Making of a modern-day mantra-śāstra pamphlet

Originally, all mantra instructions were oral, keeping with the spirit of the śruti. But over time, starting probably around the Mauryan age, written manuals began to supplement the oral teachings. Thus, through the ages practitioners of the mantra-lore have made pamphlets for self-use, use by students or more general distribution. Until the English conquest, the Hindu, despite having familiarity with the printed mode of text dissemination, preferred the written mode. However, one of the signs of Hindu modernity was the production of printed and subsequently electronic texts. These mantra-manuals were originally written entirely in Sanskrit but with the decline of the use of ārya-speech outside the ritual, bilingual manuals started making their appearance. What we present here is one such bilingual manual. It is made taking advantage of TeX, the famous and innovative typesetting system of Donald Knuth, which allows you to make nice-looking documents. The TeX distribution used is MiKTeX2.9 and the typesetting engine is XeLaTeX. The text is input using the TeXworks environment, which comes with MiKTeX for editing and typesetting. The Devanagari and IAST conversion from an ITRANS input was achieved using the Sanscript converter (http://www.learnsanskrit.org/tools/sanscript). The font for the Devanagari script is the Siddhanta font of Mihail Bayaryn. The manual presented here is a reasonably comprehensive discourse on “The Sāvitrī and the upāsanā of the Deva Savitṛ” (pdf format). It is primarily aimed at a serious student with some familiarity with the śruti.

The source .tex file may be obtained here.

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Some words on mathematical truth, scientific conviction and the sociology of science

Sometime in the bronze age more than one group of humans, including our own Aryan ancestors, discovered that the squares of the two legs of a right triangle sum up to the square of the hypotenuse. This is the famed bhujā-koṭi-karṇa-nyāya, which remains true to this date in Euclidean space. In contrast, only a few of the scientific theories of the bronze age have survived in any form close to how they were originally proposed. Coeval with this momentous mathematical discovery, in the bronze age, most civilizations thought that the sun and the planets go round the Earth. Then a few millennia later the counter-hypothesis that the earth and the planets go around the sun took birth. But it took a long time for the older hypothesis to be falsified and the new one to take root. The new one stood the test of all subsequent falsifications but its actual form underwent many further modifications. This flow of the scientific process has been presented in its idealized formed by the Jewish intellectuals Popper and Kuhn. However, it should be kept in mind that the actual process of science rarely follows the post-facto idealized presentation. In any case, the primary lesson from this abstraction of the scientific process is that science is rather different from the mathematics in one matter.

A mathematical truth once discovered remains pretty much the same. This truth is established by what is termed as a proof in mathematics, which itself is based on an underlying set of axioms (for now we shall set aside the big issue of Gödel’s theorems). The form of the statement of such a mathematical truth, a theorem, might change over time due to the concept of “mathematical rigor” affecting the nature of the proof which is supplied for it; nevertheless, its essence remains pretty much the same. However, unless a scientific matter can be trivially reduced to an underlying mathematical theorem, there is no such truth in science as there is in mathematics. Instead, there are only falsifications and attempted falsifications. A scientific statement which survives all subsequent falsification attempts may be considered a scientific “truth”. More correctly, it may be considered a scientific conviction because, for the most part, it is established in a way quite different from the mathematical truth arrived at by the device of a proof.

Donkey
Figure 1

Yet, there is a basic similarity of a key process used in both mathematical and scientific discovery. The investigation begins with a body of observations. For example, one observes that whatever triangle one draws or conceives the sum of two of its sides is always greater than the third. This can be easily proved under the axioms of Euclidean geometry as in Figure 1 thereby becoming the mathematical truth, the Donkey’s theorem. In science too we begin in the same way by gathering a mass of observations. Then one makes a proposal to explain that mass of observations, which may be termed the scientific hypothesis. Here is where things get different between mathematics and science. The proposal is considered truly scientific only if it offers a specific “prediction”, which can then be tested usually by another set of observations. If these new observations falsify the original proposal, then the hypothesis is no longer considered as a valid one and a new proposal has to be sought to explain the observations. Now, scientific conviction regarding a hypothesis gets established by a large body of supporting empirical observations. This is quite contrary to mathematical proof. A large body of empirical observations supported Fermat’s last theorem, which was then finally proved. All observations within our current reach support the hunch that the logarithmic integral \textrm{Li}(x)> \pi(x) but Littlewood proved it to be false. Similarly, the Mertens conjecture regarding the value assumed by the Mertens function has been proven to be false but no current empirical observation has reached the point where it is really false. Thus, mathematical truth is very different from scientific conviction – a corresponding body of observations as those ‘testing’ the \textrm{Li}(x)> \pi(x) or Mertens conjectures would have made for a strong scientific hypothesis yet that body contributed nothing to the truth of the respective mathematical statements. In this regard it might be pointed out that the mathematicians tend to term their hunches or even well-tested but unproven convictions as conjectures. Some of these which are supported by a large body of downstream evidence but still remain unproven are dignified by the term ‘hypothesis’, e.g. the Riemann hypothesis regarding the connection between the Zeta function and the prime numbers. Finally, it should be stated that even when scientific conviction is established upon successful hypothesis-testing, underlying it is a probabilistic statement. This usually takes the form that given the body of testing observations, the chance of an alternative hypothesis as opposed to the chosen one explaining the observations is some low value.

Often, getting a valid body of observations is itself a limiting factor in science because one may or may not have had the technology in the first place to generate such observations. Further, even with the technology in place, the observation collection might have other practical roadblocks like the capacity of the human or machine observers. Thus, a big part of science is the collection of a clean body of observations – this is often overlooked in narratives privileging the hypothesis-creation step. The availability of technology again plays a central role in the testing of the hypothesis. The observation of gravitational waves or the Higgs boson are classic examples of this. The specific predictions were made a long time ago by the respective hypotheses in these examples. However, we needed all this time for technology to catch up to make the test of the hypothesis.

The role of the idea of proof in establishing mathematical truth, pioneered by the yavana thinkers, played a huge role in their thought process and also that of the traditions which borrowed from them like the Mohammedans and the later Europeans. Among the Hindus, a parallel concept of proof from a set of axioms developed from the linguistic tradition culminating in the work of the sages Pāṇini, Kātyāyana, and Patañjali. The great Pāṇini, after an expansive data-collection foray, created the clean data set of the gaṇapāṭha. This formed the basis of developing a system of proof for a linguistic observation based on certain axioms. As an example, let us take the word mahoraskaḥ meaning ‘he who has a broad chest’, which is a bahuvrīhi compound. How do you “prove” the formation of this compound word from the constitutive root words mahat and uras. Following Pāṇini you get the below proof.

mahat~su+uras~su-> mahat~su+uras~su+ka~p-> mahat+uras+ka-(ānmahataḥ…)-> mahā+uras+ka-> mahoraskaḥ |

Here,’~su’ is a Pāṇinian meta-element, much like the construction of the circle in the above proof of the Donkey’s theorem. It is indicated by Pāṇini’s sūtra: anekam-anya-padārthe | (2.2.24). Likewise, the ending is specified by a samāsānta-sūtra. In this case the uras~su triggers the samāsānta-sūtra: uraḥ prabhṛtibhyaḥ kap | (5.4.151), which brings in the ending and the meta-element ‘+ka~p’ for the ending. Once that has been docked to the terminal one applies the sūtra concerning the meta-elements: supo dhātu-prātipadikayoḥ | (2.4.71), which directs the deletion of the meta-elements. This then triggers a transformation of one or both of the combined elements by a samāsāśrayavidhiḥ. In this case, it is: ān-mahataḥ samānādhikaraṇa-jātīyayoḥ | (6.3.46) which causes a transformation of the mahat to mahā. Then it triggers the sandhi-sūtra-s, which in this case are akaḥ savarṇe dīrghaḥ | (6.1.101) and ādguṇaḥ | (6.1.87) which finally result in mahoraskaḥ (Footnote 1)

Thus, this system provides a means of “proving” the formation of a compound as per the Pāṇinian axioms.

