I’m sure most of you have watched the movie The Matrix. Do you remember the scene where the simulation is paused while Morpheus is walking in the street with Neo?

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.