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.