It talks about relation between Mathematics and Divine

1. Mathematical Pathways in
Mystical Expedition
Pr es ent ed by
Mayamma Joseph,
Faculty, Department of Mathematics
Christ University, Bangalore
MwB 2011, B
- angalore, January 08,
2011

2. 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

3. 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.

4. 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.

5. 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.

6. 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.

7. Construction of Sacrificial Altar- Example
"He who desires heaven is to construct a fire-altar in the form of a
falcon…”

8. 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.

9. ‘Mathematics and the Divine’
Mathematics and the Divine-
Perspectives of
Perspectives of
Mathematicians
Mathematicians

10. 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)

11. 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.

12. Platonic solids

13. 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.

14. 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!!!

15. 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.

16. 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!

17. 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

18. Math blended with mysticism- Goethic arches

19. Mathematical concepts for
vision
beyond comprehension

20. Mobius Transformations- Truly mystical!

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

22. Isn’t this mystical?

23. Infinite curve with finite length!
Logarithmic spiral (r = aeθ cot b)

24. Already…not yet !!! Curve appears to meet (0,0);
but not really
The topologist’s sine curve
f(x)=sin(1/x)

25. A to A’; B to B’…N to infinity???
Stereo Graphic Projection

26. Meditate Mathematically!!!
Sriyantra

27. 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)

28. Journey from Simplicity to complexity made simple!

29. .

30. 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)

31. Golden Ratio Spiral Orbit Trap

32. Diffusion Limited Aggregation (DLA) Fractal

33. Apollonian Gasket

34. Fractals in nature

35. We all stand on one ground-
The Holy Ground
That holds us together and
That cannot be separated

36. 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.

37. Thank you