You know education is always a hot topic. Everybody has a share and interest in education: Parents, educators, government officials, elected public representatives, nonprofit organizations, religious groups, and so on. Besides, there are many aspects of education, such as finance, law, administration, etc. However, do you also think that teachers have long been overlooked regarding their authority in their own classes? Haven’t they been overlooked regarding their professional development, even though they are the backbone of the education system and they need some assistance in their efforts more than ever? It is good if you also think like me. It is even better if you don’t. Thus, you can share your recommendations with me once you see this school where things are done differently than many other contemporary schools. These teachers at this school see themselves as a learning society. They criticized the status quo they were in some time ago, and renewed themselves collectively. I want you to see the product. Let’s start our journey, shall we?

Room 114

We enter through the big doors of the school. Physically, it is not the best school you would ever see, but it can definitely be said that vitality and cleanliness of the halls stand out. The atmosphere is very welcoming; something is different, but it is hard to tell what it is. I point to the very first door on the right hand side. With pride, I ask if you would like to see this classroom. The sound coming from room 114 is not different from the sound that bees make when they work. As we approach the door, I understand that this is Ms. Williams and her eighth-grade students studying math. She welcomes us and tells us, “please come on in; there are some empty chairs over there,” pointing to two empty chairs, and we make our way towards to the back of the class and have our seats. This is a math class, but I don’t see any numbers on the board or equations being discussed. Soon, we understand that they finished the chapter on expressing roots of cubic equations last week, and today they are discussing a person named Omar Khayyam (1044 – 1123 C.E.). Ms. Williams knew that Omar Khayyam was a philosopher, mathematician, astronomer, and poet. She had formed her students into four groups, and assigned each of his professions to a group, so that they could do some research and present their findings to the whole class. I understand what you mean from the way you look at me: you’re sure this is a math class? you seem to be saying. You’re right to ask. Don’t they need to finish a chapter and pass to the next one immediately, and stay focused on the possible topics that are asked at the end-of-year tests?

Let’s hold on to that question; we can ask this after the class is dismissed. The first three groups already talked about his contributions to philosophy, astronomy, and poetry. We feel very fortunate not to miss at least the last group, who talk about him as a mathematician. I can see brightness in students’ eyes, and their excitement to share everything they have learned with their classmates. The first student, Omar who coincidentally happens to share a name with Omar Khayyam, begins. Omar Khayyam was a scholar among many others of his time. He is best known today for his poetry, but his contribution to mathematics was great. He showed how to express roots of cubic equations by line segments obtained by intersecting conic sections… Another student, Margaret continues: his work on algebra was known throughout Europe in the Middle Ages, and he also contributed to calendar reform. The algebra (from Arabic al-jabr) of Khayyam is geometrical, solving linear and quadratic equations by methods appearing in Euclid’s Elements… April inserts that Khayyam also gave important results on ratios giving a new definition and extending Euclid’s work to include the multiplication of ratios.

They continue their discussion, asking how these scholars tried to seek knowledge and an answer to who they were; and the great contributions they made towards peace and understanding among people of different origins. However, I am currently very much impressed and I lose myself in the beauty of these students’ discussion that I have found myself holding my breath and forgot about your presence next to me. It was no different than waking up from a dream when the bell was ringing. Sure, I will introduce you to Ms. Williams. I understand that you want to learn more about the uncommon method she adopted for this class: Ms. Williams, do you have a minute, please?

Of course, she says: “I have some time. It’s break time now.” You ask her about the topic and the way she formed her teaching method: Why would you choose a person from old history to discuss? Why do you want your students to talk about poetry or astronomy or philosophy in a math class? Do you think they know enough math to solve the questions on the end-of-year test? Why did you choose a Muslim scholar? Is this an Islamic or a secular school? Would you… and you realize you have bombarded her with your questions. You halt and smile.

She smiles back and says: “Yes, this is a secular state school. We try to practice secularism in a true sense that we can talk about religion, and bring religious materials to school to study, but we never try to impose these ideas into our students’ minds. They actually study very different kinds of perspectives, and they enjoy it very much. We take each student as a whole. Since our topic was roots of cubic equations, and Omer Khayyam was the father of the idea, you happened to see this example.”

