New developments in scientific thought and hypotheses testing are changing the ways we think about truth and certainty.

Not only do scientific decisions rely heavily on tools and procedures of quantitative analysis, but also on social life and daily decisions. I want to start with a story from the judiciary system about the misuse of an important quantitative tool of the scientific community, Hypothesis Testing. It’s about the overturned Amanda Knox case in Italy. In 2007, she was accused of killing her roommate and sent to jail. The odds of DNA matching in the case of accidental death were reported as incredibly rare by a forensic data analyst, who examined data with regard to incidents of crime, concluding that the accident is not statistically an accident. The judges then made their decision based on this rareness, a single value, named p-value. Later, this was overturned (New York Times, March 27, 2013), because of a misinterpretation by the judges and lawyers, of the rare probability. What factors forced the judiciary system to make this decision based on a single value alone? Honestly, this is the story of the century and a story of Hypothesis Testing. This article introduces the logic behind their decision, draws attention to the misuse of data, and mentions alternative logics to this traditional approach on the matter (in March 2013, Italy’s Supreme Court ordered a retrial and found the two suspects guilty on January 30, 2014).

What is Hypothesis Testing?

We live in a data-driven world. Statistical inference is the process of drawing conclusions or decisions from data. The Hypothesis Testing procedure in the Neyman-Pearson paradigm is one of the procedures of this process, widely used in the scientific community for over a century. In this mindset, two complementary hypotheses are first defined: one is the conventional thesis (the null hypothesis) accepted to be true by default; the other is the alternative thesis (the alternative hypothesis) that needs evidence from data to falsify the null hypothesis. It results in a single value, called p-value, which is calculated from the data using theoretical model assumptions. A p-value is a measure – in probability sense, ranging from 0% to 100% – of how much evidence you have against the null hypothesis. The smaller the p-value, the more evidence you have against the null hypothesis. One may combine the p-value with the significance level to make decisions on a given hypothesis. This is also called significance testing. It has an accept-reject mindset (dichotomous decision or binary logic); in such a case, if the p-value is less than some threshold (usually .05) then the null hypothesis is rejected. Basically, it is a black or white decision, without considering the contrasts between them. This interpretation has been widely accepted in the twentieth century of scientific research, and many scientific journals routinely publish papers using this interpretation for the results of hypothesis tests, even though there are current tendencies not to use this. Let’s take a look at the history of this black-white logic.

History

The history of the process of hypothesis testing starts with Fisher (1890-1962), around the early twentieth century. Later, the contributions of Pearson (1857-1936) and Neyman (1894-1981), who were early fathers of statistics, about the interpretation of the hypothesis tests were integrated into present-day applications. Fisher’s approach focuses on inductive inference about a single hypothesis, whereas the Neyman-Pearson approach informs future behavior based on a test using two complementary hypotheses (Newman 2007).

These approaches are strongly influenced by Popper’s logic of falsification. Falsification can be defined as the act of disproving a proposition, hypothesis, or theory. This logic asserts that sufficiently improbable events can be considered impossible. A p-value suggests whether a null hypothesis is sufficiently improbable to be considered practically falsified in the sense of a logical refutation (Newman 2007). When we come back to the Amanda Knox case, the decision the judges made and justified is followed from this falsification logic in hypothesis testing. Another criticism with this mindset is: how fair it is to rely on a single value that yields a rare result? Are we going to generalize one lucky drawing of raffle tickets to believe all tickets are winners?

Logics and proofs

We use logical flaws and biases in our daily life. Finding the logical flaw in scientific papers is something most readers aren’t interested in. Instead, the results are believed as a fact. One of the widely used flaws/biases is in seeking or interpreting evidence in ways that are partial to existing beliefs or a hypothesis in hand (Mercie and Sperber, 2011). We tend to prove or show the accessibility or superiority of arguments by bringing evidence. Failure to prove that a treatment – say, a low fat diet – is effective is not the same as proving it is ineffective.

Let me give another example to clarify this: The New York Times editorial news reported on February 9, 2006, “Millions of Americans have tried to reduce the fat in their diets, and the food industry has obligingly served up low-fat products. Yet now comes strong evidence that the war against all fats was mostly in vain.” Actually, in the hypothesis testing mindset, the evidence collected from research should either reject the null hypothesis which is “diet is ineffective,” or fail to reject it. The mindset in testing is not about finding evidence to support the null statement. It is also not about proving the alternative hypothesis, “diet is effective.” Rather, it is basically looking for evidences to falsify the null statement. In the report, the “evidence” the reporter meant should be against the null hypothesis, not against the alternative hypothesis. Data is collected to falsify the null hypothesis, not to prove its truthiness, because the null is already accepted to be true unless convincing evidence is collected against it.

