Contribution of Vedanta Desika to Mnemonics and Combinatorics

Abstract - Śrī Vedānta Deśika is well known as a poet, as a philosopher and as a logician of the 13th century. Less known is the fact that he was also an expert in diverse areas of science and technology. He not only possessed great poetic skills but also had expertise in solving complex combinatorial mathematical problems and devised several methods for compression of information. This paper is aimed at bringing out the contribution of Vedānta Deśika to mnemonics and combinatorics. Keywords – Mnemonics, Information compression, Katapayadi System, Prosody

I. INTRODUCTION

Mnemonics is any learning technique that aids information retention or retrieval (remembering) in the human memory. Mnemonics make use of elaborative encoding, retrieval cues, and imagery as specific tools to encode any given information in a way that allows for efficient storage and retrieval. Mnemonics aid original information in becoming associated with something more accessible or meaningful—which, in turn, provides better retention of the information. Mnemonics make use of elaborative encoding, retrieval cues, and imagery as specific tools to encode any given information in a way that allows for efficient storage and retrieval. Symbols or Maxims are used in the place of technical terms of Grammar. They help use information already stored in long-term memory to make memorisation an easier task.

II. APPLICATIONS OF MNEMONICS

Commonly encountered mnemonics are often used for remembering

• Lists
• Numerical sequences
• Foreign-language acquisition
• Medical treatment for patients with memory deficits lists and in auditory form, such as short poems, acronyms, or memorable phrases, but mnemonics can also be used for other types of information and in visual or kinaesthetic forms. Their use is based on the observation that the human mind more easily remembers spatial, personal, surprising, physical, humorous, or otherwise "relatable" information, rather than more abstract or impersonal forms of information.

III. TYPES OF MNEMONICS

A wide range of mnemonics are used for several purposes.

• Music mnemonics
  • Example - Children remember the alphabet by singing the ABC's.
  • Maheshwara sutras in Panini's Ashtadhyayi not only reminds us of this method but also useful in compressing the information.
• Name mnemonics
  • The first letter of each word is combined into a new word.
  • For example: VIBGYOR for the colours of the rainbow.
  • Adivaravu in the Santhai tradition followed for the last 1000 years by the Srivaishnavas
• Expression or word mnemonics
  • The first letter of each word is combined to form a phrase or sentence
  • E.g. "Richard of York gave battle in vain" for the colours of the rainbow.
  • The anthadi style of composition reminds one of the next stanza
• Ode mnemonics
  • The information is placed into a poem
  • But in the philosophic tradition, the information is provided in the form of shloka, mostly in Anushtup chandas, which is very easy to remember
• Note organization mnemonics
  • The method of note organization can be used as a memorization technique.
  • Sangraha-vistara bhava used in the Shastras are far superior to this method.
• Image mnemonics
  • The information is constructed into a picture.
  • There are ample evidences of Communication through pictures in our tradition
• Connection mnemonics
  • New knowledge is connected to knowledge already known.
  • Sangati used in the Sribhashyam and other works of Ramanuja are standing examples of this methodology.

IV. VEDANTA DEŚIKA'S CONTRIBUTION TO MNEMONICS

Vedanta Deśika has composed over a hundred works in languages such as Tamil, Sanskrit, Prakrit and Manipravala (a mixture of Sanskrit and Tamil) that revealed his ingenuity, creativity, logic, linguistic expertise, devotional fervour and erudite scholarship.

He is renowned for combining poetry with logic in his philosophical works such as Shata-dushani, Mimamsa-paduka and Tattva-mukta-kalapa. Appaya Dikshita, the great mediaeval scholar appreciated Deśika by composing a verse in Sanskrit:

"tam vichintyas sarvatra bhavaah santi pade padhe kavi tarkika simhasya kavyeshu laliteshvapi”

Meaning:

"Even in the simple and soft compositions of this lion of poetry and lion of logic, there is poetic excellence evident at every step he took, and indeed in every word he wrote.”

