UPDATE : For people who want to skip the theorem can look at an easier illustration of Reductio ad Absurdum in the bottom of the post.
Reductio ad Absurdum or Proof by Contradiction is one of the usual techniques followed by mathematicians to prove a theorem in mathematics. To give you a flair of this technique, let me illusrtate the celebrated classical theorem of Euclid.
Theorem : There are infinitely many prime numbers.
Proof : Assume the contrary that there are only finite number of prime numbers and let k denote the largest prime number. Thus, the prime numbers can be enumerated by 2,3,5,7,...,k. Consider the product P = 2x3x5x7x...xk. Now, P is clearly divisible by the prime numbers 2, 3, 5, etc. upto k. And P+1 is not divisible by any of these numbers and so P+1 is a prime number greater than k. This is a contradiction to our hypothesis that k is the largest prime number. Hence, there cannot be any largest prime number and so prime numbers are infinite.
This method of proof is called Reductio ad Absurdum and believe me it really works. The key idea is to assume a contradiction and we come up with another contradiction to deduce a truth. But, the other way doesn't work. If we assume the truth of a statement we can never arrive at a contradiction. We can come up with another truth but that doesn't prove anything.
Now, let's come to philosophy. There are several philosophies that are founded based on Vedas and Upanishads. Now, if we assume the authority of the Vedas/Upanishads and propound a philosophy based on our authority, then we cannot claim Vedas/Upanishads are consistent just because our philosophy is in sync with them. We have already assumed their truth. On the other hand, if we start with a philosophy and prove the validity/consistency of the Vedas/Upanishads then we can claim the true nature of these scriptures. Do any of the philosophies Advita, Visishtadvaita, Dvaita follow Reductio ad Absurdum ?
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Another illustration of Reductio ad Absurdum
Theorem : Zero is the smallest non-nagative number.
Proof : Suppose not. Let x be the smallest non-negative number other than zero. Then x/2 is a smaller number than x which is also non-negative. This is a contradiction to the hypothesis that x is the smallest non-negative number. Therefore, zero is the smallest non-negative number.
Reductio ad Absurdum or Proof by Contradiction is one of the usual techniques followed by mathematicians to prove a theorem in mathematics. To give you a flair of this technique, let me illusrtate the celebrated classical theorem of Euclid.
Theorem : There are infinitely many prime numbers.
Proof : Assume the contrary that there are only finite number of prime numbers and let k denote the largest prime number. Thus, the prime numbers can be enumerated by 2,3,5,7,...,k. Consider the product P = 2x3x5x7x...xk. Now, P is clearly divisible by the prime numbers 2, 3, 5, etc. upto k. And P+1 is not divisible by any of these numbers and so P+1 is a prime number greater than k. This is a contradiction to our hypothesis that k is the largest prime number. Hence, there cannot be any largest prime number and so prime numbers are infinite.
This method of proof is called Reductio ad Absurdum and believe me it really works. The key idea is to assume a contradiction and we come up with another contradiction to deduce a truth. But, the other way doesn't work. If we assume the truth of a statement we can never arrive at a contradiction. We can come up with another truth but that doesn't prove anything.
Now, let's come to philosophy. There are several philosophies that are founded based on Vedas and Upanishads. Now, if we assume the authority of the Vedas/Upanishads and propound a philosophy based on our authority, then we cannot claim Vedas/Upanishads are consistent just because our philosophy is in sync with them. We have already assumed their truth. On the other hand, if we start with a philosophy and prove the validity/consistency of the Vedas/Upanishads then we can claim the true nature of these scriptures. Do any of the philosophies Advita, Visishtadvaita, Dvaita follow Reductio ad Absurdum ?
********************
Another illustration of Reductio ad Absurdum
Theorem : Zero is the smallest non-nagative number.
Proof : Suppose not. Let x be the smallest non-negative number other than zero. Then x/2 is a smaller number than x which is also non-negative. This is a contradiction to the hypothesis that x is the smallest non-negative number. Therefore, zero is the smallest non-negative number.
22 comments:
Dear KASTHURI,
the question borders on the rhetorical as it is a well known fact that Sabda or scriptural testimony is taken for granted by all three schools as a valid source of knowledge or pramaana! however Ramanuja in the SriBashya mentions that if scriptures come into contradiction with genuine perception (let us keep this as perception substantiable through consensus as opposed to experiences like haluucinations), then scriptures cannot carry greater force. this could have been a milestone in Vedanta ( as all other schools proclaim the validity of scriptures even against direct perception)but the system itself got submerged in the battle of wits that ensued to protect the Shruthi and Smritis conforming to it.