“But why would you touch on different areas of study in a math class?” You say without taking a second breath.

Ms. Williams smiles again. “This is very common in our school. I am sure Mr. Wraga next door will also get into some math in his literature class since he knows my course syllabus. We try to keep our teachings interdisciplinary as much as possible. I receive feedback that kids enjoy it greatly. I understand your concern about the tests. However, testing is the very last thing we think of. Most of our students do well at those tests anyway, and those who cannot do so well are seen to be still successful mostly as artists or musicians…”

There is a long pause and silence now, but hurricanes are happening in our minds. How is it possible? What kind of philosophy is this school following? Is it the principal who leads these teachers like a conductor leading an orchestra? As Ms. Williams answered our questions, more questions kept popping up in our minds. The bell rings again, and the break is over. We thank her for letting us sit in the class. We make our way back to the hall again, and I am already elated with the intelligence of these students. You look at my eyes without saying a word, and smile: Room 115?

Concepts like the number zero, negative numbers, complex numbers, matrices, and spatial geometry are inventions of mathematicians which were studied earlier and presented especially for the use of physicists. When mathematicians theorized the “Group Theory,” they were reported to have said, “Finally we have developed something that physicists cannot use.” However, this theory, which stemmed from intangible algebra, was found to be useful in investigating the symmetry of physical systems and had serious applications in particle physics.

If there were no mathematical advances, would physics and other sciences have developed as far as they have? What does it mean that sciences are found in such an interrelated state, and thus support each other?

The perfect relationship between physics and mathematics, as in the expression of many physical laws via simple mathematical equations, is truly amazing. The laws that describe the physical world – from equations expressing the laws of motion (like X=V.t, V=a.t, F=m.a h= (gt2)/2), to basic electrical equations (like V=I.R, P=I.V, E=V/d), going all the way to the equations that define gravitational forces and the expansion of the universe (like F=G.m1.m2/d2, V=H.d) – can easily be expressed through mathematics. Furthermore, this simplicity and plainness in creation of the universe fascinates many scientists. Einstein expressed this fascination when he said, “The most incomprehensible thing about the universe is that it is comprehensible.”

One of the most important relations between mathematics and physics is that the independent study of intangible works of mathematics unexpectedly became one of the best tools to describe the physical world. Numbers, for example, are one of the greatest inventions of humanity. Humans have used them to quantify their properties. For thousands of years, people used numbers, but then the concept of negative numbers developed for seemingly no reason. For centuries, negative numbers were seen as nonsense. Of course there is not a square with a negative side length, a circle with a negative value area, or a classroom with negative number of students. However, negative numbers were found to be applicable in many areas of physics; they are now accepted as being as real as positive numbers. For instance, it is almost impossible to graph position-time, speed-time, acceleration-time, and the momentum-speed relation of objects without negative numbers.

Ellipses, parabolas, and hyperbolas (plane sections of cones cut in different shapes) were studied by Apollonius (BC 262-200), who was a contemporary of Archimedes. Interestingly these shapes were one day used by Kepler and Newton to describe the orbits of heavenly bodies like planets. Three dimensional pentagonal and hexagonal patterns, like those on the surface of a soccer ball, were also investigated by Archimedes. This shape has been found to be in exact configuration of a special carbon molecule composed of 60 atoms.

Likewise, the number zero, which was introduced by Muhammad bin Ahmad, in 967, led to many innovations in mathematics, as well as physics. Did al-Khwarizmi (780-850) know, when he found and utilized 1st and 2nd degree equations, that he was working on something mathematicians and physicists could one day never do without?

Imaginary numbers, as proposed against the main principles of arithmetic, also provides a very good example for this topic. We cannot think of a number whose square is negative in normal conditions. In other words, when a number is multiplied with itself, the resulting number is always a positive number. But mathematicians thought of a number that is negative when squared and continued various studies accordingly. Again, these studies have proven to be an important tool, especially in understanding electrical circuits by physicists.

