PROBABILITY IN ANCIENT INDIA

PROBABILITY IN ANCIENT INDIA

C. K. Raju

INTRODUCTION: MATHEMATICAL PRE-REQUISITES

Permutations and Combinations

A first course on probability (at the high-school level) typically begins with an

account of the theory of permutations and combinations needed for calculating

probabilities in games of chance, such as dice or cards.

Like many other aspects of mathematics, this theory of permutations and combinations

first developed in India, although an account of its history is usually

missing in stock presentations of combinatorics.

In fact, the theory of permutations and combinations was basic to the Indian

understanding of metre and music. The Vedic and post-Vedic composers depended

on combinations of two syllables called guru (deep, long) and laghu (short). The

earliest written account of this theory of metre is in Pi˜ngala’s Chandahs¯utra (−3rd

c. CE), a book of aphorisms (s¯utra-s) on the theory of metre (chanda). To calculate

all possible combinations of these two syllables in a metre containing n syllables,

Pi˜ngala gives the following rule1 (which explicitly makes use of the symbol for

zero). “(Place) two when halved;” “when unity is subtracted then (place) zero;”

“multiply by two when zero;” “square when halved.” In a worked example, Dutta

and Singh2 show how for the G￣ayatr￣ı metre with 6 syllables this rule leads to the

correct figure of 26 possibilities.

That this rule basically involves the binomial expansion is made clear by Pi˜ngala’s

commentator the 10th c. CE Halayudha. Thus, in a 3-syllabic metre with two underlying

syllables, guru and laghu, 3 guru sounds will occur once, 2 gurus and

1 laghu will occur twice, as will 1 guru and 2 laghus, while 3 laghus will occur

once. Symbolically (g + l)3 = g3 + 3g2l + 3gl2 + l3. To generalize this to the

case of n underlying syllables, Halayudha explains the meru-prast¯ara (pyramidal

expansion) scheme for calculation,3 which is identical to “Pascal’s” triangle which

first appeared in Europe about a century before Pascal (on the title page of the

1Chandahs¯utra viii.28. Ed. Sri Sitanath, Calcutta, 1840. Cited by Dutta and Singh, vol. 1,

p. 76.

2B. B. Dutta and A. N. Singh, History of Hindu Mathematics, Asia Publishing House, 1962,

vol. 1, pp. 75–76.

3A. K. Bag, Mathematics in Ancient and Medieval India, Chaukhambha Orientalia, Varanasi,

1979, pp. 189–93.

Handbook of the Philosophy of Science. Volume 7: Philosophy of Statistics.

Volume editors: Prasanta S. Bandyopadhyay and Malcolm R. Forster. General Editors: Dov M.

Gabbay, Paul Thagard and John Woods.

c_

2010 Elsevier BV. All rights reserved.

2 C. K. Raju

Arithmetic of Apianus) and in China in the 14th c.4 An example, using the G¯ayatr¯ı

meter is also found in Bhaskara’s L¯ıl¯avat¯ı.5 The accounts found in stock Western

histories of mathematics (such as that by Smith6) incorrectly state that no attention

was paid in India to the theory of permutations and combinations before

Bhaskara II (12th c. CE).

Although this theory is built into the Vedic metre, the earliest known written

account relating to permutations and combinations actually comes from even before

Pi˜ngala, and is found in the −4th c. Jain Bhagwat¯ı S¯utra. Permutations were

called vikalpa-gan. ita (the calculus of alternatives), and combinations bhanga. The

text works out the number of combinations of n categories taken 2, 3 etc. at a

time.

Incidentally, this throws up large numbers of the sort that cannot easily be

written in Greek (Attic) and Roman numerals. It should be noted that while the

Yajurveda7 already used a place value system, and gives names for numbers up

to 1012, the Jain literature typically runs into very large numbers, such as 1060,

and the Buddha when challenged (perhaps by a Jain opponent) names numbers

up to 1053. Large numbers have an intimate connection with the philosophy of

probability, as examined in more detail, later on.

From the earliest Vedic tradition, there is a continuous tradition linking the first

accounts of permutations and combinations with those of Bhaskara II (12th c.), and

later commentaries on his work, up to the 16th c. CE, such as the Kriy¯akramkar¯ı.8

Thus,9 the surgeon Su´sruta (−2nd c. CE) in his compendium (Su´sruta-samhit¯a)

lists the total number of flavours derived from 6 flavours taken 1 at a time, 2 at

a time, and so on. Likewise, Var￣ahamihira (6th c.) who reputedly wrote the first

Indian text on astrology (Br. hat-J¯ataka) states in it the number of perfumes that

can be made from 16 substances mixed in 1, 2, 3, and 4 proportions.

Similar examples are found in the P¯at.¯ıgan. ita (Slate Arithmetic) of ´Sridhar (10th

c.), a widely used elementary school-text, as its name suggests, Mahavira’s (8th c.)

4Joseph Needham, The Shorter Science and Civilisation in China, vol. 2 (abridgement by C.

A. Ronan). Cambridge University Press, 1981, p. 55.

5Bhaskara, L¯ıl¯avat¯ı, trans. K. S. Patwardhan, S. A. Naimpally, and S. L. Singh, p. 102. The

verse is numbered differently in different manuscripts. K. V. Sarma in his critical edition of the

16th c. southern commentary Kriy¯akramakar¯ı (VVRI, Hoshiarpur, 1975) on the L¯ıl¯avat¯ı, gives

this as verse number 133, while the other cited source has given it as verse number 121.

6D. E. Smith, History of Mathematics, Dover Publications, 1958, vol. 2, p. 502.

7Yajurveda xvii.2 gives the names for the first 12 powers of 10, the first five being more or

less similar to what they are today.

8This has an interesting connection with the history of the calculus. Fermat’s challenge problem

is identical with a solved exercise in Bhaskara II. So this is one of the texts that travelled from

Cochin to Rome, and Bhaskara was probably Pascal’s source. B¯ıjagan.

ita of Sr¯ı Bh¯askar¯ac¯arya,

ed. Sudhakara Dvivedi, Benares, 1927 (Benares Sanskrit Series, No. 159), chapter on cakrav¯ala,

p. 40. An account of Bhaskara’s cakrav¯ala method may be found, for instance, in Bag, cited

above (pp. 217–228). For a formalised account of Bhaskara’s cakrav¯ala method, see I. S. Bhanu

Murthy, A Modern Introduction to Ancient Indian Mathematics, Wiley Eastern, New Delhi,

1992, pp. 114–21. (Bhanu Murthy’s book has a typo here.) For Fermat’s challenge problem and

“Pell’s equation”, see D. Struik, A Source Book in Mathematics 1200–1800, Harvard University

Press, Cambridge, Mass., 1969, pp. 29–30.

9A. Bag, Mathematics in Ancient and Medieval India, cited above, p. 188.

Probability in Ancient India 3

Gan. ita S¯ara Sam˙ graha, and Bhaskara II (L¯ıla¯vat¯ı) etc. Bhaskara mentions that

this formula has applications to the theory of metre, to architecture, medicine, and

khan.d.

ameru (“Pascal’s triangle”). In these later texts, one finds explicitly stated

formulae for permutations and combinations.

For example, to calculate _n

r_ values, ´Sridhar, in his text on slate-arithmetic10

(P¯at.igan. ita), gives the following rule.