V. USE OF PROSODY FOR EASE OF REMEMBRANCE

Vedanta Deśika composed his poems in various poetic metres. Vedic literature is written in the form of hymns set rhythmically to different metres, called 'chandas'. Each metre is governed by the number of syllables specific to it. Poets are expected to conform to these norms in their compositions. It is interesting to observe that in Sanskrit poetry, the concept of binomial coeficients, Fibonacci numbers and binary numeration has been in use right from the days of Pingala who was the first to write a treatise on Chandas-shastra relating to metres in Sanskrit poetry.

In Sanskrit poetry, we have stanzas with four quarters. Each quarter may have the same number of syllables or the same number of time units, a short one being assigned 1 time unit and a long one 2 time units. There are metres in which the odd quarters have the same number of syllables or time units, while the even quarters have a different number of equal units.

A Sanskrit stanza or padya consists of four padas or four quarters, which are regulated by

• The number of syllables in each quarter, or
• The number of syllabic time units or matras, a short sound being assigned one unit of time and the long one . two units of time.

Vedanta Deśika has employed 22 metres in the 862 verses he composed on presiding deities of various temples in India. Because of its metre and rhythm, poetry is easier to commit to memory than bland lines of prose. Hence, Desika used poetry to expand on what Ramanuja had established.

VI. COMBINING LOGIC AND POETRY IN HIS MAGNUM OPUS, TATTVA MUKTA KALAPA

For easily remembering the philosophical and logical concepts, Vedanta Desika has authored his logical works in the form of poetry. His book, Tattva mukta kalapa, is one of the four magnificent philosophical gems. The other three are Satha Dhushani, Nyaya Parisuddhi and Nyaya Siddhanjana. Tattva mukta kalapa is categorized as a Prakarana grantham, a manual or independent treatise to present and defend Visishtadhvaithic doctrines and to evaluate critically the deficiencies in the rival philosophical systems.

He has combined his extraordinary poetic skills with those of his dialectic skills to write 500 verses brimming with such elegance about the serious topics of philosophy.

All of the subtle points about response to this query are housed in the form of a poem in the Sragdhara metre chosen for this work.

VII. COMBINING LOGIC AND POETRY IN ADHIKARANA SARAVALI - COMPRESSION OF INFORMATION

In Adhikarana saravali, the contents of the Brahma Sutras are summarised by Vedanta Deśika. Each chapter in Brahma Sutra has sections called paadas and each paada has sub-sections called the Adhikaranas. Each Adhikarana has one or many sutras.

Each Adhikaranam has five components:

• Vishayam (Subject),
• Samshayam (Doubt),
• Poorva paksham (Opponent view),
• Sidhantham (Established truth with proof) and
• Prayojanam (Benefit of establishing the truth with proof).

All the information about each Adhikarana is condensed in one verse in this work. When he has to indicate the number of sutras that are being explained or the number of adhikaranas in his work, he does not do it in a bland way using numerals. He uses letters according to the Katapayadi Sankhya system to represent numbers in the form of words.

VIII. COMPRESSION OF INFORMATION IN DRAMIDOPANISHAD TATPARYA RATNAVALI AND DRAMIDOPANISHAD SARAM

Tiruvaimozhi is a tamil work consisting of 1000 verses. Dramidopanishad Tatparya ratnavali is a poetry containing 130 Sanskrit verses which describes the contents of Tiruvaimozhi.

Each decad (usually 10 to 11 verses) of Tiruvaimozhi has been summarized in one Sanskrit verse.

Dramidopanishad Saaram is another brilliant work consisting of 26 Sanskrit verses which give the essence of the ten centums (One hundred verses) of Tiruvaimozhi.

IX. VEDANTA DEŚIKA'S CONTRIBUTION TO COMBINATORICS - THE KNIGHT'S TOUR PROBLEM

A Knight's Tour is a sequence of moves of a knight on a chess board such that each and every cell is covered only once. If the starting point can be reached from the last point of the tour by a knight's move, then the Knight's Tour is said to be closed; otherwise it is said to be open.

