An Indian solution to ‘incompleteness’

Abstract

Kurt Gödel’s Incompleteness theorem is well known in Mathematics/Logic/Philosophy circles. Gödel was able to find a way for any given P (UTM), (read as, “P of UTM” for “Program of Universal Truth Machine”), actually to write down a complicated polynomial that has a solution iff (=if and only if), G is true, where G stands for a Gödel-sentence. So, if G’s truth is a necessary condition for the truth of a given polynomial, then P (UTM) has to answer first that G is true in order to secure the truth of the said polynomial. But, interestingly, P (UTM) could never answer that G was true. This necessarily implies that there is at least one truth a P (UTM), however large it may be, cannot speak out. Daya Krishna and Karl Potter’s controversy regarding the construal of India’s Philosophies dates back to the time of Potter’s publication of “Presuppositions of India’s Philosophies” (1963, Englewood Cliffs Prentice-Hall Inc.) In attacking many of India’s philosophies, Daya Krishna appears to have unwittingly touched a crucial point: how can there be the knowledge of a ‘non-cognitive’ mokṣa? [‘mokṣa’ is the final state of existence of an individual away from Social Contract]—See this author’s Indian Social Contract and its Dissolution (2008) mokṣa does not permit the knowledge of one’s own self in the ordinary way with threefold distinction, i.e., subject–knowledge-object or knower–knowledge–known. But what is important is to demonstrate whether such ‘knowledge’ of non-cognitive mokṣa state can be logically shown, in a language, to be possible to attain, and that there is no contradiction involved in such demonstration, because, no one can possibly express the ‘experience-itself’ in language. Hence, if such ‘knowledge’ can be shown to be logically not impossible in language, then, not only Daya Krishna’s arguments against ‘non-cognitive mokṣa’ get refuted but also it would show the possibility of achieving ‘completeness’ in its truest sense, as opposed to Gödel’s ‘Incompleteness’. In such circumstances, man would himself become a Universal Truth Machine. This is because the final state of mokṣa is construed as the state of complete knowledge in Advaita. This possibility of ‘completeness’ is set in this paper in the backdrop of Śrī Śaṅkarācārya’s Advaitic (Non-dualistic) claim involved in the mahāvākyas (extra-ordinary propositions). (Mahāvākyas that Śaṅkara refers to are basically taken from different Upaniṣads. For example, “Aham Brahmāsmi” is from Bṛhadāraṇyaka Upanisad, and “Tattvamasi” is from Chāndogya Upaniṣad. Śrī Śaṅkarācārya has written extensively. His main works include his Commentary on Brahma-Sūtras, on major Upaniṣads, and on ŚrīmadBhagavadGītā, called Bhāṣyas of them, respectively. Almost all these works are available in English translation published by Advaita Ashrama, 5 Dehi Entally Road, Calcutta, 700014.) On the other hand, the ‘Incompleteness’ of Gödel is due to the intervening G-sentence, which has an adverse self-referential element. Gödel’s incompleteness theorem in its mathematical form with an elaborate introduction by R.W. Braithwaite can be found in Meltzer (Kurt Gödel: on formally undecidable propositions of principia mathematica and related systems. Oliver & Boyd, Edinburgh, 1962). The present author believes first that semantic content cannot be substituted by any amount of arithmoquining, (Arithmoquining or arithmatization means, as Braithwaite says,—“Gödel’s novel metamathematical method is that of attaching numbers to the signs, to the series of signs (formulae) and to the series of series of signs (“proof-schemata”) which occur in his formal system…Gödel invented what might be called co-ordinate metamathematics…”) Meltzer (1962 p. 7). In Antone (2006) it is said “The problem is that he (Gödel) tries to replace an abstract version of the number (which can exist) with the concept of a real number version of that abstract notion. We can state the abstraction of what the number needs to be, [the arithmoquining of a number cannot be a proof-pair and an arithmoquine] but that is a concept that cannot be turned into a specific number, because by definition no such number can exist.”.), especially so where first-hand personal experience is called for. Therefore, what ultimately rules is the semanticity as in a first-hand experience. Similar points are voiced, albeit implicitly, in Antone (Who understands Gödel’s incompleteness theorem, 2006). (“…it is so important to understand that Gödel’s theorem only is true with respect to formal systems—which is the exact opposite of the analogous UTM (Antone (2006) webpage 2. And galatomic says in the same discussion chain that “saying” that it ((is)) only true for formal systems is more significant… We only know the world through “formal” categories of understanding… If the world as it is in itself has no incompleteness problem, which I am sure is true, it does not mean much, because that is not the world of time and space that we experience. So it is more significant that formal systems are incomplete than the inexperiencable ‘World in Itself’ has no such problem.—galatomic”) Antone (2006) webpage 2. Nevertheless galatomic certainly, but unwittingly succeeds in highlighting the possibility of experiencing the ‘completeness’ Second, even if any formal system including the system of Advaita of Śaṅkara is to be subsumed or interpreted under Gödel’s theorem, or Tarski’s semantic unprovability theses, the ultimate appeal would lie to the point of human involvement in realizing completeness since any formal system is ‘Incomplete’ always by its very nature as ‘objectual’, and fails to comprehend the ‘subject’ within its fold.