The scene depicts an ordinary day in a metropolitan city. People are crossing the streets, perhaps going to work. There are traffic lights and so on. However, it is just a simulation program; all the actions of the people and all the events in the environment were predetermined. We can imagine questions popping into Neo’s mind: “Is the universe really a kind of simulation? Does God interfere with the universe or does it operate like a machine?” and followed by “Is it possible to compute someone’s fate? Does free will exist?”

Neo would neither be the first nor the last to ask similar questions. Until the 1930s, the answers to these questions were sometimes influenced by the idea of scientific determinism which considered the universe like the one in The Matrix. A paradigm shift took place in 1930s when two brilliant scientists proposed their ground breaking studies on non-determinism. Gödel’s Incompleteness Theorem and Heisenberg’s Uncertainty Principle considered the phenomenal aspects of non-determinism in the universe. First, we will have a short journey through the idea of scientific determinism. Then, we will investigate non-determinism and its consequences.

The idea of scientific determinism

The term ‘scientific determinism’ is defined to be the computability of future states, given the current state and a computer that has sufficiently large computation capacity. Suppose we take a snapshot of the universe at a particular time, or press the pause key and freeze the universe such as the example from The Matrix. What scientific determinism says briefly in simple terms is that by looking at that entire picture of the moment, the picture of the next second can be calculated using the laws and formulas of physics. On a large scale and in the long term, this idea leads to an exact computation of a moment of the universe given the initial conditions and its governing laws at the time of its creation. This is the key point for some scientific determinists who exclude the Divine work in the universe; if God exists, He only sets the initial conditions and the physical laws of the universe. Then the universe works on its own like a machine.

The first modern idea of scientific determinism was articulated by Newton in 18th century. Newton believed that all the laws governing the universe could be deduced from formulas, and these formulas could be derived by following the scientific method. First of all, science requires consistency among all known formulas and theories. This led the German mathematician David Hilbert to tackle the problem of proving the consistency of arithmetic in 1900s. One way of showing the consistency of arithmetic was to prove or disprove infinite number of statements in arithmetic, which was practically impossible! Instead, a clever idea was to develop a methodology that will generate the proof or disproof of a given arbitrary statement in arithmetic. If that methodology existed and was shown to be correct, that would be the happy ending of the story. Motivated by this problem in 1913, Alfred Whitehead and Bertrand Russell wrote their work, Principia Mathematica, a three-volume work on the foundations of mathematics. It was an attempt to derive all mathematical truths without the requirement of human intuition. They utilized symbolic logic, since it has a profound methodology for proving statements.

Later, this work would have played a fundamental role for proving the correctness / incorrectness of any statement in the universe. As an ultimate goal, by deducing all the information that belongs to a certain time of the universe, it would have been possible to compute the exact situation of the universe at an arbitrary time, which we mentioned before.

The fall of scientific determinism

At that time, Principia Mathematica left two questions open:

• Could a contradiction be derived from the Principia’s axioms (the question of inconsistency)?

• Does a mathematical statement which could neither be proved nor disproved in the system (the question of completeness) exist?

These questions were the heart of the discussion, and they had to be both answered as ‘no’ by Principia Mathematica to be deterministic. We have already explained the necessity of consistency before. On the other hand, some states cannot be calculated in an incomplete system which violates the determinism.

In 1931, Gödel published his famous article “On formally undecidable propositions of Principia Mathematica and related systems.” This work claims to refute the claims of scientific determinists about deriving all mathematical truths from logical systems. In his article, Gödel’s first incompleteness theorem showed that sufficiently complex systems (such as the ones described in Principia Mathematica and the physical laws of the universe) could not be complete and consistent at the same time. Gödel proved his theorem using a genius idea, which was by showing that a similar statement to “This statement cannot be proved” can be expressed in any sufficiently complex logical systems. Hence, one who proves it will create an inconsistency in the formal system, or it will be considered as improvable, violating the completeness rule of the formal system.

Another stroke to scientific determinism came from Heisenberg in 1927 with his famous Uncertainty Principle. Heisenberg asserted that both the velocity and the position of a particle cannot be accurately measured at the same time. Measurement requires interfering with the particle in terms of position or velocity. Observation becomes a part of the outcome which is in fact not known to be the real outcome. In short, uncertainty principle implies that the particle positions can only be calculated as a probability distribution.

One can object that this effect may be a result of our lack of the knowledge needed to find out the facts about particle position and velocity. However, Copenhagen’s interpretation of Quantum mechanics is commonly accepted, and it states that the problem of uncertainty is not epistemological but ontological; i.e., the problem is not due to the limits of scientific knowledge but depends on the constitution of the universe . From a mathematical aspect, Gödel’s theorem strongly asserts the same fact.

Consequences of non-determinism

Non-determinism in the Universe brought back earlier concepts categorized as meta-physical, such as ‘fate’ and ‘free will,’ into the debate about scientific knowledge for further investigation, and made a phenomenal change in the perspectives on the existence of God.