India is known to be the original home of the game of chess, called caturaṅga (Murray 1917). Both the game and the board were often referred to aṣṭāpada. There are several allusions to the game in the early literature. The earliest known description of a Knight's Tour is in the celebrated work of Alaṅkāraśāstra, the Kavyālaṅkāra of the Kashmiri savant Rudraṭa (c.850). In the fifth chapter of this work, dealing with citrālaṅkāra, the picturesque figures of speech, Rudraṭa gives a verse, which has a special symmetry, namely turagapādabandha. The verse is reproduced when the syllables are permuted in a specific sequence of the movements of a horse, which sequence also constitutes a solution to the Knight's Tour problem on half of the chess board. The same tour can be trivially extended to the entire chess board.

In his Pādukāsahasra, Vedanta Deśika has come up with a much more sophisticated poetic solution to the Knight's Tour problem. As we shall discuss below, in the thirtieth section of the mahākāvya, known as citrapaddhati, Vedanta Deśika composed two verses of great poetic beauty, in the caturaṅgaturaṅgabandha. These two verses are to be read together for getting their import (ekavākyatā). They also have the special symmetry that the second verse is obtained by permuting the syllables of the first verse following a specific sequence of the movements of a horse, which sequence also constituted a solution (the same solution given by Rudraṭa) to the Knight's Tour problem.

In the modern period, it was Leonard Euler (1707-1783) who pioneered the investigation of general algorithms for constructing Knight's Tours (Euler 1759). In contemporary mathematics, the problem of constructing a Knight's Tour is a particular case of a problem in graph theory, known as the Hamilton path problem. It is only recently that Conrad et al (1994) have shown that a Knight's Tour exists for all n x n Chess boards if and only if n≥5 and similarly a closed Knight's Tour exists if and only if n is even and n≥6. They have also shown that unlike the general Hamilton path problem, the Knight's Tour problem can be solved in polynomial time. The number of Knight's Tours on a 8x8 chess board has been recently estimated to be of the order of 13.27 x 1012.

X. SOLUTION OF KNIGHT'S TOUR PROBLEM IN PADUKĀSAHASRA OF SWĀMI DEŚIKA

Pādukāsahasra is a great stotra in praise of the pādukā of śrī Raṅganātha, the presiding deity at śrīraṅgam, composed in the mahākāvya style. It has thirty-two sections, paddhatis, with 1008 verses in all. The entire work is said to have been composed by Vedanta Desika in one night. It is also said that Śrī Raṅganātha and Devī Raṅganāyikã were so pleased with this great mahākāvya that they honoured the ācārya by bestowing the titles, kavitārkikasihma and sarvatantrasvatantra.

The thirtieth paddhati, citrapaddhati, has forty verses (verse nos. 911-950) which are characterised by various captivating symmetries (citrabandha) displaying interesting sound patterns (śabdacitra).

स्थिरागसां सदाराध्या विहताकततामता। सत्पादुके सरासा मा रंगराजपदन्नय॥९२९॥ sthirāgasāṁ sadārādhyā vihatākatatāmatā । satpāduke sarāsā mā raṅgarājapadannaya ॥929॥ स्थितासमयराजत्पागतरा मादके गवि। दुरंहसां सन्नतादा साध्यातापकरासरा॥९३०॥ sthitāsamayarājatpāgatarā mādake gavi । duraṁhasāṁ sannatādā sādhyātāpakarāsarā ॥930॥ The second verse is obtained from the first by following the sequence of the steps of a horse (pūrvasmin turaṅgapadakrameṇa uddhārya śloka:). These two verses have to be taken together as a single sentence for getting the import. (anayo: ślokayorekavākyatayā anvaya ityācāryairukta iti sampradāya:). The actual sequence of the steps of the horse that lead from the verse 729 to the verse 730 are indicated in Table 1 [Kesava, Pādukāsahasram, 1951, Devakottai, p.243]. We see clearly that this corresponds to a Knight's Tour. However here we have a highly ornate poetic verse (the verse 929) which is transformed by a permutation of its syllables via the sequence of steps of a horse in a Knight's Tour to another equally ornate and poetic verse (the verse 930).

