Is mathematics a "real" science? if it is, why and how?
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.
Theories about the origin and essence of mathematics
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 socalled
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.
The language of numbers
Formalists assert that numbers came into existence when human beings began to
count. A wellknown 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 ten
fingers on 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 arithmetic.
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 1 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.
A way of checking the correctness of an operation of addition
In order to check or prove an 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 want 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.
Numbers have many characteristics only some of which have been discovered
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 mathematician study.
There have always been correct mathematical expressions
Most of the distinguished mathematicians follow the view of Plato. Kurt G�del
is one of them. Before G�del, it was almost a generally accepted view that
mathematics is a function of the working of man�s brain consisting in the
collection of the logical rules which we establish between the symbols of two
sets. G�del 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.
What flowers reveal
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, 1 and 1 make 2, and 1 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 anticlockwise 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 nearly 4/3, which 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 AllKnowing, the AllWise
and the AllBeautiful, 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.
