## Wednesday, September 5, 2012

### Vedics roots of modern mathematics. (part -2)

अनन्ता वै वेदाः = Vedas are infinities

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(1) Origin-This has been stated in a story of Taittirīya Brāhmaŋa (3/10/11)-Bharadvāja spent 3 lives with Brahmacharya to study Vedas by grace of Indra. At end, he became very old and bed-ridden. Then Indra asked what he would do if offered a fourth life of 100 years. He stated that he will complete his study of Vedas. Indra showed him 3 mountains as form of 3 vedas and took fistful of dust from each-telling that Bharadvāja had got knowledge equal to that.

भरद्वाजो वै त्रिभिरायुर्भिर्ब्रह्मचर्य्यमुवास तं जीर्णि स्थविरं शयानं इन्द्र उपब्रज्य उवाच भरद्वाज ! यत्ते चतुर्थमायुर्दद्यां, किमेनेन कुर्य्या इति ? ब्रह्मचर्य्यमेवैनेन चरेयमिति होवाच तं त्रीन् गिरिरूपानविज्ञातानिव दर्शयाञ्चकार तेषां हैकैकस्मान्मुष्टिमाददे होवाच, भरद्वाजेत्यमन्त्र्य वेदा वा एते "अनन्ता वै वेदाः" एतद्वा एतैस्त्रिभिरायुर्भिरन्ववोचथाः अथ इतरदनूक्तमेव (तैत्तिरीय ब्राह्मण /१०/११)

Here, it has been stated-
अनन्ता वै वेदाः = Vedas are infinities. It is not told in singular that Veda is infinity. Plural form means that there are at least 3 types of infinities, 2 types would have made it dvivachana.

(2) Zero and Infinity-Firstly, we may see definition of infinity. In calculus or analysis, infinity is defined as the limit of 1/x when x tends to zero. This is dependent on zero, which is defined in same manner-it is limit of 1/x when x tends to infinity. 1/x tends to zero means that its value can be made smaller than any small number ε which can be imagined. Similarly, tends to infinity means that its value can be made larger than any large number G which can be imagined. Both definitions are stated in single sentence-

अणोरणीयान् महतो महीयान्, आत्मास्य जन्तोर्निहितो गुहायाम्।
तमक्रतुः पश्यति वीतशोको, धातुप्रसादान् महिमानमात्मनः॥
(
कठोपनिषद् //२०, श्वेताश्वतर उपनिषद् /२०)

(Brahma) is smaller than the smallest and greater than the greatest. It is ātmā of this being (jantu) and enclosed in a cave. That form is akratu = detached from action, or a person detached from work and sorrow can see the mahimā of Brahma, by His grace.

There are many lines indicating Brahma as smaller than the smallest or śūnya-
अणिष्ठे वाङ्गेऽङ्गे समानयति। (मैत्रयणी उपनिषद् /)
अणीयान् ह्यतर्क्यमणुप्रंआणात्। (कठोपनिषत् //)
अणीयान् ब्रीहेर्वा यवाद्वा। (छान्दोग्य उपनिषद् /१४/)
अणु कोटर विस्तीर्णे त्रैलोक्यं जगद्भवेत्। (तेजविन्दु उपनिषद् /८७)
अणोरणीयानहमेव तद्वत्। (कैवल्य उपनिषद्, २०)
अणोरणीयांसमनुस्मरेद्यं (गीता /)
अणोरप्यणवं ध्यात्वा। (मैत्रायणी उपनिषद् /३८)

Līlāvatī of Bhāskarāchārya-2 gives rules for division by zero which are equivalent to notion of limit in calculus-

योगे खं क्षेपसमं वर्गादौ खं भाजितो राशिः। खहरः स्यात् खगुणः खं खगुणश्चिन्त्यश्च शेषविधौ॥
शून्ये गुणके जाते खं हारश्चेत् पुनस्तदा राशिः। अविकृत एव ज्ञेयस्तथैव खेनोनितश्च युतः॥
Reverse limits of zero and infinity are explained in his another book-Bījagaṇita-
अस्मिन् विकारः खहरे राशावपि प्रविष्टेष्वपि निःसृतेषु।
बहुष्वपि स्याल्लयसृष्टिकालेऽनन्तेऽच्युते भूतगणेषु यद्वत्॥

(3) Grades of Infinity-Cantor's Set theory describes grades of infinity. Smallest infinity is countable. Counting of a set of objects means we arrange them in a sequence and match each with natural (= counting) numbers 1, 2, 3, 4, …… Any set having infinity numbers can be counted with natural numbers.