Let me give another example of this paradigm. As Newfoundland (2013) described, a person is innocent until proven guilty by bringing convincing evidence. The jury can’t say “he is innocent,” instead, the jury uses evidence that produces reasonable doubt to reject his innocence.

Developments in inferential paradigms

The accept-reject paradigm in inference represents a conventional wisdom. There are other, or currently developing, paradigms in inference, too. One of the dominating paradigms is the Bayesian-likelihood approach. After enjoying much wider acceptance in social and natural sciences, the Bayesian method suggests different views of hypothesis testing. The Bayesian approach is a method of data analysis in which subjectivity, conditionality, or past information is used to update the hypothesis as additional evidence is acquired. This approach in hypothesis testing offers a dynamic structure in probability calculation so hypotheses, and decisions, are not seen as a static truth; instead, they are updated with current data and the decision is stated in the sense of likelihood.

In our court room example, the Bayesian inference is applied to all evidence presented, with the past information being combined with the current evidence. The benefit of a Bayesian approach is that it gives all historical information so the decision is unbiased all along as the past data (prior knowledge) is used correctly. This paradigm changes the way statistics are calculated and how the result and inference are interpreted. Currently it is widely appreciated in quantitative data analysis, especially after convenient software exists for its implementation. However, the criticism to this approach, made by many, is found in its subjectivity. Objectivity and handling prior knowledge is a concern here so that different people, having different opinions, may arrive at different results.

Another developing paradigm: Fuzzy Logic

Fuzzy logic is another method in quantitative decision making. In contrast to binary logic (yes-no, or accept-reject), fuzzy logic can be thought of as gray logic, which allows a way to express in-between data values. It emerged in the development of the theory of fuzzy sets, by Lotfi Zadeh (1965). Fuzzy logic is mostly seen as a branch of artificial intelligence that deals with reasoning algorithms used to emulate human thinking and decision making. It handles the concept of partial truth using linguistic variables, where the truth value may range between completely true and completely false. The concept of partial truth would be subjective and would depend on the observer. For example, for the temperature of the weather, we want to determine when to say cold, warm, and hot. The meaning of each of these concepts can be represented by a certain membership (called a fuzzy set). The concept of each would be subjective. One might consider cold for all values up to 40 0F. In Figure 1, the meanings of the expressions cold, warm, and hot are represented by functions mapping a temperature scale to “truth values” ranging from 0 to 1. A point on that scale has three truth degrees, one for each of the three functions (expressions). The vertical line in the figure represents a particular temperature that the truth values (see three arrows) are measured. Since the red arrow points to zero, this temperature may be interpreted as “not hot.” The orange arrow (pointing at 0.2) may describe it as “slightly warm,” and the blue arrow (pointing at 0.8) “fairly cold” (Fuzzy Logic, 2013, para. 6). A rule could then be adopted, like for example, if “slightly warm” then stop the fan.

Picture was obtained from

Fuzzy logic uses truth degrees as a mathematical model of the vagueness phenomenon, and it summarizes data analysis in facts with truth degrees, and leaves the decision to the observer. Its advantage is its ability to deal with vague situations with respect to linguistic variables. While the significance testing for a hypothesis declares one accept or reject, fuzzy logic allows for degrees of acceptance or rejection. However, in fuzzy logic, the notion of truth doesn’t fall by the wayside, but it is expressed in degrees and offers possibilities for different situations. Regarding inference, fuzzy logic uses the mindset ‘everything is a matter of degree and open to interpretation,’ and this mindset is also adopted to machine learning, which is considered a more suitable mindset with human reasoning instead of binary logic. The constraints in fuzzy logic are found in its tools and interpretations when complex inputs are considered.

