4. Hawking, Stephen. God created the integers. London:2005.

5. Hofstadter, Douglas R. Godel,Escher,Bach: an Eternal Goladen Braid. New York: Basic Books, 1999

6. Joseph, George G. The crest of the peacock: Non- European Roots of Mathematics. Princeton: Princeton UP, 2000.

7. Koetsier T. and Bergmans L. Mathematics and the divine: A Historical Study.Amsterdam: Elsevier , 2005.

8. Woods Richard, ed. Understanding Mysticism. New York: Image Books , 1980.

1. http://nrich.maths.org/6485

2. http://www.intmath.com

Thank you