􀀁􀀂􀀃􀀄􀀅􀀆􀀇􀀈􀀉􀀈􀀊􀀋􀀃􀀌􀀇􀀍􀀌􀀈􀀉􀀎􀀏􀀃􀀍􀀐􀀌􀀈􀀉􀀑􀀒􀀓􀀔􀀒􀀌􀀕􀀅􀀖􀀏􀀐􀀔􀀗􀀌

􀀘􀀙􀀉􀀚􀀛􀀌􀀘􀀇􀀜􀀌􀀝􀀓􀀞􀀒􀀌􀀟􀀘􀀃􀀠􀀡􀀢􀀏��􀀤􀀥􀀇􀀐􀀦􀀉􀀧􀀨􀀐􀀔􀀗􀀌

This translates as follows (P¯at.igan. ita, 72, Eng. p. 58)

Writing down the numbers beginning with 1 and increasing by 1 up to

the (given) number of savours in the inverse order, divide them by the

numbers beginning with 1 and increasing by 1 in the regular order, and

then multiply successively by the preceding (quotient) the succeeding

one. (This will give the number of combinations of the savours taken

1, 2, 3, ..., all at a time respectively.)11

Thus, in the case of 6 savours, one writes down the numbers 1 to 6 in reverse

order

6, 5, 4, 3, 2, 1

These are divided by the numbers in the usual order, to get the quotients

6

1,

5

2,

4

3,

3

4,

2

5,

1

6.

Then, according to the rule, the number of combinations of savours taken 1 at

a time, 2 at a time, etc., up to all at a time are respectively

6

1 ,

6

1

×5

2 ,

6

1

×5

2

×4

3, etc.

Although, the formulae are mostly stated in identical terms, they are applied

most flamboyantly by Bhaskara II. For example, to illustrate one of his formulae,

Bhaskara asks for the total number of 5 digit numbers whose digits sum to 13.12

􀀘􀀜􀀢􀀩􀀪􀀃􀀋􀀈􀀩􀀪􀀔􀀣􀀇􀀜􀀂􀀣 􀀏􀀥􀀄􀀄􀀒􀀕􀀫􀀏􀀒􀀡􀀞

􀀂􀀈􀀔􀀬􀀐􀀡􀀃􀀌􀀬􀀉􀀐􀀭􀀍􀀜􀀮􀀏􀀃􀀌􀀏􀀠􀀡􀀌􀀉􀀐􀀈􀀭􀀍􀀌􀀈􀀋􀀕􀀄􀀔􀀃􀀓􀀗

He then adds in the next verse that although this question involves “no multiplication

or division, no squaring or cubing, it is sure to humble the egotistical

and evil lads of astronomers”.

10´Sridhar, P¯at.igan.

ita, 72, ed. & trans. K. S. Shukla, Dept. of Mathematics and Astronomy,

Lucknow University, 1959, Sanskrit, p. 97.

11P¯at.igan.

ita of ´Sridhar, trans. K. S. Shukla. As he points out, similar articulations are found

in the Gan. ita S¯ara Sam. grah of Mahavira, vi.218, Mah¯aSiddh¯anta of ￣Aryabhat.a 2, xv, 45–46 etc.

12L¯ıl¯avat¯ı of Bh¯askar¯ac¯arya, trans. Patwardhan et al., p. 181. They give the number of this

verse as 276, whereas, in K. V. Sarma’s critical edition of Kriy¯akramkar¯ı, a commentary on the

L¯ıl¯avat¯ı, this is at 269, p. 464.

4 C. K. Raju

Weighted averages

The notion of simple average was routinely used in Indian planetary models, where

each planet had a mean motion, and a deviation from it. Unlike Western planetary

models, there was no belief in any divine harmony nor any faith in divine “laws”

of any sort involved here, just an average motion and deviations from it in a downto-

earth empirical sense. While the deviations from the mean were not regarded as

necessarily mechanically explicable, neither were they regarded as quite “random”,

for it was believed that the deviation could be calculated in principle, at least to a

good degree of approximation (required for the Indian calendar, which identified

the rainy season, and hence was a critical input for monsoon-driven agriculture in

India).

In various elementary mathematical texts, in the context of computing the density

of mixtures or alloys, one also finds the usual formula for weighted averages,

which is so closely related to the notion of “mathematical expectation” in probability

theory. For example, we find in verse 52(ii) of the P¯at.¯ıgan. ita (Eng. p. 36)

the following:

“The sum of the products of weight and varn. a of the several pieces

of gold, being divided by the sum of the weights of the pieces of gold,

give the varn. a (of the alloy).

That is, if there are n pieces of gold of weights w1.w2, . . . , wn, and varn. as v1, v2, . . . , vn,

the varn. a v of the alloy is given by v = w1v1+w2v2+···+wnvn

w1+w2+···+wn

.

(The term varn. a is analogous to the term “carat”, with pure gold consisting of

16 varn. as. ) Understandably, this topic of “mixtures” is given special emphasis

in Jain texts like those of Mahavira.

The relevance of these weighted averages to gambling was understood. It is in

this very context of “mixtures” that the Gan. ita S¯ara Sam. graha13(268.5, 273.5)

gives the example of “Dutch bets” mentioned by Hacking.14 This is “a rule to

ensure profit (in gambling) regardless of victory or loss”, a method of riskless

arbitrage, in short. The text illustrates the rule with an example, where “a great

man knowing mantra and medicine sees a cockfight in progress. He talks to the

owners of the birds separately in a mysterious way. He tells one that ‘if your

bird wins, you give me the amount you bet, and if it loses, I will give you 2

3 of

that amount’. Then he goes to the owner of the other bird where on those same

conditions he promised to pay 3

4 of the amount. In either case, he earned a profit

of only 12 pieces of gold. O mathematician, blessed with speech, tell me how much

money did the owner of each bird bet.”

13Gan. itas¯ara-Sam.graha, (Hindi trans. L. C. Jain), Jain Samskrti Samraksha Sangh, Sholapur,

1963, pp. 159–60.

14Ian Hacking, The Emergence of Probability: A Philosophical Study of Early Ideas about

Probability, Induction and Statistical Inference, Cambridge University Press, 1975, pp. 6–9.

Hacking uses the English translation of Gan. ita S¯ara Sam. graha, trans. M. Rangacharya (1912),

pp. 162–3.

Probability in Ancient India 5

Precise fractions

Apart from the ability to work with large numbers, and to calculate permutations

and combinations, and weighted averages, there is also needed the ability to work

with fractions having large numerators and denominators. Such ability, indicative

of greater precision, is not automatic. Such precise fractions with large numerators

and denominators are certainly found in Indian mathematical texts from the time

of ￣Aryabhat.

a.15 Their use is also reflected in the Indian calendar.

By way of comparison, the Romans had only a few stock fractions to base 12

(each of which had a separate name). Hence they had a wrong duration of the

year as 365 1

4 days, just because it involved such an easy-to-state fraction. And

they retained this wrong duration even after the repeated calendar reforms of the

4th to 6th c., and, for over a thousand years, until 1582. The Greek Attic numerals

were similar, and the Greeks did not work with such fractions in any text coming

actually (and not notionally) from before the 9th c. Baghdad House of Wisdom.16

The significance of the Baghdad House of Wisdom is that Indian arithmetic texts

travelled there, were translated into Arabic, and some of these were further translated

from Arabic into Greek17 like the Indian story book, the Pa˜ncatantra.18

(All known Arabic and Greek manuscripts of “Ptolemy’s” Almagest, for example,

post-date the Baghdad House ofWisdom—most also post-date the Crusades—and

are decidedly post-9th c. accretive texts as is clear, for example, from their star

lists. Therefore, it would be anachronistic to attribute, uncritically and ahistori-

15¯A ryabhat.iya, G¯ıtik¯a 3-4, trans. K. S. Shukla and K. V. Sarma, INSA, New Delhi, 1976, p. 6.