The developments of paradigms in statistical inference have similar fates as in the developments of mathematics, geometry, and physics. According to the Euclidean parallel postulate, in space, there exist no two parallel lines that intersect each other. However, after Non-Euclidean geometry was developed in the nineteenth century, a wider geometrical and mathematical reasoning stemmed from it; accordingly, the geometry of the physical universe and particles came to be understood better. The development of Riemannian geometry, offering that distance properties might vary, resulted in the synthesis of diverse results concerning the geometry of higher dimensional surfaces and the behavior of geodesics on them. It also made Einstein’s theory of general relativity justifiable. Likewise, as time passes, we witness wider paradigms or logics that abandon or correct former approaches in scientific decision making tools. This is a good reason why teachers and professors should update their current teachings as to be consistent with convincing trends, as well as to train students to be ready for wider paradigms in the future.

Conclusion

In today’s scientific community, the way decisions are made and fallacies proved, are changing as new paradigms emerge. In order to make better decisions or to validate claims, many aspects and methodologies should be considered. One way to avoid mistakes as much as we can would be to expect fallacy points to exist in human mental processing during decision making, and improving and seeking better alternatives with well-established wisdoms.

(Thanks Ugur Sahin for reviewing the preliminary copy of this article.)

References

• Fuzzy Logic. In Wikipedia. Retrieved August 1, 2013, from http://en.wikipedia.org/wiki/Fuzzy_logic
• Kass, Robert E. 2011. Statistical Inference: The Big Picture 1. Statistical Science, 2011, Vol. 26, No. 1, 1-9, DOI: 10.1214/10-STS337, Institute of Mathematical Statistics.
• Mercier, Hugo, Dan Sperber. 2011. “Why do humans reason? Arguments for an argumentative theory.” Behavioral and Brain Sciences. 34, 57-111.
• Newfoundland, Jason. 2013. “The Cell Phone-Brain Cancer Controversy.” The Fountain Magazine, Jan-Feb, Issue 91.
• Newman, Michael C. 2008. “‘What exactly are you inferring?’ A closer look at hypothesis testing.” Environmental Toxicology and Chemistry, Vol. 27, No. 5, pp. 1013-1019, 2008.

Selflessness is when one relinquishes oneself from certain personal desires and aspirations, forgoing certain goals associated with property or wealth, and even values associated with one’s self-honor and dignity, all for the sake of lofty aims and noble goals. When a person is able to actualize such renouncements in their life, then that person can be considered to be a selfless person.

For example, when one, either through individual or collective efforts, relentlessly endeavors to exalt faith and spirituality, and in their pursuit builds centers and institutes for learning, establishes schools, or when dormitories and youth centers are built to cater to the needs and demands of the younger generations, when all these are enabled, either through financial support or through one’s own voluntary work, then such acts are considered to be sacrifices. Given that the intent behind is purely to serve an exalted aim that is greater than the self, then the only criteria is to make the doer of these acts a selfless person, or a person of sacrifice.

A closer examination will reveal that there is no contradiction between selflessness and reason. In other words, selflessness, or sacrifice, is not a mere emotional expression that is incongruent to reason. In fact, not only does reason justify selflessness, arguably it necessitates it. At an immediate and rather superficial glance, a contradiction between the two may be observed. Let us take, for example the numbers of selfless people who, while working tirelessly in educational institutes and campaigns, make great self sacrifices, although fully aware that none of the fruits of their genuine labor will show immediate results. Yet they devote their entire lives and health to these campaigns. It is as if God has bestowed on them special favors, for their work becomes far more efficient and productive then reason would allow. The cycle continues, as they in turn use these favors for furthering their efforts.

For those onlookers who may view the matter with little reflection, such attitudes can be considered to be apparent contradictions to reason. To selflessly strive for some cause and sacrifice thousands of other truths for the sake of one truth may appear to them to be a sacrifice that unnecessarily conflicts with reason.

However, if the heart has attained enlightenment beyond the discernment of the inner workings of the universe, then the depth in appreciation takes on a different aspect. If one realizes that everything in creation flows towards the Hereafter and if the compelling beauty of The Most Beautiful One is felt within one’s conscience, then thousands of years of happiness in this world cannot equate a single minute of life in Paradise and thousands of years of life in Paradise cannot equal a moment’s view of the beauty of Almighty God; here then the tradeoff between the fleeting pleasure of this world to that of the unending Hereafter will not be even as precious as the wing of a small insect. A person with such depth and foresight can perfectly exercise reason and will willingly make apparent sacrifices for real returns that will be attained from the transcendent realms.

