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.

We have a book written in golden letters lying in front of us. The decoration of the pages is attractive and impressive.. the small diamonds that are spread out over the pages are so shiny.. the cover is attractive.. the pages are velvety like a rose.. the colors are lively and bright.. the letters have been written by a talented calligrapher and are so harmonious; each one is a work of art.. While one may be interested in the book and its material another will find understanding what has been written and its content more compelling…

We recognize and perceive that which exists and evaluate that which is happening around us by using our natural senses, intelligence, and experience; as a result we are able to make inferences about the world through methods of deduction or induction. Positivism claims that the only true and valid knowledge is that which is based on actual sensual experience. According to materialism, the only thing that can be truly proven to exist is matter. In general, in analyzing matter and physical phenomena the deterministic approach is employed; all events are thought to be unavoidable consequences of the preceding sufficient causes. Sometimes the deterministic descriptions and formulations may end up in chaos. Chaos theory deals with the chaotic character of such formulations qualitatively. The quantum theory could also be interpreted as a deterministic tool that describes physical phenomena, more specifically, the behavior of matter and energy, within a range of probabilistic likelihood.

Some philosophers tend to include other factors in the cognitive faculties and the means that shape human perception and cognition: for example, emotions and intuition (e.g. Kant, Bergson). This apparently is a diversion from materialism and the deterministic point of view. The human being is not only an elegant combination of atoms; we also have a part that is not a function of matter or time: and this is the soul. If we restrict ourselves only to the natural senses and matter, as the positivist and materialist approaches do, then severe limitations on human perception are imposed, as demonstrated below. In general, it is not possible to perceive the truth completely without using all of one’s cognitive faculties. The question is then, how can a new program of reasoning be systematized to combine the senses and mind along with the emotions and intuition in a coherent manner? This novel program, which is called qualitative inference, is defined in this article using examples; however, first there is a brief introduction to a related topic, known as qualitative reasoning.

Qualitative reasoning is a relatively new approach in the estimation of possible solutions to real-world problems through non-quantitative reasoning. This approach provides a program to reason about the behavior of physical systems, such as the physiology of the human body, automobiles, etc., especially in the case of inexact or incomplete data, or when there is a lack of the precise quantitative information that is required by conventional methods. Qualitative models are more able to express states of incomplete knowledge about continuous mechanisms than traditional models which have been developed for the analysis of physical phenomena, as the former guarantees the ability to find all possible behaviors consistent with the knowledge in the model.

To get a flavor of what is meant by qualitative reasoning, consider this everyday scenario: You are confronted with a water-filled pan on a lit stove. You can easily predict that the pan will warm up, which will warm the water. At some point, the water will start to boil, and the pan might eventually become empty. To make these predictions, the exact values of the variables involved, such as the amount of water, the temperatures of the stove and the water, or the boiling temperature need not be known.

Observing pouring rain and a river’s steadily rising water level is sufficient to make a prudent person take measures against possible flooding-without knowing the exact water level, the rate of change, or the time the river might flood.

Qualitative inference is a program of qualitative reasoning that extends inference to the non-physical aspects of physical information, including reason, the relation between subjects and objects, etc. The universe itself is a great set of information, but this may not be exact or complete enough for metaphysical inductions. Needless to say, any such possible inductions should be consistent with known physical information. Consequently, in qualitative inference, the two key points we are referring to, namely the use of incomplete or inexact data and consistency with physical knowledge, are crucial.

While the deterministic approach could be considered to be the interpolation of existing physical knowledge and information in order to justify what is known or to discover what is unknown, qualitative inference can provide a tool for the extrapolation of existing physical knowledge and information. It can be used to reach what is unknown, physically unobservable, or even impossible to validate, regenerate, repeat, comprehend, or completely embrace by using standard scientific techniques.

Qualitative inference

With some examples we will explain how qualitative inference can be applied to everyday, real life experiences for inferences and inductions that are beyond our physical world.

A coincidence or a gift?