This verse gives revolution numbers for various planets, and requires us to calculate fractions

such as 1582237500/4320000. Up to a hundred years ago, Western historians, who subscribed to

the view that the world was created a mere 6000 years ago, invariably described these figures as

fantastic cosmological speculations, and meaningless by implication. However, the above fraction

leads to a surprisingly accurate figure of 365.25868 days for the length of the sidereal year

(compared to its modern value of 365.25636 days). In contrast, the value attributed to Ptolemy

(365.24666 days) is significantly less accurate, and we know that the length of the (tropical) year

on the Roman calendar was less accurate than the figure attributed to “Ptolemy”. This led to

a slip in the date of Easter, and hence necessitated the repeated attempts at Roman calendar

reform in the 5th and 6th centuries.

16The difference between actual and notional dates is important. The only clear way to check a

notional date is through the non-textual evidence, and it is hard to believe that the crude Greek

and Roman calendars, despite repeated attempts by the state and church to reform them, could

have co-existed for centuries with relatively sophisticated astronomy texts, such as the Almagest

attributed to Ptolemy (which attribution makes its notional date 2nd c., although the actual

manuscripts of it come from a thousand years later, and are accretive).

17C. K. Raju, Cultural Foundations of Mathematics, Pearson Longman, 2007. This text examines

in detail the points made in note 16 above. Also, C. K. Raju, Is Science Western in

Origin?, Citizens International and Multiversity, Penang, 2009, contains a brief account.

18Edward Gibbon, The Decline and Fall of the Roman Empire, Great Books of the Western

World, vols 37–38, Encyclopaedia Britannica, Chicago, 1996, vol. 2, note 55 to chp. 52, p. 608.

Others have assigned the date of 1080 to Simon Seth’s Greek translation of the Pa˜ncatantra from

Arabic. The Arabic translation Kalilah va Dimnah by Ibn al Muqaffa (d. 750), was long before

the formation of the House of Wisdom, and the movement called the Brethren of Purity (Ikhwan

al-Safa) derives inspiration from this text. This translation was from the Pahlavi translation

which was from the Sanskrit, and done by Burzoe himself, the vazir of Khusrow I, Noshirvan,

according to the Shahnama of Firdausi.

6 C. K. Raju

cally, the use of sexagesimal fractions in such a late text to a mythical Claudius

Ptolemy of the 2nd c.19)

The game of dice in India

Thus, Indian tradition had all the arithmetic tools needed for the calculation of

discrete probabilities. But was there a concept of probability? Stock mathematics

texts draw their examples from a range of sources, and we also occasionally find

some examples related to games of chance such as dice. The same slate-arithmetic

text (P¯at.¯ıgan. ita, 99–101, p. 145) gives a long and complex rule for calculating

whether one has won or lost in a game of dice. However, this is given as an

application of the formulae for arithmetic progression. The substance of these

formulae is explained as follows.

Suppose that two persons A and B gamble with dice, and that they alternately

win p1, p2p3p4 casts. If the stake-moneys of the casts be in the arithmetic progression

a, a + d, a + 2d, . . . then the amount won by A = [a + (a + d) + (a + 2d) +

. . . (p1 terms)] + [a + (p1 + p2)d + a(p1 + p2 + 1)d + . . . (p3 terms)]. Likewise, the

amount won by B = [a + p1d + a + (p1 + 1)d + . . . (p2 terms)] + [a + (p1 + p2 +

p3)d++(p1 +p2 +p3 +1)d+. . . (p4 terms)] and this leads to the enunciated rule.

This suggests that the game of dice might not have been played the same way

in India, as it is played today, and also that a common strategy followed was

(somewhat like martingale bets) to go on increasing the stake as the game went

on. But this still does not give us enough information. (Accounts of the game of

dice are not found in the Gan. ita S¯ara Sam. graha.20 Presumably, the Jains did not

want their children to start thinking about such things!) The text naturally takes

it for granted that the readers are familiar with this game of dice, but we do not

seem to have adequate sources for that at the moment, or at least such sources

are not known to this author.

The hymn on dice in the R. gveda

Games of chance, such as dice, certainly existed in Indian tradition from the

earliest times. We find an extraordinary aks.a s¯ukta or hymn on dice in theR.

gveda

(10.2.34). The long hymn begins by comparing the pleasure of gambling with the

pleasure of drinking soma!

“There is enjoyment like the soma in those dice”.21 (som-y˜v mOjvt-ym"o)

It goes on to describe how everyone avoids a gambler, like an old man avoids

horses, even his mother and father feign not to recognize him, and he is separated

from his loving wife. Many times the gambler resolves to stay away, but each

time the fatal attraction of the dice pulls him back. With great enthusiasm he

19This is argued in more detail in C. K. Raju, Cultural Foundations of Mathematics, cited

above. See also notes 15 and 16 above.

20Mahavir, Gan. ita S¯ara Sam. graha, Hindi trans. L. C. Jain, Jain Samskrti Samrakshaka

Sangha, Sholapur, 1963.

21R.

gveda, 10.2.34.1

Probability in Ancient India 7

reaches the gambling place, hoping to win, but sometimes he wins and sometimes

his opponent does. The dice do not obey the wishes of the gambler, they revolt.

They pierce the heart of the gambler, as easily as an arrow or a knife cuts through

the skin, and they goad him on like the ankus, and pierce him like hot irons. When

he wins he is as happy as if a son is born, and when he loses, he is as if dead.

The 53 dice dance like the sun playing with its rays, they cannot be controlled

by the bravest of the brave, and even the king bows before them. They have no

hands, but they rise and fall, and men with hands lose to them. The gambler’s

wife remains frustrated and his son becomes a vagabond. He always spends his

night in other places. Anyone who lends him money doubts that he will get it

back. The gambler who arrives in the morning on a steed leaves at night without

clothes on his back. [Such is the power of dice!] O Dice, I join the ten fingers of

my hands and bow to the leader among you!

THE NOTION OF A FAIR GAME AND THE FREQUENTIST

INTERPRETATION OF PROBABILITY

Fair and deceitful gambling in the Mahabharata

This tells us a great deal about the social consequences of gambling, but still

very little about how exactly the game was played. (This is naturally assumed

to be known to all.) But we do find something interesting from a philosophical

perspective on probability. For, from the earliest times, there was also a notion of

what constitutes a “fair game”, a notion which is today inextricably linked to the

notion of probability.

This can be illustrated by a story from the Mahabharata epic. A key part of

the story, and the origin of the Mahabharata war, relates to the way the heroes

(Pandava-s) are robbed of their kingdom by means of a game of dice (Ŋ`t ĞFXA).

They cannot very well refuse the invitation to play dice because the game involves

risk, and a Kshatriya is dishonoured by refusing to partake in an enterprise involving

risk. However, at the start of the game, Yudhisthira, the leader of the

Pandavas, makes clear that he knows the dice are loaded, or that the game will

involve deceit. He says: “EnEĞEtd˜‚vn\ pAp\ n "A/o/ prAĞm,” (Mbh, Sabh￣a Parva,

59.5). That is, “Deceitful gambling is sinful, there is no Kshatriya valor in it.”

