Thinking about thought and understanding our own understanding encapsulates an inherent difficulty: self reference. Self reference is the ability of a complex system to reflect upon its own existence. No matter how it might be the basis of our consciousness, it brings inconsistencies. With self reference we are able to build sentences like “This sentence is wrong.” This very well known paradoxical statement is the point in our immense potential that makes us different from all other creatures, and it is the point where we start to perceive our weakness at comprehending our own selves. Self reference is the means of knowing our own reality, but it only gives a blurry glimpse of it. A mathematical theorem shows that we will never be able to capture our own reality in a clear picture, there will always be inconsistencies. The theorem is known as Gödel’s Incompleteness Theorem after the famous mathematician Kurt Gödel (1906–1978). Gödel used mathematical reasoning in exploring mathematical reasoning itself. In this article the implications of this theorem for self consciousness will be investigated.

The scientific study of mathematical reasoning begins with the concept of formal systems. In formal systems there are axioms and rules. The axioms are the very basic statements that are accepted as true by default (e.g. any two points in space can be joined by a straight line). The rules are common notions that are consistent with the rules of the nature (e.g. things that equal the same thing also equal one another). Formals systems apply the rules to axioms in order to reach true mathematical propositions (valid theorems).

New valid theorems are built on the existing ones and the formal system is iterated for exploring mathematical reality; hence, the formal system learns the mathematical truth with formal iteration. The structure and the development of formal systems are similar to that of human beings: babiesare born with some built-in knowledge of the environment (axioms), and new knowledge is acquired through learning in the guidance of physical laws (rules). Just like human beings, a formal system that is rich and complex enough can make reference to its theorems, as in “This theorem is right” or to its own self as in “This formal system is consistent.” Gödel’s theorem universally states that the formal systems that can make self reference suffer from an inherent incapability to comprehend the “self.” A formal system cannot be sure that it is consistent because the theorem “This formal system is consistent” cannot be proven inside the system, which makes the system incomplete. In fact, the existence of such a theorem is the sole source of inconsistency. Consistency and completeness are required in a formal system to reach reality. Therefore, with the ability to reflect upon itself, a formal system cannot decide on the true nature of its own reality and lacks a complete understanding of itself. The Gödelian argument applies only to systems that are rich enough to have self reference, and interestingly richness of the systems brings about its downfall. It is analogous to the concept of “critical mass” in nuclear physics. A radioactive substance will blow up only beyond a critical mass; otherwise it will stay stable.

Though the system is intrinsically incapable of having a complete understanding of its own self, intelligence outside of the system can decide on the system’s consistency and can fully comprehend it. The need for an outside agent to comprehend reality is exemplified by the analogy of an ant walking on a Mobius strip (Fig. 1). A Mobius strip is a two-dimensional surface with only one side, unlike a regular strip which has two sides. It can be easily made by taking a paper strip and giving it a half-twist, and then merging the ends of the strip together to form a single strip.

If you start drawing a line from a point on the Mobius strip, you will reach the point you started. This is a characteristic of a round three dimensional shape, like cylinder (Fig. 2). When an ant walks on a Mobius strip, it will cross the location that it started at (Fig. 3). The same will happen when it walks on the surface of a cylinder. An ant on these geometric shapes will not be able to differentiate between the two-dimensional Mobius strip and three-dimensional cylinder, and hence will not comprehend the true natural geometry of the system that it is in. Only an agent outside of these shapes can understand the true geometry of the shapes, just like an intelligent agent outside the formal system fully comprehending the system. The question is, can an ant ever get out of the geometry that it is in? Sure, it can step outside of the Mobius strip, but yet again it will be on another geometric shape that it cannot comprehend. Similarly, new theorems can be added to the formal system that patch the existing holes and make the system step outside itself, but this improved formal system still suffers from the same problems caused by self reference.

The futile struggle to jump outside of the system is beautifully illustrated in another painting by Escher (Fig. 4). Escher explains his painting Dragon (1952): “However much this dragon tries to be spatial, he remains completely flat.Two incisions are made in the paper on which he is printed. Then it is folded in such a way as to leave two square openings. But this dragon is an obstinate beast, and in spite of his two dimensions he persists in assuming that he has three; so he sticks his head through one of the holes and his tail through the other.” Hofstadter adds in his seminal book Gödel, Escher, Bach: An Eternal Golden Braid, “No matter how cleverly you try to simulate three dimensions in two, you are always missing some ‘essence of three-dimensionality.’ The dragon tries very hard to fight his two-dimensionality. He defies the two-dimensionality of the paper on which he thinks he is drawn, by sticking his head through it; and yet all the while, we outside the drawing can see the pathetic futility of it all, for the dragon and the holes and the folds are merely two-dimensional simulations of those concepts, and not a one of them is real. But the dragon cannot step out of his two-dimensional space, cannot know it as we do.”

