It is said that on the door to Aristotle’s dwelling was written: ‘One who does not know mathematics cannot enter.’ I do not know whether this means that those who did not know mathematics would not be able to understand Aristotle or if it was simply a way to urge people to study mathematics. We do know that mathematics has had an important place in the thinking and life of people from the most ancient times. Pythogaras’ famous theorem about the square on the hypotenuse etc is still taught in primary and secondary schools. Every century has contributed something of its own to mathematics, which is now a universal ‘language’ studied throughout the world.
There are two major theories about the origin or essence of mathematics. One of these theories is attributed to Plato, and the other to the so-called Formalist school. According to Plato, mathematics exists independently of man. What man does is to discover its objective reality, just as other ‘laws of nature’, which we tend to call ‘Divine laws of nature’, are discovered. The Formalist school by contrast asserts that mathematics is a product of human thinking. In order to understand the difference between these two schools, we may cite as an example their view of prime numbers (that is, numbers like 7, 17, 41 which can only be divided exactly by themselves and the number 1). Platonists argue that the prime numbers exist independently of us: before we discovered their existence, they existed in infinite number. Whereas, Formalists are of the opinion that the prime numbers exist because we have defined them as such, and it is meaningless to think about whether they are of infinite number or not.
Formalists assert that numbers came into existence when human beings began to count. A well-known account of how this happened is that of a shepherd who used to put a stone in his bag for each of his sheep and by matching a stone with a sheep could find out whether any of his sheep had been lost or not. Later on, people began to call numbers each by a different name and since there were two fingers in the two hands, they found it easier to make calculations by the decimal system. This was followed by the operations of addition and subtraction.
According to the Formalists, even the simplest mathematical operations like the four basic ones consist in some logical rules based on certain axioms. They say that we do mathematics by expressing certain rules with certain symbols. That is, we take, say, 5 and 7, a couple of signs whose meaning in the physical world we do not know, and put between them the plus sign, a third sign whose meaning in the physical world we do not know, followed by an equals sign. And we know we must write 12 after the equals sign because that is a requirement of the axioms and rules of logic we are using. This is just what a calculating machine does, that is, it goes through the operation required of it without knowing what it is doing.
Let us suppose that an adding operation consists only in applying axioms or certain logical rules, and has nothing essential to do with the physical world. If we were to take our number signs and apply them to physical objects like stones and sheep, we should be surprised, amazed even, as if by a miracle, that 5 and 7 stones or sheep added together (according to the same rules as 5+7) make 12 stones or 12 sheep. We would come to know that the abstract, conceptual realities in our mind correspond to physical realities in the outer world. According to Paul Davies, the renowned physicist, if we lived in a universe where different physical realities prevailed, in a space where, for example, there were not any countable things, we would not be able to make most of the calculations we make today. David Deutsch claims that counting emerged as the result of experiences. According to him, we can do arithmetic because physical laws allow the existence of physical models convenient for arithmetics.
Richard Feynman, regarded as the greatest physicist after Einstein, says about mathematics that the problem of existence is a very interesting and difficult problem. When you take the third power of certain numbers and then add them with each other, you obtain interesting results. For example, the third power of I is 1, of 2 is 8, and of 3 is 27. The addition of these numbers gives the result of 36. The addition of 1, 2 and 3 is 6 and the second power of 6 is also 36. When you add to this the third power of 4, which is 64, the result is 100. The addition of 6 and 4 is 10 and the second power of 10 is also 100. Added to this the third number of 5, which is 125, the result is 225. 225 is the second number of 10 plus 5, i.e. 15. And so on. According to Feynman, we may not have known this typical characteristic of numbers before but when we do come to know such characteristics of numbers, we feel that they exist independently of us, and that they existed before we discovered them. However, we cannot determine a certain space for their existence. We feel their existence as conceptions only.