Let us take the example of a package sent to a person in the mail. Let us specify that it is a shirt. Let us further specify that the color of the shirt is the favorite color of the receiver, it fits him very well, the design is his preference, and the brand and the fabric of the shirt are those that he generally chooses. Based on this information, as it is incomplete, it would be difficult to answer questions about the possible relationship between the sender and receiver, the purpose of this shipment, etc. from a positivist or deterministic point of view. Through qualitative inference one could conclude, however, that the sender is someone who knows and is close to the receiver, who would like to please him likely because the sender likes him, and that the shirt has been mailed as a gift. This inference is obviously consistent with the known information. If we were to exclude the sense of love, enjoyment, pleasure, and closeness between two people it would not be possible to make these conclusions.

The apple

The apple is one of the most abundant types of fruit on the earth. “An apple a day keeps the doctor away,” goes an old saying. Doctors in modern times agree that the apple is important as a nutrient. An apple with its skin includes about 4 grams of dietary fibers. Potassium, calcium, phosphorus, magnesium, and selenium are minerals that can be found in an apple in different amounts. It also contains small amounts of iron, manganese, copper and zinc. The vitamins contained in an apple include vitamins A, C, E, and folic acid, which is an important nutrient during pregnancy. While eating an apple, one also strengthens the gingiva gums often without being aware that this is happening. The different colors of apples are very pleasant, as are their smell. How many artists have used the apple as their subject, due to its compelling beauty? It leaves a pleasant taste on the tongue and in the mouth. Many cuisines would be severely handicapped without the apple as an ingredient. The size is just right to grasp in the hand, and its flesh is succulent to bite. Different types of apples give the feeling of experiencing a new thing, yet remaining within the limits of our acquaintance. Apples are attractive and important sources of nutrients for animals as well. There is so much more to say about the apple and its importance to human and animal life, about how it is a very good source of nutrients that matches our needs, and addresses our tastes and pleasures.

With this information, the positivist perspective looks at the apple itself, its attributes, nutritional value, material properties, as well as the physical functions of the human being, and their need for the nutrients that are naturally included in the apple. This physical information, although valuable, is incomplete in understanding the perfect correlation between not only the need of the human being, but also our sense of taste and the apple, as we have briefly summarized above. Through qualitative inference, however, one could conclude that apple is a gift not only to the human being, but to most other living creatures, much like the shirt that was sent as a gift in the example above. This conclusion does not include any contradictions with the known, but incomplete, physical information. Although the sender of this gift is not known, one can infer that such a being exists and this being knows the human very well and is able to make and send this gift via the trees; this being possibly likes human beings and others who benefit from the apple, and likes to please them. However, without reference to the senses of love, enjoyment, and pleasure, it would not be possible to derive such conclusions.

Robot vs. mother

Now, we will consider another event, that in which a mother prepares food for her child. If she is feeding the baby with a formula, the steps she is likely to follow are sterilizing the bottle, boiling the water, refrigerating the water or waiting for the water to reach the appropriate temperature to mix the formula with the water, and stirring it carefully and thoroughly to make this a pleasant experience for her baby. From time to time, the order, attention of the mother, and duration of the steps may change without a clear difference in the underlying conditions. If one were to study this case scientifically, from a positivist perspective, then one would study the temperature of the water, the hygiene of the bottle, the actions of the mother, etc. for some possible inferences. But, these would be far from being a complete account of the situation and cannot reflect the entire picture or truth. The main point that a prudent person can derive from these scenes is clearly the mercy, compassion, and love of the mother towards her baby. This is the fact and main motivation that ensures that the mother take the actions mentioned above voluntarily, willingly, carefully, and repeatedly.

The Earth’s circulatory system

With this example in mind, let’s now review what we learned in high school about the earth’s circulatory system, a wonderful mechanism that purifies, cleans, and processes the water, the most important factor of the life. Human beings can survive for only a few days without water. Water is stored under the ground in natural reservoirs, whose formation alone is the topic for a series of scientific articles. Although the temperature increases as one approaches the center of the earth, sometimes the water comes to the surface at very cold temperatures; such cold clear water is the most desired substance on a hot, sunny day. Rain drops in their natural form are pure water and do not have a pleasant taste. These rain drops and the water used by people or animals is processed, that is, the water is cleaned, filtrated, and purified under ground. Minerals essential to human and animal life and some chemicals that naturally exist in the soil mix with the water. The addition of all these substances is good not only for the water’s nutritional value, but also creates a more pleasant taste. It is with these constituents that water becomes an essential nutrient for human beings and animals. Some of these additions are fluorine that strengthens the bones, fluoride, which is beneficial in low concentrations, sodium, potassium, and chloride, all of which play a role in the metabolism of the body. When the water comes to the surface as hot water, it can be used as a treatment for some diseases. The water is not processed only under the ground, as we have described here, but surface water is also atomized for the ultimate purification. When water evaporates, the vapor rises to a certain location in the atmosphere, accumulates there and forms clouds (which are a subject matter for poets, due to their beauty, especially when they appear on a sunny day). When water is condensed and catalyzed by the wind, the vapor returns to the earth as rain drops to complete the cycle.