Consider an intelligent creature living on a two-dimensional plane, and we are observing its world from above, but it can not see us since there is no such thing as “above” in its world. In his free time, the creature wanders on the plane to seek new realities (theorems) of its world and learn them. We accidentally touch the plane that he is living in with one of our fingers; obviously the creature will only see the two-dimensional projection of the adjacent finger surface. I am sure it will be surprised to experience an object appearing out of nowhere, and even more surprised to see the shape suddenly disappear when we withdraw our finger. It will question the consistency of the system that it is living in and ask, “Is my world inconsistent?” Then, it might go crazy while figuring out what has happened or simply patch the informational hole in its world by saying, “Sometimes things of the shape that I saw might suddenly appear and disappear in my world.” Our creature will be happy again after resolving the unfortunate inconsistency, but can it be sure that the system of knowledge (theorems) is complete? We will not let our unlucky creature rest and we will put a cup on him; since the two dimensional projection of the touching surface of the cup draws a circle, the creature will be surrounded by a circular prison again appearing out of nowhere. It will be surprised but this time devastated also. Then, there will come another patch to its knowledge about the system: “Sometimes, a circular prison can suddenly appear and enslave me.” As you can imagine, there are infinitely many operations that we can apply to the two-dimensional world of our poor creature, and each one of them will lead to a patch in the system of knowledge of its world. However, it will never be able to generalize the newly added theorems (due to anomalies) since neither the rules nor the existing two-dimensional theorems are well suited to grasping the three-dimensional operations that are applied. The two-dimensional world of the creature will never be complete: there will always be phenomena that it cannot explain with existing theorems. Let us put our creature to the hardest test of all time: pick it up from its plane, take it to our three-dimensional world, let it experience the extraordinary “third dimension” and then let us put it back on the plane where it belongs.1 I cannot imagine the struggle of the creature to tell its fellows what it experienced “out there.” Could its folks ever understand? The reader might have pitied the two-dimensional creatures with their inability to understand the “real” world. Well, it might be that we are, as human beings, just one more dimension better than them: what if there is a fourth dimension?2

Mathematical reasoning suggests that a formal system can talk about itself but it cannot jump out of itself. Even it speciously achieves that by adding new theorems, it becomes an improved formal system with a new boundary but the same inability. Can we jump out of ourselves and self-transcend? If we could do it, we would be able to comprehend our own physical reality, but even then we are inside another reality beyond the usual that we can’t fully comprehend. We are always limited by a boundary that is set by the “self.”

The exploration of the “self” is tightly related to the query of the soul or the spirit. The Qur’an addresses this issue in Isra 85: “They ask you about the spirit. Say: ‘The spirit is of my Lord’s Command, and of knowledge, you have been granted only a little.’” The verse is a declaration of the inability of human beings to comprehend their own spirit. More importantly, it indicates that the reality of us as human beings will be communicated to us through information sources that can jump out of the system3. Can we ever be able to fully comprehend the true nature of humanity and of our reality without knowledge from the divine?

Notes

1.This allegory might be compared with the concept of ascension. See the verses on the Ascension of the Prophet (Qur’an 4:14) and angels (Qur’an 70:4).

2.See the verses about the higher dimensional operations on our 3D world (Qur’an 21:82 and 27:38–39).

3.Messengers of the Creator. See Qur’an 2: 136, 213.

Reference

Hofstadter, Douglas R. Gödel, Escher, Bach: An Eternal Golden Braid, 1999, Basic Books.

]]> https://fountainmagazine.com/all-issues/2004/issue-46-april-june-2004/artificial-intelligence-vs-the-mind/ Thu, 01 Apr 2004 00:00:00 +0000 http://107.21.79.195/all-issues/2004/issue-46-april-june-2004/artificial-intelligence-vs-the-mind/ In the last fifty years, computer technology has led a new discussion centered on Artificial Intelligence (AI) vs. the mind. The main aim in AI is to construct systems which behave in ‘logical’ ways as far as possible. While a hundred years ago, the question, “can a system with artificial intelligence be more advanced than the human mind?” could not have been imagined, it is one of the most frequently discussed subjects of recent years. AI supporters claim that in the near future there will be advanced systems which possess better decision and evaluating mechanisms than humans. On the other hand, many scientists think that this will not be possible.