In our time we have sent out rockets to explore the speculative possibilities of cities in space. Let us take this as a reality and for a moment imagine that such a source of comfort, which is beyond both the sight and hearing of human beings, has actually been discovered; here, life does not resemble our way of living and is beyond our comprehension. Large amounts of funds are spent to transport humanity there. In this process, there is a good probability that some will fail to perceive the possibility and may question the motives for such a prospect; this is because they are unfamiliar with these worlds and the goal may sound unfathomable to them. Objections may be raised with arguments like: “Many people are dying of hunger in Africa, but you are traveling to outer space with shuttles for the sake of adventure and are wasting large amounts of money.”

This approach is the outcome of superficial reasoning. What if, indeed, such a world was discovered? What would happen if one day a happier world outside this contaminated and muddled earthly life was to be found and somehow we could all be moved there, where our lives-with God’s grace-could continue happily for years to come? That is, a time will come when everyone will realize that this service has good and valuable implications for all of humanity and is certainly compatible with reason.

The innate ability to fathom far-reaching and multi-dimensional reasoning surely surpasses the limits of our logic; yet, it is clearly conceivable on a logical basis, beyond the influence of five sensory shackles, that every act of sacrifice made to this end is necessary. Therefore, endeavors to achieve greater future benefits cannot be considered to be illogical.

Similarly, a believer’s quest can be likened to the above. In order to attain a life of genuine happiness in Paradise and the privilege of meeting with the Prophet, peace and blessings be upon him, and in order to be mesmerized by the beauty of Almighty God Himself, all efforts or selfless acts made in the process only confirm the need for such a logical investment.

Most of the time, the criticisms that are hurled at those people of sacrifice come from no deeper an analysis than the above; they are usually based on shallow reasoning. In fact, such criticisms stem from two streams of different perspectives of the world. Hasan al-Basri once touched on this difference when he profoundly remarked: “If you had seen the Companions of the holy Prophet, you would call them ‘mad.’ … and if they had seen you, they would hesitate and ask, ‘are they believers?’”

Today is no different; some people may regard those who are engaged in sacrificial services as “mad” and question “why should you work so hard without any material gain?” This perception is revealing and delineates those who contemplate personal return and benefit in every act. In other words, for those mindsets who carry the proverbial attitude: “what’s in it for me?” it is not possible to understand why an elderly or infirm person would want to continue working just like an ordinary person, for example in the construction of a building or campaigning for donations for their good work, or for that matter, allocate all their physical and/or mental efforts towards a cause. Indeed, such mindsets cannot conceive the possibility of making all these sacrifices just to please God. As a result of their failure to grasp this, they will continue with their misguided considerations and false accusations of such selfless work.

On the other hand, for those of us with the insight to see beyond the materiality of things, selfless acts are perfectly normal and the most intelligent person is the one who is prepared to give up everything they own to serve God by serving humanity. If a person with such breadth and depth of heart is not seen to sacrifice all of their wealth all at once, then chances are high that they are most probably cherishing the noble thought: “Let me hold back part of my wealth so that I can invest it to earn more therefore continue to spend in the path to God.”

To sum up, it is clear there is no essential contradiction between selflessness and logic. I can safely and confidently assert that logic entails sacrifice. Whoever can grasp this fine point will use their logic to make more profound sacrifices.

Let us consider the case of the “heap” to understand the idea of vagueness in fuzzy logic. What is a heap? What do we mean when we are talking about a heap of dresses, for example? Assume that we have a heap of clothes and we begin to put them into a wardrobe by taking each item of clothing separately out of the heap. We take first the socks, then the shirts, and so on. In doing this, we observe that the heap disappears slowly. Finally, when we take the last of the items of clothing, there is nothing left. So, at what stage of this activity does the heap disappear? Does the heap disappear when we have taken away half of the clothes, or one third of them, or all of them? These questions are very important in terms of the logical characterization of this case. Classical logic, which depends on two truth values, namely true-and-false, does not answer these questions because there is no clear boundary between being a heap of clothes and not being a heap of clothes in this case. If we consider the constantly changing flow of events in the universe, we face similar cases all the time. Another outstanding example is the spectrum of light. There are no clear boundaries between colors. Where exactly is the boundary between yellow and light yellow? The indeterminacy of boundary can be observed in social area as well. Political ideologies range from the most radical right wing to the most radical left wing, and we can observe many different variations of ideologies in this range. For example, sometimes we may have difficulty in identifying the position of a particular political movement if it is between moderate right wing and moderate left one.

