Friday, 2 March 2018

Nagarjuna, Gödel’s Theorem and Russell’s Paradox



Unconventional knowledge
by Kenichi Morita


Distinguishing between “conventional computing” and “unconventional computing” is not so easy, since the notion of unconventional computing is rather vague. Some scientist may want to give a rigorous definition of it. But, if he or she does so, then unconventional computing will become less attractive. The very vagueness of the concept stimulates one's imagination, and thus is a source of creation.

In this short essay, related to such a problem, we consider thinking styles of the West and the East. We examine several possibilities of ways by which we can recognize various concepts in the world, and acquire enlightenment from the nature. At first, we begin with the two categories of knowledge in Buddhism. They are “discriminative knowledge” and “non-discriminative knowledge” (however, as we shall see below, discrimination between “discriminative knowledge” and “non-discriminative knowledge” itself is not important at all in Buddhism). Although it is very difficult to explain them, in particular  non-discriminative knowledge, by words, here we dare to give some considerations on them.

Discriminative knowledge is just the set-theoretic one. Namely, it is a knowledge acquired by classifying things existing in the world.  For example, the discriminative knowledge on “cat” is obtained by distinguishing the objects that are cats from the objects that are not cats. Therefore, what we can argue based on discriminative knowledge is a relation among the sets corresponding to various concepts, e.g., the set of cats is contained in the sets of animals, and so on. Knowledge described by an ordinary language (or a mathematical language like a logic formula) is of this kind, since “words” basically have a function to distinguish certain things from others.

Non-discriminative knowledge, on the other hand, is regarded as the true wisdom in Buddhism. But, it is very difficult to explain it in words, since words can be used for describing discriminative knowledge. Therefore, the only method by which we can express it is using a negative sentence like “Non-discriminative knowledge is not a knowledge that is obtained by distinguishing certain things from others.”

Actually, non-discriminative knowledge is recognized neither by words, nor by thinking, nor by act. Moreover, it is not even recognizable. This is because all acts such as recognizing, thinking, and explaining some objects necessarily accompany discrimination between the self (i.e., actor) and the object. In Buddhism, everything is empty, i.e., it has no reality in the world in its essence. Hence, there is nothing to be discriminated, and there is a truth that can be gotten without discriminating things. Furthermore, such a truth (non-discriminative knowledge) itself is also empty, and thus does not exist. It may sound contradictory, but this is caused by explaining it by ordinary words.

There is no doubt that discriminative knowledge brings practical convenience to our daily life. Today's science also relies on discriminative knowledge. There, objects to be studied are clearly identified, and their properties are described precisely. By this, science brought us a great success. However, discrimination is considered as a kind of “biased view” in Buddhism. Thus, we should note that such a knowledge is a “relative” one. Namely, when we state a scientific truth, we can only say like “If we assume a certain thing is distinguishable from others based on some (biased) viewpoint, then we can conclude so-and-so on it.” We should thus be careful not to overestimate the descriptive power of languages.

It is well known that from the end of 19th century the foundation of mathematics has been formalized rigorously with the utmost precision. It is, of course, based on discriminative knowledge. However, at the same time, problems and limitations of such a methodology were also disclosed. A paradox by Bertrand Russell on the set theory is the most famous one, which first appeared in Nachwort of the Frege's book (Frege, 1903). Russell's paradox is as follows:

Let R be the set of all sets each of which does not contain itself as a member.

Is R a member of itself or not? In either case, it contradicts the definition of R.

Due to this paradox, the naive set theory had to be replaced by some sophisticated ones such as the type theory. The incompleteness theorem by Kurt Gödel (1931) also shows a limitation of a formal mathematical system. He proved that in every formal system in which natural numbers can be dealt with, there exists a “true” formula that cannot be proved in this system. He showed it by composing a formula having the meaning  “This formula is unprovable.”

Nagarjuna is a Buddhist priest and philosopher who lived in India around 150 - 250 AD. He is the founder of Madhyamaka school of Buddhism, where he developed the theory of emptiness. In his book Vigrahavyavartani (The Dispeller of Disputes) (Westerhoff, 2010), he pointed out “very logically” that false thinking will be caused by relying only on discriminative knowledge. This book is written in the following form. First, philosophers of other schools who believe every concept has a substance (here, we call them philosophical realists) present objections against those of Madhyamaka school. Then, Nagarjuna refutes all of them.

While philosophers of Madhyamaka school assert every concept has no substance (but they assert “nothing” as we shall see below), the opponents (philosophical realists) say as follows (Westerhoff, 2010):

If the substance of all things is not to be found anywhere, your assertion which is devoid of substance is not able to refute substance. (Verse 1)

Moreover, if that statement exists substantially, your earlier thesis is refuted. There is an inequality to be explained, and the specific reason for this should be given. (Verse2)

Nagarjuna says:

If I had any thesis, that fault would apply to me. But I do not have any thesis, so there is indeed no fault for me. (Verse 29)

To that extent, while all things are empty, completely pacified, and by nature free from substance, from where could a thesis come? (Commentary by Nagarjuna on Verse 29)

That is, without saying “all things are empty,”all things are empty by nature, and hence the Nagarjuna's assertion itself is also empty.

We can see that the observation “If all things are empty, then the assertion ‘all things are empty’ cannot exist” resembles the second incompleteness theorem:

“If a formal system in which natural numbers can be dealt with is consistent, then consistency of the system cannot be proved in the system” by Gödel (1931).

However, methodologies for obtaining the above observations are quite different. In the former case, non-discriminative knowledge played the crucial role, and thus the observation itself is again empty.

Nagarjuna launches a counterattack against philosophical realists, who claim “all things have substances", by the following objection:

The name “non-existent” what is this, something existent or again non-existent? For if it is existent or if it is nonexistent, either way your position is deficient. (Verse 58)

It is clear that the above argument is analogous to Russell's paradox. By this, Nagarjuna pointed out that philosophical realists who rely only on discriminative knowledge have a logical fault. However, as stated in Verse 29, Nagarjuna asserts nothing in his book.

It will be reasonable to regard discriminative knowledge as conventional knowledge. Then, how is non-discriminative knowledge? Although this kind of knowledge has been argued by philosophers and Buddhists for a very long time, we can say neither conventional nor unconventional. Probably, it is meaningless to make such a distinction. Instead, we consider a question: Can we use non-discriminative knowledge for finding a new way of scientific thinking, and for giving a new methodology of unconventional computing? Since current scientific knowledge is very far from non-discriminative knowledge, it looks quite difficult to do so. However, it will really stimulate our imagination, and may help us to widen the vista of unconventional computing.

I have been studying reversible computing and cellular automata (Morita, 2008) for more than 30 years. Through the research onthese topics, I tried to find novel ways of computing, and thus I think they may be in the category of unconventional computing. Besides the scientific research, I was interested in Buddhism philosophy. In 1970's and 80's, I read Japanese translations of several sutras and old texts of Buddhism. They are, for example, Prajnaparamita Sutra (Sutra of Perfection of Transcendent Wisdom), and Vimalakirti-nirdesa Sutra (Vimalakirti Sutra), as well as Vigrahavyavartani (The Dispeller of Disputes). All of them discuss emptiness of various concepts and things in the world, but assert nothing. I was greatly impressed by these arguments, which themselves are empty. Although my research results are, of course, given in the form of discriminative knowledge, and thus in the purely Western style, I think such a thought somehow influenced me on my research when exploring new ways for unconventional computing.

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