The theorem published in 1931 by the 25 year old Austrian scientist, Kurt Godel, made a great impact on the scientific circles. Not only did it destroy the hopes of many scientists, but it also initiated a new point of view concerning AI and the mind. This theorem is one of the most important ones to be proven this century, ranking alongside Einstein’s Theory of Relativity and Heisenberg’s Uncertainty Principle. However, very few people know about it. In this article, we will examine in detail the effects of Godel’s theorem on AI.

What is Godel’s Incompleteness Theorem?

As a formal definition, proof is a sequence of well-formed-formulas (wff), each of which is either an axiom or a wff that is derived from preceding wff’s. Godel’s contemporary Hilbert, one of the most famous mathematicians, thought that all proofs in mathematics can be obtained in an automated way (with an axiomatic system) and he started to work on this project. He believed that if he derived all wff’s in basic arithmetic from its own axioms, then he could derive all facts in mathematics using these axioms.

Unfortunately, Godel demonstrated the impossibility of this. First of all, he found a method of translating the syntax of a formal system into arithmetic. Then he formulated the statement, “This formula is improvable in the system,” (G) in arithmetic. Using the same method, he also formulated the negative of the statement G (“This formula is provable in the system”). For the next step, he showed that if the truth value of G was calculated, the truth value of negation of G could also be calculated, causing a contradiction. At the end of his calculations, Godel arrived at two very important consequences:

1. If a formal system that contains minimal arithmetic is consistent, then it is incomplete.

2. Consistency of any formal system containing minimal arithmetic is not internally provable (by using the system’s own rules and formulas).

Surprisingly, even if G were added as a further axiom into the system, a new Godel sentence could be easily found. In other words, no matter how many axioms we add, one can find a Godel sentence that will make the truth value undeterminable.

What Does the Theorem Imply for Artificial Intelligence vs. the Mind?

By examining Godel’s Theorem, one can determine very important consequences for artificial intelligence. An English mathematician, Turing, described an abstract machine called the “Turing Machine.” This is an abstract machine which has an unlimited amount of storage space and which can go on computing forever without making any mistakes. This machine can compute any type of algorithmic problem. According to the Turing Theorem all computers are Turing equivalents. After proposing this, Turing went on to observe that some type of problems have no algorithmic solutions. In the meantime, “the Halting Problem” emerged – the problem of deciding those situations in which a Turing Machine action fails never comes to a halt because of the consequences of the Godel’s Incompleteness Theorem.

It has been proven that a halting problem is computationally insoluble. This leads us to an important conclusion; a computer cannot be the same as the human mind because the non-computational physics of the mind is not available for Turing equivalent machines and the nature of the algorithms is not compatible with the thinking process due to the halting problem.

The argument of the Godelian Case problems made great sense to AI supporters. Godel’s Theorem started a great debate between supporters of AI vs. those of the human mind.

Reviews of the Theorem on AI vs. Mind

Penrose claims that the human mind cannot be compared to artificial intelligence. Penrose bases his claim on Godel’s Incompleteness Theorem. By appealing to the results obtained by Godel (and Turing), mathematical thinking (and hence conscious thinking generally) is something that cannot be encapsulated within any purely computational model of thought. This is the part of Penrose’s argument that his critics have most frequently taken issue with. In addition, he states that there are certain classes of problems that do not have any algorithmic solutions (R. Penrose, 1994, p.29). In fact, Turing described this as the halting problem. Penrose gives an example of the completely deterministic, but non-computable “tiling problem” (R. Penrose, 1994, p.30-33).

Penrose asserted that some mathematical relations required long chains of reasoning before they could be perceived with certainty. But the object of a mathematical proof is to provide such chains of reasoning that each step is indeed something that can be perceived as being “obvious.” He concluded that the endpoint of such reasoning is something that must be accepted as being true, even though it may not, in itself, be at all obvious. One might imagine that it would be possible to list all possible “obvious” steps of reasoning once and for all, so that from that time on everything could be reduced to computation. But, what Godel’s argument shows is that this is not possible. There is no way to eliminate the need for new “obvious” understandings. Thus, mathematical understanding cannot be reduced to blind computation (R. Penrose, 1994, p.56).

Penrose claims that the results of Godel’s

theorem established that human understanding and insight cannot be reduced to any set of computational rules (R. Penrose, 1994, p.65). In the chapter entitled “The Godelian Case” of his book Shadows of the Mind, Penrose supported his idea with Turing’s Halting Problem and showed sound examples on non-computability. At the end of the chapter he answered possible technical objections to his idea based on Godel’s Theorem in details (R. Penrose, 1994, p.64-116).

