Since all monotheistic scriptures come from the one and only God, we should view other scriptures as potentially enriching our understanding of our own scripture. But in the middle ages, almost all readers thought of revelation as a zero sum sport like tennis rather than a multiple win co-operative sport like mountain climbing.

In a zero sum game, any value or true spiritual insight I grant to another scripture somehow diminishes my own. This was the result of the influence of Greek philosophy’s emphasis on the logic of the excluded middle. Something is either true or it is false. There is no other option. If two propositions contradicted one another, one or both of them must be false. If my religion is true, yours must be false.

In modern terms, light could not be both a particle and a wave at the same time. Yet we now know that light is indeed both a particle and a wave at the same time.

This situation did not improve much in modern times. In the last two centuries, university academics have written many studies of comparative religion which they claim are objective and not distorted by their religious beliefs. Unfortunately, academics who treat other religions academically usually do not believe that other scriptures are actually Divinely inspired. Indeed, many academics do not believe that even their own scriptures are Divinely inspired.

They use the same kinds of explanation to understand a revealed religion that they would use to explain secular history and literature. I follow a different model, one I learned from Prophet Muhammad. For example, the Mishnah (an early third century compilation of the oral Torah), states, “Adam was created as an individual to teach you that anyone who destroys a single soul, Scripture imputes it to him as if he destroyed the whole world” (Mishnah Sanhedrin 4:5).

And the Qur’an states, “one who kills a human being, unless it be for murder or for spreading mischief in the land, would be as if he slew the whole people, and if any one saved a life, it would be as if he saved the life of the whole people” (Qur’an 5:32). Academics explain the similarity of the two statements by assuming that since the Jewish statement is several centuries earlier than the Qur’an, Muhammad must have heard it from a Rabbi or other educated Jew in Medina.

But I believe Muhammad is a prophet of God who confirms the Torah of Prophet Moses. Muhammad has no need to learn this statement from another human being. Academics might reply that the statement is not found in the written Torah; it appears in the oral Torah, written by the Rabbis in the Mishnah more than 1000 years after Moses.

But the Rabbis maintain that the Mishnah is part of the oral Torah that was passed down from Moses through many generations, just as the Prophet Muhammad’s sayings have been passed down through the generations. Indeed, the Qur’an itself introduces this statement as follows, “It is because of this that We ordained for the Children of Israel ‘one who kills a human being …'” (Qur’an 5:32).

No prophet of God needs to be informed by another human what should be written in Holy Scripture. God is the source of all Divine inspiration. There are several verses in the Qur’an that mention things from the oral Torah. My perspective is that prophets and Holy Scriptures cannot in reality oppose one another because they all come from one source. Prophets are all brothers; they have the same father (God) and different mothers (motherlands, mother tongues, nations, cultures, and historical eras).

All of these produce different rituals and legal systems, but their theology can differ only in small and unessential details. As the sage of Konya, Jalal al-Din al-Rumi says, “Ritual prayer might differ in every religion, but belief never changes” (Fihi Mafih 49). Religions differ because the circumstances of each nation receiving them differ. Where the Scriptures differ, they cast additional light on each other. My belief is based on an important Hadith of Prophet Muhammad: a disciple of Muhammad named Abu Huraira relates, “The people of the Book used to read the Torah in Hebrew and then explain it in Arabic to the Muslims. Allah’s Apostle said (to the Muslims). “Do not believe the people of the Book, nor disbelieve them, but say, ‘We believe in Allah, and whatever is revealed to us, and whatever is revealed to you.'”

Following Muhammad’s teaching, I too neither believe nor disbelieve the Qur’an. If I believed in the Qur’an, I would be a member of the Muslim ummah (community). But I cannot disbelieve in the Qur’an because I believe that Muhammad was a prophet and I respect the Qur’an as a revelation to a kindred people, in a kindred language. In fact, the people, the language and the theology are closer to my own people, language, and theology than that of any other on earth.

How does this perspective affect my understanding of their Qur’an and my Torah? I practice “hosgoru.” Unlike those in the past who played the zero sum game, I do not seek some verse in the Qur’an I can dispute or object to. Indeed, this is what the Qur’an itself teaches. “For every community We have appointed a whole system of worship which they are to observe. So do not let them draw you into disputes concerning this matter” (22:67).

One of the major differences between the Qur’an and the Torah is the Torah’s attention to details (names of people and places) and the Qur’an’s emphases on universals. The Torah has long lists of geographical locations and of genealogies that many people today, especially non-Jews, find boring.

The Qur’an rarely identifies locations, and often omits the names of the people it does mention, for example the prophet who appointed Talut to become the first king of Israel (2:246). Indeed, Muslim commentators disagree about many of these details. Some say the prophet who appointed Talut king of Israel was Samuel and others think it was Joshua or Simeon.

These disagreements occur because many commentators do not use the Bible to fill in the details for the generalities of the Qur’an. Sometimes the Qur’an refers to nameless Messengers sent to a nameless town and no additional details can be added because there is no parallel in the Torah (36:13-29). In this case, even the punishment is only a vague “single blast and all became corpses” (26:29).

This lack of detail points to the Qur’an’s universal and ongoing Divine call for all human societies to repent and reform themselves.

On the other hand, many Rabbis get caught up in the details of the Torah and even expand them into super details. Thus, the rules relating to dietary observance of Passover and prohibited work on Shabbat have multiplied endlessly.

We need to learn from the Sunnah of the prophet as narrated by Abu Huraira: The Prophet said, “Religion is very easy and whoever overburdens himself in his religion will not be able to continue in that way. So you should not be extremists, but try to be near to perfection and receive the good tidings that you will be rewarded” (Bukhari Volume 1, Book 2, Number 38).

Another important lesson from the Prophet’s Sunnah, as narrated by his wife Aisha, says, “Whenever the Prophet was given an option between two things, he used to select the easier of the two as long as it was not sinful; but if it was sinful, he would remain far from it” (Bukhari Volume 4, Book 56, Number 147).

This is the path that I and most Reform Rabbis have taken in the last two centuries. If Orthodox Jews in the time of Muhammad had followed the prophet’s teaching, Reform Judaism (the largest of several different religious groups of Jews in North America) would have begun 14 centuries ago, instead of only two centuries ago.

The Qur’an was also far ahead of its time in many other ways. One of the most important ways is the Qur’an’s oft repeated statement that believers (Muslims) should believe in all the messengers of God. This message of religious pluralism and toleration is sorely needed in the 21st century.

The Qur’an states: “They say that none will enter Paradise unless he be a Jew or a Christian. That is their wishfulness. Say ‘Produce your proof if you are truthful'” (2:111).

At the time of Muhammad, both Orthodox Rabbis and Catholic Priests did claim that only their own believers would enter Paradise. The Qur’an instructs Muslims that this claim is not based on the Jewish or the Christian scriptures but only on the desires of those people who make these claims. In truth, nowhere in the Torah of Moses, the Zabur of David, or anywhere else in the Hebrew Bible is this claim that only Jews will enter Paradise asserted.

The great Sage Hillel, who lived in the first century prior to the birth of Jesus, taught that, “The righteous of all nations have a place in Paradise” (Tosefta Sanhedrin). Jesus also taught, “In my Father’s house are many dwelling places; if not so, I would have told you. I go to prepare a place for you” (John 12:2).

