In the universe one
world beneath cloud
foliage. In that world
a town. In the town
a house with a child,
who is blind, staring
over the edge of the universe
into the depth of love..
R.S. Thomas
I came across brilliant under noted article by William L. Hosch – on Blog Britannica, which I am reproducing with all the hyperlinks. Essentially it encapsulate underlying profound & mysterious connections between metaphysical poetry of William Black with Mathematics, calculus & physics..
Contemplation of Benoit Mandelbrot’s famous fractal oft brings to my mind the opening stanza of William Blake’s Auguries of Innocence:
To see a World in a Grain of Sand
And a Heaven in a Wild Flower,
Hold Infinity in the palm of your hand
And Eternity in an hour.
No matter how much one zooms in within the Mandelbrot set, intricate patterns can be discerned that bear a striking similarity to the whole set. (The selfsimilarity in the Mandelbrot set can be seen in this amazing video.) This inability to reach some simple structure or quiescence reminds me of the search for some final fundamental particle or superstring in physics. The application of ever greater amounts of energy seems to create just more particles out of the void.
Of course, one cannot really zoom forever in actuality. Both computer power and time constraints form natural limits. Such practical considerations, however, do not limit the mathematical contemplation of infinity. It wasn’t always so. For the ancient Greeks, infinity was anathema and they went to elaborate lengths to avoid it. When it did come up, as in Zeno’s paradox, it was often used to demonstrate some perceived fundamental philosophical flaw with the concept of infinity. In fact, the problem of infinity led Aristotle to differentiate between potential infinity (such as being able to count indefinitely) versus actual infinity (which he denied existed). Aristotle’s distinction lasted right down to the 20th century, with the exception of those who allowed for an infinite divinity.
Galileo was one of the first to notice that the set of counting numbers could be put in a onetoone correspondence with the apparently much smaller set of their squares. He similarly showed that the set of counting numbers and their doubles (i.e., the set of even numbers) could be paired up. Galileo concluded that “we cannot speak of infinite quantities as being the one greater or less than or equal to another.”
Such apparent contradictions did not long deter mathematicians from developing the study of instantaneous change and irregularlyshaped spaces, or the calculus (with infinitesimal increments). Of course, this created many new paradoxes involving infinity. For example, Gabriel’s Horn (also known as Torricelli’s Trumpet) is formed by rotating the curve y = 1/x for x > 1 about the xaxis. Remarkably, the volume of the resulting threedimensional figure is infinite, while the area of the twodimension surface of the horn is finite. In other words, Gabriel’s Horn could be completely filled with a finite quantity of paint, but no amount of paint would suffice to paint the horn’s surface. A modern example, known as the Banach–Tarski paradox, comes from set theory and requires the axiom of choice. In 1924 it was shown that a solid sphere can be decomposed into a finite number of pieces (originally 6, subsequently 5) that, when suitably moved about by rigid motions, can be reassembled into two new solid spheres, each of which is the same size as the original sphere. In essence, to put it another way, a billiard ball could be cut apart and put back together to fill the same space as the sun (or any other object, for that matter).
Contemplation of infinity always reminds me of the joke about the relative who thinks that he is a chicken. When asked why his family doesn’t get him some treatment, the response is that they need the eggs.
William L. Hosch – January 12th, 2007 (C) open.britannica.com
Blake, reason and the passions by W.B. Yeats The reason, and by the reason he meant deductions from the observations of the senses, binds us to mortality because it binds us to the senses, and divides us from each other by showing us our clashing interests; but imagination divides us from mortality by the immortality of beauty, and binds us to each other by opening the secret doors of all hearts. He cried again and again that everything that lives is holy, and that nothing is unholy except things that do not live lethargies, and cruelties, and timidities, and that denial of imagination which is the root they grew from in old times. Passions, because most living, are most holyand this was a scandalous paradox in his timeand man shall enter eternity borne upon their wings.
William Butler Yeats (18651969), from ‘William Blake and the Imagination
Blake’s Visions by W.H. Auden
Selfeducated WILLIAM BLAKE
Who threw his spectre in the like,
Broke off relations in a curse
With the Newtonian Universe.
But even as a child would pet
The tigers VOLTAIRE never met,
Took walks with them through Lambeth, and
Spoke to Isaiah in the Strand,
And heard inside each mortal thing
Its holy emanation sing.
Wystan Hugh Auden (190773), from ‘New Year Letter, Jan. 1, 1940
And God said, I will build a church here And cause this people to
worship me, And afflict them with poverty and sickness
In return for centuries of hard work
And patience. And its walls shall be hard as Their hearts,
and its windows let in the light Grudgingly, as their minds do,
and the priest’s words be drowned By the wind’s caterwauling.
All this I will do,Said God, and watch the bitterness in
their eyes Grow, and their lips suppurate with Their prayers.
And their women shall bring forth On my altar,
and I will choose the best Of them to be thrown back into the sea.
And that was only on one island.
R.S.Thomas

Connecting the dots..