Let us take another example. Ibrahim Haqqi of Erzurum, a Turkish Sufi, religious scholar and scientist of the 18th century, discovered a way of checking the correctness of an operation of addition which may still be unknown to modern mathematicians. In order to check or prove the addition, we first add up the digits of each of the two numbers we are going to add up. Let us say, we are going to add 154 to 275, for which we get the answer 429. Adding the digits of each of the first two numbers, we get 1+5+4 = 10 and 2+7+5 = 14. The next step is to subtract 9 from each of these two sums, giving us 1 and 5 respectively. The third step is to add these two results together, 1+5 = 6. Now we do the same thing with the digits of the answer we are wanting to check, namely 429, and again subtract 9: 4+2+9 = 15, 15-9 = 6. The fact that we end up with the same number (i.e.6) means that our addition was correct. This way of checking an addition exists independently of us. We did not create it, we discovered it.
As water had the force of lifting objects of certain weight before Archimedes discovered it and, again, objects thrown into air or a fruit disconnected from its branch fell before Newton discovered the law of gravity so also numbers have many characteristics only some of which have been discovered.
Heinrich Herzt, a physicist, says that we cannot help but feel that the mathematical formulas discovered so far exist out there independently of us. We know that these formulas existed before we discovered them but we cannot determine a space for them. Rudy Rucker, a mathematician, is of the opinion that there is, besides the physical space, a space of mind, which he calls ‘mindspace’ and it is that that mathematician study.
Most of the distinguished mathematicians follow the view of Plato. Kurt Godel is one of them. Before Godel, it was almost a generally accepted view that mathematics is a function of the working of mans brain consisting in the collection of the logical rules which we establish between the symbols of two sets. Godel persuasively argued that there have always been correct mathematical expressions even though their correctness cannot always been proved. Another Platonist mathematician, Roger Penrose, believes that beyond the thoughts of mathematicians there are profound truths or realities in mathematical conceptions. Human thought is directed to extend into these eternal realities and they are there to be discovered as mathematical facts by any one of us. Penrose mentions complex numbers as an example for his argument. According to him, there is a profound, timeless truth in complex numbers. Penrose cites the set of Mandelbrot as another example to prove his argument. The reality this set reveals is the fact that even the lines, twists and shapes of mountains and clouds were or are formed according to certain mathematical formulas.
Almost everyone has heard of the series of Fibonacci. This series, named after the famous mathematician, Leonardo Fibonacci, progresses as 1,1, 2,3,5,8,13,21,34,55,89,144, and so on, each term being equal to the addition of the previous two. That is, I and I make 2, and I and 2 make 3, and 2 and 3 make 5, and 3 and 5 make 8, and so on. This is the series found in nature. For example, when we count the spirals formed of the seeds in a sunflower, we find that those arranged clockwise are 55 and the others arranged anti-clockwise are 89. Both of these figures are among the consecutive terms in the Fibonacci series. These figures may vary according to the size of the sunflower: we may find the figures of 34 and 55 in a relatively small flower, and 55 and 89 in a normal sized one, but the arrangement is always as consecutive numbers in the Fibonacci series. The spirals are arranged in pine cones in 5 to 8. We may encounter the same figures in the arrangement of tobacco leaves. Another extremely interesting characteristic is found in the numbers of petals of flowers. A lily has 3 petals, while a buttercup has 5, a velvet 13, a dahlia 21, and a daisy 34 or 55 or 89, varying according to its family. It is impossible to attribute this miraculous arrangement to chance or ignorant nature. If the DNA of a sunflower or a pine cone determines random numbers for its petals or spirals, how can you explain their correspondence with the terms of the series of Fibonacci? The ratio between the consecutive terms in the series of Fibonacci is quite near what is called the golden ratio’ and known in classical art as the ratio most pleasing to human eye. In order to explain the origin of this miraculous reality, you have to either accept that flowers know what is most pleasing to human eye or that the ‘Hand’ of One, the All- Knowing, the All-Wise and the All-Beautiful, is working in nature.
In short, what Fibonacci did is to discover this characteristic in nature. This means that the universe has a mathematical order or mathematics is the branch of science studying the miraculous order of the universe, the order which the Absolute Orderer and Determiner, One Who determines a certain measure for everything, has established.