But generations after the death of Jesus, claims were made in his name, that only those who believed Jesus was the son of God, who died on the cross to save all humans from going to Hellfire, would be able to enter Paradise. In reaction to these polemical Christian claims, some Talmudic Rabbis began to counter-claim that only Jews would enter Paradise.

Yet even then the Rabbis did not think that eternal punishment was the fate of all those excluded from Paradise. Gehenna-Hellfire was conceived of as a temporary abode generally believed to last a maximum of 12 months.

The great sage, Rabbi Akiba, stated, “The punishment of the wicked in Gehenna lasts 12 months” (Mishnah Eduyyot 2:10). This is repeated in the Talmud (Shabbat 33b), and elsewhere it is stated that sinners, both Jewish and non-Jewish, are punished in Gehenna-Hellfire for (up to) 12 months (Rosh HaShanah 17a).

Thus the Qur’an accurately states, “They say, ‘the Fire will not touch us except for a fixed number of days” (2:80). The Qur’an instructs Muslims to say to both Jews and Christians, “If the abode of the Hereafter with God is reserved for you alone, excluding other people, then long for death … but they will never long for it” (2:94-5).

So, the answer to those who claim that, “none will enter Paradise unless he is a Jew or a Christian,” is “No! Rather, whoever submits his whole being to God as one devoted to doing good, aware that God is seeing him, his reward is with his Lord, and all such will have no fear, nor will they grieve” (2:112).

Thus, the Qur’an affirms that those Rabbis who strayed from the words of Hillel, “The righteous of all nations have a place in Paradise,” were wrong. Those followers of Jesus who abandoned his teaching, “In my Father’s house are many dwelling places,” and instituted the doctrine that Saint Augustine espoused, “No one can find salvation except in the Catholic Church. Outside the Catholic Church one can have everything except salvation,” are also wrong.

Tragically, during many centuries of Medieval debate between the three religions, everyone thought that religion was a zero sum game (one winner or one truth vs. many winners or many truths, i.e. soccer or tennis vs. hiking or mountain climbing).

Thus, some Muslim commentators also began to take the same exclusionary view condemned by Prophet Muhammad by adding specific details of theological belief to the statements in the Qur’an that stress a simple and firm belief in the one and only God: “No! rather, whoever submits his whole being to God as one devoted to doing good, aware that God is seeing him, his reward is with his Lord, and all such will have no fear, nor will they grieve” (2:112).

And even more explicitly, “Those who believe (Muslims), those who advocate Judaism, Christians, Sabeans, whoever truly believes in God and the Last Day, and does good righteous deeds, surely their reward is with their Lord, they will not fear, nor will they grieve” (2:62).

Thank God, in 21st century America the majority of most religious groups now believe the teachings of the Qur’an cited above (2:112 and 2:62).

A survey of over 35,000 Americans in 2008 found that most Americans agree with the statement: many religions – not just their own – can lead to eternal life. Among those affiliated with a religious tradition, seven-in-ten say many religions can lead to eternal life. This view is shared by a majority of adherents in nearly all religious traditions, including 82% of Jews, 79% of Catholics, 57% of evangelical Protestants and 56% of Muslims (from the U.S. Religious Landscape Survey, Pew Forum on Religion & Public Life, 2008, Pew Research Center).

Thus, in the 21st century United States most Christians, Jews, and Muslims have rejected the zero sum mind set and believe in the Qur’an’s teachings, “If God had so willed He could surely have made you a single community, but (He didn’t) in order to test you by what he has given you. Strive then, in competing in good works” (5:48). For ultimately, “On the Day of Resurrection God will judge between you about what you differed” (22:69).

Only those who reject God by disbelief or by unrepentant evil activities will be the losers when Judgment Day comes. But many, perhaps most, theologians will learn that they might not be as smart as they thought they were.

In conclusion, it is very important to understand that “religious pluralism is the will of God” is very different from religious, moral, or cultural relativism. Relativism teaches that all values and standards are subjective, and therefore there is no higher spiritual authority available for setting ethical standards or making moral judgments.

Thus, issues of justice, truth or human rights are, like beauty, just in the eye of the beholder. Most people, especially those who believe that One God created all of us, refuse to believe that ethics and human rights are simply a matter of taste. Religious pluralism as the will of God is the opposite of cultural or philosophical relativism.

The fundamental idea supporting religious pluralism is that religious people need to embrace humility in many areas of religion. All religions have always taught a traditional anti self-centered type of humility. Religious pluralism also opposes a religious, philosophical, and self righteous intellectual egoism that promotes a tendency to turn our legitimate love for our own prophet and Divine revelation into universal truths that we fully understand and know how to apply.

Religious pluralism teaches that finite humans, even the most intelligent and pious of them, cannot fully understand everything the way the infinite One does.

This is true, for every human being, even for God’s messengers themselves. When Prophet Moses, “who God spoke with face to face, as a person speaks with a friend” (Exodus 33:11) asks to see God face to face, he is told, “You cannot see My face, for no man can see My face and live” (33:20).

Similarly, in the Qur’an, Prophet Jesus admits to God, “You know everything that is within myself, whereas I do not know what is within Yourself” (7:116). When Prophet Jesus was asked, in private, by his disciples, “What will be the sign for your coming (back) and the end of the age?” (Matthew 24:3) Jesus warns his disciples about all kinds of upheavals and false Messiahs that will come. Then Jesus concludes by saying, “But about that day and hour no one knows, not even the angels in heaven, not even the son: only the Father” (24:36).

A similar statement was made by Prophet Muhammad when he was asked, “Tell me about the Hour.” He said: “The one questioned about it knows no better than the questioner” (Muslim book 1 Hadith 1&4).

God taught the general principle of epistemological humility through his Prophet who taught his followers, “I am no novelty among the messengers. I do not know what will be done to me, or to you” (Qur’an 46:9).

Only on Judgment Day will the answer to the theological differences that divide us now be resolved. “To each among you have we prescribed a law and an open path. If God had so willed, He would have made you a single people, but (He desires) to test you in what He has given you: so (compete) strive in all virtues as in a race. The goal of you all is Allah. He will (on Judgment Day) show you the truth of the matters in which you dispute” (5:48).

In this article we will review these three concepts and the relations among them by giving examples from the realms of both physical and metaphysical concepts.

Even though there is no universally accepted definition of analogy, or metaphor, or literal similarity, the purpose of their use is unanimously defined as “throwing light on an unfamiliar concept or situation using a familiar one.” The past few decades have witnessed several attempts of redefining these terms or at least extending a definition that is somewhat agreed-upon to an all-encompassing one. In so doing, researchers tended to employ mathematical structures that are by no means simple. What seems to differ from one approach to another is the method used in the process of establishing a correspondence between the familiar and the unfamiliar. Our purpose is not to give a full account of these approaches here, as there are so many of them, but rather to adapt one that is more convenient for conveying our examples.

Analogy: The word analogy is derived from the Greek word analogia. Originally, this was a mathematical (actually a musical) term before becoming a grammatical and linguistic one, meaning “proportion” (Szabo 1978, 23). An example of Aristotelian analogy is “spine is to fish as bone is to animal.” This simple analogy is generally formulated as A:B = C:D. We say, in this case, A and C are analogous. The sense in which A and C are analogous could be a property they have in common as well as the similarity between their relations to B and D, respectively. In the spine and bone example above we observe both. In the analogy “feet are to animal as wheels are to automobile,” however, feet and tires have virtually nothing in common, but they are analogous with respect to their functions within the bodies that they are part of, i.e., both are used as a means of transportation. Therefore, common relations are essential to analogy, but common objects are not. (Gentner-Markman 1997, 46).

