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MATHEMATICS
IS REAL: 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 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.
The
language of numbers
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 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 which was
discovered by a Turkish Sufi scholar of
18th century and may still be unknown to
modern mathematicians
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.
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 that mathematician study.
There
have always been correct mathematical
expressions
Most
of the distinguished mathematicians
follow the view of Plato. Kurt Gödel is
one of them. Before Gödel, 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. Gödel 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
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 nearly 7/4, 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
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.
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