While, as we saw above,  there is a distinction between scientific conviction and mathematical proof, the “hidden hand” of geometry underlies the establishment of a scientific conviction. In physics this is actually not so hidden – it might be directly operating via the reduction of the physics to an underlying mathematical expression. Alternatively, the types of hypotheses that can be created are seriously constrained by underlying geometric truths. This latter expression is also seen in chemistry to a great extent. In biology too we find that the geometric constraints of hypotheses to be a serious player, often but not always relating to the underlying chemistry. In fact, we go as far as to say that the geometric constraints layout even part of the basic axioms from which biology should be built. However, we posit that in biology a second underlying element is critical in constraining the hypothesis that can be formed. This takes the form of the grammatical structures similar to those analyzed by the school of Pāṇini in the analysis of the Sanskrit language. One may see this earlier note for some details (section: An ideal realm with a syllabary?). In conclusion, having an eye for these underlying geometric constraints and the parallel “linguistic” constraints allows one to formulate hypotheses that can produce genuine scientific convictions, especially in biology.

In practice, such an understanding regarding hypothesis-formation, while widespread among physicists and in large part among chemists, is not common among biologists. They have neither a clear idea of the foundational axioms nor the foundational theories of their science. They can still be effective at gathering data, but the pressure from the funding agencies for “hypothesis-driven science” has resulted in a fetish for poorly framed hypotheses or pseudo-hypotheses that are not really capable of producing genuine scientific convictions. However, biology, particularly its study at a molecular level, has drawn a lot of money due to its direct relationship to the human condition via the promise of medical advances. This money, like most other monetary incentives, is available in a competitive manner to biologists. With the competition for money comes the opportunity for winners to lead a life of mores, or even a larger than life existence with wide-ranging world travel at public expense. There are other non-monetary benefits – fame, and adulation via vanity articles in the popular press (e.g. note the vanity article on Voinnet, a French fake researcher in RNA biology in the Science magazine prior to his suspension for faking. He was also conferred some big award and one of his commenders even felt he should have been given the Nobel prize). The display of success in order to win the next round of funding is typically achieved through publications in certain prestige venues, like what the Chinese and the Koreans call CNS (the Cell journal and the magazine Nature and Science). Sometimes just raking up a large number of publications in other respected venues might also do the trick. The availability of big money also allows investigators in this field to run labs like sweatshops and lowers the bar for the employment, thereby letting in a body of less-discerning and/or less-intelligent people into the field. In fact, the widespread lack of foundational knowledge has allowed such individuals to even prosper widely – almost the equivalent of having physicists or engineers with a poor understanding of Newtonian mechanics. Moreover, the widespread lack of foundational knowledge leads to a tendency of it being better to be “vague rather than wrong” – an inverse of the correct scientific attitude (voiced by mathematical think Freeman Dyson): “it is better to be wrong than vague.” This manifests in molecular biology and allied fields like immunology in the form of an emphasis on phenomenology and vague models rather crisp biochemical predictions (of course on the other side there is also physics-envy manifesting in the form of worthless mathematicization that yields little biological insight). With such a system in place, we are left with an explosive situation – an unsurprising call to the only too human urge to cheat.

This cheating has taken two major forms: 1) rampant plagiarism; 2) production of fake results. The first is primarily a sociological problem arising from the urge to sequester all the spoils for oneself. However, it also feeds the extensive misrepresentation of scientific results and inflation of particular findings in order to gain an edge against competitors. Not surprisingly, it creates a rather unhealthy social system within science. The second is fundamentally damaging to the science itself for it fills the field with noise. This is compounded by both the drive to publish a large number of worthless papers and the fetish of peer review orchestrated by cartels which work as echo-chambers. As a result, it becomes difficult for the inbuilt corrective mechanisms of science to clean up the mess in piling mass of literature. While I have taken molecular biology as the centerpiece here, it appears that this is a more general problem. It might actually be even more rampant in fields like psychology and also the area of applied medical and nutritional research. This should not be just a cause of concern for the scientists in the field because 1) a lot of research is done on public money; 2) a lot of this research informs medical practice which directly impinges on the health of people; 3) unscrupulous practice in publicly funded science will seep through (via cartel formation) to commercial medical research and practice leading to more suffering for the patients – a striking example in recent times is that of the Italian ‘celebrity’ doctor who claimed to perform tracheal transplants only to end up consigning several of his patients to gruesome deaths; he was prone to faking his scientific results and credentials.

Is there a way out of this? At this moment that does not look easy to me. Very powerful people in Euro-American science are part and parcel of the problem. Those who have read this story of ours before will get a hint. The whole attitude within Euro-American science need to change and some of that has deep connections to the Abrahamistic undergirding of their culture. Sadly, the negatives are worsened by either the ‘gaming’ of or the imitation of the Euro-American system to different degrees by all the eastern nations (China, Korea, Japan, and India being the chief among them). In all this, we see the wisdom of father Manu that the brāhmaṇa’s ethic is needed for such pursuits and that the brāhmaṇa should keep a low-profile staying away from this business of feasting on adulation.


Footnote 1: This example was taken from a learned paṇḍitā Sowmya Krishnapur’s lecture on the bahuvṛīhi compound.

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A note on the least prime divisor sequences of 2p plus or minus 1

Let p be the sequence of prime numbers: 2, 3, 5, 7… Define the sequences q such that q[n]=2p[n]\pm 1. Then sequence f_1 is defined such that f_1[n] is the lowest prime divisor (LPD) of q[n]=2p[n]+1 and sequence f_2 is defined so that f_2[n] is the LPD of q[n]=2p[n]-1.

f_1: 5, 7, 11, 3, 23, 3, 5, 3, 47, 59…
f_2: 3, 5, 3, 13, 3, 5, 3, 37, 3, 3…

primes2p_plus_1_1Figure 1. A plot of the first 100 terms of f_1, f_2

We observe that for f_1 the successive record values (i.e. successive maxima), M_{2p\pm 1}, are what are called safe primes and the corresponding p[n] is a Sophie Germain prime. For f_1[n] \ge 11 these M_{2p\pm 1} values are primes of the form 12n-1. In the case of f_2 when f_2[n] \ge 13 the successive M_{2p\pm 1} values are primes of the form 12n+1. From Figure 1 we observe that though the record values keep rising for these sequences for most part they assume low values. Obviously, the lowest value it can take is 3. We also observe that frequency of the occurrence of the n^{th} prime in these sequences from 3 upwards keeps decreasing. Below we tabulate the frequencies for the first 10 primes in f_1, f_2 for n \le 25997. The 3th column has the frequencies of the first 10 primes in the sequence of LPDs of odd numbers 2n+1 up to some large n.

primes2p_plus_1_tab

primes2p_plus_1_2Figure 2. Frequencies of the first 100 primes in f_1, f_2 (blue and red). The frequencies of the first 100 primes in the sequence of LPDs of odd numbers up to some large n (cyan). The curve y=\tfrac{1}{2x^2} is shown in green for comparison.