One of the recent approaches to analogy which is widely used is the structure-mapping theory (SMT) (Dedre Gentner, 1983). It presumes a mapping between the familiar (domain or base) and the unfamiliar (target). The domain and the target consist of objects and relations among objects. SMT establishes a one-to-one correspondence between the objects of the domain and the target in a way that preserves the relations between objects (Those who are familiar with the category theory in mathematics will notice the resemblance of this mapping to a “covariant functor” between two “categories”). This is a “structural alignment” between the domain and the target (Gentner-Markman 1997, 47). In the “feet and wheels” example the mapping occurs between a body and an automobile. It maps feet to wheels and the relation TRANSPORT (feet, body) to the relation TRANSPORT (wheels, automobile). One can find more matching elements between a body and an automobile. For instance, we can match the heart of the body to the engine of the vehicle and observe that the relations RUN (heart, body) and RUN (engine, automobile) are preserved.

One might ask where do the arms get mapped? The answer is “nowhere.” We do not expect this mapping to match every object in the domain to an object in the target. The “power of the analogy” is not in the number of objects that are matched or the common attributes that the objects share. It is rather the degree of matching among relations that makes an analogy more powerful (Gentner 1982, 110).

Having introduced the concept of analogy by using some simple physical examples let’s now consider a not-so simple metaphysical situation.

In the 1920’s two mathematicians proved a very interesting but at the same time very puzzling theorem, known as the Banach-Tarski Paradox. In plain English, the theorem states that it is possible to divide a solid ball into a few pieces and reassemble those pieces together to make two balls, each of which has the same size as the original ball that was divided. A more striking consequence of their theorem is (you may want to sit down before you read this) a solid ball the size of a small pea can be cut into a number of pieces and reassembled into a new ball the size of the sun! Strange as it may sound, it is a valid mathematical argument, not a myth. (We have to note that it is not something one can do at home using a knife and a cutting board, because some of the pieces have no volume!) The analogy that we would like to establish under SMT is to map that solid ball of the theorem to a small amount of water which could quench the thirst of an army of about 30,000 in Tabuk in year 631. This was a miracle given to Prophet Muhammad, peace be upon him. A similar miracle of Prophet Jesus, peace be upon him, is described in the Bible, Matthew 14:21 (Volker Runde, in the Sky 2 (2000), 13–15).

Metaphor: The essence of metaphor is understanding and experiencing one kind of thing in terms of another. (Lakoff & Johnson 1980, 104) Simple examples are “What a sweet baby!” “Your car is a lemon, let’s take mine.” Obviously babies are not candies, nor are vehicles some sort of fruit. We use this sort of “identification” in order to communicate our ideas/feelings in a more striking way. Here are more examples of metaphors: “he is the apple of my eye,” “it’s raining cats and dogs,” “he has a golden heart,” “she cried rivers” etc. As these examples show, the distinguishing characteristic of metaphors is substitution, for example, rivers taking the place of tears in the last example. And this substitution takes place between different domains. If we say “a nightingale is a bird,” that’s not a metaphor; it’s a literal categorization, as both nightingale and bird signify the same domain (Gentner 2005, 200).

As long as the correspondence is not purely attributional, but some relations are also preserved we can also talk about structural alignment for metaphors (Gentner 88, 49). From this perspective, many metaphors are in fact analogies, but it’s the form of the language that helps us differentiate them. For example, “the engine is the heart of an automobile” is a metaphor, but it uses the same structural alignment of the analogical comparison that we mentioned above.

Nonetheless, not every metaphor is of this sort. Consider, for example, the verse (48:10) “God’s hand is over their hands” from the Qur’an; this was revealed in connection to some 1,400 believers’ pledging allegiance to Prophet Muhammad, peace be upon him, under a tree in Hudaybiyah in year 628 by giving their hands to him. Whatever “God’s hand” in this verse refers to, whether it is to His power, His victory, His protection and blessings for the believers, or His acceptance of their pledge, it is far from being a physical hand. Therefore, we can not imagine a relation between dissimilar objects (Gentner 1982, 109; Gentner 2001, 204) that is mapped to the relation (Hand, God) under a structural alignment using SMT.

Literal similarity: If there is a considerable number of common attributes that are shared by the objects of the domain and the target then the comparison is more likely to be a literal similarity (Gentner 1982, 110). For example the comparison in the following verse of the Qur’an is a literal similarity: “Indeed, the example of Jesus to God is like that of Adam. He created him from dust; then He said to him, “Be,” and he was” (3:59). In this example there are a great number of attributes that Jesus and Adam (peace be upon them) share but only one of them is the subject of the comparison here, which is that both Jesus and Adam had no father. If Jesus also had no mother this would be an “identity,” not a similarity (Gentner 1982, 110).

The statement “Jesus is like Adam” makes much more sense than “Jesus is Adam,” even if we make it clear in what sense we use this identification. Typically metaphors use the word “is” and literal similarity comparisons (similes) use either of the words “like” or “as.” Mathematically speaking, if metaphors are equalities, then similes are approximations (Casnig).

Even though metaphors are substitutions, they are not necessarily “two-way” identifications. In other words, most of the times they are asymmetric; i.e., there is a sense of direction in the “identification.” For example, we never substitute a car in the place of a lemon and say “this lemon is a car.” This is because the first and foremost requirement of any comparison is that it be informative. What information do we obtain from the statement “rose is love” or “iron is fist”? In that respect many metaphors are irreversible, like similes: “a butcher like a surgeon” is quite different than “a surgeon like a butcher.” One is a compliment but the other is not! (Gentner 2001, 224). Perhaps irreversibility can also be taken as a distinguishing character of (at least attributional) metaphors.

Mathematical and metaphysical examples: Although there is no consensus on how mathematical ideas come about, it is certain that they are introduced into our world of knowledge by symbolism. Whether we associate a symbol to something physical, such as the symbol 70 to the weight of an object or the letter g to gravitational force, or to an abstract mathematical concept, such as the letter i to √–1, we are substituting one thing to mean another thing. We even associate symbols to concepts we cannot even describe, such as the concept of infinity and the symbol ∝ that is used to represent what it signifies in mathematics. From this perspective, mathematical symbols can be viewed as metaphors (Pimm, David 1981).

On the other hand mathematical formulas preserve the relations among the objects or concepts that they represent. In that respect, it is possible to see them as analogies as well. Therefore, they can be subjects of structural alignment. In the next example we will use SMT to form an analogical comparison between infinity and human life in order to better understand a Qur’anic verse.

In verse (5:32) of the Qur’an reads: “…he who kills a soul unless it be (punishment) for murder or for causing disorder and corruption on the earth will be as if he had killed all humankind; and he who saves a life will be as if he had saved the lives of all humankind…”

In a sense this verse means “one equals many.” How can that be? How can one be equal to a million? Or a billion? Those who know something about mathematics with infinities will remember that it is a quite different thing from mathematics with ordinary numbers. For example, if we add two infinities we still get infinity, i.e., ∝+∝=.∝=∝, something that never happens with any finite number, except for 0 (0+0=2.0=0). Likewise, if we add any number of infinities together we still get infinity: ∝+∝+∝+…∝=∝.Therefore, for any positive number m if we write m.∝=∝ it would be a perfectly acceptable mathematical statement.