Penrose believes that there is something beyond computation in the human mind. In Chapter 3 of Shadows of the Mind, he examines the thinking process and non-computability in mathematical thought carefully and uses formal representations (R. Penrose, 1994, p.127-209). Godel’s theorem states that in any sufficiently complex formal system there exists at least one statement that cannot be proven to be true or false. Penrose believes that this would limit the ability of any AI system in its reasoning. He argues that there will always be a statement that can be constructed which is unprovable by the AI system. However, Penrose believes that somehow the human mind can see the truth of such Godel statements directly (R. Penrose, 1989).

Along the same lines as Penrose, Lucas believes that Godel’s theorem seems to prove that the idea of “Mechanism” is false, that is, that minds cannot be seen in terms of machines. He claims that Godel’s theorem must apply to cybernetics, because the essence of being a machine is that it should be a concrete instantiation of a formal system. It follows that for any given machine which is consistent and capable of doing simple arithmetic, there is a formula which it will be incapable of producing as being true (i.e., the formula is improvable in the system but which we can see to be true). It follows that no machine can be a complete or adequate model of the mind, that minds are essentially different from machines. This does not mean that a machine cannot simulate any piece of the mind; it only says that there is no machine that can simulate every piece of the mind. Lucas says that there may be deeper objections. Godel’s theorem applies to deductive systems, and human beings are not confined to making only deductive inferences. Godel’s theorem applies only to consistent systems, and one may have doubts about how far it is permissible to assume that human beings are consistent. Godel’s theorem applies only to formal systems, and there is no a priori bound to human ingenuity which rules out the possibility of our contriving some replica of humanity which is not representable by a formal system (J. Lucas, 1970).

Chalmers examines the situation when a formal system F, which understands the consequences of Godel’s Theorem, is given. According to his claim, F may not be sound, so Godel’s theorem cannot be applied. He specifies that the crucial point of Godel’s argument is not to know “a formal system is sound”; but to determine “if we know that our system is sound.” It follows that we perhaps have a sound system, but we can not conclude that “we know that we have a sound system” (D. J. Chalmers, 1995).

Like Chalmers, McCullough claims that not only artificial intelligence, but also the human mind is tightly related with Godel’s theorem. Godel argument did not prove that human reasoning had to be noncomputable – it only proved that if human reasoning was computable, then it had to either be unsound, or it had to be inherently impossible for a human to know both what a human’s own reasoning powers were and to also know that they were sound. And adds, Penrose dismisses the possibility that a human knows its reasoning powers, but does not know that they are sound. In his paper, McCullough also examines the appliability of Godel’s theorem on non-computable systems and the human mind. According to him, both are possible, by the way he asserts that Penrose’s idea is wrong. Consequently, McCullough agrees with Penrose that human reasoning cannot be formalized in some sense, because humans do not understand their reasoning system well enough to formalize it. This limitation is not due to a lack of human intelligence, but is inherent in any reasoning system that is capable of reasoning about itself. (D. McCullough, 1995).

As a short conclusion, it seems that the discussion between AI vs. mind will last for a long time. But, considering the present situation, AI has a long way to the go in order to achieve the expected skills.

References

• Chalmers, D.J. (1995). “Minds, Machines, and Mathematics”. http://psyche.cs.monash.edu.au/v2/psyche-2-09-chalmers.html

• Godel, K. “On Formally Undecidable Propositions of Principia Mathematica”. http://www.ddc.net/ygg/etext/godel/

• Lucas, J.R. (1970). “Minds, Machines and Godel”. The Freedom of the Will, Oxford: Oxford University Press. http://users.ox.ac.uk/jrlucas/mmg.html

• Maudlin, T. (1995). “Between The Motion And The Act…”. http://psyche.cs.monash.edu.au/v2/psyche-2-02-maudlin.html

• McCarthy, J. (1995). “Awareness and Understanding in Computer Programs”. http://psyche.cs.monash.edu.au/v2/psyche-2-11-mccarthy.html

• McCullough, D. (1995). “Can Humans Escape Godel?”. http://psyche.cs.monash.edu.au/v2/psyche-2-04-mccullough.html

• Penrose, R. (1989). The Emperor’s New Mind. New York: Oxford University Press.

• Penrose, R. (1994). Shadows of the Mind. New York: Oxford University Press.

• Penrose, R. (1996). “Beyond the Doubting of a Shadow”. http://psyche.cs.monash.edu.au/v2/psyche-2-23-penrose.html

• Pysche (1995). An Interdisciplinary Search of Consciousness, Vol. 2, Symposium on Roger Penrose’s Shadows of the Mind. http://psyche.cs.monash.edu.au/psyche-index-v2.html