Mathematicians and Dreams

Some time ago I wrote about "Cybernetics and Dreams" and how the cipher to John Wesley's coded journal was revealed to Nehemiah Curnock in a dream.

One of the most extraordinary stories about mathematical revelations through dreams was celebrated recently at the 125th anniversary of the birth of the Indian mathematician, Srinivasa Ramanujan. Ramanujan, a mathematical genius and prodigy.

Photo: Wikipedia

Wikipedia says...

Srinivasa Ramanujan FRS (22 December 1887 - 26 April 1920) was an Indian mathematician and autodidact who, with almost no formal training in pure mathematics, made extraordinary contributions to mathematical analysis, number theory, infinite series, and continued fractions. ...
Born at Erode, Madras Presidency (now Tamil Nadu) in a poor Hindu Brahmin family, Ramanujan's introduction to formal mathematics began at age 10. He demonstrated a natural ability, and was given books on advanced trigonometry written by S. L. Loney that he mastered by the age of 12; he even discovered theorems of his own, and re-discovered Euler's identity independently. He demonstrated unusual mathematical skills at school, winning accolades and awards. By 17, Ramanujan had conducted his own mathematical research on Bernoulli numbers and the Euler-Mascheroni constant.

This remarkable man died at the age of 32. A Daily Mail article,  reported...

While on his death-bed in 1920, Ramanujan wrote a letter to his mentor, English mathematician G. H. Hardy, outlining several new mathematical functions never before heard of, along with a hunch about how they worked. 

Decades later, researchers say they've proved he was right - and that the formula could explain the behaviour of black holes.
....
Ramanujan, a devout Hindu, thought these patterns were revealed to him by the goddess Namagiri.

The Durban Sunday Tribune elaborated, crediting the Daily Mail report...

Genius saw formulas in dreams 

Researchers have finally solved the cryptic deathbed puzzle renowned Indian mathematician Srinivasa Ramanujan claimed came to him in dreams...outlining several new mathematical functions never before heard of... Ramanujan, a devout Hindu, thought these patterns were revealed to him by the goddess Namagiri.
...
Emory University mathematician said..... "No one was talking about black holes back in the 1920s when Ramanujan first came up with mock modular forms, and yet, his work may unlock secrets about them,"
...
The findings were presented... at the Ramanajun 125 conference at the University of Florida, ahead of the 125th anniversary of the mathematician's birth on December 22.

- Tribune Herald, 6 January 2013

Hinduism Today featured an article posted by Dr Jai Maharaj in which he wrote:

.... Ramanujan possessed supernatural intelligence, a well of genius that leaves even brilliant men dumb-founded.  Ramanujan was a meteor in the mathematics world of the World War I era.  Quiet, with dharmic sensibilities, yet his mind blazed with such intuitive improvisation that British colleagues at Cambridge -- the best math brains in England -- could not even guess where his ideas originated.  It irked them a bit that Ramanujan told friends the Hindu Goddess Namagiri whispered equations into his ear.  Today's mathematicians -- armed with supercomputers -- are still star-struck, and unable to solve many theorems the young man from India proved quickly by pencil and paper.   

Ramanujan spawned a zoo of mathematical creatures that delight, confound and humble his peers.  They call them "beautiful," "humble," "transcendent," and marvel how he reduced very complex terrain to simple shapes.   

In his day these equations were mainly pure mathematics, abstract computations that math sages often feel describe God's precise design for the cosmos.  While much of Ramanujan's work remains abstract, many of his theorems are now the mathematical power behind several 1990's disciplines in astrophysics, artificial intelligence and gas physics.

Who is Namagiri?

In Hindu mythology Namagiri is the consort of Narasimha, the lionman avatar of Vishnu. Narasimha symbolises the omnipresence of God. Like the Sphynx of Giza, he is the everpresent guardian, an Aslan figure.

Srinivasa Ramanujan attributed his mathematical findings to the goddess Namagiri. According to Ramanujan, she appeared in his dreams, proposing mathematical formulae, which Ramanujan would then have to verify. One such event was described by him as follows:

"While asleep, I had an unusual experience. There was a red screen formed by flowing blood, as it were. I was observing it. Suddenly a hand began to write on the screen. I became all attention. That hand wrote a number of elliptic integrals. They stuck to my mind. As soon as I woke up, I committed them to writing."

Dr. Jai Maharaj wrote:

"Debate still lingers as to the origins of Ramanujan's edifice of unique ideas.  Mathematicians eagerly acknowledge surprise states of intuition as the real breakthroughs, not logical deduction.  There is reticence to accept mystical overtones, though.... many can appreciate intuition “in the guise” of a Goddess.  But we have Ramanujan's own testimony of feminine whisperings from a Devi and there is the sheer power of his achievements.  Hindus cognize this reality."

Namagiri, the goddess of creativity, is akin to the Greek Muse, Urania, the muse of astronomy, science, and mathematics. As such she is an Anima figure inspiring, revealing, and unveling the mysteries of nature. Namagiri would be the feminine, guiding aspect, who can be thought of as a counterpart of the Jewish Hokmah or the Christian Sophia.

The genius of  dreams....let us pay attention to our dreams!

Also see: http://www.legacyoframanujan.com/index.html and http://www.hinduwisdom.info/quotes321_340.htm

Picture Credits: Wikipedia and Gustave Moreau, Hésiode_et_la_Muse

©Colin G Garvie HomePage: http://www.garvies.co.za

Ecstatically Factorial!

"Following in the way of the Ancients, we are in complete agreement with them in saying that, since there is no other approach to a knowledge of things divine than that of symbols, we cannot do better than use mathematical signs on account of their indestructible certitude."