Now, everyone will agree that there is no value one can associate to human life. Therefore, it’s fair to say that the value of human life is infinite. Let’s map “∝” to “saving the life of one person” under a structural alignment and let counting people in the target correspond to adding infinities in the domain. Now, if m represents the total number of lives on earth then what element should we associate to saving them altogether? The answer is m infinities added together, in other words m.∝ But seeing that m.∝ is equal to ∝ we can say that saving the lives of all people on earth is no different than saving the life of one individual! That’s how one can equal many. Now, if we map “killing an innocent person” to “–∝” then we clearly see that the mathematical statement m.(-∝)=-∝translates into “killing all innocent people on earth is as grave a sin as killing one.”

Note that this “one equals many” situation is not a violation of the requirement that structure mapping be one-to-one (Gentner-Markman 1997, 47). It’s a relation that occurs between the objects of the domain and the objects of the target and that relation is preserved under a one-to-one structure mapping.

Another relation between infinities is that “∝-∝” is indeterminate. In particular “∝-∝” is not necessarily 0. Using the analogy we just constructed we can translate this as “killing one innocent person and saving the life of another does not balance out.” Then, “∝-∝” is indeterminate” translates as “we cannot know if God will forgive that person for saving a life or punish him/her for taking one; it depends on which ∝ is greater!” Indeed, those who have taken calculus will remember that “∝-∝” sometimes turns out to be a positive number and sometimes a negative number; as well as 0 or ∝ or -∝

One could ask, what about m.0=0? Isn’t this also true? Yes, indeed “the sum of m zeros is equal to zero” is another mathematically accurate statement. Then, what meaning could this equation be given? Perhaps this would be the “murderer’s” version of the Qur’anic principle that we mentioned above: “When the life of an individual has “no value” in your heart, killing one person or a million people should be equally disheartening (!)”

Another mathematical tool that would be helpful in interpreting the above-mentioned principle is a technique that is used in mathematical proofs. When proving a fact about a set of elements in mathematics one proves it for an arbitrarily chosen member of the set and it is automatically generalized to the rest of the elements in the set. For example, in order to prove the statement “the square root of every prime number is irrational” we choose a prime number, say p, arbitrarily and prove the statement for it. Once we do that it is as if we proved that √2 is irrational, √3 is irrational, √5 is irrational etc. It covers all the numbers in the form √p, where p is a prime number. In the same way, when one kills an innocent person the message that it is acceptable to kill “any” innocent person is given, as the choice of person has been made arbitrarily. Therefore, this can be likened to killing all of mankind because being able to kill one person is simply generalized to all people who could be in that person’s shoes.

Summary: In this article we have given examples of analogy, metaphor, and literal similarity and have tried to indicate some distinguishing characters that set them apart. We included some uncommon metaphysical phenomena which have been placed in analogical correspondence with mathematical quantities with the intention of showing that mathematics can be used to shed light on purely religious concepts as well.

Yusuf Ziya Gurtas is an assistant professor of mathematics in Queensborough Community College, CUNY.

References

• Casnig, John D. 1997–2009. A Language of Metaphors. Knowgramming.com. Kingston, Ontario, Canada.
• Gentner, Dedre. 1982. “Are scientific analogies metaphors?” in David S. Miall (Ed.), Metaphor, Problems and Perspectives, Brighton, England: Harvester Press, pp. 106–132.
• –. 1983. “Structure-Mapping: A Theoretical Framework for Analogy.” Cognitive Science, 7, pp. 155–170.
• –. 1988. “Metaphor as Structure Mapping: The Relational Shift.” Child Development, 59, 47–59.
• Gentner, Dedre & Markman, Arthur D. 1997. “Structure Mapping in Analogy and Similarity.” American Psychologist, 52, 45–56.
• Gentner, Dedre, Bowdle, B., Wolff, P., & Boronat, C. 2001. “Metaphor is like analogy” in D. Gentner, K. J. Holyoak, & B. Kokinov (Eds.), The Analogical Mind: Perspectives from Cognitive Science, Cambridge, MA: MIT Press, pp. 199–253.
• Gentner, Dedre & Bowdle, Brian F. 2005. “The Career of Metaphor.” Psychological Review, 112, pp. 193–216.
• Lakoff, G., & Johnson, M. 1980. Metaphors We Live By. Chicago: University of Chicago Press.
• Pimm, David. 1981. “Metaphor and Analogy in Mathematics.” For the Learning of Mathematics, 1, 47–50.
• Runde, Volker. 2000. “The Banach-Tarski Paradox or What Mathematics and Miracles Have in Common.” in the Sky, 2, 13–15.
• Szabo, A. 1978. The Beginnings of Greek Mathematics. Dordrecht, Holland: Reidel.

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.

First, we should emphasize that Prof. Zewail’s statements resonate very well with what famous scholar Ibn-al Haytham said 10 centuries ago:

“The seeker after truth is not one who studies the writings of the ancients and, following his natural disposition, puts his trust in them, but rather the one who suspects his faith in them and questions what he gathers from them, the one who submits to argument and demonstration, and not the sayings of a human being whose nature is fraught with all kinds of imperfection and deficiency. Thus the job of the man who investigates the writings of scientists, if learning the truth is his goal, is to make himself an enemy of all that he reads, and applying his mind to the core and margins of its content, attack it from every side. He should also suspect himself as he performs his critical examination of it, so that he may avoid falling into either prejudice or leniency.”

Every human being is given the ability to think. That is what makes all the progress of humanity possible. When we think we process the information we receive based on what we already know and we usually make judgments. Nobody can deny that all humans go through this process many times every day. But we can question the outcome of our thinking process, that is, our judgments. Many times, we can be wrong in our judgments, and this is where “critical” thinking might help.

Today, most people learn how to think systematically at schools, particularly during university education. In fact, according to Derek Bok, the former president of Harvard University, faculty members in the US almost unanimously agree that “teaching critical thinking” is the principal aim for undergraduate education. Although many have great conviction that critical thinking is very important, there is no universal definition for it. The American Philosophical Association gives the following definition:

“Purposeful, self-regulatory judgment which results in interpretation, analysis, evaluation and inference as well as explanation of the evidential, conceptual and methodological considerations on which a judgment is based.”

Based on this definition and what we quoted from Ibn-al Haytham, we can identify the following points about critical thinking:

1) When you read or hear a statement, do not accept it without question.

2) Try to find other information that can support or disprove the statement, try to come up with ways to demonstrate validity or falsehood of the statement, and test the statement.

3) As you perform your critical examination, be aware of your own deficiencies as a human. Consider your own biases, numerous possibilities of making mistakes in your judgments, or prejudices you can assume without knowing.

4) Be aware of the method you use for evaluating a claim, and think about both its strengths and weaknesses.