Each step in the mechanism of circulatory system mentioned above can be analyzed scientifically, from a positivist perspective, to attain useful and important information. On the other hand, these results will never allow us to make inferences about the reason behind the perfect correlation between the needs of human beings and other living creatures and the availability of water in its most useful form as an essential nutrient with a pleasant taste, or the continuous availability of clean, fresh water for the living creatures who have no control over never-failing mechanism of the circulatory system, or the availability of water with the perfect chemical combination that allows coexistence of water in three different physical forms (ice, water, and vapor). On the other hand, using such incomplete physical information, namely the perfect match between the nature, properties, and availability of water, and the needs, the sense of taste of living creatures, a sensible person could easily infer that there is a hidden mercy and compassion acting behind the scenes, just as with the case of the mother preparing food for her baby. Without this qualitative inference, which is consistent with the known physical information and refers further to the sense of mercy and compassion, the entire picture of this phenomena and the truth can never be captured, regardless of scientific developments, the technology used in labs, or the computer systems that can carry out extremely complicated computations.

Conclusion

Qualitative inference leads us to arrive at conclusions about the non-physical or metaphysical aspects of physical phenomena. This program of reasoning considers physical phenomena and the universe as an incomplete source of information for understanding the metaphysical aspects of the physical world, helps us to make inferences that are consistent with this incomplete physical information, and to use all our cognitive faculties, including intelligence, senses, intuition, and even emotions, to arrive at non-physical inferences. Qualitative inference leads one to consider the apple as a gift and the availability of the fresh water through the natural water cycle as an indication of universal compassion and mercy for all living creatures; no living creature would be able to carry out such a process on their own. Without this reasoning, as exemplified above, only a very limited portion of natural phenomena can be understood sufficiently; the main points will always be missing or overlooked.

Through the same program of reasoning, a gift necessitates a loving sender, universal compassion and mercy requires a being that is compassionate and merciful. Just like the case studies presented above, all real life events have similar messages that can be understood by those who do not restrict themselves to physical nature. All physical events, for example, indicate a universal consciousness that implies the existence of someone who is conscious of everything. Such a being should be able to do all these things according to His will, so He is All Mighty… These examples shed light onto some of His attributes, such as His mercy, compassion, will, might, and consciousness. When other phenomena are analyzed in a similar manner, one can discover other essential attributes of this Supreme Being. As a result, qualitative reasoning not only proves the existence of the Supreme Being, it provides a useful tool in understanding His attributes.

Similarly, the rebirth of trees, insects, and the earth as a whole every spring after their death in the winter is a physical phenomena that we have all experienced. This information, through qualitative inference, can lead us to understand the Hereafter, that is, the resurrection of all human beings in another world. All the attributes of the Supreme Being mentioned above also support this inference. Mercy cannot be absolute and universal, for example, if the oppressed people and tyranny are treated the same; unfortunately, this is what seems to be happening in this world.

In brief, positivist and materialist approaches impose severe limitations on human perception. In reality, all the cognitive faculties of the human being play roles in our evaluation of our being, our position and role in this world, and our relation to others. The truth can only be obtained by a person through the systematic use of all such faculties. Qualitative inference suggests a program for doing so. While the positivist approach puts mankind into the position of being an observer of the things they can touch or see by invalidating all other sources of knowledge, qualitative inference allows one to address the non-physical or metaphysical aspects of the physical phenomena by freeing all of our cognitive faculties, and in particular allows to be a believer while observing the universe in a similar manner.