Let’s finish our examples with ones from modern physics. Riemann (1826) was a mathematician who studied spatial geometry and proposed the concept of space curves. Mathematical equations designed by Riemann, pertaining to spatial geometry, were used by Einstein in 1908 to describe and formulate the concept of general relativity. Einstein also used Minkowski’s four dimensional geo-spatial continuum when developing his theory of general relativity.

There are many more examples. The famous Russian mathematician Friedman established a mathematical model that allows the expansion of the universe by improving Einstein’s model of universal geometry. This model was also later confirmed by the physicist De Sitter in discovering universal expansion and by Hubble in formulating the expansion. In addition, well before the discovery of quantum mechanics, Davit Hilbert proposed the complex vector space with a very different mathematical purpose known as “Hilbert Space.” This concept of space with an infinite number of dimensions is today used by quantum mechanics.

Sometimes physical realities can be foreseen via these invented equations. For example, Dirac proposed the existence of a particle known as the positron (or as we call it, the twin of the electron; it just differs by the charge) through his mathematical equation that he wrote in 1928. Four years later this particle was discovered by Carl D. Anderson, as predicted. To name, James Clerk Maxwell (1831-1879), a famous physicist and mathematician, predicted the presence and speed of electromagnetic waves mathematically via his own equations. Later, these waves were detected by Hertz (1886) through experiments. Again, Maxwell calculated the speed of electromagnetic waves via his equations, and by revealing that it was equal to the speed of light, it was understood that light was also a type of electromagnetic wave. Nowadays, the particle called the “graviton,” which is supposed to be in charge of gravitational forces, and the “Higgs” particle, that theoretically fills space according to quantum theory, are waiting to be discovered.

The book of nature is written in such a way that it is expressible by mathematical language. Famous physicist Sir James Jeans (d. 1946) expressed this situation as follows: “From the intrinsic evidence of his creation, the Great Architect of the Universe now begins to appear as a pure mathematician.” Yes, the level of knowledge that is at play in the universe encompasses both physics and mathematics. The overall interconnectedness of sciences and the interdisciplinary character physics and mathematics point to an owner of this knowledge.

As a conclusion we can deduce that physics and mathematics, just like material and non-material worlds, are in fact intertwined with each other’s various dimensions. The physical face of the universe is the place where records are kept and concepts of matter like heavy or light, big or small, and soft or hard, exist. The mathematical face of the universe (as if spiritual) is the unseen side of events or materials that are hidden and intangible. In a way, this relation between physics and mathematics is a display of the material and spiritual sides of universe.

Nuri Balta is the Head of Physics department at Samanyolu Schools in Turkey. He is also pursuing a PhD degree in Physics at Middle Eastern Technical University, Ankara, Turkey.

In this article I address four questions regarding the importance of mathematics in our lives:

1. Why is mathematical thinking significant to comprehend the universe and how it runs?

2. What are the problems in the new millennium and what solutions to these problems are expected from scientists?

3. How important a role does mathematics play in today’s world?

4. What is the relationship between defense industry and space research

In contemporary scientific research methodology, mathematics is the most objective tool which can be used to draw a general conclusion from outcomes obtained. Mathematics is considered to be an expression of the knowledge of the All-Knowing. This characteristic of mathematics recognized by Muslim scholars in the Middle Ages was emphasized by well known scholars such as Ghazali, Al-Biruni, Nasiruddin Tusi, Al-Hujandi, and Al-Khwarizmi. Following in the path of Muslim scholars and being considered one of the pioneers of modern science, Galileo stated in his second book published in 1623, Il Saggiatore, that “it is impossible to understand the universe without learning the real logic of the universe and decoding its characters. The universe was created in mathematical logic and it is impossible for us as human beings to comprehend its words without mathematics.” Galileo’s statement points towards an important truth-that even though it is possible partially to explain the intricate perfection in the universe through the mathematics that has been developed so far, we are not skilled enough to produce or comprehend any mathematical systems or formulas that can express the whole universe despite the complexity of occurrences that go on in the universe.