In The Matrix, remember the scene wherein Neo says “Deja vu” after he sees the black cat for the second time at the stairs of the apartment. Then Trinity tells him that it happened due to a change in the Matrix. The free will of Neo and his friends causes the failure of the deterministic nature of the Matrix; the Matrix cannot make exact calculation of their actions and foresee the future. Instead, it alters the simulation program based on their actions. Now, the concept of fate is not a predetermined destiny in the Matrix; it involves the human free will, hence nobody knows what will happen in the future, including the Matrix itself.

Free will requires the ability to make choices independent from deterministic constraints. A friend of yours offers you to take one of two identical apples, and you take one of them. If it was possible to rewind time and get back to the moment of the offer, this time you might choose the other apple. Note that the states are precisely the same; the entire history of the events in the universe is exactly the same as the one in previous scenario. This is the key point: in two equivalent states of the universe; i.e., all conditions are exactly the same, free will allows you to choose different options in these two states. This situation contrasts with a deterministic universe however well suits to a non-deterministic one.

Another major consequence is about the Creator’s role in the universe. As mentioned before, scientific determinism can reduce the concept of ‘God’ into the following statement [4, 6]:

• If there is a God, He created the universe, set the initial conditions and the laws, and left it the way it works. He doesn’t interfere with its execution.

By non-determinism, this statement can no longer be regarded as the absolute truth. Instead of clearly determined outcomes in the future time frames, Quantum mechanics introduced the notion of probabilities. In each decision point (i.e., quantum time frames or every single moment), there are many possible outcomes. That means either there is an entire ‘randomness’ or there is a Decision Maker that decides at these decision points and controls the flow of all actions in the universe. In case of randomness, it is always possible to reach a ‘failure state’. However, the universe has never failed for billions of years. Here, the failure state is not the ending of the universe or Armageddon, because these states are internally consistent. Failure state means to get the ‘blue screen’ as in Windows; suddenly everything stops or perishes. ‘Failure’ can be described as an unexpected error that leads unrecoverable / unhandled error which crushes the operating system. At this point, the universe needs to restart in order to start everything from the beginning. The idea of randomness in the course of actions is strongly opposed by Einstein as stated in his famous saying “God doesn’t play dice.”

So the question is “how can the concepts ‘God’, ‘fate’ and ‘free will’ be related to each other?” Now, we may say the following statements about fate: it is not computable, it is not random (due to the definition of fate), and free will is possible. In this case, we have two options:

• there is no free will – God controls all the events and free will is not involved in it,

• there is free will – God incorporates free will in his creation of the upcoming states

According to the second idea, we have been given the right to choose among limited amount of options and then God creates the chosen option. For instance, I would like to move my arm and show the tendency to move it. So, God creates necessary conditions such as the pulse from the brain that orders the arm to move, and the arm moves. Of course, He always preserves the right not to create the things according to my will. In that case, the arm doesn’t move – I might have a stroke or might not have sufficient strength or something else might happen.

Consequently, in case of free will, fate becomes all the outcomes of God’s will by including free will in the universe. But “God knows everything, everything that will be in the future. Then, doesn’t that mean the universe is deterministic?” As we mentioned above, determinism requires the computation of future states. However the Creator knows the fate with His infinite knowledge by overseeing the entire system but not computing. Here, computation and knowledge are two different concepts; knowledge may not be acquired by computation. If somebody shows us the result of a multiplication of two numbers, we learn the result by seeing it but not computing it.

Non-determinism in the universe might be considered as the beginning of a re-marriage between science and meta-physics. Non-determinism implies that it is not reasonable to absolutely reject Divine work in the operation of the universe and the idea of free will. The concepts that are considered to be currently meta-physical such as fate and free will need further investigation in the physical sciences.

Fatih Gelgi has a PhD in computer science. He is currently the computer coordinator of Accord AMSP team in Los Angeles.

References

[1] K. Devlin. “Kurt Gödel – Separating Truth from Proof in Mathematics,” Science, 298: 1899-2000, 2002.

[2] F. Gelgi. “Implications of Gödel’s Incompleteness Theorem on Artificial Intelligence vs. Mind,” The Fountain Magazine, 46, 2004.

[3] K. Gödel. “Uber formal unentscheidbare Satze der Principia Mathematica und verwandter Systeme I,” Monatshefte für Mathematik und Physik, 38: 173-98, 1931.

[4] S. Hawking. Brief History of Time, Bantam Dell Publishing, Ed. 10, 1988.

[5] E. Nagel, J. R. Newman. Gödel’s Proof, New York University Press, 2001.

[6] C. Taslaman. Kuantum Teorisi Felsefe ve Tanri, Istanbul Yayinevi, 2008.

[7] Wikipedia, “Principia Mathematica,” retrieved on Feb 27, 2010 from http://en.wikipedia.org/wiki/Principia_Mathematica.

Notes

1. Interested reader may refer to [5] for the detailed and understandable explanation of Gödel’s proof.
2. For details see “Chapter 4: Uncertainty Principle” in [4].
3. Different interpretations of Quantum physics can be found in [6] or at: http://kuantum.gen.tr.

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