## Some visions of infinity from the past and our times

The great Hindu mathematician and astronomer Bhāskara-II’s work preserves a high-point of Hindu knowledge. His work contains ideas that are often seen as characterizing “modern” scientific understanding i.e. what in the west would seen as starting with Leibniz and Newton going down to our times. About 40 years from the time of Bhāskara’s death around 1185 CE we see the first irruptions of the monstrous Mohammedans in south India, initially spear-headed by the ghāzi sufi-s and then by then by the main army of the Sultanate. The first of these invasions were successfully repulsed by the yādava monarchs. We hear in their inscriptions from around that time titular statements such as:

turuṣka-kopa-pralaya-mahārṇava-magna-medinī-samuddharaṇa-mahā-varāha |
The great boar [the yādava monarch] who lifted up the world sunk by the deluge of the Islamic irruption.

gambhīrābhīra-pracaṇḍa … prarājya-rājya-turuṣkopaplava-medinī-samuddharaṇa-[kartṛ] |
The mighty fierce warrior of the cowherd race who lifted up the kingdom, frontiers, and the world from the Islamic irruption.

Despite their initial successes it appears that the yādava monarchs did not entirely understand the extent of the threat they were facing. This conspired with internal fissures in their polity and about 50 years from that point in time their kingdom came to a cataclysmic end at the hands of the Army of Islam. This marked the drastic cessation of creativity in many of the deeper aspects of Hindu knowledge. Thus, the high-point reached by Bhāskara-II remained that. Though Bhāskara continued to be studied and commented upon the creative expansion of the foundations laid by him mostly ceased in Hindudom other than in the heroics of the nambūtiri-s and their students in the south. This was mainly because of the Vijayanagaran empire reviving Hindudom in the south after overthrowing Mohammedanism in a series of brilliant campaigns. While this provided a refugium for the survival of Hindu knowledge, the spread of ideas and participation across the subcontinent which drove earlier Hindu creativity was no longer possible due to the truncation caused by the horrors of Mohammedanism.

In Bhāskara’s thought we see facets of something which fascinated us since childhood, though we never really understood it in completeness: the idea of infinities. Yet we talk about it for after all life is full of things that we can see and visualize but perhaps hardly “understand”. As the old Hindus would say we experience their pratyakṣa-s but may not realize their parokṣa. Hence, we use Bhāskara’s work as the starting point of some musings on this matter. In Bhāskara’s work we explicitly or implicitly encounter at least two forms of infinity that a typical student of the sciences grapples with by the time he enters the second decade of his life. One of these is the infinity associated with calculus and its counterpart the infinitesimal. The development of the apparatus of calculus had a long history in Hindu mathematics that is not yet fully known. We already see use of what we today call derivatives in the calculation of the motion of the Moon by the great astronomer Mañjula the Bharadvāja in his surviving work the Laghumānasa written around 932 CE. The rationale for this is presented by Bhāskara-II along with other mathematical operations that imply the knowledge of infinity of calculus and limits. This comes out clearly in his commentators like Munīśvara and the nambūtiri-s and their students. A second infinity is an algebraic infinity that Bhāskara lays out in his bīja-gaṇita on the six-fold operations of zero.

vadha-ādau viyat khasya kham khena ghāte kha-hāro bhavet khena bhaktaś ca rāśiḥ | 2.18
In multiplication etc of a number by zero the result is zero, in the multiplication by zero of a number the result is zero [the transitivity of zero multiplication]. The division of a number by zero results in [an infinity called] khahara.

dvi-ghnaṃ tri-hṛt khaṃ kha-hṛtaṃ trayaṃ ca śūnyasya vargaṃ vada me padaṃ ca | 2.19
$0 \times 2; \frac{0}{3}; \frac{3}{0}; 0^2; \sqrt{0}$ tell me [the answers]

asmin vikāraḥ kha-hare na rāśāv api praviṣṭeṣv api niḥsṛteṣu |
bahuṣv api syāl laya-sṛṣṭi-kāle anante acyute bhūta-gaṇeṣu yad-vat || 2.20

In this number khahara there is no change from addition to or subtraction from it of many quantities even as at the time of destruction and emission there is no change in Ananta (Saṃkarṣaṇa) or Acyuta (Vāsudeva) [by the absorption or emission] of material beings.