In the history of science, the structure and the mechanism of the universe have been explained to some extent using mathematics. Physicists have developed equations to demonstrate the structure of matter and forces in nature. An engineer who designs an artificial heart considers the equation that governs the bloodstream in a vein. An astronaut at NASA utilizes equations that describe the motion of satellites or the orbit of a spacecraft. In our contemporary world, the crucial role of mathematics is the main reason why Landon Clay, a millionaire philanthropist and the founder of the Clay Mathematics Institute, came up with a list of seven “millennium problems” and promised seven million dollars to the first person who found the solution to each of them. They have not yet been solved.

Many of us remember traditional mathematics classes as boring because they were not apparently related to real life. Only once symbols and equations become meaningful and solutions are found, does mathematics become pleasurable. Despite the stress endured, true success is hidden in the process of writing the correct equation. An equation developed to solve a specific mathematics problem becomes an invention when it is practically used in life, for example, to build a spacecraft or design a medical device.

However; in order to make an invention, the correct equation for that invention needs to be developed, or a pre-developed equation that works for the invention needs to be determined. The next step is to solve it. Even if the solution to an equation cannot be found, an approximate solution can always be discovered and used to build an invention.

The equations for two of the millennium problems come from physics. One of the problems involves finding a general solution to the Navier-Stokes equations governing fluid dynamics. These equations were first formulated in the 1820s to describe the motion of fluids and gasses. Examples include the flow of water around a boat, air over the wings of a plane, and blood pumped from the heart to the vessels. At first glance, the Navier-Stokes equations resemble equations taught at the undergraduate level in the fields of science and engineering. However, the way they look is deceptive because no one has ever come close to finding the general solution to these equations. Even though a general solution to these equations does not yet exist, the Navier-Stokes equations do help one comprehend the aforementioned problem. Therefore, they do not help naval architects to construct better marine vehicles, aerospace engineers to build better aircrafts and spacecrafts or biomedical engineers to build artificial organs.

Another millennium problem involves finding a solution to the set of equations formulated by Chen-Ning Yang and Robert Mills in 1954 that describe the fundamental forces of nature. This set of equations reveals the description of the raw material out of which everything in the universe has been created. None of these equations have been solved so far. Physicists have gained accurate results and made calculations tested in laboratories based on Yang-Mills equations that could be solved by using computers as in Navier-Stoke equations. Even though these kinds of equations provide physicists with almost all the necessary information, no one has ever been able to solve the Yang-Mills equations by known methods. What is important is not to solve equations; it is to figure out what the solution means instead. Using numbers and making calculations based upon these equations remain secondary despite their importance.

As a result of positivist and materialist approaches to knowledge and science, most people today are interested in science and technology for the sake of their own material wealth and comfort. If this degrading approach continues, worldwide degeneration cannot be prevented. However, mathematics is a universal language generated by mathematical thinking. This type of thinking is one of the qualities of the “inheritors of the earth.” In The Statue of Our Souls (2005), M. Fethullah Gulen says,

In the past the people in Central Asia and later on in the West achieved their renaissances by means of the laws of mathematical thinking. Man discovered and brought to light many uncertain and unknown things in the mysterious world of numbers. Without going to the extremes of the Hurufis,^[1] what we say is that without mathematics it is not possible to understand the relations of humanity and natural phenomena with one another. It illuminates our roads like light on the line that stretches from the universe to life; it indicates to us what is beyond the human horizon, even the depths of the world of contingencies, which is very difficult to think upon; and it makes us meet with our ideals.

On the other hand, being mathematical does not mean knowing everything related to mathematics. It is to think mathematically, to think within mathematical laws, and to be aware that it permeates everything from man’s thoughts to the depths of existence, from physics to metaphysics, from matter to energy; from body to soul, from law to Sufism. In order to comprehend existence completely, we have to accept a dual method of Sufi thinking and scientific research. The West essentially lacks essence, and has tried to compensate for this loss, as far as it can, by taking refuge in mysticism. In our world, which has been always intimate with the soul of Islam, there is no need to look for anything strange or foreign, or to take refuge in anything. We have all our sources of power within our system of thought and faith. That suffices as long as we comprehend that source and spirit with its original richness. Then we will see some of the mysterious relations in existence, how harmoniously such relations run, and reach a different knowledge of observing and taking pleasure in everything.

