Citadel of the Self: Incompleteness Theorem

2008-01-01 00:00:00

What is consciousness? Is it established in the brain, or distributed throughout the whole body? Is it physical? Can I jump out of my own consciousness? These are the questions that have occupied the minds of thinkers for centuries. This article does not promise answers but more questions.

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 GÃ¶del’s Incompleteness Theorem after the famous mathematician Kurt GÃ¶del (1906–1978). GÃ¶del 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.” GÃ¶del’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 GÃ¶delian 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 GÃ¶del, 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. GÃ¶del, Escher, Bach: An Eternal Golden Braid, 1999, Basic Books.

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