There are two aspects that are striking in this account of the khahara. The first of these relates to how it is handled in algebra. We encounter this in a simple equation which Bhāskara provides to test the student for this concept:

kaḥ khena vihṛto rāśiḥ koṭyā yukto (a)thavo(ū)nitaḥ |
vargitaḥ svapadenāḍhyaḥ khaguṇo navatir bhavet ||
Which number divided by zero, then increased or reduced by crore, then squared, then increased by its own square root, then multiplied by zero becomes 90.
Now let us unpack this using $\circledcirc$ for kha and $\bowtie$ for khahara of Bhāskara:
$x\\ \frac{x}{\circledcirc}=x\cdot \bowtie \\ x\cdot \bowtie \pm 10^8 = x\cdot \bowtie\\ \left (x\cdot \bowtie \right)^2=x^2 \cdot \bowtie \\ x^2 \cdot \bowtie + \sqrt{x^2 \cdot \bowtie} = (x^2+x) \cdot \bowtie\\ (x^2+x) \cdot \bowtie \cdot \circledcirc = (x^2+x) \cdot 1 = 90\\ x^2+x-90=0\\ \therefore x=9; x=-10$

The positive root is $x=9$ is the preferred answer of the Hindus as rāśi is taken to mean positive quantity. If one observes this operation carefully, the reduction to an ordinary quadratic would not happen if one were to treat the kha and the khahara as the zero and infinity of limits as in calculus. It would blow up to infinity and not leave behind a solvable quadratic residue. This happens only if one strictly adheres to Bhāskara’s prescription regarding khahara. A further discussion of such problems can be found in Avinash Sathaye’s work on Hindu infinities.

This brings us to the second striking aspect of the account of khahara: Its philosophical implication(s), especially given that Bhāskara goes out of his way to use pāñcaratrika vaiṣṇava terminology to explain khahara. Pāñcaratrika tradition uses the concept of large numbers to illustrate the idea of the emission of the universe from the Lakṣmī-Nārāyaṇa continuum. For instance the Lakṣmī-tantra postulates that the whole universe has emerged from a particle that is merely $10^{-18}$ the quantity of the Lakṣmī-Nārāyaṇa continuum. Thus, it tries to illustrate that emission of something as huge as the universe does not make much of a change in the continuum. Bhāskara too uses such huge numbers but he sees them like other good Hindu astronomers of the age as merely large finite numbers. For example he states a hypothesis that the diameter of the universe is $4.69947571\times 10^{16}$ km. He is not sure if the hypothesis is correct and adds that it might merely be the distance that light from the sun has travelled rather than the actual size of the universe (while this is a large number it is still short of the modern estimate of the estimate of the diameter of the visible universe at $8.8 \times 10^{23}$ km). Bhāskara’s concept of khahara is not merely a huge number but a special kind of infinity where operations like addition, subtraction, squaring etc have completely eroded. Thus, it may be visualized as the entity that restores the balance caused by the introduction of zero — a principle of conservation to the number system which is shattered by the “black hole” of zero i.e. kha, viyat, śūnya. This conservation principle almost has a paradox in it because after all the example he uses is of things being emitted and absorbed by Viṣṇu without making any difference to the god. But once one sees the other pole of this system as the abyss of kha it becomes consistent. Moreover, Bhāskara was not the only scientist to see philosophical implications for zero and infinity. Such an allusion is also made by the mathematician Nārāyaṇa paṇḍita as pointed out by Sūryadāsa:

evam etat prasaṃgena sāhityo(u)ktyā nārāyaṇo’api svakṛta-bīje nirūpayāṃ cakāra yathā |
In this way Nārāyaṇa had also defined [note use of periphrastic perfect] this in his own algebra by means of a poetic utterance:

śūnyābhyāsa vaśāt-khatām upagupto rāśiḥ punaḥ svaddhṛto
‘apy āvṛttiṃ punar eva tanmayatayā na prāktanīṃ gacchati |
ātmābhyāsa-vaśād ananyam amalaṃ cid-rūpam ānandaṃ
prāpya brahma-padaṃ na saṃsṛtipathaṃ yogī garīyān ive(a)(i)ti |

Under the operation of multiplication by zero the number vanishes (literally: is concealed by) into zeroness. But again when divided by zero it gets absorbed in that (khahara) and does not return to its previous (finite) state, even as a respectable yogin through the practice of self-realization having attained the unique bliss of pure-consciousness, free from impurities, does not return to the path of union (with matter).

Nārāyaṇa’s analogy is one of yoga. We posit that implicit in this again is a philosophical conservation principle: The destruction of finite number by zero is offset by the its absorption into the khahara. In the sense of a yogin the reduction of this material “mala” to zero (state of amala) is offset by his becoming infinite by oneness with the universal consciousness.