Critical thinking, if we develop the habit of performing it, can help us every day. Consider how you receive information in the twenty-first century. Typically, you read a newspaper or watch news on TV. If you just accept what you hear and see, then you can sometimes be manipulated or fooled. In an ideal world we can assume that objective presentation of news is possible. In the real world, however, those who transfer news bits to the population are also humans who are prone to similar biases. Many times a newspaper or a TV station represents certain values, ideologies, or supports some political positions. Besides, news reporting is so much commercialized that news editors and reporters also think about how to attract readers, and might use catchy words deliberately. Therefore, everyone should ask questions, clarify certain interpretations, and need to fight their own biases before reaching conclusions whenever they encounter a media report. An increased number of critical thinkers will help create a healthy debate about controversial issues.

We can expand this critical approach with respect to news media to our formal education experiences. At schools, our teachers deliver new information every day. We also read our textbooks. Typical student tendency might be to just accept what they hear and what they read, though this may depend on the student’s intellectual development and the culture of the society. If we perform critical thinking in the classroom and while reading textbooks, we can grow intellectually much more than otherwise. We can retain what we learn more easily. In addition, the purpose of schools and universities is to prepare us for life. We all know that in real life problems are never well defined, and most of the time you cannot readily find an authoritative figure or a manual to consult. Therefore, it is imperative that we teach students to question what they hear and what they read. In order to encourage students to perform more critical thinking in the classroom, however, we might need to find alternative modes of teaching and move away from conventional method of active teacher – passive students.

Although we must strive to teach critical thinking, we should also recognize the fact that there are roughly three distinct phases in our lives with respect to thinking, and not everyone is in the same phase at a given time. When we are young we usually think that every question has a certain answer, and it is just a matter of finding the authority who can give that answer. As we grow up, however, we realize that sometimes many authoritative figures may disagree. This observation may make one a relativist; that is, we might start thinking that depending on who you ask the answer varies, and it is all relative. As we further grow in our intellectual development, we usually notice that although people disagree on certain topics, we should always make some decisions, and some points of view have more support than others. By performing critical thinking we can filter out less reliable information, and come to better conclusions. In some complex cases, however, we might realize that there is no single correct judgment, and circumstances might lead us down different paths.

In the context of these three different phases, we can safely claim that on the average you expect the ability to think critically improve, as we get older. Elementary school children are most likely to think that there is a correct answer to every question. In middle and high school, many students will realize that there may be different answers to the same question depending on whom you ask. By the time of their graduation we expect university students to learn how to critically evaluate information. This final outcome, though very significant for a healthy society, highly depends on our teaching strategies. That is, if rote memorization is more valued than critical evaluation, then university graduates might stay as naive relativists or even worse.

Critical thinking is extremely crucial to separate the truth from myth. A well known example is about our brains. In many societies, it is said that people typically use about 10% of their brain, and geniuses like Einstein are able to use more of their brains. Some questions to ask are as follows:

1) What does it mean to use 10% of the brain?

2) Who provides this information? Is s/he a neuroscientist?

3) How do people measure brain usage?

4) Einstein lived long time ago. Did they measure his brain activity the same way as the control group to conclude that he used more of his brain?

When we search for answers to these questions, we typically find that there is no basis for this claim. It does not mean much. Maybe the statement originally meant that many do not use their brain’s full potential, which might be true. But, when the statement is compared to what we know about brain science, then it becomes a myth.

While everyone and every profession benefits from critical thinking, some jobs require critical thinking more than others. Scientists, for example, have an obligation to be extremely critical of others’ work. Let us remember that Ahmad Zewail and Ibn-al Haytham are both scientists. In science, there is a process called “peer review,” which is extremely useful. Any research carried out becomes scientific knowledge after it is submitted to the critical reviews of other scientists in the field. Any contribution submitted to a scientific journal is tested for correctness and originality. At the end of the critical review, the reviewers submit a report to the editors of the journal, who then make a decision. Many times the authors of the article are asked questions or are asked to make modifications, to which they may respond. In short, we can safely state that a scientist must be a critical thinker, or otherwise s/he cannot perform his or her research duties, and s/he is prone to become a repeater of someone else’s ideas, and may go awry at times.

From what we explained so far, it is probably obvious that we need some background knowledge about a subject in order to become a real critical thinker. It is clear that if someone knows nothing about urban planning, for example, it is very hard for him to critically examine any work on urban planning. This straightforward observation suggests that most people cannot be critical thinkers even if they want to be. Fortunately, this conclusion is only partially true because critical thinking is not a single discrete outcome but rather a continuum. Although knowledge of a topic makes us better thinkers in a field, there are always some minimal questions one can ask. For example, there is a controversial topic and someone makes some crucial statements about it. Even if you don’t know about the topic, you can ask whether the speaker is an expert on the topic or not. Or you can ask about what kind of evidence he is using. Does the evidence presented come from a reliable source? Why may this person take this side of the argument, but not other alternatives? Are there any other alternative approaches to the same issue? That is, at minimum, the answers to these questions can help us develop sound judgment.

A great case in which many feel lost is the global warming debate. By doing a simple web search on global warming you can find articles both supporting and denying human-caused global warming. But which point of view should you believe? Although there is consensus that global warming is happening, there is disagreement about whether it is caused by our technological conveniences, or whether it is just a result of natural temperature fluctuations in geological time scales (hundreds of thousands of years).

Indeed, the question “which point of view should you believe?” may not be the correct question. Instead, perhaps we should ask why people feverishly debate about this issue. Why do people care a lot about various aspects of global warming and atmospheric science research? A little investigation shows that there is a lot at stake. Accepting that excessive use of energy by humans, which leads to increase in carbon dioxide levels in the atmosphere, causes global warming has many economic, political, and social consequences. If you agree that man-made global warming can trigger many other events which can eventually make the climate on the Earth fall off-balance, then it follows that many people in the world must change their lifestyles, and this may also mean that some big companies must make changes which may hurt their profits, or that some governments might not use certain political leverage in international relations.

Obviously, it will not be easy to accept such findings, even if they come from some prominent scientists. There are, in fact, scientists on both sides of the aisle. This is because atmospheric science is extremely complex, and even with our best computers modeling the changes in the atmosphere it remains a very hard task. Further investigation also shows that there are different types of evidence people use. Some point to reports prepared with sponsorship from some governments or companies. Some expand on disagreements among scientists, and conclude that no action is needed until all the claims are proven. As we mention above, there are also scientific articles published in “peer-reviewed” journals. The Intergovernmental Panel on Climate Change (IPCC) puts out consensus statements. In short, it is a complex issue, and the credibility of evidence varies. We leave further investigation on climate change to practice critical thinking to the interested reader.

Critical thinking requires effort, sometimes a lot of effort. Lazy personalities and those who do not like to engage their mental abilities are less likely to perform critical thinking. In this regard, critical thinking is also very different from rote criticism. Rote criticism does not require much effort; typically the statements of the person performing the critique do not have a sound basis. While critical thinking is necessary for a healthy society to clarify especially certain controversial topics, mere criticism of one another can be a means to create unnecessary enmity among individuals.

Critical thinking does not mean that we need to become a skeptic and reject everything. In fact, descriptions of critical thinking involve paying special attention to certain criteria and standards. In practice, if one does not have any reference it becomes very hard to move forward in thought. What needs to be emphasized is that the world is not black and white. It displays a vast variety of colors. We can give more weight to certain colors than to others. What we cannot claim is that there is a single color, and all others are false. We can certainly have certain beliefs, paradigms, and values. These may become the basis for some of the criteria and standards against which we evaluate new information. At times, great events and findings might force us to evaluate certain paradigms we take for granted, too. However, such sharp turns do not happen frequently in the life of an individual or society.

