-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/
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