The philosophical interpretations of the algebraic infinity in Bhāskara and Nārāyaṇa may be linked comparable concepts from earlier Hindu tradition. In the context of Bhāskara’s khahara his commentator viloma-kavi Sūryadāsa cites the verse of Bhīṣma from the Mahābhārata regarding the Vāsudeva the supreme god of the pāñcaratrika-s:
yataḥ sarvāṇi bhūtāni bhavanty ādi yugāgame |
yasmiṃś ca pralayaṃ yānti punar eva yuga-kṣaye ||
From whom all entities come into being at the begining of the first yuga and also into whom all go to dissolution at the end of the yuga-cycle.

Further one may also point to a much earlier, famous mantra from the Śukla-yajurveda that contains a comparable concept:
oṃ pūrṇam adaḥ pūrṇam idaṃ pūrṇāt pūrṇam udacyate |
pūrṇasya pūrṇam ādāya pūrṇam evāvaśiṣyate ||
That is the whole; this is whole; from the whole comes out a whole.
When the whole is removed from the whole what is left is also the whole.

We had learned this mantra as the concluding chant of the only upaniṣat of the Śukla-Yajurvedin-s whose recitation our teacher taught. Perhaps we were subliminally influenced by what we had just learned as we wandered to ironically attend a class on partial derivatives in our college. The attendance of that mathematics class was always sparse but that particular teacher, a good Hindu nationalist, was one of the few who had a soft-corner for us; hence, we saw it as an obligation to return the favor by sitting in his class. As we waited for the class to begin we stared out from terrace of the mathematics department into an open patch of untamed wild life that still existed in those days of our youth. That patch often filled us with a remarkable visions of natural selection action. But that day had begun early for us and we lapsed into a hypnogogia: We witnessed two visions. One was somewhat tame: We found ourselves on an endless staircase, with a serially increasing integer on each step. We kept walking up that staircase much like the escalator at the train station next to our workplace. We kept walking and walking but the stairs and the numbers on them never seemed to end. If we looked back we could see the stairs endless leading down to a great dark abyss whose bottom could not be seen. If we looked up the stairs endlessly led to the realm of skyblue light that never seemed closer how much ever we walked. Our legs seemed to hurt but with manly intent we labored on and on with no respite. It was then that we realized that this is how the ordinary infinity of arithmetic manifests — the endless realm of numbers with neither a beginning nor an end. If the old jñānātman Bhāskara compared it to Viṣṇu, the vision that came to us was that of the śaiva-s — of manifestation of their prime god Rudra as the liṅgodbhava-mūrti. Suddenly we snapped out of that vision. A new vision filled our eyes. We saw a stairway in the distance with an oddly distributed size-pattern for the stairs it contained. We looked at it more closely and each stair seemed to be made up of smaller stairs arranged like the whole stairway itself. Now when we looked at each stair of those miniature stairways on a single stair of the stairway we saw more miniature stairways in them. Our vision kept zooming and there was a whole in every whole and whole within it. The physicality of the vision faded and it looked more and more like a plot of a peculiar 2D mathematical function. We raked our minds to think what might be the equation of such a function but it evaded us. Since, after all a mathematics class lay before us, we asked the teacher after it was over if he might have some suggestions. He lent us a Russian book and asked us to see if might help us in our quest.

With aid from that volume we arrived at the function known as the Devil’s staircase. Our versions of it were slightly different from the traditionally defined one which we learnt off. Let us define various $\textrm{Dscf}(x)$ thus:
$\textrm{Dscfa}(x)=\displaystyle\sum_{n=1}^\infty\dfrac{\lfloor nx\rfloor}{a^n}$; where $a>1$
The classic $\textrm{Dscf}(x)$ which we encountered in books is based on this version where $a=2$,
$\textrm{Dscf2}(x)=\displaystyle\sum_{n=1}^\infty\dfrac{\lfloor nx\rfloor}{2^n}$
We also defined other versions for $a=\phi$ where $\phi$ is the Golden ratio. Finally, we also define a variant version of the function $\textrm{Dscr}(x)$ where we used the round function $\lfloor x \rceil$ which rounds to the nearest whole number instead of the floor function in the above definition. This was what we used in our original definitions because the $\lfloor x \rceil$ was the intuitive way of producing the infinite staircase:
$\textrm{Dscfa}(x)=\displaystyle\sum_{n=1}^\infty\dfrac{\lfloor nx\rceil}{a^n}$; where $a>1$

In figure 1 we illustrate some of these functions which clearly showing the part as a whole concept inherent in these functions. A remarkable result pertaining to one of these functions is,
$\textrm{Dscf2}(\phi)=2+\cfrac{1}{2^{F_0}+\cfrac{1}{2^{F_1}+\cfrac{1}{2^{F_2}+\cdots}}}$

where $F_{0:n}= 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55...$, i.e. the meru-prastha series.