I feel that the vedas and upanishads can be approached from a mathematical-scientific angle only if Ramanuja's singular assertion is lifted out of the SriBashyam and applied in its proper context. It does not matter that we will have to ask difficult questions as there can be no philosophy/religion higher than truth.
the consistency of each system that you have mentioned can be checked only against the PrasthanaTraya. the epistemological considerations of each school differ in the number of pramaanams they are willing to accept as authoritative.
therefore in the present context it cannot be claimed that any Vedantic standpoint is provable using objective knowledge which is not scripturally sanctioned as all acharyas insist that the Shruthi cannot be subjected to logical or any other analysis and has to be accepted without the slightest alterations to its content.
Vasudeva-Krishna has repeatedly criticised this in the Gita and has argued in favour of contemplation at the same time cautioning against immature skepticism which leads to no conclusion.
I 'll take your argument into account when I put up my next post(Narayana has to give the insights anyway).
thanks for this post. gave me the inspiration I needed to come up with a new concept.
SARVAM SRIKRISHNAARPANAMASTHU!
Kasthuri, wish I had read these posts before deciding to stay away from Math, all the concepts are so logical and easy to comprehend when you present them. I must confess that this was a bit of brain teaser to me :)-
Anand, Thank you so much for your valuable comment. I could understand something however I wish to mention that I am not questioning the consistency of Vedantic system such as Advita etc. I am questioning the consistency of Upanishads and Vedas from the stand point of these systems. In other words, if I accept the systems as true (let's say I accept Advita), then how far I will be able to prove the consistency of Upanishad, say Mundaka Upanishad. Please let me know your opinion on this.
hmm,
sure its a brain teaser :)
late a vandhaalum latest a vandhuteenga!..
as usual, i will be an active listener to the comments and will give my 2 hoots if i have them!
oh wait.. this reductio ad absurdum.. i feel it is very similar to occam's razor principle?!.
Too late...If i had known before...Maths la nalla mark eduthirupean....
Kasthuri,
Don't you think the answer to your question is a simple "NO"!
Anand,
Scriptures too are someone's (or some group of people's) perception isnt it? So what Ramanuja says is very much in the scientific spirit - replacing modern perception with an older perception given the right justifications.
If the other philosophies disagree with that, then, on that count alone, I think they become inferior. Giving unquestionable authority to a set of scriptures is just as good as giving unquestionable authority to the Bible or Koran.
I don't know if the other philosophies "proclaim the validity of scriptures against perception". Is that true? Is there evidence for that?
And yeah, the proof for infinite primes is a little incomplete...
Just because P+1 is not divisible by any of 2, 3, 5,...k does not mean P+1 is prime.
It just means, either P+1 is prime or it has a prime factor greater than k - and both scenarios are contradictions to the assumption.
Right?
@ paavai : Probably I have complicated the idea by giving an involved proof. I have updated with another illustration. Thanks.
@ ganesh : :-)
@ tt_giant : I guess Occam's razor principle is different from the proof by contradiction. The former tells us to have a restricted set of assumptions while the latter procceds by wrong assumption.
@ rags : :-)
@ Swami : I don't know whether its a simple no or a complicated no. This is beacuse Sri Ramanuja uses reductio ad absurdum in refuting certain ontological statements. Nagarjuna (C 150-200) one of the famous buddhist thinkers also uses this technique as a main tool. The famous phrase "Neti-Neti" expounded by Sri Adi Sankara is a classic example of reductio ad absurdum. Well, there are several examples like this. I would like to comment a little on questions to Anand.
While Sri Ramanuja's views have scientific spirit in them, scriptures are not just perceptions, if we think perception as an awareness of the elements of environment through physical sensation. They(scriptures) also have certain rules of construct (for example, Panini's grammer) and so cannot be merely restricted to just perception alone. So, it is not that easy to disregard other philosophies as inferior. When you say "Giving unquestionable authority to a set of scriptures" you need to consider who is giving such authority. That becomes very important in these philosophical debates. Okay, that's from me. I'll leave some space for Anand.
Reg. Theorem : You are right. There are also some other things that I didn't want to bring in which will make the theorem complete only then (such as the fundamental theorem of arithmetic).
Kasthuri,
In perception, I also include "intelligent thinking" - like conceptualizing the general from the particular...
And I am sure Ramanuja included this when he talked about contradiction with "genuine perception" like Anand says.
Obviously, not all thought can be attributed to sensory perceptions.
So coming back to my point - if a philosophy disagrees with the idea that perception should override the scriptures if both are found to be contradictory - then in my opinion that philosophy becomes immediately inferior.
However, I do not believe that these interpretations (Advaita, Dwaita) give such irrefutable authority to the Vedas and Upanishads. If they do, then they are no different from Christianity and Islam. (Which is okay, only that I do not think it is so!)
Reductio ad Absurdum or for that matter any tool can just prove an interpretation of the philosphy, for these concepts are so generic yet very 'coded'[Marai porul].
