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Self-similarity should be thought of as a portrait of a self-referential entity. A self-referential entity refers to itself before it exists, and this process is essential in its existence. For example, if we think about the self-similar shapes discussed in this chapter, all we have seen from them is simply a snapshot of a certain level, their true and complete selves existing only in infinity. Self-referentiality in science is a new idea. Cantor's set theory is probably responsible for its recent developments. The many paradoxes which Cantor's set theory created were first thought to be pathological cases. Notably, Henri Poincaré called Cantorism ``a sickness from which mathematics would have to recover'', while Hilbert thought that Cantor had created a new paradise in mathematics[7, page 1]. However, once Gödel published his paper, ``On Formally Undecidable Propositions Of Principia Mathematica And Related Systems'' in 1931[15], self-referentiality was taken very seriously.

The basic idea behind Gödel's paper is that no formal system can be complete and consistent at the same time, or in other words, no formal system, no matter how rigorous, can cover the whole truth. There are perhaps many interpretation of Gödel's work, and it is generally understood that a full comprehension of the paper has not yet been reached. Gödel's paper is a completely rigorous mathematical work. However, the basic idea is very simple and intuitive[18, page 17]. He was inspired by the Richard paradox, which is a self-referential paradox in number theory, showing that any meta-mathematical statement which is about a formalized calculus can indeed be formalized within the system itself[30, page 66]. In this way a system can create undecidable propositions. There is really no need to think about mathematics to understand Gödel's work; all one needs to do is to try to decide if the following statement is true:

This statement is wrong.
This statement can neither be true or false. Once one applies a truth value to this sentence, the sentence itself reverses its truth value. This situation arises since the statement refers to itself (``this statement'') before it is completed. Such statements are deep within the system of our thought and senses. For example, the idea of seeing is not an issue unrelated to what we see. As a child what we see creates the idea of seeing; any new visual information can change our concept of vision. If we take this idea on the path of evolution, we may ask: ``Was something seen first before an eye was evolved, or is it the other way around?'' There is really no substance in such questions, except that they make us aware of the self-referential issues in evolution. Dawkin treats the paradoxical issue of survival in being selfish or altruistic to our own or other species in ``The Selfish Gene''[8], and for that he almost takes the consciousness away from living beings to the gene level.

These types of questions inevitably take us on the path of philosophy. Self-referentiality is one of the strongest elements in the philosophy of Zen Buddhism and Taoism[23]. Self-refentiality is especially found in the poetry of many of the eastern cultures. The 20th-century western literature and philosophy of the absurd is mainly concerned with questions of authority and power, which, once questioned, become self-referential entities. Many of the works of Kierkegaard deal with issues like paradoxes and ironies of life. One of his most influential works ``Fear and Trembling'', deals with the paradox of faith. He says[21, page 55]:

Faith is namely this paradox that the single individual is higher than the universal -- yet, please note, in such a way that the movement repeats itself.
What Kierkegaard meant as repetition, is actually understood by us now as recursion. He further says:
Faith is precisely the paradox that the single individual as the single individual is higher that the universal, is justified before it, not as inferior to it but as superior -- yet in such a way, please note, that it is the single individual who, after being subordinate as the single individual to the universal, now by means of the universal becomes the single individual who as the single individual is superior, that the single individual as the single individual stands in an absolute relation to the absolute.
Kafka's work which now is hailed as a masterpiece of 20th century modern literature is also deeply based upon self-referentiality. The following is one of his short paradoxes called ``On Parables'' which has many levels of self-referentiality[19]:
Many complain that the words of the wise are always merely parables and of no use in daily life, which is the only life we have. When the sage says: `Go over', he does not mean that we should cross to some actual place, which we could do anyhow if the labor were worth it; he means some fabulous yonder, something unknown to us, something too that he cannot designate more precisely, and therefore cannot help us here in the very least. All these parables really set out to say merely that the incomprehensible is incomprehensible, and we know that already. But the cares we have to struggle with every day: that is a different matter. Concerning this a man once said: Why such reluctance? If you only followed the parables you yourselves would become parables and with that rid of all your daily cares.

Another said: I bet that is also a parable.

The first said: You have won.

The second said: But unfortunately only in parable.

The first said: No, in reality: in parable you have lost.
And here is the shortest self-referential statement we have arrived at:
Nothing exists3.5.

next up previous contents
Next: Self-similarity in Sound and Up: What is Self-similarity? Previous: Fractional Dimensions   Contents
Shahrokh Yadegari 2001-03-01