Philosophically speaking, there clearly are two concepts here: “deceitful gambling”

as opposed to “fair gambling”. The notion of a fair game, though not

explicitly defined, is critical to the story, and involves some notion of probability

implicit or explicit. The Mahabharata narrative itself brings out the unfairness of

the game both through Yudhishthira’s statement, at the beginning of the game,

and by telling us about a long series of throws in each one of which Yudhishthira

loses, without ever winning once. Of course, there is room to argue, as the devil’s

advocate might, and as Duryodhana’s uncle Shakuni does, that it is a case of

gambler’s ruin or a long run of bad luck with a finite amount of capital howsoever

large. Apart from the Yudhisthira-Shakuni dialogue, the Mahabharata narrative

8 C. K. Raju

itself carries the depiction of this unfairness to the utmost. Yudhisthira goes on

losing, and in desperation, he even stakes his wife, Draupadi, who is then deemed

to have been won, and publicly stripped and insulted by Duryodhana.22

Interestingly, a little later in the same epic, Yudhisthira, now banished to the

forest, recounts his woes: how he was cheated out of a kingdom and his wife

insulted. To console him, a sage recounts to him the celebrated love story of Nala

and Damayanti. King Nala too lost his kingdom—once again through deceitful

gambling. Having lost also his last remaining clothes in the forest, he covers

himself with half of Damayanti’s sari, and abandons her in the forest. He takes

up a job as a charioteer with king R.

tuparn.

a of Ayodhya. His aim is to learn the

secret of dice from R.

tuparn.

a , for he has been advised by a Naga prince that

that knowledge will help him win back his kingdom. He gets an opportunity, at a

tense moment, while rushing from Ayodhya to Vidarbha for the announced second

marriage (swayamvara) of Damayanti whom R.

tuparn.

a too wants to wed. Nala

stops near a Vibh¨ıt.aka tree23—it is significant that the nuts of this very tree, with

five faces, were used in the ancient Indian game of dice—andR.

tuparn.

a can’t resist

showing off his knowledge of mathematics (gan. ita vidy¯a) by saying: the number

of fruits in the two branches of the tree is 2095, count them if you like.

Nala decides to stop and count them. R.

tuparn.

a, who is apprehensive of being

delayed and knows that he cannot reach in time without Nala’s charioteering skills,

suggests that Nala should be satisfied with counting a portion of one branch.

Then the king reluctantly told him, ‘Count. And on counting the leaves

and fruits of a portion of this branch, thou wilt be satisfied of the truth

of my assertion.’24

Nala finds the estimate accurate and wants to know how it was done, offering

to exchange this knowledge with his knowledge of horses. “And king R.

tuparn.

a,

having regard to the importance of the act that depended upon Vahuka’s [Nala’s]

good-will, and tempted also by the horse-lore (that his charioteer possessed), said,

‘So be it.’ As solicited by thee, receive this science of dice from me...” (ibid ). (The

story has a happy ending, and Nala does get back his wife and his kingdom.)

From our immediate point of view, the interesting thing is the way the Mahabharata

text conflates sampling theory with the “science of dice”. (It is legitimate

to call this “science”, for there is a clear relation to a process of empirical verification,

by cutting down the tree and counting.) This connection of sampling

theory to the game of dice is mentioned by Hacking,25 who attributes it to V. P.

Godambe. They have, however, overlooked the other aspect of the story, which is

that this knowledge was regarded as secret, andR.

tuparn.

a parts with it only under

22Eventually, it is decreed that Yudhisthira incorrectly regarded her as his property to be

staked, however, one of the heroes, Bhim, swears that he will break Duryodhana’s thigh, on

which he seats Draupadi, and also drink the blood of his brother Dusshasana who forcibly

dragged Draupadi to the court, both of which promises he fulfills years later in the battlefield.

23Mahabharata, van parva, 72, trans. K. M. Ganguly, 1883–1896, Book 3, pp. 150–51.

24ibid, p. 151.

25Ian Hacking, The Emergence of Probability, cited earlier, pp. 6–9.

Probability in Ancient India 9

extraordinary conditions; therefore, one should not expect to find this knowledge

readily in common mathematical texts.

“Fair gambling”, the law of large numbers

There are, however, repeated references to “deceitful gambling”. So, one can

ask: what exactly did “fair gambling” mean? As the notion is understood today,

in an ultimate sense, the fairness of the game cannot be established merely by

the law of large numbers, for it is well known that the frequentist interpretation

of probability fails because relative frequency converges to probability only in a

probabilistic sense. Shakuni is right that a long streak of bad luck is always possible

(and must almost surely occur in the long run). Therefore, also, the randomness

of a set of random numbers can ultimately be established only by reference to

the process which generated those random numbers and not merely by post-facto

statistical tests of randomness. Note the emphasis on “ultimate”. Given a series of

random numbers even a simple χ2 test could show,26 for most practical purposes,

whether the series is concocted; the question is whether it can settle “all” doubt,

for there is always some residual probability that the test gave a wrong result,

and this problem of what to do with very small numbers takes us back to the old

problem of the law of large numbers. One cannot rule out the occurrence even of

probability-zero cases, and the best one can say is that they will “almost surely”

not occur.

Law of large numbers and the notion of convergence as a supertask

At this point, it is probably a good idea to understand two key differences between

the Indian philosophy of mathematics and the Western philosophy of mathematics.

The contemporary notion of a fair game already involves some notion of the

law of large numbers, hence a notion of convergence in some sense (such as convergence

in probability). It does not matter how “weak” this notion of convergence

is (i.e. whether Lp convergence or convergence in measure): the point is that, as

conceptualised in the Western philosophy of mathematics, any notion of convergence

involves a supertask—an infinite series of tasks. Such supertasks can only

be performed metaphysically, within set theory; however, barring special cases, a

supertask is not something that can be empirically performed (in any finite period

of time).

Supertasks and the clash of mathematical epistemologies

Historically speaking, the transmission of the calculus27 from India to Europe in

the 16th c. led to a clash of epistemologies. The situation is analogous to the

26E.g., C. R. Rao, Statistics and Truth: Putting Chance to Work, CSIR, New Delhi, 1989.

27C. K. Raju, Cultural Foundations of Mathematics, Pearson Longman, 2007, chp 9, “Math

wars and the epistemological divide in mathematics”.

10 C. K. Raju

clash of epistemologies which occurred when the Indian arithmetic techniques,

called the Algorismus (after al Khwarizmi’s latinized name), or “Arabic numerals”,

were imported into Europe and clashed with the abacus or the native Roman

and Greek tradition of arithmetic. With regard to the Indian calculus, Western

philosophers did not understand its epistemology, and felt that it involved supertasks.

The practical usefulness of the calculus (in calculating trigonometric values,

navigational charts etc.) then created the need to justify such supertasks. In the

Western philosophy of mathematics, this mistaken belief about supertasks arose

because of two reasons, (1) a belief in the perfection of mathematics, and (2) a

distrust of the empirical as means of proof.

Thus, for example, reacting to the use of the new-fangled calculus-techniques

by Fermat and Pascal, Descartes28 wrote in his Geometry that comparing the

lengths of straight and curved lines was “beyond the capacity of the human mind”.

Galileo’s reaction, in his letters to Cavalieri, was similar, and he ultimately left it

to his student Cavalieri to take credit or discredit for the calculus.

Descartes’ reaction seems idiosyncratic and excessive, because any Indian child

knew how to measure the length of a curved line empirically by using a piece of

string, or ´sulba or rajju, and this was done in India, at least since the days of the

´sulba s¯utra [−500 CE]. Perhaps, Descartes thought that the European custom of

using a rigid ruler was the only way of doing things. (European navigators certainly

had a serious problem for just this reason.) In any case, Descartes proceeded from

the metaphysical premise that only the length of a straight line was meaningful. On

this metaphysical premise, he thought that the length of a curved line could only

be understood by approximating it by a series of straight-line segments. However

large the number of straight line segments one might use, and howsoever fine the

resulting approximation might be, it remained “imperfect”, hence was not quite

mathematics, for mathematics (Descartes believed) is perfect and cannot neglect

the smallest quantity.