This is a special case of indefiniteness which we call “fuzziness” or “vagueness,” but there are different types of indefiniteness. For instance, when we use the term “bank” unqualifiedly, we can either mean a financial institution or a place to sit. We call such cases “ambiguous.” Whether it will rain next week or not is another indefinite case related to our ignorance of future.

The indefiniteness, which we deal with in fuzzy logic, is vagueness. The problem of vagueness is as old as humanity. It is discussed under the name of “sorite paradoxes”in ancient Greek and under the term “fuzzy logic” today.

Fuzzy logic proposes a systematic model to solve problems in vague cases. This model depends upon mathematical symbolization. It seems that fuzzy logic presents a better way to deal with vague cases than the way suggested by classical logic. Let us see why this is so.

Classical two-valued logic

Classical logic has two truth values, namely true and false. Both Aristotelian logic and symbolic logic have two truth values even though there are important differences between them. For that reason, both of them are treated under the name of “classical logic.” Aristotelian logic analyzes sentences by dividing them into subject and predicate. Consider the following sentence: Tom played tennis with John. “Tom” is the subject in this sentence and “played tennis with John” is the predicate. On the other hand, in symbolic logic, this sentence is analyzed in a different way. According to symbolic logic, there are two subjects in this sentence, namely Tom and John. The predicate of the sentence is a relation between these two subjects, namely “played football.” Furthermore, since Aristotelian logic is very close to metaphysics, it has a very strong existential import. For instance, if we assert the sentence “All dogs bark,” we also imply that there are dogs, according to Aristotelian logic. However, there is no such existential import in symbolic logic. In the view of the latter, we cannot say anything about the existence of dogs when we assert the above-mentioned sentence. What we mean by that sentence is that if something is a dog, then it barks. Nothing more, nothing less.

Although there are some differences between Aristotelian and symbolic logic, both of them are two-valued logics. However, we cannot call classical logic the logic of “white-and-black” because it has two values. This is a common mistake stemming from not appreciating exactly to what these two values refer. White-black duality ignores the grey tones between white and black. However, classical logic covers all tones between white and black. The duality here is between whiteness and non-whiteness or between blackness and non-blackness. So non-whiteness encompasses the grey tones.

The problem in classical logic is not exclusion of the grey tones but its insufficiency in representing them adequately. We have a problem in determining the boundary between whiteness and non-whiteness, and this is the problem of vagueness. Therefore, classical logic is inapplicable to vague cases, but clear cases lie in the area to which classical logic is applicable. Think of mathematics. A number is either even or odd, and this is clear for any number in question. There is a clear boundary between odd and even numbers. In comparison to classical logic, fuzzy logic claims that it has wider range of applicability. Let us examine fuzzy logic in detail.

Fuzzy logic

There is a discussion as to whether a logical system for vague concepts can be established or not. Gottlob Frege, one of the founders of symbolic logic, thought that logic cannot be applicable to fuzzy concepts and restricted logic to the area including clear concepts. The fundamental problem in vague concepts is determining the boundary between a concept and its negation. Assume that this problem cannot be solved by a two-valued logical system. Does it mean that the problem in question cannot be solved by three- or many-valued logical systems either?

Now, let us examine a three-valued logical system and see whether it solves the problem or not. In 1920, a Polish logician, Lukasiewitcz, presented a three-valued logical system whose values are true, false and indeterminate.

Assume that “This object is white” is true, “This object is black” is false and “This object is grey” is indeterminate in this logical system. Does presenting the grey area as indeterminate solve the problem of vagueness or make it more complex? In fact, the problem becomes more complex. The complexity arises from the question as to whether there is a certain, clear boundary between white and grey or between black and grey. Again here, we cannot claim that there is such a clear boundary. Vagueness is observed not only between black and non-black but also between black and grey. The problem of vagueness does not disappear if you increase the number of truth values in a logical system. For instance, postulating twenty truth values, instead of three, does not bring a solution to this problem. However, if you establish a logical system that includes infinitely many truth values, then the problem of vagueness seems to disappear. In that case, all colors with their tones can be analyzed by a scale of real numbers between 1 and 0. The infinitely many real numbers in this scale correspond to a truth value and so we can postulate a truth value for every tone of a color. For instance, “1” refers to pure blackness, “0” refers to pure whiteness, “0.5” refers to whiteness and blackness, “0.2” refers to a color whose whiteness overweighs its blackness. Since there are infinitely many truth values, you can give a truth value for any place in the light spectrum. In this way, we can get rid of determining clear boundaries.

