And in answer to Euclid’s question of the ‘number of points in a line ?’ Galileo concludes:-
…..I shall ask you to tell me whether, in your opinion, a continuum is made up of a finite or of an infinite number of finite parts….My answer is that their number is both infinite and finite ; potentially infinite before division and actually finite after division ; because parts cannot be said to exist in a body which is not yet divided or at least marked out…..I think there is, between finite and infinite quantities, a third intermediate term which corresponds to every assigned number…..I grant, therefore to the philosophers, that the continuum contains as many finite parts as they please and I concede also that it contains them, either actually or potentially, as they may like…
◊◊◊◊ The amazing Argentinian writer Jorge Luis Borges [pronounced Bŏrhez](1899 – 1986) sums up our musings, so far, in an interesting short story titled ‘The Book of Sand’….. He comes to the point right at the beginning “The line is made up of an infinite number of points; the plane of an infinite number of lines; the volume of an infinite number of planes; the hypervolume of an infinite number of volumes……No unquestionably this is not – more geometrico – the best way of beginning my story.”
Borges uses the metaphor of an infinite book acquired by a salesman from India whose pages are uncountable like grains of sands….
“ ‘ I acquired the book in a town out on the plain in exchange for a handful of rupees and a Bible. Its owner did not know how to read. I suspect he saw the Book of Books as a talisman. He was of the lowest caste; nobody but other untouchables could tread his shadow without contamination. He told me his book was called the Book of Sand, because neither the book nor the sand has any beginning or end.’
The stranger asked me to find the first page.
I laid my left hand on the cover and, trying to put my thumb on the flyleaf, I opened the book. It was useless. Every time I tried, a number of pages came between the cover and my thumb. It was as if they kept growing from the book.
‘Now find the last page.’
Again I failed. In a voice that was not mine, I barely managed to stammer, ‘This can’t be.’
Still speaking in a low voice, the stranger said, ‘It can’t be, but it is. The number of pages in this book is no more or less than infinite. None is the first page, none the last. I don’t know why they’re numbered in this arbitrary way. Perhaps to suggest that the terms of an infinite series admit any number.’ ”
Borges touches upon space and time when he says…
“If space is infinite, we may be at any point in space. If time is infinite, we may be at any point in time.”
The author’s dilemma of analyzing this Book of Sand leads him to insomnia and he keeps looking for patterns until he realizes that there is no end to infinite possibilities. He then just ‘loses’ the book in a library.
◊◊◊◊◊ Few decades later, John Milton (1608 – 1674) as if gazing through Galileo’s invention, the telescope, wrote in ‘Paradise Lost’ :-
The golden Sun in splendor likest Heaven
Allur’d his eye : Thither his course he bends
Through the calm Firmament ; but up or downe
By center, or eccentric, hard to tell,
Or Longitude, where the great Luminarie
Alooff the vulgar Constellations thick,
That from his Lordly eye keep distance due,
Dispenses Light from farr ; they as they move
Thir Starry dance in numbers that compute
Days, months, and years, toward his all-chearing Lamp
Turn swift their various motions, or are turnd
By his magnetic beam, that gently warms
The Univers, and to each inward part
With gentle penetration, though unseen,
Shoots invisible virtue even to the deep :
So wondrously was set his Station bright.
Their lands the Fiend, a spot like which perhaps
Astronomer in the Sun’s lucent Orbe
Through his glaz’d Optic Tube yet never saw……
Here, the poet gazing at ‘the calm Firmament’ and its ‘center, or eccentric, hard to tell’ outlines the Lordly assignations between these two limits.
● And now we will see how bindu addresses this duality of nature :-
(1) We first look at the Brahmabindu Upanisad which literally means “esoteric instruction on the bindu that signifies the (higher) Brahman” . In the first shlöka the duality is said to emanate from the mind – the very seat of consciousness :-
The Mind, they say, is twofold, either impure or pure.
Impure, when it imagines desires; Pure, when it is free from desires.
The connotation is that desire is the first disturbance in consciousness and this can have two aspects, viz. desire in which the mind is involved, and the other is, desire in which the mind is not involved. So here this shlöka is pointing towards the idea of niśkāmkarm, or action without worrying about its fruit or end-result. We can also see the first hint of the ‘order-disorder’ dichotomy. (note: talk about ‘rhythms of nature’)
The literal word for dimension in Sanskrit is parimāṇ ( pirmaaNa ) . The suffix pari ( pir ) means ‘extensive’, ‘surrounding’ or in terms of quantity ‘substansive’ and mā again is ‘measure’. Thus parimān could apply to space, weight or in general to any measurable entity and this is to be achieved by division.
This is a natural stop for ‘Dimension – I’. We will look at the unfolding of the dimensions in the next talk and see how the concept of bindu first leads us to a duality and then a multiplicity that eventually encompasses this entire conscious universe – the māyā .