Nice timely post
have taken Analysis of algorithms this sem..:) and we are going thru stuff like this ..
yeah..it would defintely intersting see proof by contradiction when applied to vedanta philosophy and the shruthis.
but if we are convinced that shruthis have divinity as the source, then should we ever attempt to prove it thru anymeans.;) Even if one attempts to prove, it would defintiely be a gargantuan task that might take years...may be u can do it in months/ may be days..;)
Dear Kasthuri,
Logic in India particularly as used by the Vedantins is not formal. to prove the consistency of the Upanishads you have to resort to some other means as something which is derived from the Upanishads (advaita, dvaita etc) cannot possibly lead to Upanishadic conclusions as if that were the case only one logically admissible position would exist that is derivable from the Upanishads- however this is not the case therefore the Upanishads cannot be established inductively (I hope I amright about this) from any vedantic standpoint.
SARVAM SRIKRISHNAARPANAMASTHU!
SWAMI,
all vedantic acharyas accord divine status to the scriptures of the Shruthi and the instance I have cited has something to do only with contradiction with direct perception. in this case Ramanuja has this view but advaitins say that perception is affected by Maya( nescience) at the root and can never be trusted against scriptural testimony.
apart from this question the philosophers belonging to vedantic schools never question the logical consistency of the Shruthi.
Swami,
I would prefer using the terms used by Popper when it comes to comparing philosophies- either refutable or not,
this dichotomy of inferior-superior is not exactly required here- such is my humble opinion.
SARVAM SRIKRISHNAARPANAMASTHU!
Anand,
If you agree with the "Acharyas" according divine status to the scriptures, then you probably also agree with me that Hinduism (or whatever other name you wish to call it) is no different from the other religions like Christianity and Islam (which also accord "divine" but more importantly irrefutable status to a book).
More interestingly for the argument though, you point out that perception is affected by maya. I want to ask whether the people who wrote the scriptures were unaffected by maya and if yes, how so?
You might probably counter this with the usual theistic argument that scriptures were not written by "people" but by somebody else (usually named God).
In that case, my question to you would be to give a chronology of when these scriptures came into existence and in what order - so that we can possibly debate whether God had mechanisms of delivering it to Humans in some way.
swami,
I merely informed of the status quo. it does not reflect upon my stance which is diametrically opposite. I accept the valditiy of the Gita as it is the direct knowldege imparted by Krishna (if you want to call him God fine I have better terms)
yet I try to prove these statements and can prove them if the need arises (I am not inclined to do so on the blogosphere)
your issues will be put to rest in due course of time if I become a full time academic instead of being an employee blogging during the day.
Hinduism is not a real religion but a politico-religious myth.
I discourage use of the term Hindu with respect to vedantic philosophy.
In India people refer to their beliefs as constituting a SAMPRADAAYAM OR TRADITION and refer to the philosophical position (adwaita etc) as MATHAM OR religion.
thus a clear understanding of issues is required before attempting a comparitive study of religions.
I do accept that the VEDAS are not written they are captured by men in states of yoga,
these men articulate these truths in a language known to them while being in a state of Yoga.
(heard of Bhaava Samadhi?)
I have refuted the MayaVada in detail. therefore I cannot claim that perception is affected by it. find some hardcore advaitin to argue that one over.
the scriptures could not have been written by God or Brahman but only delivered as intuitve knowldege,
my belief lies with the assumption that even science and maths are insights offered by this reality.our understanding of reality cannot and does not rest solely on scriptures or science but a composite system which combines the best of the two. there is still time for reaching that level of philosophical accuracy.
chronological listing is not my job, I am not into hair-splitting and twaddle, if you need guys for that kind of debate there are plenty in the world.
SARVAM SRIKRISHNAARPANAMASTHU!
You have been Tagged... :D
Kasthuri - I read this blog of ur b4 - but read it again properly..awesome man!! I like the theoram :)..I shall think about ur question too :)..
Good work!R u a mathematician? :)
Srini: TJ mundikondu vittaar.. You have been tagged by me too.. !!
@ tj : Very true. It may take a long while to have that philosophical accuracy just like Anand says. Also, thanks for tagging :-) I'll try to untag myself asap.
@ krishna : Yes, it will be gargantuan task if someone accomplishes. I am less than a novice when it comes to Vedanta.
@ arjuna : I was supposed to be a mathematician. Now I donno who I am - to say in u'r style :-) Thanks btw.
@ tt_giant : Thanks for tagging. I'll see whether I can come up with my seven.
@ satya : I saw that u tagged me yesterday - but I failed to acknowledge. I *do* visit u'r site often. Btw, are u in town already ?
Mathematically Philosophical or philosophically mathematical ?
so what happens to 0 and 1 now ? :P
@ arvind : Both mathematically and philosophically...are they different ?
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