Thus set of all natural numbers is equal to set of all even numbers or all squares-
1, 2, 3, 4, 5, ------ up to infinity
2, 4, 6, 8, 10, --------
1, 4, 9,16,25,------------
Even the set of all fractions is equal to all natural numbers. It can be shown by arranging all fractions serially, so that they can be serially matched with natural numbers-
1/1 1/2 1/3 1/4 - - -- - -
2/1 2/2 2/3 2/4- - -- - -
3/1 3/2 3/3 3/4 - - -- - -

We arrange the first row with numerator as 1 followed by denominators as 1, 2, 3, 4, … In second row numerator is 2, third row has 3 and in all rows numerators are 1, 2, 3, 4, .

From top left corner we count diagonally, up then down, up etc. Thus in sequence 1, 1/2/, 2/1, 3/1, 2/2, 1/3….., we count all fractions and it is same as infinity of natural numbers.

Cantor showed that real numbers cannot be arranged in a sequence which can be counted by natural numbers. Whatever sequence is made, we can always find a number between 0 and 1 which does not come in any sequence. Take the first number and its first digit, it can be 0 if decimal ends, take 9, otherwise take 0 in first decimal place of new number. At second place, take a digit different from second digit of that number and so on. Thus, we get a number which is not in that sequence and real numbers are definitely bigger infinity than natural numbers. Cantor further showed that, set of real numbers is equal to number of all sub-sets of natural numbers. In each subset, a number can be taken or left. So, if infinity of natural numbers is N0, then count of real numbers is N1 = 2^(N_0 ). Set of all subsets of real numbers itself will be still higher infinity equal to N2 = 2^(N_1 ).

(4) Indian definitions-A jain author Vīrasena in his book Vīra-dhavalā has classified infinities of 3x3 = 9 kinds-samkhyāta, asamkhyāta, ananta-each divided into 3 categories (See Tao of Jain Sciences by Lakshmi Chand Jain-Appendix-2)

Viṣṇu sahasranāma gives many names meaning infinity-Ananta (659, 886), Anantajit (307) Anantarūpa (932), Anantaśrī (933), Anantātmā (518), Aniruddha (185, 638), Anirdeśyavapu (177, 656), Anekamūrtti (721), Apāmnidhi (323), Avyaya (13, 900), Aprameya (46), Aprameyātmā (248) Amānī ((747), Amitavikrama (516, 641), Ameyātmā (102, 179), Ambhonidhi (517), Asamkhyeya (247), Asammita (108), Nidhih Avyayah (30), Naikah (726), Naikakarmakṛt (469), Naikajah (890), Naikamāyah (302), Naikarūpah (271), Naikaśṛngah (763), Naikātmā (468), Paramātmā (11), Parameṣṭhī (419), Parardhih (389) Parigrahah (420), Paryavasthitah (931), Pūrṇah (685), Bṛhat (836), Brahma, (663, 664), Brahmavivardhanah (665), Brahmaṇya (669), Brāhmī (668) Mahat (841), Mahardhi (350), Mahākramah (671), Mahānidhi (806), Mahāmāyah (170)
Mahārhah (522), Viśvam (1), Sarvah (25). In addition, many other words also may mean infinity-Sahasra (1000, infinity), Vīra (boundary, brave, Akabar = without cover or infinite in Persian).