From the above we see that the frequencies of the n^{th} primes in the sequences f_1, f_2 are very similar and likely to asymptotically converge to the same value. We can easily calculate the exact frequencies of the n^{th} prime in the sequence of LPDs of odd numbers in general: e.g. 3 will occur at fr=1/3; 5 will occur at fr=(1-1/3)\times 1/5=.13333; 7 will occur at fr= (1-.\overline{3}-.1\overline{3})\times 1/7 =0.07619048; 11 will occur at fr= (1-.\overline{3}-.1\overline{3}-0.07619048)\times 1/11 =0.04155844 and so on. Thus, we observe that the frequencies of the n^{th} prime in f_1, f_2 notably differ from the frequencies of the same in the sequence of LPDs of odd numbers in general. We have not figured out if there is a means of exactly calculating the frequencies of the n^{th} prime in f_1, f_2. Strangely, the first few frequencies are close to reciprocals of the sequence 2, 8, 16, 32, 41, 78, 90, 128, which relates to a certain co-primality triangle. While this make no sense at all to us, it is unclear if it is all chance or some relationship exists (see postscript).

We then investigated how exactly the record values of f_1[n], f_2[n] grow with n. This is shown in Figure 3.

primes2p_plus_1_3Figure 3. f_1, f_2 plotted to 25997 terms

Visual examination of the plot showed that the record values M_{2p\pm 1} grow very similarly in but f_1 and f_2 and they are bounded by a smooth curve that appears to be of the form y=k x \log(x), where k is some constant. The original Gaussian form of the prime number counting function can be written as (using the asymptotic notation):
\pi(x) \sim \dfrac{x}{\log(x)}
From this we can write the expression for the n{th} prime p_n thus:
p_n \sim n \log(n)
The record values of the LPDs of f_1,f_2 will be primes of the form 2p_n \pm 1. From this we can infer that that the record values of the two sequences M_{2p\pm 1} will be fitted by the curve:
y= 2x \log(x)

In Figure 3 this is plotted as the cyan curve. While this reasonably captures the behavior of of the bounding curve of M_{2p\pm 1}, it systematically falls short of it. As we have seen before, the above Gaussian form of the prime counting function is only a crude approximation, which Gauss and Dirichlet eventually replaced with the logarithmic integral \textrm{Li}(x). In this regard Rosser had proved long ago that p_n \ge n\log(n); hence, what we see is a direct consequence of this. Inspired by the work of Chebyshev and Riemann, the obscure Russian village mathematician I.M. Pervushin (Pervouchine) investigated an exact formula for the n^{th} prime using a table of 25997 primes (for numbers \le 3 \times 10^5), which is coincidentally the same as the number we used in our investigation. Consequently he arrived at the remarkable formula:

p_n \approx n\left(\log(n)+\log(\log(n))-1 +\dfrac{5\log(n)}{12}-\dfrac{1}{24\left(\log(n)\right)^2}\right)

This formula inspired Ernesto Cesàro to discover the more correct formula for the n^{th} prime:

p_n=n\Bigg(\log(n)+\log(\log(n))-1 +\dfrac{\log(\log(n))-2}{\log(n)}-\dfrac{\left(\log(\log(n))\right)^2-6\log(\log(n))+11}{2\left(\log(n)\right)^2}\\ + o\left(\dfrac{1}{\left(\log(n)\right)^2}\right) \Bigg)

Here, the small-o notation can be interpreted to mean that the final error term is negligible compared to \tfrac{1}{\left(\log(n)\right)^2}

Searching the literature, we found that recently Pierre Dusart had proved that

p_n \le n\left(\log(n)+\log(\log(n))-1 +\dfrac{\log(\log(n))-2}{\log(n)}\right), \; n \ge 688383

Thus, for large n the first 4 terms are sufficient. Hence, based on Cesàro’s formula we arrived at the approximate function for the behavior of M_{2p\pm 1}:

y=2x\left(\log(x)+\log(\log(x))-1 +\dfrac{\log(\log(x))-2}{\log(x)}\right)

This uses only the first 4 terms of Cesàro’s formula but gives us a good fit as seen by the red curve in Figure 3. In numerical terms for the largest prime in both f_1, f_2 for the first 25997 terms this approximation gives an error fraction of .002 suggesting that it is indeed a good one.


After we posted this note on Twitter we rather quickly heard back from an acquaintence on that forum about his solution for exact form of the frequencies of the n^{th} prime in the above sequences. You can read his excellent post here.

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A note on āmreḍita-s in the Ṛgveda and issues of word distribution

sa darśataśrīr atithir gṛhe-gṛhe
vane-vane śiśriye takvavīr iva ।
janaṃ-janaṃ janyo nāti manyate
viśa ā kṣeti viśyo viśaṃ-viśam ॥ RV 10.91.2 by Aruṇa Vaitahavya

He, with visible auspiciousness, a guest in house after house,
in forest after forest lurking like a hunting falcon,
people after people, no people are overlooked by him,
The clan among the clans, he dwells in clan after clan.

The āmreḍita or the iterative is a duplicated compound like those seen in the above ṛk: gṛhe-gṛhe etc. While the variety of āmreḍita-s seen in the RV is no longer seen in modern Sanskrit, some forms have persisted from that time e.g. punaḥ-punaḥ or anyam-anyam. Their variants are also seen in other extant Indo-Āryan languages. This form of a compound is attested to my knowledge in the earliest branching lineage of Indo-European, Anatolian. It is also seen in Tocharian which probably branched off next. Forms equivalent to the “pra-pra” (forward and forward) found in the RV are also attested in Homeric Greek and are termed prepositional complements of verbs. However, the āmreḍita, with the involvement of all elements of speech, is most developed only in Indo-Iranian. Old Iranian in the form of Avestan displays forms like:
nmānē-nmānē = Skt: dhāmni-dhāmni; house after house (i.e. every household)
vīsi-vīsi = Skt: viśe-viśe; clan after clan (i.e. every clan)

Given their parallels in Sanskrit, it is clear that this expanded system of āmreḍita -s was already present in the ancestral Indo-Iranian tradition. Here we briefly examine the āmreḍita-s in the Ṛgveda. Given the temporal position of the RV, it likely presents a picture of the usage of such compounds close to the beginning of the Indo-Iranian tradition.

There can be anywhere between 1 to 4 āmreḍita-s in a given ṛk. The number of ṛks-s with each number of āmreḍita-s per ṛk is shown in Table 1.

AMreDita_tabTable 1

The maximum number of āmreḍita-s in the RV is four, which is seen in a single ṛk, the one given in the opening lines of this article. By far the most common āmreḍita is dive-dive (day after day), which occurs 47 times in the RV (the next most common, the ancient pra-pra, occurs only 12 times). A related usage ahar-ahaḥ meaning the same as dive-dive occurs 6 times. Thus, the āmreḍita used in the sense of ‘daily occurrence’ seems to be a characteristic feature of the Vedic language. Another temporal āmreḍita is yuge-yuge (6x), which famously occurs in later literature in the Bhagavadgītā. Whereas in that text it means ‘eon after eon’, in the RV it likely means some version of the pañcasaṃvatsara-yuga, which is explained by Lagadha in the Vedāñga Jyotiṣa. Some other temporal āmreḍita-s are also used on rare occasions in the RV: māsi-māsi (every month) and parvaṇā-parvaṇā (every fortnight).