Table 1: Caturaṅgaturagabandha of Vedanta Deśika

स्थि sthi 1	रा rā 30	ग ga 9	सां sāṁ 20	स sa 3	दा dā 24	र raā 11	ध्या dhyā 26
वि vi 16	ह ha 19	ता tā 2	क ka 29	त ta 10	ता tā 27	म ma 4	ता tā 23
स sa 31	त्पा tpā 8	दु du 17	के ke 14	स sa 21	रा rā 6	सां sāṁ 25	मा mā 12
रं raṁ 18	ग ga 15	रा rā 32	ज ja 7	प pa 28	द da 13	ञ्ञ ñña 22	य ya 5

Following is a brief translation of the verses 929-930 (which have to be read together as we noted before):

satpāduke: The sacred pāduka of śrī Raṅganātha!

sthirāgasāṁ sadārādhyā: you are to be worshipped all the time by those who habitually commit violations

vihatākatatāmatā: you put an end to the sequence of the sorrows and the undesirable ends

sarāsā: you have a pleasing sound

samayarājatpā: you protect those who are steadfast in dharma

āgatarā: you are golden and bestow wealth

mādake gavi sthitā: you dwell in the mesmerizing light of the Sūryamaṇḍala

duraṁhasāṁ sannatādā: you save even the terrible sinners from their pitiable state

sādhyātāpakarā: you produce brightness bereft of unbearable heat

āsarā: you move around everywhere

satpāduke mā raṅgarājapadannaya; you, sacred pāduka of śrī Raṅganātha ! Lead me to the feet of śrī Raṅgarāja.

Clearly what Swāmi Deśika has presented in verses 929 and 930 of Pādukāsahasra is an extraordinarily poetic solution of the Knights Tour problem.

XI. KATAPAYADI SYSTEM

Vedic Scholars adopted a different but more convenient system of letter-notation for numbers, called “Katapayadi” or “Vedic Numerical code”.

In this system of representation, the Sanskrit consonants (vyanjanas) beginning with क, ट, प and य represent the digits from 1 to 9 ending with 0 (i.e. letters from क to ञ denote 1 to 9 and 0 respectively and so on)

• The nasals ञ and न denote 0;
• In the case of conjunct consonants, the number denoted by the last consonant only is taken;
• The vowels following consonants have no value; and
• The vowels not preceded by consonants represent 0. The arrangement of the digits is from right to left.
• The vowels not preceded by consonants represent 0. The arrangement of the digits is from right to left.

The entire representation is shown in the form of a table as below:

Table 2: Vedic Numerical Code

1	2	3	4	5	6	7	8	9	0
क	ख	ग	घ	ङ	च	छ	ज	झ	ञ
ट	ठ	ड	ढ	ण	त	थ	द	ध	न
प	फ	ब	भ	म
य	र	ल	व	श	ष	स	ह	ळ	क्ष

Vedanta Desika also used Numbers ingeniously in his already pellucid prose & innovative poetic works. His tryst with numbers, in many of his inspiring works, demands a further research, from mindful readers, mathematicians, cognitive psychologists & neuro biologists.

Using the Katapayadi Sankhya system (where letters are used to represent numerals), Vedanta Desika uses the word 'subhaasee', in the introduction to Adhikarana Saravali; This word indicates auspiciousness & also represents the number 5-4-5, which also sums upto 545 verses of this work.

Vedanta Desika devised an easy way to remember this work's title, by just knowing the number 545.

XII. CONCLUSION

Vedanta Deśika was a Sri Vaishnava guru/philosopher and one of the most brilliant stalwarts of Sri Vaishnavism in the post-Ramanuja period. Across Deśika's vast range of styles and themes, there is clear underlying unity and consistency of vision. What is conveyed in moving mystical poetry is expressed in systematic, rigorous philosophical language. His contribution to the field of information compression, mnemonics and combinatorics is remarkable and worth looking into. More research needs to be conducted in these areas in the coming future.

XIII. REFERENCES

[1] Varadācārya. K.S., Tatvamuktākalāpa of Śrīmad Vedānta Deśika with the commentary Sarvankașa, 2010, Arsha Grantha Prakashana Bangalore, p 115

[2] Kesava, Padukāsahasram, 1951, Devakottai

- Prof. M. A. Lakshmithathachar