At 2 places, Śankarāchārya has given 2 meanings of word 'Ananta'. At serial 659, he tells-

व्यापित्वान्नित्यत्वात् सर्वात्मत्वात् देशतः कालतो वस्तुतश्चापरिच्छिन्नः, अनन्तः सत्यं ज्ञानमनन्तं ब्रह्म (तैत्तिरीय उपनिषद् /) इति श्रुतेः। गन्धर्वाप्सरसः सिद्धाः किन्नरोरगचारणाः। नान्तं गुणानां गच्छन्ति तेनानन्तोऽयमव्ययः। (//२४)इति विष्णुपुराण वचनाद्वा अनन्तः।

At 886, he tells-
नित्यत्वात् सर्वगतत्वात् देशकालपरिच्छेदाभावात् अनन्तः शेषरूपो वा।

At both places, these meanings are indicated-eternal, all pervading, soul of all, not bound by time and space, satya (truth, sameness) of three types etc. At first place, it occurs after words-Anirdeśya-vapu = indeterminable body, Viṣṇu = enclosing
all, so it means infinite in time and space. At second place, it occurs with Hutabhuk, Bhoktā =consumer etc. Here, it may mean infinite consumption or work.

Asamkhyeya means not measurable with cardinal numbers.

Aprameya is numbers not defined with algebraic formula.

Ambhonidhi is collection of continuous numbers like spread of water in 3 dimensional space.

Naikah means not measurable with cardinal numbers starting with 1.

Kātyāyana śulba sūtra defines infinity as greater than any standard-(greater than any large number imagined)-

अपरिमितं प्रमाणाद् भूयः (कात्यायन शुल्ब सूत्र /२३)

(5) Vedic usage-Tulasīdāsa starts mangalācharaṇa of Rāmacharita-mānasa with-
वर्णानामर्थसङ्घानां रसानां छन्दसामपि। मङ्गलानां कर्त्तारौ वन्दे वाणी विनायकौ॥

Here, collection of letters, syllables is countable with natural numbers. Their meanings are more and partly abstract-they too are countable. Bhāva or rasa is not countable. Countable perception of world is Gaṇeśa (where Gaṇana or counting can be done). Abstract concept is Rasa-vatī or Sarasvatī. It is similar to 2 broad divisions of noun in English grammar-countable and abstract. Countable is further divided into proper, common, collective nouns. In mathematics, it is discrete or continuous theory. These are equivalent to set of natural and real numbers. In 1981, I had given example of analog and digital computers to explain this concept to Prof Abdul Salam of Pakistan who had come after getting Nobel prize and was my guest for 2 days. He was very happy with the description and made a remark in some journal that controller of Digital computer (Gaṇeśa) should be called a mouse (his vehicle). That name has become popular now.

Reverting to original context of Taittirīya brāhmaṇa, the three infinities are-
Ṛg veda (collection of forms = mūrtti)- Natural number infinity.

Yajurveda (motion, creation)-Real number
Sāmaveda (field of influence = mahimā) =Transcendental numbers.

Atharva veda (Brahma, reference) = Indeterminate.
ऋग्भ्यो जातां सर्वशो मूर्त्तिमाहुः, सर्वा गतिर्याजुषी हैव शश्वत्।
सर्वं तेजं सामरूप्यं शश्वत्, सर्वं हेदं ब्रह्मणा हैव सृष्टम्॥ (तैत्तिरीय ब्राह्मण /१२//)
Īśāvāsyopaniṣad also hints at triple infinities-
पर्यगात् शुक्रं अकायं अव्रणं अस्नाविरं शुद्धं अपापविद्धम्।
कविर्मनीषी परिभूः स्वयम्भूः याथातथ्यतो अर्थान् व्यदधात् शाश्वतीभ्यः समाभ्यः॥८॥

This explains 2 complementary aspects of world-Vāk and Artha. In physical world creation from abstract is by creating a boundary or cover (paryagāt), body, internal and external links, defects (gradients of density etc), separation (pāpaviddha). Abstract thought in mind remains there up to 3 stages of vāk-Parā, PaśyantI, madhyamā. When it is expressed out by speech or writing it is broken into letters, syllables, words, clause, sentence and paragraphs logically linked internally and externally and separated by comma, full stop etc. When transformation from abstract internal thought is exact, the creation (kāvya) becomes eternal. There is a similar verse about start of kāvya by Vālmīki.

Collection of letters, words, sentences in speech or writings= natural number set.
Meanings linking internal (paśyantī) and external steps of vāk = real number set.
Parā vāk = Transcendental number.

jayasree said...