Other āmreḍita-s tell us about the old Āryan way of life and religion. Several of them indicate the sense ‘in every household’: dame-dame (6x), gṛhe-gṛhe/gṛhaṃ-gṛham (7x), vastor-vastoH (2x). Others denote the sense ‘in every clan’: viśe-viśe and variants (7x), jane-jane/janaṃ-janam/jāto-jātaḥ (5x). These meanings are shared with the most common Iranian āmreḍita-s, suggesting that they were significant for the Indo-Iranians at large. Specifically, these terms indicate that the households and the clans were the primary organizational units of society. There is no mention of towns or even villages in any āmreḍita-s although forests and trees are alluded to vane-vane/vṛkṣe-vṛkṣe. This is reflective of the semi-mobile life in the forest-steppe zone with its characteristic confederation of clans.

As we have noted before, the people of the RV were a warlike people. Thus, the āmreḍita-s meaning something like ‘in every battle’ or ‘in every contest’ are common: bhare-bhare (6x), vāje-vāje (4x), raṇe-raṇe (2x). There is also a śatroḥ-śatroḥ, i.e. ‘of every enemy’. This indicates that frequent military encounters were a feature of the lives of the early Indo-Āryans.

We also have many terms relating to the Vedic religion as would befit a primarily religious text. First, we have devaṃ-devam/devasya-devasya/devo-devaḥ (10x) implying the worship of every god in the pantheon. We also here of groups or troops of gods: gaṇaṃ-gaṇam/śardham-śardham (3x). These are clear expression of the polycentric polytheism of the Ārya-s. Second, the soma- pressing and drinking sessions, which are part of the high Vedic rite, are alluded to in the terms made-made/sute-sute/some-some (7x). Third, rites and related terms are referred to multiply. The fire ritual: yajñe-yajñe and variants (8x); the ritual fires: agnim-agnim (3x); Ritual observances and actions: vrātaṃ-vrātaṃ (2x), karman-karman and variants (3x); ritual offerings: havir-haviḥ, samit-samit (2x); Incantations, recitations and invocations: dhiyaṃ-dhiyaṃ, girā-girā, have-have, brahma-brahma (5x)

To better understand the distribution of āmreḍita-s in the RV we shall first take a detour to look at some basic statistics of the RV:

RV_stats_tabTable 2. Some basic numbers for RV including number of āmreḍita-s occurring per maṇḍala

Figure 1 shows the number of words per hemistich (ardhark) with the alternate hemistichs colored in blue and red. The maṇḍala boundaries are marked with vertical dotted lines.

AMreDita_words_half_RkFigure 1

This plot shows that the RV has a generally consistent average behavior across most maNDala-s keeping with the relatively tight repertoire of Chandas (meters) used by most clans. However, the anomalies are also immediately apparent. The maṇḍala-8 dominated by Kāṇva-s clearly stands out as having a distinct structure. Moreover, this pattern is also seen in the initial part of maṇḍala-1 where multiple Kāṇva-s are represented. Further, in maṇḍala-1 we see an anomalous spike corresponding to the ultra-long meter, the atyaṣṭi used by prince Parucchepa Daivodāsi. In maṇḍāla-9, the soma-maṇḍala we again see an anomaly. Here there is a strong separation of the short meters (gāyatrī-s) in the first part and the long meters closer to the end. This reflects this distinct aggregation history of the maṇḍala-9, which unlike the family books tends to collect the sūkta-s used in the soma ritual from different clans around a Kāśyapa core. These anomalies again come out clearly in the first two panels of Figure 2, which show the average number of words per hemistich and the average number of words per sūkta across the 10 maṇḍala-s.

AMreDita_plotsFigure 2 shows some of the statistics for the RV with relation to the āmreḍita distribution.

These features are related in part to peculiarities discussed by in our earlier analysis of related issues pertaining to the RV. We observe that the maṇḍala-s 8 and 9 have the lowest average number of words per hemistich, keeping with the dominance of the shorter meters in these maṇḍala-s (panel 1). However, we can see that Kāṇva-s tend to compose long sūkta-s; hence, they figure a higher number of words per sūkta on an average (Panel 2). However, in contrast, maṇḍala-9, which also has the lower average number of words per sūkta, has no such compensation and is dominated by short sūkta-s. These are peculiar to the soma ritual and the sāman-s composed on them. We also note that the Atri-s and Vasiṣṭha-s tend to compose more short sūkta-s than the others. Barring these anomalies, the RV is quite uniform, especially in terms of the average number of words per hemistich. This gives us the general background to investigate the distribution of āmreḍita-s.

Panel 3 for Figure 2 shows the āmreḍita-s in a sliding window of 2000 words through the length of the RV. The maṇḍala boundaries are marked by a vertical dotted line. We observe that the āmreḍita-s are not uniformly distributed. There are whole regions with a low count and others with notable spikes. We find that the maṇḍala-s differ in their use of āmreḍita-s (Panel 4 of Figure 2): maṇḍala-s 3 and 6 of the Vaiśvāmitra-s and Bhāradvāja-s are rich in āmreḍita-s, whereas those of Vāmadeva, the Vāsiṣṭha-s and the soma maṇḍala are particularly poor. In the case of the soma maṇḍala, the metrical structure with a low average number of words per hemistich probably discriminates against āmreḍita-s. However, in the case of the other maṇḍala the difference in āmreḍita counts is in spite of their mostly average behavior in terms of word count per hemistich (compare panel 4 and panel 1).

To understand if this difference might have any significance, we simulated the distribution of āmreḍita-s as a random process using the total number of āmreḍita-s in the RV (Table 2). We created 10000 artificial sets corresponding to the size of each maṇḍala, checked the number āmreḍita-s reached in each replicate and computed the Z-scores for the observed number of āmreḍita-s and the probability of getting the observed number or more/less by chance alone (Figure 3).

AMreDita_simulationFigure 3

This experiment suggests that maṇḍala-s 1 and 10 have more or less the average number of āmreḍita-s one would expect by chance alone. This probably reflects their composite nature rather than being the product of one dominant clan. However, maṇḍala-s 3 and 6 have greater than expected number āmreḍita-s (p=0.016 and 0.014 respectively), whereas maṇḍala-s 7 and 9 have lower than expected number of āmreḍita-s (p=0.035 and 0.037 respectively). This observation suggests there was possibly a conscious difference in the poetic styles of the Vaiśvāmitra-s and Bhāradvāja#-s on one hand and the Vāsiṣṭha-s on the other, with the former showing a predilection for the use of āmreḍita-s. This makes one wonder if the reduced use of āmreḍita-s by the Vasiṣṭha-s, who had some links to the Iranian side, represents a regional tendency also seen in the Avesta, which also uses a low number of āmreḍita-s. As noted above the unique structure of the soma-maṇḍala probably accounts for its low āmreḍitacount.