In short, being mathematical is necessary to describe the universe we live in and the principles of how it runs. This tool will be considered triumphant in as much as it can remove blockages from the individual’s eyes and exhibit the truth. Only if scientists who have attained the harmony of heart and mind penetrate into the secrets of existence, utilizing science and its fruits for the benefit of humanity, will justice be done to their profession.

As a relatively new and exciting science, chaos science grew very slowly during its infancy. Yet in the last decade, due to active research in many areas, it has became one of the hottest topics in academia as well as the popular science press. Many books have been published, and millions of Internet pages have been designed full of fractal pictures.1 Given that chaos is associated with disorder or confusion in a system or condition, why does it continue to attract so many people?

History of chaos

To understand chaos, one first has to understand the Newtonian worldview. Sir Isaac Newton’s (1642-1727) development of the calculus and laws of classical mechanics began a scientific revolution in seventeenth-century Europe that caused all subsequent scientists to view nature from a profoundly different perspective. Now that they finally could determine the dynamics of bodies by simple equations, they believed that they had found the ultimate eternal rules that shape the universe.

French physicist Pierre-Simon Laplace (1749-1827), who based his work upon Newton’s work, is credited with the following famous quotation (often referred to as Laplace’s Demon): ‘We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at any given moment knew all of the forces that animate nature and the mutual positions of the beings that compose it, if this intellect were vast enough to submit the data to analysis, could condense into a single formula the movement of the greatest bodies of the universe and that of the lightest atom; for such an intellect nothing could be uncertain and the future just like the past would be present before its eyes.’2

Laplace’s Demon states the idea of determinism, that the past completely determines the future. One can clearly see why determinism was so attractive to scientists at that time. However, in Laplace’s word everything was predetermined: no chance, no choice, no uncertainty. A solid, inevitable destiny had frozen the events in every corner of the past and is continuously spreading out to the future to do same there. Determinism apparently invokes the idea that whole universe is like a clock. God set it in motion at the beginning of creation and then removed Himself, for everything had been planed before. The ideas that there was no place for free will and that God could not interfere killed the belief in a soul and, consequently, in spirituality. Philosophers and scientists have discussed this for many years. Determinism affected many philosophies and triggered the major ideological movements of during eighteenth century, especially in Europe.

Toward the end of the 1800s, mathematicians and scientists began encountering some very difficult equations, some of which we know today are unsolvable. The most troublesome are various nonlinear differential equations. Even though it looks like such a simple and totally deterministic system, the problem of three bodies attracting each other with purely gravitational forces (e.g., the sun, Earth, and moon triple) turns out to be missing an exact solution. At first, such problems were cast-off as special cases and largely ignored.

One reason for this also might come from the fascinating world of quantum mechanics, which dazzled even the great physicists, and the lack of fast computers at that time. When these equations finally were studied in detail, a fundamental change that would ultimately overthrow determinism began to occur in mathematics and science. An indication of the science that would be come to be known as ‘chaos’ began to appear.

Why does chaos interest people?

In contrast to its common usage, chaos does not actually mean disorder or confusion. By definition, it should occur in well-defined orderly systems. However, most natural physical systems often can exhibit an unpredictable or intractable behavior in the long run, even though the system is defined by clear-cut orderly mechanisms. In this sense, chaos can be defined as unpredictability rather than disorder.

For example, meteorologists use 12 sets of well-defined equations to forecast the weather. They relate such atmospheric parameters as pressure, temperature, flow speed, and time to each other. One can make a computer program that calculates the parameters’ final values by taking any initial conditions as the run’s starting point. In principle, therefore, if we know the initial temperature, pressure, and time values that describe today’s weather conditions, it is possible to derive tomorrow’s weather conditions by running a computer program, which is nothing more than a chain reaction of numerical iterations.