In conclusion, critical thinking skills are essential for everybody. We should be aware of our own shortcomings, biases, prejudgments, as well as the sources, agendas, worldviews, paradigms and the information channels we use when we process new information. Critical thinking cannot be done instantly; it requires effort and courage to come face-to-face with one’s own errors. Teaching critical thinking is not instant, either. Formal and informal education institutions must strive to develop critical thinking skills in students, and such an endeavor might require modification of our approach to teaching. In the final evaluation, it is extremely crucial that we take the necessary steps to increase the number of critical thinkers in the world for the benefit of all humankind.

Dr. Ertan Salik is an Assistant Prof. of Physics at California State Polytechnic Univ, Pomona. As well as teaching and conducting physics research Dr. Salik is currently involved in many education programs.

Notes

1. The Fountain magazine, interview with Ahmad Zewail by Nuh Gedik, Jan-Feb 2009, issue 67.
2. Steffens, Bradley. Ibn-al Haytham: First Scientist, Morgan Reynolds Publishing. Also see Book Review in The Fountain Magazine, issue 63, May-June 2008.
3. Bok, Derek. Our Underachieving Colleges, Princeton Univ Press, 2006.
4. Critical Thinking: A Statement of Expert Consensus for Purposes of Educational Assessment and Instruction, American Philosophical Association Report, 1990.
5. IPCC consensus statements can be downloaded from http://www.ipcc.ch/.

Given the emphasis by the Qur’an on logical reasoning, it is understood that its audience (all of mankind) is expected to have a basic understanding of logical principles. It is our purpose in this article to go over a few basic principles of logic and give examples of how they are used in the Qur’an.

Logical principle 1: Proof by contradiction. An important aspect of logic concerns methods of proving or disproving statements. This is one of the basic principles of proving statements, and it is commonly used to prove mathematical statements. To show that a statement is true using this principle, suppose its negation is true, that is, that the statement itself is false. If you can then obtain a contradiction, something that is clearly false, this shows that the original statement is true. We shall illustrate this principle with an example. The reason we choose this example is that it is a classical result in mathematics, and the argument is beautiful. According to Hardy, this is an example of a first-class theorem about a significant mathematical fact (“a real mathematical theorem”) that is accessible to a general, educated audience (Hardy, 1967). There is a fundamental theorem in number theory which says that “there are infinitely many prime numbers.”1 There are many other proofs of this theorem but the most elementary and elegant proof is given by Euclid and it is by contradiction. Hardy claims that “both the statement and the proof can be mastered in an hour by an intelligent reader, however slender his mathematical equipment” (Hardy, 1967). Here is that proof: Suppose there is only a finite number of primes. Call them p1 , p2 , … pn. Now, consider the number N = p1 , p2 , … pn +1. Clearly, N is not equal to any of the primes in the list. Therefore, N itself is not a prime, and hence N must be divisible by one of the primes p1 , p2 , … pn (this is by “the fundamental theorem of arithmetic” which states that every integer greater than 1 can be written as a product of prime numbers). However, that is not the case: when N is divided by each one of pi there is a remainder of 1. This is a contradiction, and it arises from the assumption that there are only finitely many primes. Hence, the assumption cannot be true, and we conclude that there are infinitely many primes.

Proof by contradiction, or reductio ad absurdum in Latin, is one of the mathematician’s finest tools. There are many examples of theorems in mathematics where the simplest proof is by reductio ad absurdum. Another classical example is to show that √ is not a rational number (Hardy, 1967).

Now, we would like to give examples from the Qur’an where this logical principle is used. Consider the verse: “Had there been in the heavens and the earth any deities other than God, both would certainly fallen into ruin” (Anbiya 21:22). Here the argument is: suppose there were gods other than God. Then the earth and the heaven could not maintain their order (because the powers and wills of different gods would interfere with each other). Since this is clearly not the case (because we see perfect order in the universe), this contradicts the logical consequence of the initial assumption. Therefore, the initial assumption (i.e., there are more than one god) is wrong.

For another example, let us consider the verse: “Do they not contemplate the Qur’an? Had it been from any other than God, they would surely have found in it much inconsistency” (Nisa 4:82). Here the statement to be proven is “The Qur’an is the word of God, and it is from God.” To prove this statement by reductio ad absurdum, suppose for a moment the opposite statement is true, that is, that the Qur’an is not from God. A logical consequence of this assumption would be it is the word of a human being, but then there would be inconsistencies in it because the Qur’an talks about such great matters that if a human being had written it he would have made mistakes and inconsistent statements (for more on this point see Unal, 2007, page 228, and Nursi). However, we see no such statements in the Qur’an, thus contradicting the initial assumption. Therefore the Qur’an is the word of God.

The final example we would like to give related to this principle is expressed in the verse: “If you are in doubt about the Divine authorship of what We have sent down our servant (Muhammad) then produce just a sura (chapter) like it and call for help to all your supporters” (Baqara 2:23). Here the statement to be proven is the same as the one in the previous example. The argument goes like this: If it was possible for anybody other than God to produce the Qur’an, you should be able to make something like it with all of your supporters, if not an entire book, just a chapter of it. The Qur’an presents disbelievers with a challenge. However, they have not been able to meet the challenge. They chose to fight with the sword because fighting with words was not possible. Therefore the assumption that “the Qur’an could have been written by a human” is wrong.

Logical principle 2: The next principle we would like to discuss is the equivalence of an implication to its contrapositive. A statement of the form “If P, then Q” is called an implication. In such an implication P is called the premise, or the hypothesis and Q is called the conclusion. Such a statement asserts that whenever P is true (or satisfied) then Q is guaranteed to be true. However, if/when P is not true, it does not say anything about the truth value of Q. An example of an implication from daily life is: “If it is rainy, then it is cloudy.” Here the premise is “it is rainy” and the conclusion is “it is cloudy.” Another example would be: “If you are in New York City, then you are in the US.” A mathematical example is: “if a number n is divisible by 6, then it is even.” An implication can also be read as “P implies Q,” or written “P => Q.” The contrapositive of an implication “if P then Q” is “if not Q then not P.” For example, the contrapositive of the first implication above is “if it is not cloudy then it is not rainy,” of the second one, “if you are not in the US, then you are not in NY city,” and of the third one, “if n is odd (not even) then n is not divisible by 6.”

Here are two examples from the Qur’an where we can apply this principle. The first one is the verse we discussed above: “If you are in doubt about the Divine authorship of what We have sent down our servant (Muhammad) then produce just a sura like it and call for help to all your supporters.” We can rephrase this statement as “if the Qur’an is not the word of God, then humans can produce a like of it,” and its contrapositive is “if humans cannot produce something similar to the Qur’an, then it is the word of God.” Remember that these two statements are logically equivalent. Now, we know that humans cannot produce anything comparable to the Qur’an despite much effort and motivation, therefore it follows that it is the word of God.

The second example we want to consider is the verse: “Say (to them, O Messenger) if you indeed love God, then follow me, so that God will love you and forgive your sins” (Al Imran 3:31). The contrapositive of this statement can be stated as (slightly simplified) “if you are not following his messenger, then you really do not love God.” So, we obtain a clear measure of whether one really loves God or not. Following the Prophet is a necessary condition to attain the love of God.