The first serious study on fuzzy logic was carried out by Lotfi Askerzade Zadeh in 1965. He published an article called “Fuzzy Sets” when he was a professor of electrical engineering in the USA. Fuzzy logic was then improved by Zadeh, Joseph Gougen and many other scholars.

Fuzzy logic has a wide area of applicability in technology today ranging from washing and dishwasher machines to computers. Since it brings the advantage of flexibility, it increases productivity. Fuzzy logic is also used in the research program of artificial intelligence and modeling the human mind in computers.

If we come back to the internal structure of fuzzy logic, we observe that some of the fundamental principles of classical logic, such as the principle of non-contradiction and the law of the excluded middle, are not valid in fuzzy logic. For instance, if the proposition that this man is bald is true to a degree of 0.3, we say that this man is neither completely bald nor completely non-bald. The position of this man is the intersection point of baldness and non-baldness to a degree of 0.3. This man belongs to the set of baldness to a degree of 0.3 and to the set of non-baldness to a degree of 0.7.

An Islamic evaluation of vague cases

Said Nursi was one of the outstanding Muslim scholars of the twentieth century. He considers vague cases which give rise to relative truths to be important in terms of understanding divine activity in nature and knowing God’s Names and Attributes:

. . . God, the Eternal All-Wise and as the requirement of His Eternal Grace and Wisdom, has created this world as a testing arena, a mirror to reflect His Beautiful Names, a vast page upon which to write with the Pen of His Destiny and Power. People are tested here to develop their potentialities and to manifest their abilities. This emergence of abilities causes relative truths to appear in the universe, which, in turn, causes the Majestic Maker’s Beautiful Names to manifest their inscriptions and make the universe a missive of the Eternally-Besought-of-All.1

So, relative truths play an important role in knowing God. They are relative because they do not correspond to clear and fixed cases. As we remember from the analysis of fuzzy logic, these truth values are between 1 and 0. All values between 1 and 0 are relative because the case represented with such values belongs to two different sets with different values and the value depends on the perspective from which you are looking at it. If the truth value of “This man is bald” is 0.3, then the truth value of “This man is not bald” is 0.7.

According to Nursi, God gives rise to vague cases by mixing opposites. He mixes goodness with badness, beauty with ugliness, and so on. By this mixture, there occur infinitely many levels of beauty, goodness, and other qualities.

On the other hand, there is a famous saying of the Prophet Muhammad, peace be upon him, which suggests a criterion for human actions in fuzzy cases: “What is permitted is clear, what is forbidden is clear as well. There are doubtful cases between these two cases. Whoever avoids the doubtful things, his religion remains clean. Whoever approaches the doubtful cases, he approaches the forbidden cases as well.” Prophet Muhammad points out the vague cases between forbidden and unforbidden cases. In Islamic jurisprudence, scholars suggested many categories to represent these vague cases. Some cases are close to the forbidden ones, and some are not. They named each case separately. Fuzzy logic can be used in this regard to represent these different levels better than classical logic’s representation. Think of the following case. Grape juice is permitted in Islam. However, wine, which is prepared from grapes, is forbidden. More interestingly, there is a stage of fruit juice which is permitted but very close to the forbidden case. This stage is called “must” (unfermented grape-juice). After this stage, it gradually becomes closer to the forbidden case and we cannot exactly determine where the boundary between permitted and forbidden cases is. The Hadith emphasizes this point and suggests that we avoid such cases.

Conclusion

Fuzzy logic represents a wider area of cases than classical logic does. By representing vague cases in a symbolic and systematic way, fuzzy logic functions better than classical logic. However, fuzzy logic is not the ultimate system of thought which has been established by human mind. We should not exaggerate its importance by claiming that it can represent any case no matter what it is. For instance, fuzzy logic is inadequate to represent the logical relations between concepts including the notion of infinity. To deal with cases of infinity in mathematics, a different type of logic is proposed, namely intuitionistic logic. All of these systems show that the human mind is powerless or unable to grasp or establish a system which represents all cases and any case no matter what it is.