Descartes was willing to make a concession and allow that infinitesimal quantities

could be neglected (a premise shared by many of his contemporaries, including,

later Berkeley, in his criticism of Newton, and a process later formalised

in non-standard analysis). Therefore, the only situation Descartes was willing to

contemplate as acceptable was a situation where the neglected error was infinitesimal.

But this required an infinity of straight line segments, each of infinitesimal

length. But that created another difficulty: to compute the length of the curved

line, one now had to sum an infinity of infinitesimals. And this, thought Descartes,

was a supertask only God could perform.29 (This was in the days before formal

mathematics. Descartes thought of summing an infinite series by actually carrying

out the sum.)

28Ren´e Descartes, The Geometry, Book 2, trans. David Eugene and Marcia L. Latham, Encyclopaedia

Britannica, Chicago, 1990, p. 544.

29For more details, see C. K. Raju, “The Indian Rope Trick”, Bhartiya Samajik Chintan 7(4)

(New Series) (2009) pp. 265–269. http://ckraju.net/papers/MathEducation2RopeTrick.pdf.

Also, Cultural Foundations of Mathematics, cited earlier, for more details.

Probability in Ancient India 11

That is, with the European way of thinking about mathematics as “perfect”,

the problem of neglecting small numbers got converted into the problem of supertasks

or convergence. This got related to the difficulty of summing an “infinity of

infinitesimals”. It is these philosophical problems (supertasks, convergence) which

are deemed to have been resolved in the Western tradition of mathematics today,

through formal set theory, which enables the supertasks needed for the definition

of formal real numbers and limits to be carried out formally, and which are needed

for the current notion of a “fair game”.

To put matters in another way, the philosophical issue which blocks the frequentist

interpretation of probability is the Western belief that mathematics is

“perfect”, and hence cannot neglect even the smallest quantity, such as 10−200.

In practice, of course, this frequentist understanding is commonly used in physics

and works wonderfully well with a gas in a box where a “large” number may mean

only 1023 molecules.

In Western thought, however, what is adequate for physics is inadequate for

mathematics, because empirical procedures are suspect and regarded as inferior

to metaphysical procedures which are regarded as certain and “ultimate”. This

attitude is, in fact, at the core of almost all Western philosophy: deduction, a

purely metaphysical process, is believed to be surer than induction, which involves

an empirical process.

(To simplify the discussion, we are not getting here into another fine issue:

the very division of physical and metaphysical in terms of Popper’s criterion of

refutability already involves inductive processes, and their relative valuation, for

logical refutability does not guarantee empirical refutability, which latter requires

an inductive process. Popper claimed that he had solved the problem of induction,

30 since probabilities, defined the Kolmogorov way, are not ampliative and are

left unaffected by any new experiments. However, Popper overlooked the fact that

empirically one can only obtain estimates of probabilities, never the probabilities

themselves, and estimates of probabilities may be ampliative. So, Popper did not

succeed in throwing any new light on the problem of induction.)

Suffice to say that these difficulties regarding “the fear of small numbers” and

“inferiority of the empirical” are closely related to the understanding of the ”law

of large numbers” on the one hand, and , on the other hand, to the notion of

“convergence” and “limits” that developed when Europeans tried to assimilate

the calculus imported from India in the 16th c. CE.

Zeroism: beyond the clash of epistemologies

The hate politics that prevailed for centuries in Europe during the Crusades and

Inquisition had a decisive impact not only on the Western history of science, but

also on its philosophy: both had to be theologically correct (on pain of death).

The way this was exploited by colonialists and racists is made abundantly clear

30K. R. Popper, Realism and the Aim of Science, PS to LscD, vol 1, Hutchinson, London,

1982.

12 C. K. Raju

in various ways. For example, Frank Thilly’s text on philosophy, used as a text in

pre-independence India, starts with the dismissal of all non-Western traditions as

“non-philosophy”. This belittling of anything non-Western as inappropriate and

not-quite-science still represents a typical Western attitude today.

However, if we abandon the current Western historical myths about mathematics

in Indian tradition, and look at the tradition per se, that gives us a fresh angle

on the philosophical problems related to small numbers, and supertasks and the

alleged unreliability of empirical procedures, which philosophical problems are intimately

linked to the notion of convergence, and the related issue of the frequentist

interpretation of probability.

First, let us take note of the differences in the cultural context. Basically, in

India, unlike Europe, there was never a monopoly or hegemony of any single religion,

and in this pluralistic (and largely secular) Indian environment, where many

different brands of metaphysics were prevalent, there was no question of treating

any particular brand of metaphysics as certain. Different people may (and did)

have conflicting beliefs even about logic from the earliest times.31 The Lok￣ayata

rejected deduction as an inferior technique which did not lead to sure knowledge.

The tradition in India, since the time of the Buddha, was to decide truth by

debate, and not by the forcible imposition of any one brand of metaphysics.

Since metaphysics carried no certitude, the empirically manifest (pratyaks.a)

was the primary basis of discourse for two persons with conflicting metaphysical

beliefs (even about logic etc.). This was the one (and only) means of proof which

was accepted by all philosophical schools of thought in India. All this is in striking

contrast to Western tradition where mathematics is regarded as metaphysical, and

this metaphysics is simply declared, by social fiat, to be “universal” and certain

like a religious belief. This attitude emerged naturally in the West, where mathematics

was linked to religion and theology, and was spiritual and anti-empirical

since the days of Plato and Neoplatonists like Proclus. (Indeed the very name

“mathematics” comes from mathesis, meaning learning, and learning, on the wellknown

Platonic/Socratic doctrine, is recollection of knowledge obtained in past

lives, and so relates to stirring the soul. Mathematics was thought of as being

especially good for this purpose, since it involved eternal truths which moved the

eternal soul by sympathetic magic.) It is this belief—that mathematics contained

eternal truths—which led to the belief that mathematics is perfect and hence

cannot neglect the smallest quantity.

We have seen that the philosophical issue which blocks the frequentist interpretation

of probability is just this Western belief in the perfection of mathematics.

A number, howsoever small, such as 10−200 cannot be neglected and set equal to

zero. On the other hand, Indian tradition allowed this to be done, somewhat in

the manner in which rounding is done today, but with a more sophisticated philosophy

known as ´nyav¯ada which I have called zeroism,32 so as to emphasize the

31B. M. Baruah, A History of Pre-Buddhistic Indian Philosophy, Calcutta, 1921; reprint Motilal

Banarsidass, Delhi, 1970.

32C. K. Raju, “Zeroism and Calculus without Limits”, 4th dialogue between

Probability in Ancient India 13

fruitful practical aspects of the philosophy, and avoid sterile controversies about

the exact interpretation of Buddhist texts.

Now it so happens that the formula for the sum of an infinite (anantya) geometric

series was first developed in India (we find this in the ¯Aryabhat.iya Bh¯as. ya

of the 15th-16th c. N￣ılakan.

t.

ha33). This naturally involved the question of what

it means to sum an infinite series. A procedure existed to test the convergence

of an infinite series (of both numbers and functions), and I have explained it

elsewhere in more detail.34 This is similar to what is today called the Cauchy

criterion of convergence, except for one aspect. Thus, given an infinite series of

numbers, Σan, and an _ > 0, the process checked whether there was an N beyond

which __N+m

n=N an_ < _. The supertask of actually summing the residual

partial sums for all numbers m could obviously not be carried out except in some

special cases (such as that of the geometric series). However, if S(k) = _k

n=1 an

denotes the partial sums, the usual process was to check whether the partial sums

became “constant”, beyond some N.35 Obviously, the partial sums S(k) would

never literally become constant, and when successive terms are added, there would

always be some change (except when all the terms of the series are zero beyond

N, so that the infinite sum reduces to a finite sum). So this “constancy” or “no

change” was understood to hold only up to the given level of precision (_) in use.