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The amazonian banana republic: the strī-rājya in Hindu tradition

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The śaiva tradition shows a dichotomy with respect to the role of the sex in ritual and purity. The earlier antimārga or pāśupata tradition focused on abstinence and the so-called “upward flow” or ūrdhvaretas. This indeed the underlying idea behind the ithyphallic depiction of Lakulīśa, the founder of one of the key atimārga traditions. However, within the śaiva tradition there was another ambivalent practice with earlier roots in the shared pool of ascetic practices, which were also inherited by the vaiṣṇava-s (e.g. Vaikhānasa-gṛhyasūtra). This was the asidhārā-vrata. Here the practitioner engages in kissing and coital contact with his wife or another beautiful and sexually active woman without spilling his seed. Successful practice of this for a certain fixed period is said to confer rewards on the practitioner. This practice continued within the śaiva-mantra-mārga both in the saiddhāntika (e.g. in the Mataṅga-pārameśvara tantra) and bhairava (e.g. Brahma-yāmala) streams. Thus, the practice was likened to walking on the sword-edge. Unlike this practice, which still emphasized the non-spilling of seed, among the practices within the bhairava-srotas of the mantramārga the full-fledged sexual ritual with actual ejaculation developed with many variations in doctrine and praxis. The founder of one of central traditions within this stream, namely the kaula tradition, was the siddha Matsyendranātha. A successor of his was another siddha Gorakṣa, who in certain late manifestations of the tradition is portrayed as superseding Matsyendra himself. This manifestation seemed to have been accompanied by a reversal to more abstinent practices and explicitly castigated the sexual activities of Matsyendra.

This is portrayed in famous story we narrated earlier, which is widespread in the eastern reflexes of the nātha tradition. Here, Matsyendra is described as going to a kingdom where only women existed, ruled by a female chief. It was termed the strirājya or Kadalirājya (the banana-kingdom). There Matsyendra engaged in sex with the queen and was about to die from total loss of vīrya, when his student Gorakṣa comes and saves him. This was the first time we learned of the strīrājya. A tale similar to this one of Matsyendra was also incorporated into the hagiography of the advaitācārya Śaṃkara presented in the Mādhavīya Śaṃkara-digvijaya. These accounts were consistent with our next encounter with the term strīrājya in the sūtra-s of Vātsyāyana. There, in his sexual ontology, he says that the women of strīrājya like violent actions in bed and also the use of kṛtrima-liṅga-s. Since then, we kept encountering the strīrājya in a number of Hindu sources and it struck us that this was a parallel to the amazons, who are frequently mentioned in Greek lore. We had to visit an art museum, where we saw a modern imitation of a Classical sculpture of an amazon, probably one of the famous amazons featured in Greek legend (Top). The person, whom were showing the museum, remarked that the amazon had an “Indian” touch to her – whether there was any truth to that or not – it prompted us to revisit the topic leading to the current discursion on the strīrājya.

While the amazons are frequently mentioned in the Greek epic and early literature, their counterpart, the strīrājya, finds only a rare mention in the Hindu epic, the Mahābharata. Yet a closer examination suggests that the inspiration for both probably stemmed from related steppe Iranic groups:
1) While there is no consensus it is most likely that the Greek word amazon does not have a Greek etymology. Rather, it is likely to have some kind of Iranic etymology such as ha-mazon, perhaps meaning a warrior band.
2) The Greek evidence from writers such as Herodotus associate them as mixing with the steppe Iranic groups like Scythians (śaka tigracūḍa) and spawning the Sarmatians (sairima). Consistent with this they are described as being experts of horse-borne archery.
3) The Roman leader Pompey records them as being in the army of Mithradata-VI the formidable Greco-Iranian king. The later Roman writer and general Ammianus mentions them as a neighboring tribe of the Iranic Alans (Aryans).
4) The archaeologist David Anthony notes that among the “Scythian-Sarmatian” warrior kurgans about 20% contain interred women in battle-suits like their male counterparts. Consistent with this, some Greek sources record the amazons being interred in large kurgans. This can also be placed in the context of Herodotus’ account of the death of Cyrus, where he marches against an Eastern Iranic steppe kingdom of the Massagetae which was led by a queen Tomyris.

This suggests that indeed these steppe Iranics with female participation in warfare might have inspired the yavana legends about the amazons. They may have been more familiar to the early Greek sources than the Indic ones because they launched a series invasions in the direction of the Greek sphere and are even credited to have built some temples in the Greek sphere, which were subsequently centers of Greek worship.

On the Indian side of the evidence we find a further mention from the great Gupta age naturalist Varāhamihira in his Bṛhatsaṃhitā:
diśi paścimottarasyāṃ māṇḍavya-tukhāra-tāla-hala-madrāḥ |
aśvaka-kulūta-halaḍāḥ strīrājya-nṛsiṃhavana-khasthāḥ || 14.22

He places the strīrājya in the northwest along with several other tribes including the aśvaka, madra-s and the Tocharians. This is consistent with strīrājya being associated with the steppe Iranics of the Northwest. In the second reference to strīrājya by Vātsyāyana it is situated along with Bāhlika (modern Balkh) again pointing to the northwest direction. This reference also mentions the strīrājya women sequestering youths in their antaḥpura-s comparable to the a Greek tale regarding how the amazons reproduce by sequestering males from other tribes. The Chinese bauddha traveler-scholar Xuanzang and the Tang-Shu record a country Lang-ka-lo with its capital as Su-t’u-li-ssu-fa-lo which has be rendered by some as strī-īśvara. It is explicitly stated as being under Iranian rule despite using Brāhmi script and having both bauddha-s and Hindus (hundreds of deva temples) on the way to the “Western woman country”. This would suggest that all these sources recognized the same Northwestern land, likely associated with one or more steppe Iranic groups.

This position is also in line with the mention by Kalhaṇa in the Rājataraṃgiṇi of the strīrājya. He describes strīrājya as being invaded in course of the expansive conquests of the greatest Kashmirian emperor Lalitāditya to the north of Kashmir. He mentions strīrājya as being conquered prior to the Lalitāditya conquering this Uttarakuru-s would again place it to the north and potentially in the steppes. Notably, he appears to attack it from the east crossing a desert which might have meant the southern reaches of the Takla Makan. In this regard we hear from the kavi who mixes the rasa-s of warlike and the erotic:

tad yoddhān vigalad dhairyān strīrājye strījano’karot |
tuñgau stanau puraskṛtya na tu kumbhau kavāṭinām || 4.173

Then the women-folk of strīrājya made the valor of [Lalitāditya’s] soldiers melt,
by placing to fore their high breasts and not the frontal lobes of their elephants.

strī-rājya-devyās tasyāgre vīkṣya kampādi-vikriyāṃ |
saṃtrāsam abhilāṣaṃ vā niścikāya na kaścana || 4.174

Seeing the emotions of trembling and the like exhibited by the queen of the strīrājya in front of him (Lalitāditya), no one could say for certain if it was due to to fear or eros.

ekam ūrdhvaṃ nayad ratnam adhaḥ karṣat tathāparam |
baddhvā vyadhān nirālambam strīrājye nṛhariṃ ca saḥ || 4.185

By placing one magnetic gem which pulled it upwards, and another one which pulled it downwards, he (Lalitāditya) installed an idol of Nṛsimha suspended in the air without support in the strīrājya.