The exploration of $\textrm{Dscf}(x)/\textrm{Dscr}(x)$ led us to another remarkable function we learnt of in an obscure pamphlet of mathematical illustrations (sadly whose name I forget by noted Indian mathematician and politician: Badri Nath Prasad). This function was discovered by the famous mathematician Hermann Minkowski, the teacher of Albert Einstein. Minkowski was among those advocating, what we as heathens would term, a return to “Pythagorean” thinking by emphasizing mathematical beauty as a guiding principle for discovery of new physics. Fittingly, in his study on the geometry of numbers he presented the discovery of the question mark function, $\textrm{?}(x)$, which maps quadratic surds on to rational numbers! Let the number $x$ be defined as a continued fraction,
$x=a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+\cdots}}}$ then,

$\textrm{?}(x)=a_0+2\displaystyle\sum_{j=1}^n\dfrac{(-1)^{j+1}}{2^{a_1+a_2+a_3...+a_j}}$ where $n$ is a finite number if $x$ is rational and $n=\infty$ if $x$ is irrational.

Figure 2 shows that $\textrm{?}(x)$ has a structure similar to Devil’s staircases.

When our code plotted out Devil’s staircases and $\textrm{?}(x)$ successfully for the first time our vision had become a pratyakṣa and the apprehension of third type of infinity infinity was under way. This infinity, distinct from that of calculus and algebra, was one which embodied each part of the whole being a whole in itself. Thus, it seemed to be a mathematical reflex of the above mantra from the Śukla-yajurveda. While these mysterious functions were now pratyakṣa on our computer screen, thereby bringing into the apprehension of a lesser mortal like us what was only previously accessible to the mathematician, the question that confronted us was whether such an infinity had a place in the real world.

It immediately caught our attention that we could convert these monotonically increasing functions to periodic or quasi-periodic functions (Such a thing had been independently done by others for $\textrm{?}(x)$). For the Devil’s staircases we centered the functions on 2, which resulted in a saw-tooth function. Thus we get:
$\textrm{Ddiff2}(x)=2\textrm{Dscf2}(x)-4x+1$
$\textrm{Ddifr2}(x)=2\textrm{Dscr2}(x)-4x+1$
We can likewise define attenuated functions like,
$\textrm{Ddiff2},\phi(x)=2\textrm{Dscf}\phi(x)-4x+1$, etc
For Minkowski’s function we can define the periodic version simply as:
$\textrm{?dif}(x)=\textrm{?}(x)-x$
Now when we use these functions to define a polar curve e.g. $r=a\cdot\textrm{Ddiff2}(\theta)$ then we get the below fractal forms of great interest:

Figure 3 $r=a\cdot\textrm{Ddiff2}(\theta); 0 \le \theta le 3\pi$

Figure 4 $r=a\cdot\textrm{Ddifr2}(\theta); 0 \le \theta \le 2\pi$

Figure 5 $r=a\cdot\textrm{?dif}(\theta); 0 \le \theta \le 7$

Figure 6 $r=a\cdot\textrm{Ddiff2},\phi(\theta); 0 \le \theta \le 3\pi$

One notes that these forms bear resemblance to those that appear in nature in jagged leaves and petals (e.g. Dianthus), in inflorescences and petals. Thus, it appears that the biological processes that generate these natural forms are in some way trying to capture the geometry predicated by the nesting of wholes within a whole. However, as on the computer screen the biological forms are by no means totally scale-free facsimiles of the mathematical objects they resemble. The nesting has a certain lower bound beyond which the structure changes. Thus, in a sense they are approximate real world reflections of the Platonic forms of the third type of infinity — that of infinite nesting. While the actual biological process that result in the form might differ from case to case and the mechanistic process might not involve constructing and plotting a $\textrm{Ddiff2}(x)$ or a $\textrm{?dif}(x)$, the forms nevertheless converge to those dictated by such functions. Thus, this type of infinity, even more so than others, might impinge upon the real world. Indeed, perhaps, Yājñavalkya wished to capture the Platonic ideal behind such forms when he recited the above mantra of the Vājasaneyin-s. The realization of this infinity has since then had a deep influence on our way of looking at the world and experiencing beauty.