However, in practice, initial conditions cannot be measured exactly and so contain a degree of uncertainty. But since the system’s governing laws are known, one may estimate the effect of errors on future results. Hence, instead of giving the exact results, one can provide an approximate range of possibilities. This range can still be very useful, provided that the deviations do not stray too far from the actual values. In addition, knowing how the error grows in the system might help us understand and control the systems. But if we apply these error estimates to weather forecast equations, we will encounter a large problem, for errors grow exponentially in such systems. Even a tiny deviation at the beginning can create huge deviations from the actual values. It also can provide unrelated or nonsensical results.

An example of chaotic systems

This numerical behavior was first observed by the meteorologist Edward Lorenz, a pioneer in modern chaos work. Fascinated by the results he obtained, in the early 1960s he gave an interesting metaphor to explain the situation of high sensitivity to initial conditions: A butterfly’s slight wing movement (i.e., a little deviation from the initial conditions) can change the future in a way that causes some chain reaction that ultimately result in a hurricane.

Such systems that exhibit a very high sensitivity to initial conditions are called chaotic systems. Chaos comes from the mathematical properties hidden in the equations defining the system, and such unpredictability cannot be removed. Even if the measurements’ quality could be improved by minimizing errors, chaos never disappears from a chaotic system.

One may suppose that chaos occurs in complicated systems, such as weather forecast systems having 12 sets of equations. But even much simpler systems, such as billiards, can exhibit a very high degree chaos. A usual billiard system consists of many balls and a rectangular shaped table. I challenge master billiard players by requesting them to play the game in a stadium-shaped table. I am sure that they will find it difficult to do so, because such billiard tables would be chaotic systems.

If a system is defined as chaotic, this does not necessarily mean that its behavior is totally undefined all the time. As in stadium billiards, a ball has to be inside the billiards, so it should be somewhere on the table, even though sometimes we cannot foretell its exact position because of chaos. Besides, if you send the ball with a velocity perpendicular to a straight side, it will bounce back and forth between the two sides forever. Therefore, depending on which initial conditions are taken, chaotic systems also can show characteristics of regular motion.

A 3-body problem (in general n-body problems) such as the sun, Earth, and moon system, is a chaotic system. But since we can predict the motions of these celestial objects with great precision for many years in the future, why do we call this system chaotic? This triple system possesses a very special set of conditions: distance between bodies, their masses, and their velocities. These parameters cause it to exhibit near-regular behavior. It is analogous to the example of stadium billiards given above, for this triple system bounces back and forth between the table’s sides.

Conditions for chaos

Chaos also can be caused by other factors than just uncertainties measured in the initial conditions. Scientists generally define hypothetical systems by isolating them from the outside world in order to simplify them as much as possible. However, as even objects in the real world that are far apart interact with each other, no system in the real world can be isolated. Given this, a closed (isolated) system might be defined as a fluctuating approximation to its real counterpart, which is changing in an unpredictable manner all the time. In short, even though we would know the exact initial conditions, the actual system could be chaotic due to changes in the approximate system. In this sense, many physical systems have an inclination toward being chaotic.

Due to its maximum complexity, the universe is the largest chaotic system. Observable regular patterns in special parts of that system repeat themselves in time. While the rest of the flows are unpredictable, they are not totally irregular, abrupt, or disordered. Just like whirls in a flowing river, they are in a kind of free motion searching for convenient conditions in which to give birth to organized structures.

The future of chaos

Chaos gives today’s scientist a new worldview, for Newton’s concrete, cold, and deterministic one has been shaken by the uncertainty principle of quantum mechanics. No one ever thought that the Newtonian worldview could be replaced. However, now scientists are more likely to be open to chance and choice than their predecessors. The question is whether chaos theory will cause large revolutions in how we understand the universe. However, the existing excitement, expanding research and growing number of articles, and its numerous applications from economy to biology, seem to indicate that a surprise improvement might not be so far off. ‘

Footnotes

1. Fractal: A geometric pattern that is repeated at ever smaller scales to produce irregular shapes and surfaces that cannot be represented by classical geometry. Fractals are used especially in computer modeling of irregular patterns and structures in nature.
2. ‘Chaos and Fractals: Laplace’s Demon.’ Online at: www.pha.jhu.edu/ ldb/seminar/laplace.html.