Appendix B Appendix. Argument against the Gödelian argument
In 1961, J. R. Lucas presents the Gödelian argument against the existence of a ”strong” AI. His proof is based on Gödel theorem, which is stated as followed:
In any consistent system which is strong enough to produce simple arithmetic there are formulae which cannot be proved-in-the-system, but which we can see to be true. Essentially, we consider the formula which says, in effect, ”This formula is unprovable-in-the-system”. If this formula were provable-in-the-system, we should have a contradiction: for if it were provable-in-the-system, then it would not be unprovable-in-the-system, so that ”This formula is unprovable-in-the-system” would be false: equally, if it were provable-in-the-system, then it would not be false, but would be true, since in any consistent system nothing false can be proved-in-the-system, but only truths. (Lucas, 1961)
This theorem holds for all formal systems which are consistent, adequate for simple arithmetic, and shows that those formal systems are incomplete, with some fact being true, but unprovable.
It is of the essence of being a machine, that it should be a concrete instantiation of a formal system. It follows that given any machine which is consistent and capable of doing simple arithmetic, there is a formula which is incapable of producing as being true… (Lucas, 1961)
Further argued, he then comes to such conclusion that no machine can be a complete or adequate model of the mind, since ”the mind are essentially different from machines”. Lucas’s defenders, Roger Penrose, also state in his Shadow of the Mind (1994). A human mathematician, if presented with a sound formal system $F$, could argue as followed:
Though I don’t know that I necessarily am $F$, I conclude that if I were, then the system $F$ would have to be sound and, more to the point, $F^{\prime}$ would have to be sound, where $F^{\prime}$ is $F$ supplemented by the further assertion ”I am $F$”[1]. I perceive that it follows from the assumption that I am $F$ that the Gödel argument $G(F^{\prime})$ would have to be true and, furthermore, that it would not be a consequence of $F^{\prime}$. But I have just perceived that ”If I happen to be $F$, then $G(F^{\prime})$ would have to be true”, and perceptions of this nature would be precisely what $F^{\prime}$ is supposed to achieve. Since I am therefore capable of perceiving something beyond the powers of $F^{\prime}$, I deduce that I cannot be $F$ after all. Moreover, this applies to any other system, in place of $F$. (Penrose, 1992, 3.2)
By default, the argument supplemented from Penrose raised the contradiction of proof-ness. The Gödelian argument implicitly creates layers, and levels, on which one puts those languages they are abided to seem fit of their expressions on the shelf, by the order of effectiveness. Such notion then, would make the advancement of machine to human seems perpetually, unsophisticatedly, inoperable and impossible in essence. Lucas argument, just as Searle, also claim that it is all the computer can do, of which the system itself is inherently useless of hosting such entity. However, artificial intelligence, as for now, using this term since chapter 3 which is not yet here, is not a computer in its form. We say, however, for an artificial intelligent subject with computers as its existential facilities, not the computer itself. This open up the fact that the notion of computer we are having right now, are also limited to the kind of classical computer, and not taking into account of any such similar ‘computing architecture’ of framework that might differ from such understanding. If so, then CRA is only partially right. But partially wrong since the comparison is limited to a form of internal structure in a well-formed system. That is to said - we need to create the (a) construct(s) that exceed(s) such argument. The problem is, how?
B.1 Remark
Before even taking a stance on such argument, what is the meaning and interpretations, as well as ostensively why it is even important to divulge into such point? The answers might be a bit difficult.
Human is variedly different from machine, for the current time with all the knowledge at present. Truth to take, the action of writing this itself is part of the endeavour to discover one’s self, or rather, to understand $F$ with the assertion of ’I am $F$’, for now, that we can, and is doing. By the language and construction of contemporary and propositional logic, a machine cannot do that.[2]
However, if the converse situation happens, where we cannot totally perceive what we actually think, and how it is formed - per metric, being either consciousness, or one’s self, surprisingly, it does not support the previous argument from an intuitive view (Bear in mind that this is a non-rigorous study). If stays rigid as it is, not counting being dynamic as we want, the model created from a human being can only imitate and represents what directly is entailed in the human mind of interpretation and logics. But logic and interpretation is a construct of the mind, for all intents and purposes, to directly infers to the physical world, the living world. However, if one is to use such inference on itself, for example, examining the brain itself, then to a certain point, what can be deduced from such observation can only fit in the interpretation space of what its creator, the human brain itself, can contrive. Thereby, we might conclude that figuratively, even human cannot understand human itself, from certain perspective. But the quality of succinctly interesting loop is to be taken seriously. The point now is, what type of construct, even logic, would be sufficient of taking the understanding, and will it make uses of the looped behaviours? By that, we then argue superficially that anything that relies on the machine cannot model the human existence and conscience itself. There are some assumptions thereof in the argument:
- Existences and the state of the world are in fact, modelled in mathematics, for one way or another. This is to facilitate the use of formal system in the argument. Everything is a set of rules, in which things operates.
- A machine per its definition, cybernetic machines are of all expressed by the single principle that it is born out of a formal system itself.
- Truth is the finite quantity that exists in such formal system, and is absolute.
- The mind is an entity of which is inherently different from the logic of formal system.
Those fundamental, overlapping assumptions make up the bulk of the Gödelian argument, from the surface. However, is that true of all the merit? [3]
The phrase ”I am $F$” is merely a shorthand for ”$F$ encapsulates all the humanly accessible methods of mathematical proof” ↩︎
In general, we cannot even say that it is true of the truth that human actually differs from what is proposed to be perceiving $F^{\prime}$ being $F$. For human understanding of ourselves alone, we are trying to fit it into the interpretation and the rough ’understanding’ of human itself. That is, there exists the space of reason and argument of a scientist, which interpretation follows. If, supposedly, this interpretation is strong enough, then we might be able to perceive and understand ourselves from ourselves - a looped interpretability. This mechanism, if ever, is not well understood if exists. ↩︎
It turns out, however, the Gödelian argument has various proponents and opponents, and there are arguments of it being false. See Bringsjord (2000) for such argument, but it can be simplified as this. The Gödelian argument makes use of two assumptions: $G(F^{\prime})$ is true for a Gödelian statement, and $F^{\prime}\not\vdash G(F^{\prime})$ for $F^{\prime}\not\vdash G(F^{\prime})$ for $F^{\prime}$ being ”I am $F$” with added semantic. Then, the statement on $G(F^{\prime})$ is true is nothing but a satisfaction claim, of meta-mathematical assertion which can be reduced to $\mathcal{I}\vDash G(F^{\prime})$ is true for given interpretation $\mathcal{I}$. Thereby, there exists no contradiction thereof. ↩︎