-Nicholas of Cusa, Of Learned Ignorance, I.xi

Nowadays now that I am more prone to senior moments, a friend of mine speaks ecstatically about what I'll call "mathematical moments", moments when he's been raptured by the discovery of a new mathematical insight such as Mandelbrot Sets or the Golden Ratio. These are a cause of celebration for Kevin. Kevin becomes quite animated as he shares his story.

There was a young man from Trinity,

Who solved the square root of infinity.

While counting the digits,
He was seized by the fidgets,
Dropped science, and took up divinity.
                                 -Author Unknown

The other day Kevin shared one such exquisite moment when, in a flash of inexplicable insight, the elegance of factorials had dawned on him. For Kevin, it was a singular "red letter day" in his life. He cannot explain shy it should have affected him in such a profound way other than it seemed as if it was a moment in which the Universe fell into place for him. Something similar happened to me when once my lecturer in “Space Physics” remarked that the spiral was the key to creation. It was one of those rare "Aha!" moments.  Kevin admits he isn't even sure he fully understands what factorials are. Nor do I but he did get me to thinking that, if anything, such moments of awesome wonder might have something to do with the elegance and beauty of Factorials in particular and of Mathematics in general.

Davis and Hersh in their The Mathematical Experience explain that...

"The aesthetic appeal of mathematics, both in passive contemplation and in actual research pursuit, has been attested by many..." (p.168f)

Aristotle wrote:

"The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms of the beautiful." (Metaphysics, M 3, 1078 b)

Thomas Dubay says in his “have to read book”, The Evidential Power of Beauty, Science and Theology Meet, "...that mathematicians, at least the most alive of them, can burst into ecstatic joy over a newly discovered equation." (p.130)

Dirac went as far to say that it is more important to have beauty in one's equations than to have them fit the experiment! For some, such beauty might present itself in algebra. For others it could be the sublime theorems of Geometry. Trigonometry is the means of grace too. But for Kevin, it was Factorials.

The trouble with integers is that we have examined only the very small ones. Maybe all the exciting stuff happens at really big numbers, ones we can't even begin to think about in any very definite way. Our brains have evolved to get us out of the rain, find where the berries are, and keep us from getting killed. Our brains did not evolve to help us grasp really large numbers or to look at things in a hundred thousand dimensions. ~Ronald L. Graham

What is a factorial? In mathematics, the factorial of a positive integer n, denoted by n!, is the product of all positive integers less than or equal to n.  (http://en.wikipedia.org/wiki/Factorial See also: http://en.wikipedia.org/wiki/Factorial_number_system)

For example:

1! =1x1 = 1

2! =2x1 = 2

3! =3x2x1 = 6

4! =4x3x2x1 = 24

5! = 5x4x3x2x1 = 120

6! = 6x 5x4x3x2x1 = 720

Or we could express it this way:

The factorial of 4 is 4 times the factorial of 3.

The factorial of 3 is 3 times the factorial of 2.

The factorial of 2 is 2 times the factorial of 1.

The factorial of 1 is 1 times the factorial of 0.

Or:

4! = 4 x 3!

3! = 3 x 2!

2! = 2 x 1!

1! = 1 x 0!

Cause and Effect. The one is the consequence of what has gone before. This is called "recursion". Each new term is generated by recalling a particular function that has gone before. The factorial of a number is that number multiplied by the factorial of the number before.  Though factorials and fractals are not to be confused the processes of recursion  and iteration are similar. Fractals are patterns within patterns within patterns. Factorials are numbers.

"Everything in the universe exists because of a cause and effect relationship. Any thing you wish to examine exists as an effect something else that existed before it. .....you get the regression going back. But as in all recursions, the regression must stop so later "things" can exist," explains one writer. 0! can be thought of as the First Principle or Primal Cause.

Factorials occur in many business, engineering, and science calculations such as in "permutations", "combinations", and "calculus" . For example, permutations or the possible number of rearrangements of objects in relation to each other has long amused the human mind.

Wikipedia again:

The rule to determine the number of permutations of n objects was known in Hindu culture at least as early as around 1150: the Lilavati by the Indian mathematician Bhaskara II contains a passage that translates to:

The product of multiplication of the arithmetical series beginning and increasing by unity and continued to the number of places, will be the variations of number with specific figures.

Or consider this example. This is so elegant...

....where e is the mysterious, transcendental number

e = 2.718281828459045235360287471352662497757247... ...

that can never be fully resolved.

Common or Briggian Logarithms are calculated to the base 10, Natural or Napier Logarithms are calculated to the base e.

Mathematics is akin to an icon, a window into a deeper mystical essence. It could even be sacramental in a way, a means of grace...an outward and visible sign of an inward and invisible grace. As Davis and Hersh remind us, mathematics is a fit subject for thoughtful contemplation. It has evoked wonder and bliss in the hearts of many. Mathematics is a symbolic representation of the Universe of which we are an integral part.

To all of us who hold the Christian belief that God is truth, anything that is true is a fact about God, and mathematics is a branch of theology. ~Hilda Phoebe Hudson

Oh! I almost forgot, by definition, 0! ≡ 1. Go figure!

For further reading:
1. HE Huntley, The Divine Proportion













2. PJ Davis & R Hersh, The Mathematical Experience















3. CC Clawson, Mathematical Mysteries, The Beauty and Magic of Numbers














4. T Dubay, The Evidential Power of Beauty

 
 
 
 
 
 
 
 
 
 
 
 
 
©Colin G Garvie HomePage: http://www.garvies.co.za/