Logical principle 3: The final principle we discuss is false implies anything. This is actually a special case of an implication of the form “P implies Q.” Recall that such a statement asserts that whenever P is true, Q is also true, so P promises Q. However, if P is not true then it does not claim anything about Q; it may or may not be true. For example, if it is not rainy then it may or may not be cloudy. An implication is false when P is true yet Q is false. Intuitively, we can think of this as breaking a promise. The question arises as to what the truth value of an implication should be when the premise P is false. It is accepted that the implication will be true in this case, because when the premise is false, the conclusion is totally irrelevant. This is called “false implies anything.” One can make any conclusion whatsoever based on a false premise, but it really does not mean anything. For example, the statements “if 2+2=5, then there are only finitely many primes,” and “if the world is flat, then there are no wars in the world” are both true statements. These statements are true but they really do not have any substance. They are true by a convention in logic. The important point one needs to pay attention to is this: Once you believe a premise that is not true, it is possible to make any conclusion you want and the whole argument looks right. However, such an argument is quite meaningless. So one needs to examine very carefully what is being assumed in a logical argument.

One example we want to consider from the Qur’an related to this principle is the verse: “Those who deny Our Revelations and turn arrogantly from them-for them, the gates of Heaven will not be opened and they will not enter Paradise unless a camel can pass through the eye of a needle” (A’raf 7:40). With a little simplification, we can rephrase this verse as “if a camel passes through the eye of a needle, then disbelievers will enter Paradise.” Since the premise is false (obviously, a camel cannot pass through the eye of a needle), this implication is true. However, this is not good news for disbelievers because the truth of this implication has nothing to do with the truth of its conclusion. This again shows the point: One needs to examine logical arguments carefully. What looks true or convincing formally and on the surface may not mean anything in reality if it is based on a false premise.

Nuh Aydin is an associate professor of Mathematics at Kenyon College, in Ohio, USA.

Notes

1. A positive integer n that is greater than 1 is called a prime if its only positive divisors are 1 and n. 1 is not considered to be a prime. Thus the sequence of prime numbers is 2, 3, 5, 7, 11, 13, 17, 19, 23, 29…

References

• Hardy, G. H. A Mathematician’s Apology, Cambridge University Press, 1967.
• Nursi, Bediuzzaman Said. The Words (translation), The Twenty-Fifth Word, Available online at http://en.nurglobal.net/modules/mastop_publish/?tac=The_Twenty-Fifth_Word
• Unal, Ali. The Qur’an with Annotated Interpretation in Modern English, The Light Inc., 2007.

Wolf (1999) notes that a nonprofit organization is neither in the profit sector nor in the public sector but sits somewhere between the two. This position allows the nonprofit great flexibility in its operation but requires great skill in its management. Nonprofit organizations have characteristics of profit and public organizations. Wolf defines nonprofit organizations as “those legally constituted, nongovernmental entities incorporated under state law as charitable or not-for-profit corporations that have been set up to serve some public purpose and are tax exempt.” All nonprofit organizations must have the following characteristics:

• They must have a public service mission-their main goal must be to serve others, not themselves.

• They must be organized as a not-for-profit or charitable corporation-they must formally claim to be non-profit.

• Their governance structure must preclude self interest and private financial gain-this goes back to the first characteristic in that the organization’s main goal is to serve people.

• They must be exempt from paying federal tax.

• They must possess the special legal status that stipulates gifts made to them are tax-deductible.

It is obvious that the nonprofit sector has expanded dramatically over the past few years, so has public scrutiny over the sector. Governance is needed to ensure that the organization is providing services that are permitted within the legal system and the bylaws and to ensure that the organization does not permit fraud, potential abuse, and violations of the expectations of the public. Governance of a nonprofit can be defined as the ultimate authority, accountability, and responsibility for an organization (Ott 2001). Leadership, such as the board, is important in the nonprofit sector because it is governed through a complex set of functional roles and procedures that are defined in laws and tax codes that differ from those for the public and for-profit sector. The board is responsible for policies, procedures, financial outcomes, and ensuring that the mission of the organization is being implemented and accomplished to the best of the organization’s ability to serve its target population or general public.

There are some challenges that nonprofit organizations face in our modern societies. The first major challenge for a nonprofit organization is articulating a clear public service mission. A nonprofit organization is required by law to limit its activities to those articulated in the mission statement, so the mission statement must be broad enough to cover all possible activities. The mission statement must also guide the direction of the organization’s programs, services, and activities. In addition, a mission statement may require review and revision over time. For example, an arts council can begin with a mission to develop, foster, and promote the arts, and then develop a stronger mission statement that provides detailed plans for programs and activities, such as working with the school district to develop art education programs, and seeking funds from state, federal, and private sources.

The second major challenge is risk/survival analysis: the search for the proper balance between organizational extension and risk-taking, and organizational security. Engaging in risk/survival analysis means having an understanding of what should and should never be compromised, always being cognizant of the mission and whether or not daily activities or decisions are in harmony with that mission. This challenge identifies a great dilemma for nonprofit organizations: whether it is more important to ensure survival of the organization or to stay true to the mission. There is no easy answer. To meet this challenge, the organization must continually plan, analyze future options (both practical and idealistic), and debate the pros and cons of each option.

The third major challenge facing a nonprofit organization is constituency identification and involvement. Identifying and involving the constituency entails having a working knowledge of the clientele you desire to serve. This must not be too narrow or too broad. The nonprofit organization must establish a concise identification of its constituency in order to avoid problems in this area. The organization must identify the group that it seeks to serve, and then must create a structure that supports a commitment to that group. To meet this challenge, the organization must have a well-defined understanding of its constituency, seek to involve the constituency at all levels of the organization (particularly at the trustee level), and provide programs and activities that reflect a commitment to the constituency.

The fourth major challenge is testing for organized abandonment: that is, disbanding that is the result of careful deliberation, rather than unplanned events, such as financial crisis or management problems. In order to meet this challenge, the organization must have a clearly defined mission, establish criteria for evaluation of the mission, and have a formal system to determine whether or not the mission is relevant and effective. The organization must examine whether or not its mission is obsolete or if other organizations are more effective in achieving its mission.

In conclusion, nonprofit organizations go where other organizations fear to tread. Nonprofits eagerly work with the most disadvantaged members of society-the homeless, the hungry, the ill, and the befuddled. Nonprofits fill the gaps in society-consider the thousands of charter schools that have sprung up to provide a genuine alternative, most frequently for students from disadvantaged backgrounds. Nonprofits extend and even function as government service-providers: how would most governmental social service agencies function if not by contracting out to tens of thousands of nonprofit service providers. Think too of the thousands of neighborhood associations that knit together their communities, and sometimes act indistinguishably from chartered governments. Nonprofits help people find meaning-both through quiet worship and organizing millions of people to do “good deeds.” People who want to “do something, to give back to society” sometimes do so through governments, but most do so through nonprofits. Millions of volunteers contribute and share in the nonprofit sector. Finally, unlike the other sectors, the nonprofit sector does not start out by asking how much money can be made, or where the money is going to come from; no, the nonprofit sector asks what the need is and how to fulfill the need. Only then does it seek strategies to support the social needs it discovers. Although by nature humble, it is the nonprofit sector that makes our complex and frequently materialistic and even selfish society, humane, purposeful, and cohesive. The private sector may provide the engine of growth and a quality of life; the public sector may organize us and keep the private sector in line; but it is the nonprofit sector that is our heart, our soul, and our spirit as a society.

