An Indian solution to 'incompleteness'

Abstract

Kurt Godel's Incompleteness theorem is well known in Mathematics/Logic/Philosophy circles. Godel 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 Godel-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' moksa? ['moksa' is the final state of existence of an individual away from Social Contract] - See this author's Indian Social Contract and its Dissolution (2008) moksa 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 moksa 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 moksa' get refuted but also it would show the possibility of achieving 'completeness' in its truest sense, as opposed to Godel's 'Incompleteness'. In such circumstances, man would himself become a Universal Truth Machine. This is because the final state of moksa is construed as the state of complete knowledge in Advaita. This possibility of 'completeness' is set in this paper in the backdrop of Sri Sankaracarya's Advaitic (Non-dualistic) claim involved in the mahavakyas (extra-ordinary propositions). On the other hand, the 'Incompleteness' of Godel is due to the intervening G-sentence, which has an adverse self-referential element. Godel's incompleteness theorem in its mathematical form with an elaborate introduction by R.W. Braithwaite can be found in Meltzer. The present author believes first that semantic content cannot be substituted by any amount of arithmoquining, (Arithmoquining or arithmatization means, as Braithwaite says, "Godel'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 - Godel invented what might be called co-ordinate metamathematics") Meltzer (1962 p. 7). In Antone (2006) it is said "The problem is that he (Godel) 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 Godel's incompleteness theorem, 2006). ("it is so important to understand that Godel'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 Sankara is to be subsumed or interpreted under Godel'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.