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As I discuss in this question, by far the most popular school of Hindu philosophy is the Vedanta school. But there are five other Astika or orthodox schools of Hindu philosophy: Purva Mimamsa, Samkhya, Yoga, Vaisheshika, and Nyaya. My question is about the Vaisheshika school, according to which atoms are the cause of the Universe. As I discuss here, the second oldest work of the Vaisheshika school is Prashastapada's Padartha Dharma Sangraha. Now in this excerpt of the Padartha Dharma Sangraha, Prashastapada discusses the nature of numbers:

Number forms the basis of such usages as "one" and the rest. It inheres in one and many substances. The number inhering in one substance has its eternal and transient manifestations as those of the color etc. of the atom of water and the rest. The number inhering in many substances begins with "Two" and ends with "Parardha" (100,000,000,000,000,000).

I'm interested in the part in bold. The fact that a Parardha equals 100,000,000,000,000,000, i.e. 100 quadrillions, is a parenthetical note given by the translator, but it's confirmed in this chapter of the Vishnu Purana:

A Parárddha, Maitreya, is that number that occurs in the eighteenth place of figures, enumerated according to the rule of decimal notation.

But my question is, why did the Vaisheshika school believe that 100 quadrillion is the largest number? The fact that they believed this seems fairly clear. Here's what this excerpt from Sridhara's Nyaya Kandali, a commentary on Prashastapada's work, says:

The number inherent in more than one substance includes all numbers from two to the highest number. "Parardha" is that at which all limitations of number cease.

But what is the reason for this belief? Now as I said, the Vaisheshika school believed in atomism, so is it possible that they thought that 100 quadrillion was the number of atoms in the Universe, and thus the most significant number? Do any Vaisheshika works shed light on this?

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    It could just be that it was the highest named number. There is no point of naming powers of 10 beyond a particular point: a name assigned to 10^(10^100) for example does not add much value to English or Sanskrit.
    – user1952500
    Oct 22, 2017 at 3:53
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    @user1952500 But then why would Sridhara's Nyaya Kandali call a Parardha "the highest conceivable number" and "that at which all limitations of number ceases"? That seems like a stronger claim than "it's the highest named number".
    – Keshav Srinivasan ♦
    Oct 22, 2017 at 4:05

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The Nyāya-Vaiśeṣika texts don't mention the specific numerical value of parārdha, what values you probably saw in the Praśastapāda's Padārthadharmasaṅgraha (PDS) has been added by the translator, and isn't there in the original text or its commentaries, as you already figured out.
Also, an important point to note is that, the parārdha used by Praśastapāda and the commentators on his work PDS is not the same as that used by Bhagavata Purāṇa, which refers to half-life of Brahmā. And Viṣṇu-Purāṇa is not the right source to know parārdha in the Nyāya-Vaiśeṣika context.

As we see in PDS, after one, the numerical qualities run from 2 to a large but finite number parārdha. The number parārdha is mentioned in many texts as the highest decimal place name.
The precise value of parārdha varies: in the Taittirīya Saṁhitā (4.40.11.4, 7.2.20.1) and the Maitrāyanī Saṁhitā (2.8.14), it is given as 10^ (12), while the Kāṭhaka Saṁhitā records both 10^12 (17.10) and 10^13 (39.6). Among the mathematicians and the mathematics treatises such as Līlāvatī by Bhāskara II, it is always 10^(17) i.e. a hundred quadrillion (which you read).

There are names for higher decimal powers in Buddhist and Jaina texts (for e.g. Abhidharmakośa 3.93-94 names powers up to 10^ 59). Only the Nyāya-Vaiśeṣika take parārdha to be the highest number, and not merely the highest named placed value.

It's important to understand the notion of many, in order to grasp parārdha in the Nyāya-Vaiśeṣika context, Śrīdhara is said to have argued that many is itself a number, distinct from any number in the cardinal series. This is based on his view presented by Śaṅkara Miśra (1923:179)

‘Since in the case of a forest or an army, there is no definite combinatory cognition [i.e. counting up of the number of trees or men],a mere many-ness comes into being, but not the number a hundred or a thousand’ – this is the view of the teacher Srīdhara.

According to Śrīdhara, numbers are created by mental acts of counting, but the uncounted army or forest has no definite no. of men or trees. Thereby, Śrīdhara's view is that, in such a case, is of an indefinite act of counting procedures in the objects the indefinite number, many. This problems is due to clear delineation of uncounted collection of objects, which might, as in these cases, be a perceptible aggregate or it might be picked out under a description, for instance 'planets in solar system'. Thus, there seems to be inconsistency with common-sense intuition that the collection has some definite if unknown number, and the Nyāya-Vaiśeṣika thesis that number is the result of a mental act of counting up the objects in the collection.

Udayana (1956:179 & 458-459), on the other hand, insists on the validity of common sense, that a doubt about the number of men in the army only arises when one thinks of it as having a definite but unknown number. Furthermore, if many is really a number, then we would not be able to judge one group of many from the other and say which one is larger or smaller. Thus, Udayana states on the basis of his argument, a definite or precise mental act of counting is both the cause and the manifestor of the number whereas an indefinite or imprecise mental counting is the cause but not the manifestor of a number.

Uddyotakara (1985:211) links the distinct many with processes of increase while Śaṅkara Miśra (1923:180) goes ahead and points that many denotes neither a particular number in the cardinal series nor the series in totality, but rather a property (of being large in no) that co-exists with the definite number the object possesses.

Actually, there is no room for the idea of maximal finite number such as parārdha if one thinks of the number series as generated by recursive application of the successor function, unless in the case of Bhāsvarjña who attempts to construct such a number series. However, within a conception of numbers as qualities of substances, indeed, it seems that there has to be a largest number, if the number of things in the cosmos is finite. But that largest such number (parārdha) need not be definite, and that's what the case is in PDS and it's commentaries.

References & Further Reading

  • Ganeri, J. (2001). Objectivity and Proof in a Classical Indian Theory of Number. Synthese (Dordrecht), 129(3), 413–437. https://doi.org/10.1023/A:1013144411940

  • Śrīdhara: Jha, Durgadhara (Ed.) (1977), Praśastapāda with the commentary Nyāyakandalī by Śrīdhara Bhaṭṭa (D. Jha, Trans. in Hindi). Ganganatha Jha Granthamala Vol. 1.(pp. 267–214). Sampurnanand Sanskrit Vishvavidyalaya

  • Śaṅkara Miśra: Sastri, Dundhiraja (Ed.) (1923), Vaiśeṣikadarśana with Praśastapāda's Padārthadharmasaṅgraha with Śaṅkara Miśra's Upaskāra. Kashi Sanskrit Series 3 (pp. 173–180).Chaukhamba Sanskrit Series Office

  • Udayana: Vedantirtha, N.C.(Ed.) (1956) Kiraṇāvalī with Vardhamāna's Prakāśa and Vādīndra's Rasasāra. Bibliotheca Indica no. 200, fasc. 4. (pp. 441-462). The Asiatic Society

  • Gautama: Nyāyatarkatīrtha, T. & Tarkatīrtha, A. (Eds.) (1985), Nyāyasūtra with Vātsyāyana Bhāṣya, Uddyotakara's Vārttika, Vācaspati Miśra's Tātparyaṭīkā and Viśvanātha's Vr̥tti. Calcutta Sanskrit Series no. 18-19 (2nd ed). Munshiram Manoharlal

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