References

• Ott, J. Steven (ed.), Understanding Nonprofit Organizations: Governance, Leadership, and Management, Westview Press, Boulder, CO: 2001.
• Wolf, Thomas, Managing a Nonprofit Organization in the Twenty-First Century, Simon & Schuster, New York: 1999.

The Qur’an is not a book of sciences. But since sciences deal with nature and man and since sciences and technology constitute a very important aspect of man’s life and are themselves the product of man’s mind, the Qur’an, which contains ‘whatever is wet and dry’ either explicitly or implicitly or by allusion, certainly refers to sciences and scientific advancements. But, while sciences deal with nature and things for their own sake and concentrate on the question of ‘how?’, the Qur’an refers to them for the sake of God and for their most, fundamental purpose as signs of Divine Existence and Unity as the manifestations of Divine Names, and therefore as the means of obtaining knowledge of God. Second, the Qur’an seeks to guide people and inculcate in them belief and high standards of morality. The great majority of people do not have specialized knowledge about scientific facts or theories. It would be inappropriate for a book of guidance directed to all people in all ages to refer to things and natural events in the manner of sciences. If the Qur’an referred to, say, the sun as a heavenly body of such and such size, made up of gases composed of two thousand billion times billion tonnes of matter, with the remains of other elements and in which for every million atoms of hydrogen there are about 85,000 helium atoms, most people would be simply bewildered or indifferent. As the comprehensive and conclusive Revelation, the Qur’an addresses all levels of understanding and intends to be understood, with belief and action to follow understanding. Since most people judge according to their sense-impressions, the Qur’an uses the appropriate language and style. For example, while narrating the story of Dhu’l-Qarnayn, the Qur’an says that he reached the setting-place of the sun and found the sun setting in a fiery muddy spring (18.86). It is obvious that the sun does not set in a spring. But this verse, besides giving many clues to certain facts to be discovered later, considers ordinary sense-impressions. First of all, we understand from the verse that Dhu’l-Qarnayn went as far as the western end of a land adjoining water around which there was not another visible land. That is why most commentators of the Qur’an have concluded that it was the Atlantic Ocean. Second, the verse implicitly states that Dhu’l-Qarnayn did not reach the coasts of the land he conquered in the west but advanced only so far as the point from which he could see the ocean like a spring. Thirdly, when he reached that point, it was a fiery summer day and, most probably because of the vapours rising from the ocean and the marshy land adjoining the sea, it appeared from afar like a muddy spring. Fourthly, the verse contains a subtle and important point. The word translated here as spring also means eye and the sun. As the Qur’an, because of its elevated perspective, looks at the world from ‘on high’ and also there are innumerable eyes watching the world from on high, the ocean from that perspective, however large it may appear to the people in this world, appears no bigger than a spring. Further, there is a subtle allusion here to a time when those who believe in God will gain enough power and equipment to rule, at least, a considerable part of the world and, ascending the heavens, observe the world from on high.

The statement just discussed comprises only five words. All the statements of the Qur’an contain lots of information either explicitly or implicitly and allusively to satisfy all levels of understanding in all times until the Judgement Day. Another example is a statement of only four words: The sun moves (in its course) to a resting-place for it (36.38). Before elucidating other meanings and connotations of this statement, we should remember that in the past people, judging again by their sense-impressions, believed that the earth was motionless while the sun moved around it. Later developments in science and observations showed that the earth spins upon its own axis and orbits the sun which is, relatively, motionless. First of all, since people see the sun moving, the Qur’an mentions it as moving. Second, the Qur’an mentions the sun here as an illustration of the magnificent order prevailing throughout the universe as a sign of God’s Might and Knowledge, The context is as follows:

A sign for them is the night. We strip it of the day, and behold! they are in darkness. And the sun moves (in its course) to a resting-place for it. That is the measuring and ordaining of the All-Mighty, the All-Knowing. And for the moon We have appointed mansions till it returns like an old shrivelled palm-leaf. It is not for the sun to overtake the moon, nor does the night outstrip the day. They float, each in an orbit. (36.37-40)

We understand from the statement in its context that the sun has a vital function in the universal order. The Qur’an uses in the statement mustaqarr, meaning a fixed course to follow, stability and the place in which stability is secured. So, the statement can mean that the sun has a central position in the order of the universe. Second, the preposition used together with the word ‘stability1, li, has three meanings: for, to, and in. Therefore, the exact meaning of the statement comprising four words is: The sun moves following a route or course to a fixed place determined for it for the purpose of its (system’s) stability.

In recent decades, solar astronomers have been able to observe that the sun is not in fact motionless. It quivers and shakes and continually rings like a well-hit gong. These vibrations of the sun reveal vital information about the sun’s deep interior, its hidden layers, information which affects calculations of the age of the universe. Also, knowing exactly how the sun spins internally is important in testing Einstein’s theory of general relativity. Like so many other significant findings in astronomy, this discovery about the sun was totally unexpected. Having discovered the quivering and ringing sun, some astronomers have commented that it is as if the sun were a symphony orchestra, with all the instruments being played simultaneously. All the vibrations combine at times to produce a net oscillation on the solar surface that is thousands of times stronger than any individual vibration.

Commenting on the Qur’anic verse. The sun moves to a resting-place for it, several decades before this totally unexpected discovery in astronomy, Said Nursi had written:

As the word ‘moves’ points to a style, the phrase ‘in its course’ demonstrates a reality. The sun, like a vessel built of gold, travels and floats in the ocean of the heavens comprising ether and defined as a stretched and tightened wave. Although it quivers and shakes in its course or orbit, since people see it running, the Qur’an uses the word travel or float. However, since the origin of the force of gravity is movement, the sun moves and quivers in its orbit. Through this vibration, which is the wheel of its figurative movement, its satellites are attracted to it and preserved from falling and scattering. When a tree quivers, its fruits fall. But, when the sun quivers and shakes, its fruits-its satellites-are preserved from falling. Again, wisdom requires that the sun should move and travel on its mobile throne-its course or orbit-accompanied by its soldiers-its satellites. For the Divine Power has made everything moving and condemned nothing to absolute rest or motionlessness. Divine Mercy allows nothing to be condemned to inertia which is the cousin of death. So, the sun is free, it can travel provided it obeys the laws of God and does not disturb others’ freedom. So, it may actually be travelling, as its travelling may also be figurative. However, what is important according to the Qur’an is the universal order, the wheel of which is the sun and its movement. Through the sun, the stability and orderliness are ensured. [Muhakemat (Reasonings), Istanbul 1988, pp. 68-69.]

Recomended Reading

• BUCAILLE, M. (1987) The Bible, The Qur’an and Science, Taj Company, Delhi.
• JONES, B, (1992) The Night Sky, Salamander Books Ltd.
• NURBAKI, H. (1989) Verses from the Glorious Qur’an and the Facts of Science, T.D.V., Ankara.
• LEIBACHER, J., R. NOYES, J. TOOMRE & R. ULRICH (1985) ‘Helioseismology’, Scientific American (September), pp.48-57.
• BARTUSIAC, M, (1994) ‘Sounds of the Sun’, American Scientist (January-February), pp.61-68.