That is, the sum of an infinite series was regarded as meaningful if the partial

sums S(k) became constant, after a stage, up to a non-representable (or discardable)

quantity: _S(N +m) − S(N)| < _, which is just the criterion stated earlier.

What exactly constitutes a “non-representable” or “discardable” quantity (_) is

context-dependent, decided by the level of precision required, and there need be

no “universal” or mechanical rule for it.

Apart from a question of convergence, a key philosophical issue which has gone

unnoticed relates to representability. The decimal expansion of a real number,

such as π, also corresponds to an infinite series. Regardless of the convergence of

this series, it can be written only up to a given number of terms (corresponding

to a given level of precision): even writing down the terms in the infinite decimal

expansion is a supertask: this is obvious enough when there is no rule to predict

what the successive terms would be. So a real number such as π can never be

accurately represented; Indian tradition took note of this difficulty from the earliest

times, with the ´sulba sutra-s (-500 CE or earlier) using the words36 sEvf˜q (“with

something left out”) or37 sAEn(y (sa + anitya = “impermanent, inexact”), and

Buddhism and Science, Nalanda, 2008. Draft at http://ckraju.net/papers/

Zeroism-and-calculus-without-limits.pdf.

33¯A ryabhat.¯ıya of ¯Aryabhat.¯ac¯arya with the Bh¯as.ya of N¯ılakan. t.hasomas¯utvan, ed. K. Sambasiva

Sastri, University of Kerala, Trivandrum, 1930, reprint 1970, commentary on Gan. ita 17, p. 142.

34C. K. Raju, Cultural Foundations of Mathematics, cited above, chp. 3.

35ibid. p. 177-78, and e.g., Kriy¯akramakar¯ı, cited above, p. 386.

36Baudh￣ayana ´sulba s¯utra, 2.12. S. N. Sen and A. K. Bag, The ´Sulbas¯utras of Baudh¯ayana,

Apastamba, K¯aty¯ayana, and M¯anava, INSA, New Delhi, 1983, p. 169.

37Apastamba ´sulba s¯utra, 3.2. Sen and Bag, cited above, p. 103. The same thing is repeated

in other ´sulba s¯utra-s.

14 C. K. Raju

early Jain works (such as the S¯urya praj˜n¯apati, s¯utra 20) also use the term ki˜ncid

vi´ses¯adhika (“a little excess”) in describing the value of π and

√

2. ￣Aryabhat.

a (5th

c. CE) used the word38 aAsa (near value), which term is nicely explained by

N￣ılakan.

t.

ha in his commentary,39 essentially saying that the “real value” (vA-tvF\

s\HyA) cannot be given.

Taking cognizance of this element of non-representability fundamentally changes

arithmetic. This happens, for example, in present-day computer arithmetic, where

one is forced to take into account this element of non-representability, for only a

finite set of numbers can be represented on a computer. Consequently, even integer

arithmetic on a computer can never obey the rules of Peano’s arithmetic. In the

case of real numbers, or floating point computer arithmetic, of course, a mechanical

rule is indeed set up for rounding (for instance in the IEEE floating point standard

754 of 1986), and this means that addition in floating point arithmetic is not an

associative operation,40 so that floating point arithmetic would never agree with

the arithmetic according to any standard formal algebraic structure such as a ring,

integral domain, field etc.

In Indian tradition, this difficulty of representation connects to a much deeper

philosophy of ´s¯unyav¯ada. On the Buddhist account of the world, the world evolves

according to “conditioned coorigination”. (A precise quantitative account of what

this phrase means to me, and how this relates to current physics is a bit technical,

and is available in the literature for those interested in it.41) The key point is that

there is genuine novelty of the sort that would surprise even God, if he existed.

There is no rigid linkage (no Newton’s “laws”) between present and past, the

present is not implicit in the past (and cannot be calculated from knowledge of

the past, even by Laplace’s demon). Accordingly, there is genuine change; nothing

stays constant. But how does one represent a non-constant, continually changing

entity? Note that, on Buddhist thought, this problem applies to any entity, for

Buddhists believe nothing real can exist unchanged or constant for two instants, so

there is no constant entity whatsoever which is permanent or persists unchanged.

This creates a difficulty even with the most common utterances, such as the

statement “when I was a boy”, for I have changed since I was a boy, and now

have a different size, gray hair etc. The linguistic representation however suggests

that underlying these changes, there is something constant, the “I” to which these

changes happen. Buddhists, however, denied the existence of any constant, unchanging

essence or soul for it was neither empirically manifest, nor could it be

inferred: the boy and I are really two different individuals with some common

38Gan. ita 10, trans. K. S. Shukla cited earlier.

39¯Aryabhat.¯ıya bh¯as.ya, commentary on Gan. ita 10, ed. Sambasiva Sastry, cited earlier, p. 56.

40For an example of how this happens, see C. K. Raju, “Computers, mathematics education,

and the alternative epistemology of the calculus in the Yuktibhasa”, Philosophy East and West,

51(3) 2001, pp. 325–62.

41The idea is to use functional differential equations of mixed-type to represent physical time

evolution. This leads to spontaneity. See C. K. Raju, Time: Towards a Consistent Theory,

Kluwer Academic, 1994, chp. 5b. Also, C. K. Raju, “Time Travel and the Reality of Spontaneity”,

Foundations of Physics 36 (2006) pp. 1099–1113.

Probability in Ancient India 15

memories. However, while Buddhists accept the reality of impermanence, there is

a practical problem of representation in giving a unique name to each individual at

each instant. Consider, for example, Ashoka. No one, not even the Buddhists, describe

him as Ashoka1, Ashoka2, and so on, with one number for each instant of his

life, which cumbersome nomenclature would require some billion different names

even on the gross measure of one second as an atomic instant of time. Therefore,

for practical purposes, Buddhists recognize the paucity of names, and still use a

single name to represent a whole procession of individuals. This “constancy” of

the representation is implicitly understood in the same sense as the constancy of

the partial sums of an infinite series: namely, one neglects some small differences

as irrelevant to the context. That is, on the Buddhist view of constant change, the

customary representation of an individual, used in everyday parlance, as in the

statement “when I was a boy”, can be obtained only by neglecting the changes involved

(my size, my gray hair, etc) as inconsequential or irrelevant in the context,

and which changes are hence discarded as “non-representable” (for the practical

purpose of mundane conversation, in natural language).

So, from the Buddhist perspective of impermanence, mundane linguistic usage

necessarily involves such neglect of “inconsequential” things, no matter what

one wants to talk about. Note the contrast from the idealistic Platonic and Neoplatonic

belief. Plato and Neoplatonists believed in the existence of ideal and

unchanging or eternal and constant entities (soul, mathematical truths). Within

this idealistic frame, mundane linguistic usage (as in the statement “when I was a

boy”) admits a simple justification in straightforward sense that change happens

to some underlying constant or ideal entity. But this possibility is not available

within Buddhism, which regards such underlying ideal entities as fictitious and

erroneous, and can, therefore, only speak about non-constant entities, as if they

were constant. The dot on the piece of paper is all we have, it is the idealization

of a geometric point which is erroneous. (Apart from the idealist position, the

formalist perspective of set theory also fails, for Buddhist logic is not two valued.

But I have dealt with this matter in detail, elsewhere, and we will see this in more

detail below.)

Thus, ´sunyav¯ada or zeroism provides a new way to get over the “fear of small

numbers”. It was, I believe, Borel who raised the question of the meaning of

small numbers such as 10−200. On the ´s¯unyav¯ada perspective, we can discard

such numbers as practically convenient. (We have nothing better, no “ideal” or

“perfect” way of doing things.) We are not obliged to give a general or universal

rule for this, though we can adopt convenient practices.

