The Concept of Bindu

Its not that this ‘moving universe’ or jagat has been created, in the sinful, blinking of an eye, growing hair-like from the skin-pore ; then how does it emanate – from the disturbance or ruination of the void of binduh?

This last shlöka is brilliant in its metaphors and invokes the idea of infinities without any math! It resonates with the old Greek dilemma – probably best expressed by Euclid (330 – 275 BC) when he tried to answer the question – “How many points are there in a ‘line segment’ ?” Now, a line is a ‘line’; it is continuous and how can we talk about it as ‘discrete’ points ? As we have seen above the eastern philosophy treats both the ‘point’ and the ‘continuous line’ belonging to the realm of brahma, the unknown. Euclid, however, dismisses this and in the very first definition of Elements defines a ‘point’ as – that which has no part [8]. And this is the beginning of geometry. Truly speaking, this was the start of Linear mathematics that has been built meticulously on such idealistic definitions and principles. Over the centuries, as various paradoxes emerged the western mind began looking at what they call Non – linear mathematics. As we will see, the practical Universe also lends itself more favourably to the latter approach. On the other hand, the eastern mindset first looked at the Universe and its rhythms and then built a language Prākrit (Sanskrit’s mother language), based on these resonances called shrutīs.
◊◊ F. Doestovsky explains this succinctly in “The Brothers Karamazov” while explaining his idea of God to his brother Alyosha, “as stupidly as he could” goes directly to the truth of things because “while intelligence wriggles and hides itself…. stupidity is honest and straightforward”. He first expresses his dilemma – “Yet there have been and still are geometricians and philosophers, and even some of the most distinguished, who doubt whether the whole universe, or to speak more widely, the whole of being, was only created in Euclid’s geometry; they even dare to dream that two parallel lines, which according to Euclid can never meet on earth, may meet somewhere in infinity. I have come to the conclusion that, since I can’t understand even that, I can’t expect to understand about God”. And then he concludes – “I believe in the underlying order and the meaning of life; I believe in the eternal harmony in which they say we shall one day be blended. I believe in the Word to Which the universe is striving, and which Itself was ‘with God,’ and Which Itself is God and so on, and so on, to infinity”…….. “Let me make it plain. I believe like a child that suffering will be healed and made up for, that all the humiliating absurdity of human contradictions will vanish like a pitiful mirage, like the despicable fabrication of the impotent and infinitely small Euclidian mind of man, that in the world’s finale, at the moment of eternal harmony, something so precious will come to pass that it will suffice for all hearts, for the comforting of all resentments, for the atonement of all crimes of humanity, ….”

◊◊◊ Galileo Galilei (1564 – 1642) addressed these paradoxes of the continuum with great insight –
(1) the paradox of two concentric circles having the same number of infinite points, is covered in Chap.1 on Infinities.
(2) we cover here his paradox of the ‘point and the round razor’. The text is hyperlinked. Galileo constructed a cylinder with the same height as its radius and then embedded two geometrical figures in it as shown separately in Fig.1

The cone shown in red uses the same base as the cylinder and its peak rises to the centre-point of its top surface. The hemisphere shown in blue ‘cups’ into the cylinder touching its base at the centre-point of the bottom surface. If ‘ r ’ is the radius of the base, then by definition ‘ r ’ is also the height and the volume of the cylinder = π r3. That of the cone is (⅓ base x height) = ⅓ π r3 ; the hemisphere’s volume is ⅔ π r3.Galileos-Round-Razor

Galileo compared the ‘cone’ shown in red with the ‘bowl’ i.e. the volume of the cylinder in the second figure with the blue hemisphere scooped out. Now, both these geometrical figures have the same volume viz. ⅓ π r3.

If now we ‘cut’ this cylinder at any place in between its height we cut the ‘cone’ in a smaller circle and the ‘bowl’ in an annular ring as shown in Fig.2 below :-Galileos-Round-Razor2Then geometrically the area of the smaller circle is the same as that of the annular ring. The simple geometrical proof is given in the attached article. In Galileo’s words :-

And as the cutting plane (here shown in green) moves upwards, coming near the top, the two solids (always equal) as well as their bases (areas which are also equal) finally vanish, one pair of them degenerating into the circumference of a circle, the other into a single point, namely the ‘round razor’- the upper edge of the bowl and the ‘apex’ of the cone. Now, since as these solids diminish equality is maintained between them up to the very last, we are justified in saying that, at the extreme and final end of this diminution, they are still equal and that one is not infinitely greater than the other. It appears therefore that we may equate the circumference of a large circle to a single point. And this which is true of the solids is true of the surfaces which form their bases ; for these also preserve equality between themselves throughout their diminution and in the end vanish, the one into the circumference of a circle, the other into a single point. Shall we not call them equal seeing that they are the last traces and remnants of equal magnitudes ?

Note also that, even if these vessels were large enough to contain immense celestial hemispheres, both there upper edges and apexes of the cones therein contained would always remain equal and would vanish, the former into ‘round razors’ having the dimensions of the largest celestial orbits, the latter into single ‘points’. Hence, in conformity with the preceding we may say that all circumferences of circles, however different, are equal to each other, and are each equal to a single point.