Lalitāditya’s successor Jayāpīḍa is also mentioned as conquering the strīrājya:

citraṃ jitavatas tasya strīrājye maṇḍalaṃ mahat |
indriya-grāma-vijayaṃ bahv amanyanta bhūbhujaḥ || 4.587

After he conquered a large territory of the strīrājya it is a wonder that other kings considered his conquest of the field of his (Jayāpīḍa’s) senses [ever greater].

karṇa-śrīpaṭam ābadhya strīrājyān nirjitād-dhṛtam |
dharmādhikaraṇākhyaṃ ca karmasthānaṃ vinirmame || 4.588

He established the office of the court of justice and hoisted therein the auspicious silk of Karṇa, which he had seized from the conquered strīrājya.

Regarding his profligate successor Lalitāpīḍa we hear again from Kalahaṇa:

atṛptaḥ strībhir alpābhir ugrarāgaḥ sa parthivaḥ |
jaḍaṃ mene jayāpīḍaṃ strīrājyān nirgataṃ jitāt ||

The king (Lalitāpīḍa) with a raging passion and not satisfied with [just] a few women considered Jayāpīḍa impotent for having left the strīrājya after conquering it.

Thus, we see parallels of the Matsyendra story, where the king Jayāpīḍa is praised for having controlled his senses upon conquering strīrājya. Notably, the Karṇa-śrī-paṭa, while obscure in meaning, reminds one of the Greek legends of Herakles and Theseus taking away the girdle of the amazonian queen Hippolyta. A further account of the strīrājya in Hindu tradition is seen the Jaimini-aśvamedhaparvan, which presents itself as a fragment of the Mahābhārata of Jaimini. However, as it has come down to us it is much reworked text with a Vaiṣṇava focus. Here, the sacrificial horse reaches the strīrājya in course of its wanderings and is taken by Pramilā, the queen of strīrājya. Arjuna challenges her to battle and after a brief archery encounter a celestial voice advises Pramilā to give up and marry Arjuna. She releases the horse and accompanies Arjuna to Hastināpura, where she waits for him till his period of celibacy for the aśvamedha is complete. But in course of this account it informs us that there are no males in strīrājya. The females are apparently left male-less due a curse of Rudrāṇī. If the males go there they die in a month from the excessive and violent sex with the females of strīrājya in line with the Kāmasūtra-s comment in this regard (“māsamātraṃ striyaṃ prāpya paścāt prāpnoti vaiśasam | … tenaiva svena liṅgena praviśanti hutāśanam |”). This also reminds one of the Greek legend of the destruction of men by the sirens or Kirke during wanderings of Odysseus. There is indeed a reversal similar to that of Odysseus and Kirke in the Matsyendranatha tale where the women of strīrājya attack Gorakṣa when he leaves with the former, but by his yoga he turns them into birds.

Thus, we may infer both Greek and Hindu traditions had the memory of a land of females. In both traditions they were much embellished but it appears that the Greeks had much closer contact with the actual agents behind these legends. However, in both cases it seems they gave rise to a floating mass of legends, which were incorporated into various cycles in different ways. Given the relatively sparse occurrence of the strīrājya in Hindu tradition, one may ask if they really encountered them or if were merely stories borrowed from some other group like the yavana-s. In this regard we may note the following:
1) The pre-Mauryan bronze mirrors described Vassilkov of Indian origin suggest some kind of a contact between the Hindus and the steppe Iranics. Moreover the alternative account of the death of Cyrus given by Ctesias, where the Indians are said to form an alliance with a central Asia group the Dṛbika-s against the Achaemenids. These point to contacts between the Indians and the steppe long after the Indo-Aryans conquered and settled in India.
2) Stylistic similarities are seen between Northern Indian and horse-trappings (phalerae) and jewelry recovered from Sarmatian graves.
3) Recently Veeramah et al looked at ancient DNA extracted from individuals from Sarmatian and other graves from a wide swath of western Eurasia and studied their genetic affinities. Notably, one Sarmatian individual (labeled PR_10 in their study) from Russian Orenberg region (~400-200 BCE) and a Crimean individual (labeled Ker_1) with Hunnic-style deformed skull from around 200-400 CE show evidence for Indian admixture. A preliminary examination (needs more careful confirmation) does suggest that this reflects a relatively recent Indian admixture with SNPs private to greater India rather than some ancient Indo-Iranian relationship. This would imply that there was direct contact with individuals of Indian origin so as to result in gene-flow.

In conclusion, we hence believe that there was some real knowledge of the steppe Iranics with female warriors among the Hindus. They were a distant group with which the authors of the texts might not have had close familiarity. Nevertheless, the direct experience of those who had journeyed to those regions likely formed the historical core of the information presented by Hindu authors, which was then subject to poetic elaboration. It is known that among the steppe groups both Iranic and later Turko-Mongol there was some degree of participation of women in warfare (down to the Mongol times and even after their conversion in the west to Mohammedanism). This was probably the root of both the Greek amazons and the Hindu strīrājya.

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Sītā in the pyre

It was quiet, early Saturday evening. Vidrum, Jukuta, Sharvamanyu and Lootika were hanging out on the parkway of Somakhya’s house. They had assembled there for some curricular preparation on differentiation. Somakhya and Lootika had covered the chain rule and the differentiation of product functions. Buoyed up the lessons their classmate-students felt that they could sally forth for the impending tests with confidence. But Somakhya put the damper on them: “Differentiation is the easy part and one can rake up the marks on the exam like Gandulkar smashing sixes off the Rainbow Republic bowlers. The real challenges will come with the rising balls of integration.”

Lootika amplified the matter like an accompanying musician: “If you think you have mastered math, remember this tale in  Greek tradition: it was said that there was a mortal named Stentor who could yell with the voice of many men in unison. But once he attempted to contest with the god Hermes on hearing whose loud yell he died. Thus, when integration section opens you may come up against some Hermesian howlers that our classmate Hemaling spends all day integrating.”
Somakhya: “In any case we will cover that tomorrow along with some vectors.”

Vidrum flipping through the news and messages on his tablet chimed in: “I see some very disturbing news. A city man, Durjoy, has been arrested on the charges of trying to arrange aid to the socialist terrorists who have been trapped as part of the ongoing operation in Gondipura. Our classmate Manjukeshi who was supposedly taking preparatory lessons from him at that time has also been taken into custody!”
Sharvamanyu: “Good job by the security forces.”
Jukuta checking out the news herself: “That is really sad. Was this Durjoy the senior student from the St. Stephens institution? He would come to hangout outside our school and was quite good to me in those days. I do not know if was really involved in anti-national activities. But it is ridiculous on their part to arrest Manjukeshi. I believe she should be innocent.”
Lootika: “I think he was a lout. One day he accosted me outside school but I scampered away on my bike in fear, thinking he might be an acolyte of Shonit who had harassed me and my sisters in our former school. But Sharva got some interesting “stones”, from him, which he apparently found in Gondipura and gifted them to me. They turned out to be the fossil eggs and an ungual of the peculiar sauropod dinosaur Isisaurus ”
Sh: “Remember, he was always trying to seduce girls from our school. He left those fossils with me during an inter-school athletics event and never took them back. Then I gave them to you knowing you might find them interesting. Why do you think he was in Gondipura? I am sure it was not for dinosaur-hunting.”
Lootika: “Of course, now all that makes sense. I’d add Manjukeshi may not be innocent either.”
Vidrum: “Why do you think so? She is a nice girl. I know she was just taking classes from that Durjoy.”
Somakhya: “Vidrum, have you forgotten the day you first introduced her to us. She did strike us as a fighter for social justice. After all let’s not forget she used assemble for these lessons of ours whole of last semester. Until., well…I guess let it be.”
Jukuta: “I never really understood why she stopped coming. Somakhya don’t kill me for this, but she simply told me that Lootika was a horrible person. She added that despite being a girl, she was perpetuating gender stereotypes and horrible casteism. I tried to tell her that even though in school I had more than one tiff with Lootika, we are now grown up and in college and need to accept that she is truly brighter than all the girls in our class. She cut me off saying that I was either with her or with Lootika.”
Lootika: “But Jukuta you have not stopped on playing on both sides, right? Let me remind you we are under no obligation to have you here for these lessons.”
Vidrum: “Calm! Let us not get into one of those silly fights of you girls.”
Sharvamanyu: “I think it began with that article on the Rāmāyaṇa, which Manjukeshi authored for the college magazine, where she accused emperor Rāma of being a male chauvinist and a casteist. We castigated her for that and she got angry with Lootika for not supporting her silliness.”