What this amounts to is a realist and fallibilist position. All knowledge (including

mathematical knowledge) is fallible.42 Therefore, when given an exceesively

42If mathematical proof is treated as fallible, the criterion of falsifiability would need modification.

When a theory fails a test, it is no longer clear what has been refuted: (a) the hypothesis

or (b) the deduction connecting hypothesis to consequences. C. K. Raju, “Proofs and refutations

in mathematics and physics”, in: History and Philosophy of Science, ed. P. K. Sen, PHISPC (to

appear).

16 C. K. Raju

small number we may discard it, as in customary practice, or in computer arithmetic.

(Unlike computer arithimetic, where one requires a rule, with human arithmetic,

we can allow the “excessive smallness” of the number to be determined by

the context.) It is possible, that this leads to a wrong decision. If enough evidence

accumulates to the contrary, we revise our decision. It is the search for immutable

and eternal truths that has to be abandoned. Such eternal truths are appropriate

to religion not any kind of science.

Thus the traditional Indian understanding of mathematics, using zeroism, dispenses

with the need for convergence, limits, or supertasks, and rehabilitates the

frequentist interpretation of probability, in the sense that it provides a fresh answer

to a long-standing philosophical difficulty in the Western tradition.

SUBJECTIVE PROBABILITIES AND THE UNDERLYING LOGIC OF

SENTENCES

Probabilities of singular events

Of course, there are other problems with the frequentist interpretation: for example,

it does not apply to single events, for which one might want to speak of

probability. The classic example is that of a single footprint on a deserted beach

(or the origin of life). There is some probability, of course, that someone came in

a helicopter and left that single footprint just to mystify philosophers. But, normally,

one would regard it as a natural phenomenon and seek a natural explanation

for it.

In this context there is an amusing account from Indian Lok￣ayata tradition,

which is the counterpart of the Epicurean perspective in Greek tradition. Here, a

man seeking to convert his girlfriend to his philosophical perspective, goes about

at night carrying a pair of wolf’s paws. He makes footprints with these paws. His

aim is to demonstrate the fallibility of inference. He argues that by looking at

the footprints, learned people will infer that a wolf was around, and they will be

wrong. (We recall that the Lok￣ayata believed that the only reliable principle of

proof was the empirically manifest.)

More seriously, such singular events pose a serious problem today in quantum

mechanics, where the “probability interpretation of the wave function” is called

into play to explain interference of probabilities exhibited by single objects. A

typical illustration of such interference is the two-slit diffraction pattern that is

observed even when it is practically assured that electrons are passing through the

slit one at a time. Understanding the nature of quantum probabilities has become

a major philosophical problem, and we describe below some attempts that have

been made to understand this problem by connecting it to philosophies and logics

prevalent in ancient Indian tradition.

Probability in Ancient India 17

Quantum mechanics, Boolean algebra, and the logic of propositions

In the 1950’s there was a novel attempt to connect the foundations of probability

theory to the Jain logic of sy¯adav¯ada, by three influential academicians from

India: P. C. Mahalanobis,43 founder of the Indian Statistical Institute, J. B. S.

Haldane,44 who had moved to that institution, and D. S. Kothari,45 Chairman of

the University Grants Commission. Subsequently, the quasi truth-functional logic

used in the structured-time interpretation of quantum mechanics46 was connected

to Buddhist logic.47

To understand these attempts, first of all, let us connect them to the more

common (Kolmogorov) understanding of probability as a positive measure of total

mass 1 defined on a Boolean σ-algebra (usually of Borel sets of a topological space).

The common definition typically requires set theory, as we saw above, to facilitate

the various supertasks that are required, whether for the construction of formal

real numbers as Dedekind cuts, or as equivalence classes of Cauchy sequences, or

for the notion of convergence required by the law of large numbers. However,

from a philosophical perspective it is more convenient to use statements instead

of sets (though the two are obviously interconnected). Thus, instead of defining

probabilities over measurable sets, it is more natural to define probabilities over

a Boolean algebra of statements. This, incidentally, suits the subjectivist interpretation,

for the probability of a statement could then be taken to indicate the

degree of (general) subjective belief in that statement (or the objective propensity

of that statement to be true, whatever that means).

The immediate question, however, is that of the algebraic structure formed by

these statements. First of all, we can set aside the specifically σ-algebra aspect,

for we have already dealt with the notion of convergence and supertasks above.

For the purposes of this section we will focus on the Boolean algebra part. Why

should probability be defined over a Boolean algebra?

The answer is obviously that if we have a 2-valued logic of sentences, then a

Boolean algebra is what we naturally get from the usual notion of “and”, “or”,

“not”, which are used to define the respective set-theoretic operations of intersection,

union and complementation. What is not obvious is why these “usual”

notions should be used, or why logic should be 2-valued.

Quantum mechanics (and especially the problem of the probabilities of singular

events in it) provides a specific empirical reason to call the Boolean algebra into

question. With probabilities defined on a Boolean algebra, joint distributions of

43P. C. Mahalanobis, ‘The Foundations of Statistics (A Study in Jaina Logic)’, Dialectica 8,

1954, pp. 95–111; reproduced in Sankhya, Indian Journal of Statistics, 18, 1957, pp. 183–94.

44J. B. S. Haldane, ‘The Sy¯adav¯ada system of Predication’, Sankhya, Indian Journal of Statistics,

18, 1957, pp. 195–200.

45D. S. Kothari, ‘Modern Physics and Sy¯adav¯ada’, Appendix IV D in in Formation of the

Theoretical Fundamentals of Natural Science vol. 2 of History of Science and Technology in

Ancient India, by D. P. Chattopadhyaya, Firma KLM, Calcutta, 1991, pp. 441–48.

46C. K. Raju, Time: Towards a Consistent Theory, cited above, chp. 6B, “Quantum Mechanical

Time”.

47C. K. Raju, The Eleven Pictures of Time, Sage, 2003

18 C. K. Raju

random variable are assured to exist. This is, however, known to not happen

in quantum mechanics. (We will not go into details, since our primary concern

here is with Indian tradition, and not quantum mechanics. However, this author

has explained the detailed relation to quantum mechanics elsewhere, at both a

technical48 and a non-technical level.49) The Hilbert space formulation of quantum

mechanics starts with the premise that quantum probabilities cannot be defined on

a Boolean algebra, since joint distributions do not exist. The appropriate algebraic

structure is taken to be that of the lattice of subspaces (or projections) of a Hilbert

space (although there are numerous other opinions about what the exact algebraic

structure ought to be).

The usual definition of a random variable as a measurable function actually

requires only the inverse function, which is a homomorphism which preserves the

algebraic structure. In the Hilbert space context, this definition of a random

variable as a homomorphism (on a lattice, not an algebra) naturally leads one

to identify random variables with projection-valued measures (spectral measures).

By the spectral theorem, such measures correspond to densely-defined, self-adjoint

operators in this Hilbert space. Since the lattice of projections is non-distributive,

these random variables do not admit joint distributions. This corresponds to

the more common assertion (“uncertainty principle”) that dynamical (random)

variables (self-adjoint operators) which do not commute cannot be simultaneously

measured.

To return to the question of logic, unlike in India, where different types of logic

have been in existence for over 2500 years, from pre-Buddhist times,50 the West

took cognizance of the existence of logics that are not 2-valued, only from the

1930’s onwards, starting with Lukaciewicz who proposed a 3-valued logic, where

the truth values could be interpreted as “true”, “false”, and “indeterminate”.