Notes

1. Nursi, Bediuzzaman Said, The Words, Twenty-ninth Word, The Light, Inc. NJ: 2005, p. 546.

References

Williamson, Timothy. Vagueness. New York: Routledge. 1994.

Mukaidono, Musao. Fuzzy Logic for Beginners, New Jersey, World Scientific, 2001.

At the end of the 1950s, Richard Feynman (1961, pp.282-96) described the possibility of building computers that were ‘sub-microscopic’. More recently, several people have advocated the realization of massively parallel computation using the techniques and chemistry of molecular biology.

At the end of 1994 Leonard Adleman published a paper on ‘Molecular Computation of Solutions of Combinatorial Problems’ (Science, vol.266, pp.1021 – 24). He explained how a problem could be set up by synthesizing DNA molecules with a particular sequence, and solved by letting the DNA molecules react in a test tube, producing a molecule whose sequence is the answer. In the same paper he recounted how he had put this theory into practice by solving a standard problem with a DNA reaction system. Adleman called his DNA computer the TT-100, for test tube filled with 100 microlitres of fluid, which is all it took for the reactions to occur.

Since then, many advances have been proposed to refine the protocol for programming a DNA computer to reduce the complexity of the operations and eliminate errors (see Lipton, n.d.; and Boneh and Lipton, nd.). Despite their respective complexities, biological and mathematical operations have some similarities:

The very complex structure of a living being is the result of applying simple operations to initial information encoded in a DNA sequence.

The result f(w) of applying a computable function to an argument can be obtained by applying a combination of basic simple functions to w.

For the same reasons that DNA was probably selected for living organisms as a genetic material, its stability and predictability in reactions, DNA strings can also be used to encode information for mathematical systems.

Conventional computers represent information in terms of 0’s and l’s, physically expressed in terms of the flow of electrons through logical circuits. Builders of DNA computers represent information in terms of the chemical units of DNA. Calculating with an ordinary computer is done with a program that instructs electrons to travel on particular paths; with a DNA computer, calculation requires synthesizing particular sequences of DNA and letting them react in a test tube. In a scheme devised by Lipton (n.d.), the logical command AND is performed by separating DNA strands according to their sequences, and the command OR is done by pouring together DNA solutions containing specific sequences.

‘It will fill a bathtub, not the universe,’ says Lipton, ‘and it will be incredibly cheap to build.’ A pound of DNA in 1,000 quarts of fluid, about three-feet square, will hold more memory than all the computers ever made. The chemicals are inexpensive; DNA runs virtually on its own power, and the soup, with a little splicing, can be re-used from one experiment to the next. Lipton estimates that a superparallel DNA computer, offering trillions of processors working simultaneously, could be built for $100,000.

The fastest supercomputers can currently perform 1000 million instructions per second (MIPS); a single DNA molecule requires approximately 1000 seconds to perform an instruction (.001 MIPS). Obviously, if you want to perform one calculation at a time (serial logic), DNA computers are not a viable option. However, if one wanted to perform many calculations simultaneously (parallel logic), a computer such as the one described above can easily perform 1014 MIPS. DNA computers also require less energy and space. While existing supercomputers operate 109 operations per joule, a DNA computer could perform 2 x 1019 operations per joule (many times more efficient). Data can be stored on DNA at a density of approximately 1 bit per cubic nm, while existing storage media require 1012 cubic nm to store 1 bit (Adleman, 1995).

Thus, the potential of molecular computation is impressive. However, it is too early for either great optimism or great pessimism. It is possible that DNA computers will become more common for solving very complex problems and DNA computers may also become automated. In addition to the direct benefits of using DNA computers for performing complex computations, some of the operations of DNA computers already have (Adleman, 1995), and more could be, used in molecular and biochemical research.

References

• Adleman,L.(1994).’Moleculer computation of solutions to combinatorial problems’,Science,vol.266,pp.1021-24.
• Adleman,L.(1995 ‘On constructing a moleculer computer’:ftp://usc.edu/pub/csinfo/papers/adleman/molecular_coputer.ps
• Boneh,D.&Lipton,R.J.’Making DNA computers error resitant’.(Unpublished manuscript.)
• Feynman,R.P. (1961)’Minaturization’,in D.H.Gilbert (ed.)Reinhold,New York.
• Lipton,R.J.(n.d.)’Speeding up computations via molecular biology’:ftp://ftp.cs.princeton.edu/pub/people/rjl/bio.ps