Jukuta: “I did read that article. I don’t know much about the Ramāyaṇ, but if that incident of letting Sītā burn is in the book it does sound awful, like a dowry death by burning, you see. Lootika, I know you and your sisters are the only girls who know these things written by Brahmins in Sanskrit. Is it really there in the book?”
Lootika: “Jukuta, before anything else you must pronounce the Rāmāyaṇa correctly – repeat after me. When my parents first told me the tale, they left out this part. But then, when I read it myself, in order to narrate it to my sister Vrishchika, I encountered it right there in the original. I must confess I read that part with some tension, as to what was going to happen, and even surprise. It did not feel right to me then and I asked my mother about it. She gave me the explanation that Rāma was an avatāra of the god Viṣṇu and that the avātāra ended the moment he performed the superhuman task of killing the rakṣas-lord. Thus, beyond that point he behaved like an ordinary human and the act must be understood in that light. I then asked my father about it. He said that when Sītā was abducted by Rāvaṇa, the real Sītā was taken away by the gods and replaced by a māyā Sītā. As per his interpretation, at the time of the fire ordeal, the god Agni incinerated the māyā Sītā and returned the real Sītā to Rāma. So clearly, that incident was seen as bit unusual and rather than accept it without a question, people have come up with different explanations for it.”

Jukuta: “But Lootika at the bottom-line would that still not be misogynistic, as Manjukeshi had written?”
Lootika: “See, such words are neologisms, which make little sense in the context of old tradition. As a biologist I have learned not be fall prey to such emotionalistic constructs but coldly look at the hard realities on the ground. The sexes are always pitted in an inter-sexual conflict for maximizing their fitness. At same time, as we are eukaryotes, sex is institutionalized in us starting with the duplication into two paralogs of the ancient protein RecA and the HORMA protein acquired from bacteria more than 2.7 billion years ago. Like all other sexually reproducing organisms, we cannot make more copies of ourselves without the other sex. Hence, the games played by the players in this conflict should not be given any moralistic judgment. That said, if you read the original text, there is hardly anything of the kind the puerile Manjukeshi read in it. In fact, it can be read as not disallowing the option of her kind of unregulated sexuality: after all Rāma tells Sītā is that she can chose any other mate she wants. So she is offered unfettered agency in the sexual domain and it is she who chooses to remain in the union with Rāma by proving her chastity via undergoing the ordeal by fire. Moreover, you see her not caving in but giving a dignified and firm public defense of her character when it was attacked and this you can take from a brāhmaṇī who has actually read the text in the original language.”
Sharvamanyu: “Look at it from Rāma’s perspective too. There was a direct affront on his manliness as a warrior in the abduction of Sītā and he avenged it squarely. However, no man likes to be in doubt of his paternity or risk providing for his enemies’ offspring. Therefore, he had very natural reasons to be uneasy in accepting his wife who had been taken into his enemy’s custody for a while…”
Somakhya: “See, this discussion we are having here is one of the reasons for epic’s author to craft such a tale. Even though the Rāmāyaṇa, unlike our national epic the great Bhārata, is one of ideals, it has all these elements that bring out the grey shades of real life. If it were not for these, people would have had difficulty in having a discourse on dharma of the type as we are just having. When it comes to the knotted questions of dharma there are no black and white answers but only a difficult navigation though the shades in the grey zone. Incidents like this provide the framework for thinking about that path. But then the great bhārgava does not put things in the epic without a deeper metaphor, which only some grasp.”
Vidrum: “What is that deeper metaphor that you are alluding to? Does such exist in this particular incident?”
Somakhya: “It does. But I’d let this spidery girl expound it if she wishes as it concerns her ancestors.”

Lootika: “It was something I actually learned from Somakhya and that was when I came to a final understanding of the incident. Everything in the itihāsa of Valmīki is a reflection of deep elements of śruti. He has done so in order that people like you Vidrum and Jukuta can access the mysteries of religion that might be otherwise inaccessible. First, remember that the princess Sītā born of the furrow in earth is transparently the humanized form of the ancient goddess of agriculture, who was praised by my ancestor Vāmadeva Gautama in the mantra:

arvācī subhage bhava sīte vandāmahe tvā |
yathā naḥ subhagāsasi yathā naḥ suphalāsasi || RV 4.57.6

Be auspicious O Sītā, come come close to us. We worship you,
such that you confer us a good share [of things] confer the good fruits [of agriculture].

Second, my ancient clansmen, the Gotama-s, were the purohita-s of the Videgha-s, the clan of the Janaka-s to which Sītā of the Rāmāyaṇa belonged. Evidently, they transmitted the cult of Sītā to these kings. The brāhmaṇa on the Sāmidheni-s, the chants by which the fire is aroused in the śrauta rite, preserves an anachronistic ancient legend of one of the founders of my clan, Gotama Rāhūgaṇa, and his patron the founder of the Videha-s, Videgha Māthava. In that tale, it is said, alluding to the Eastward migration of the Indo-Aryans, that the land the Videgha-s eventually occupied was initially rich in water but difficult to cultivate and habit. But when Gotama uttered an incantation, the god Agni is said to have burst forth from Videgha Māthava’s mouth and burnt the land of the Sadānīra river making it habitable. It is this ancient legend that receives an epic reflection in the form of Sītā’s purification by the fire, representing that new land of the Videha-s and Kosala-s, along with the Gotama-s, to the east of the Kuru-Pāñchala, becoming fit for habitation and the furrow of agriculture. Rāma who is the earthly manifestation of the great Indra then receives that Sītā, even as my ancestor Vāmadeva says in his incantation:
indraḥ sītāṃ ni gṛhṇātu |
May Indra set his hold on Sītā (set down the furrow).”
Somakhya: “More generally, it might be seen as representing the agricultural practice of burning leftover plant material after winter (the grip of Vṛtra whose reflection in the epic is Rāvaņa) to get the fields ready for agriculture in spring under the fertilizing effect of Indra.”
Vidrum: “Interesting, but I guess such arcana might be beyond the lay user of the epics who might be satisfied with the social debate you brought up earlier.”
Lootika: “Of course this is mostly for the those who uphold the deeper language of tradition but we are just giving you a flavor of how overloaded the language of myth can be.”