Could such a 3-valued logic account for quantum probabilities? This question was

first investigated by Reichenbach, in an unsuccessful interpretation of quantum

mechanics.

The 3 Indian academics mentioned above also interpreted the Jain logic of

sy¯adav¯ada (perhaps-ism) as a 3-valued logic (Haldane), and explored 3-valued

logic as a philosophical basis for formulating probabilities (Mahalanobis), and interpreting

quantum mechanics (Kothari). Haldane’s idea related to perception.

With repeated experiments, something on the threshold of perception (such as

a sound) may be perceptible sometimes, and sometimes not. In such cases, the

“indeterminate” truth value should be assigned to the statement that the “something”

is perceptible. Mahalanobis’ idea was that this third truth value was already

a rudimentary kind of probability, for it expressed the notion of “perhaps”.

Kothari’s idea was to try and explain quantum mechanics on that basis (though

he overlooks Reichenbach’s earlier unsuccessful attempt).

48C. K. Raju, Time: Towards a Consistent Theory, cited above, chp. 6B, “Quantum Mechanical

Time”.

49C. K. Raju, The Eleven Pictures of Time, Sage, 2003.

50B. M. Baruah, A History of Pre-Buddhistic Indian Philosophy, cited above.

Probability in Ancient India 19

Buddhist and quasi truth-functional logic

While Haldane’s interpretation is clear enough within itself, it is not clear that

it accurately captures the logic of sy¯adava¯ada. Thus, the Jain tradition grew in

the vicinity of the Buddhist tradition (the Buddha and Mahavira were contemporaries).

However, Buddhist logic is not 3-valued. For example, in the D￣ıgha

Nik￣aya, the Buddha asserts the existence of 4 alternatives (catus.kot.i): (1) The

world is finite; (2) the world is not finite (= infinite); (3) the world is both finite

and infinite; and (4) the world is neither finite nor infinite.51

This logic of 4-alternatives does not readily fit into a multi-valued truth-functional

framework. Especially, the third alternative, which is of the form A ∧ ￢A, is a

contradiction within 2-valued logic, and difficult to understand even within the

frame of 3-valued logic, where it cannot ever be “true”’. The reason why 3-valued

logic is not appropriate for quantum probabilities is roughly this: in the case of

the two-slit experiment, what is being asserted is that it is true that the electron

passed through both slit A and slit B, and not that in reality it passed through

only one slit, but we do not know which slit it passed through. What is being

asserted is that we know that Schr¨odinger’s cat is both alive and dead, as in the

third alternative above, and not that it is either alive or dead, but we do not know

which is the case.

However, the 3rd alternative of the Buddhist logic of 4 alternatives (catus.kot.i)

makes perfect sense with a quasi truth-functional logic. The standard semantics

here uses the Tarski-Wittgenstein notion of logical “world”, as “all that is the

case”. On this “possible-world semantics” one assigns truth values (either true or

false) to all atomic statements: such an assignment of truth values represents the

possible facts of the world (at one instant of time), or a “possible world”. This

enables the interpretation of modal notions such as possibility and necessity: a

statement is “possible” if it is true in some possible worlds, and “necessary” if it is

true in all possible worlds (tautology). In fact, Haldane appeals to precisely this

sort of semantics, in his interpretation of Jain logic, except that his “worlds” are

chronologically sequential. Thus, A is true at one instant of time, while not-A is

true at another instant of time—there is nothing paradoxical about a cat which

is alive now, and dead a while later. However, this, as we have observed, is not

appropriate to model the situation depicted by quantum mechanics.

With quantum mechanics what we require are multiple logical worlds attached

to a single instant of time. Parallel computing provides a simple and concrete

desktop model of this situation, with each processor represented by a separate logical

world. The meaningfulness of a quasi truth-functional logic is readily grasped

in this situation where multiple (logical) worlds are chronologically simultaneous

and not sequential, for this allows a statement to be simultaneously both true and

false. That is, with multiple (2-valued) logical worlds attached to a single instant

51Brahmaj￣ala sutta of the D￣ıgha Nik￣aya. (Hindi trans.) Rahul S￣ankrity￣ayana and Jagdish

K￣ashyapa. Delhi: Parammitra Prakashan, 2000, pp. 8–9; (English trans.) Maurice Walshe,

Boston: Wisdom Publication, 1995, pp. 80–81. For a more detailed exposition, see C. K. Raju,

“Logic”, in Encyclopedia of Non-Western Science, Technology and Medicine, Springer, 2008.

20 C. K. Raju

of time, it is meaningful to say that A is true in one world while ￢A is simultaneously

true in another. So, a statement may be simultaneously both true and false,

without trivializing the theory, or making it inconsistent. From our immediate

perspective, the important thing is this: such a quasi truth-functional logic leads

on the one hand to an algebraic structure appropriate to quantum probabilities,

which structure is not a Boolean algebra.52 On the other hand, Buddhist logic

(catus.kot.i) naturally admits an interpretation as a quasi truth-functional logic.

Thus, Buddhist logic (understood as quasi truth-functional) leads to just the sort

of probabilities that seem to be required by quantum mechanics.

This quasi truth-functional logic, corresponding to simultaneous multiple worlds,

is not a mere artificial and post-facto construct imposed on either quantum mechanics

or Buddhist thought. From the viewpoint of physics, quasi truth-functional

logic arises naturally by considering the nature of time. This is best understand

through history. Hoping to make “rigorous” the imported calculus, and the notion

of derivative with respect to time required for his “laws”, Newton made time metaphysical

(“absolute, true, and mathematical time” which “flows equably without

relation to anything external”53). Eventually, this intrusion of metaphysics and

religious belief into physics had to be eliminated, from physics, through a revised

physical definition of the measure of time; that directly led to the special theory

of relativity.54 A correct mathematical understanding of relativity, shows that

physical time evolution must be described by functional differential equations (and

not ordinary differential equations or partial differential equations). The further

elimination of the theological understanding of causality in physics makes these

functional differential equations of mixed-type. The resulting picture of physical

time evolution55 is remarkably similar to the core Buddhist notion of “conditioned

coorigination”: where the future is conditioned by the past, but not decided by it.

There is genuine novelty. Thus, the relation of the quasi truth-functional logic to

the revised notion of time, in physics, parallels the relation of Buddhist logic to the

Buddhist notion of “conditioned coorigination” (paticca samupp¯ada). Note that

this last notion differs from the common notion of “causality” used in Western

thought, with which it is commonly confounded.

Of course, formal Western mathematics (and indeed much of Western philosophy)

is likely to be a long-term casualty of any departure from 2-valued logic. In

fact, the very idea that logic (or the basis of probability) is not culturally universal,

and may not be empirically certain, unsettles a large segment of Western thought,

52See the main theorem in C. K. Raju “Quantum mechanical Time”, cited above.

53I. Newton, The Mathematical Principles of Natural Philosophy, A. Motte’s translation revised

by Florian Cajori, Encyclopedia Britannica, Chicago, 1996, p. 8.

54For an exposition of Poincar´e’s philosophical analysis of the notion of time which led to the

special theory of relativity, see C. K. Raju, “Einstein’s time”, Physics Education (India), 8(4)

(1992) pp. 293–305. A proper clock was defined by postulating the velocity of light to be a

constant. This had nothing to do with any experiment. See, also, C. K. Raju, “The Michelson-

Morley experiment”, Physics Education (India) 8(3) (1991) pp. 193–200.

55C. K. Raju, “Time travel and the reality of spontaneity”, Found. Phys. 36 (2006) pp. 1099–

1113.

Probability in Ancient India 21

and its traditional beliefs about induction and deduction.