Jukuta: “But what would you say about the killing of Jambukumār, the śūdra saint, by Śri Rām? Is that not very casteist?”
Sharvamanyu: “Who the hell is this Jambukumār? Never come across such in the Rāmāyaṇa.”
Lootika: “I believe she is referring to Śambuka.”
Vidrum: “That is a tough one. What would you guys say.?
Somakhya: “Again, not reading the original but going by the words of these social justice types can give you a wrong impression. Of course, it is a long story and we could sit here arguing both sides. for a while, like all these thorny points in the epic. Would you really like a story without such shades of grey, the tensions they create and the emotions they arouse? On the other hand there are at least three messages here. First, this Śambuka was a śūdra alright but he was not saint. He was aiming to ascend to and conquer the world of the gods, like the demon Rauhiṇa; hence, it was imperative that their representative on earth Rāmacandra Aikṣvākava nip such attempts in the bud. Remember that in the early days with the Ārya-s settled in the subcontinent of the Jambudvipa what was meant by śūdra was some kind of enemy of the Ārya-s. Overtime they were defeated and absorbed into Ārya society. This leads to the second message – it is actually one reflecting this social accommodation and change. The text clearly states: bhaviṣyac chūdra-yonyāṃ hi tapaścaryā kalau yuge || It presents a doctrine that successively over the four yuga-s each varṇa acquired the capacity to do tapas. Now this Śambuka was doing it out of turn in the previous yuga and hence he was punished for breaking the rules. However, in a positive message it indicates that in our yuga the śūdra has this option and the incident merely refers to the tensions of a bygone era. Thus, in the typical futuristic format of the purāṇa-s, the text is presenting this change in social reality vis-a-vis the śūdra against the background of the previous antagonism. Third, it delivers the message of conservatism for social change. Rapid social change by accommodation of those from without the fold presents dangers. Such elements are represented by Śambuka, who wish to emulate Ārya practices, not for good purposes, but for conquering the daiva realm. Hence, such revolutionary tendencies have to be suppressed in favor more gradual accommodation over the yuga.”

Jukuta: “Your uncompromisingly conservative outlook amazes me.”
Lootika: “You may appreciate it more as you age, though part of it is in the genes and we cannot do much about it. But whether you like our message or not remember that a love for facile, feel-good messages might turn you into a Durjoy, a Shonit or a Samikaran, who in the end might do more harm than good to society.”
Sharvamanyu: “And now Jukuta I hope you don’t bring up the issues of animal rights in the incident of the killing of Vālin.”
Jukuta: “Who is Vālin?”
Lootika: “The next time we assemble at my house I’ll gift you a bāla-Rāmāyaṇa. I really think you need to be reading it more than any of this curricular stuff.”

Posted in Heathen thought, Life | Tagged , , , , , , ,

A sequence related to prime counting

The current note arose as an exploration branching off from the matter discussed in these earlier notes: this one and this one. As we saw before, Carl Gauss, while still in his teens, produced his first estimate of the prime number distribution in the form of the function:

\pi(n) \sim \dfrac{n}{\log(n)}

Here is \pi(n) is the prime counting function, which counts the number of prime numbers up to a given number n, and \log(n) is the natural logarithm of n. The \sim notation indicates that the prime counting function is asymptotic with \tfrac{n}{\log(n)}, i.e. as n\to \infty the ratio \pi(n)\big / \tfrac{n}{\log(n)} \to 1.

Subsequently, Gauss refined his fit for the prime counting function by using the famed logarithmic integral \textrm{Li}(x). We were curious if there was some arithmetic function, which was actually fitted by \tfrac{n}{\log(n)} rather it being merely a single term approximation of the \pi(n). In course of some arithmetic experiments, we stumbled upon a sequence, which we believe, without formal proof, is fitted by \tfrac{n}{\log(n)} in terms of average behavior.

This sequence f is defined thus: f[1]=1. Thereafter, add n-1 to all terms f[1:(n-1)]. Count how many of f[1:(n-1)]+(n-1) are primes. This count is f[n]. For example when n=2 we add 2-1=1 to 1 we get 2. Which is a single prime; hence, f[2]=1. Now for n=3 we add 3-1=2 to the first two terms and we get 3, 3. Thus, we have 2 primes; hence f[3]=2. For n=4, we add 4-1=3 to the prior terms and get 4, 4, 5, which yields a single prime, 5; hence, f[4]=1. Thus, the first few terms of the sequence goes: 1, 1, 2, 1, 3, 1, 4, 1, 1, 2, 7, 2, 7, 1, 1, 4, 11, 3, 9, 2, 4, 4, 11, 0, 2, 4, 4, 11, 11, 6. Figure 1 shows a plot of the first 20000 terms of the sequence.

prime_back_addition_fig1Figure 1

The blue line is the plot of this sequence and we notice right away that despite the fluctuations the average tendency is to grow with n. Via numerical experiments we were able to establish that this average growth is fitted best by the function \tfrac{n}{\log(n)} (red line in Figure 1). The green line in Figure 1 is the count of primes \pi(n). We observe that though some extreme values of f exceed \pi(n), the average behavior of f[n], i.e. \tfrac{n}{\log(n)} < \pi(n). This relates to a central development in the number theory: when Gauss conjectured the asymptotic relationship between \tfrac{n}{\log(n)} and \pi(n) the mathematical apparatus was not yet in place to prove it. This was finally developed by his last student Bernhard Riemann. Using those ideas, nearly century after Gauss’ conjecture, Hadamard and de la Vallée-Poussin proved it and it became known as the Prime Number Theorem. Further, de la Vallée-Poussin showed that \pi(n) was related to \tfrac{n}{\log(n)} thus:

\pi(n)=\dfrac{n}{\log(n)}+O\left(\dfrac{n}{\log^2(n)}\right)

Here, the second term is gives the error and is denoted using the big-O notation which was explained in an earlier note. This indicates that indeed \tfrac{n}{\log(n)} would be less than \pi(n). Thus, as can be seen in Figure 1 the average growth of f[n]<\pi(n).

We then used \tfrac{n}{\log(n)} to ‘rectify’ f[n] i.e. obtain:

f[n]-\dfrac{n}{\log(n)}

Prime_back_addition_fig2Figure 2

This rectified f[n] is plotted in Figure 2 and provides a clear picture of fluctuations in f[n] once we have removed the average growth trend. We observe right away that the amplitude of the fluctuations grows with n. To determine this growth trend of the rectified f[n], we first noticed from Figure 1 that \pi(n) tends to run close to the maxima of f[n]. Hence, we utilized the asymptotic expansion of \textrm{Li}(n), which is a better approximation of \pi(n) and captures the behavior beyond the basic \tfrac{n}{\log(n)} term:

\textrm{Li}(n) \sim \dfrac{n}{\log(n)} \displaystyle \sum_{k=0}^\infty \dfrac{k!}{(\log(n))^k}

\textrm{Li}(n) \sim \dfrac{n}{\log(n)}+\dfrac{n}{\log^2(n)}+\dfrac{2n}{\log^3(n)}+\dfrac{6n}{\log^4(n)}...

Using the first 4 terms to approximate the growth of the amplitude of rectified f[n] we get the red bounding curves shown in Figure 2. Thus, we conjecture that while f[n] grows on an average as \tfrac{n}{\log(n)}, the amplitude of its fluctuations is roughly approximated by \textrm{Li}(n)-\tfrac{n}{\log(n)} (Green bounding curves in Figure 2).

Posted in Scientific ramblings | Tagged , , ,