Nature Will Not Imagine
Mathematicians can be pig-headed. Sometimes there are problems
that admit of no solution, but the mathematician ignores common
sense and presses on regardless.
One such question would be "WHERE, ON A UNIT CIRCLE, IS A POINT
FIVE UNITS TO THE RIGHT FROM THE CENTRE"?
Pythagoras gave an answer. The hypotenuse is a unit, so you
square it. The square is still a unit. Now you look at the
position to the right. It is five units. You square it to
twenty-five and take it away.
Then you find the square root.
But one minus twenty-five is MINUS TWENTY-FOUR. What is the root
of a negative number?
There is no answer!
But common sense tells you that nothing can be on a circle if it
is further from the centre than the radius. The answer "NOT ON
THE CIRCLE" leaps to mind.
Girolamo Cardano was a mathematician who was even consulted by
Leonardo da Vinci. He was also a gambler, but used his knowledge
of probability to get the better of his opponents. He decided
that he had the solution.
You split up the problem into two parts. The root of 36 is 6.
However, one can try the root of 4 times 9 - in which case one
has the root of 4 times the root of 9. The answer is the same.
So you look for the root of -1 times the root of 24.
The root of any negative number will be the root of -1 times the
root of the absolute value of that number.
So the root of -36 is the root of -1 times six.
The root of -1 is defined as IMAGINARY, so the root of -36 is
IMAGINARY SIX. And the answer to the position on a unit circle
of the point X=5? That will be Y=IMAGINARY ROOT 24.
Cardano was scoffed at. He died. The years went by, and slowly
the mathematical community began to realise that the idea was
not quite so bad after all.
It was a NEW TECHNIQUE. To solve problems, you simply follow the
rules slavishly. These rules began to be taught in the
universities. People were taught that the old numbers - the REAL
ones - had to be kept separate from the imaginary ones. They
learned to describe a two-part number having real and imaginary
as a COMPLEX NUMBER.
The rules say that whenever you multiply real by real you get
real. Multiply real by imaginary (of imaginary by real) and you
get imaginary. Multiply imaginary by imaginary and you get MINUS
real.
Simple rules, easy to apply.
But how does this impact upon the whole of mathematics? What
happens to sines, cosines, logarithms and other things when you
use complex numbers in place of real ones? A host of great
mathematicians - including the legendary Leonhard Euler - worked
on this question. Complex mathematics was becoming established
as a standard tool.
Problems began to become apparent. The simplistic concept of
ROOT -1 was actually wrong. It should have been "ONE OF THE
ROOTS OF -1". There are actually two.
Then the problem of multiple solutions appeared. As I wrote at
http://www.wehner.org/euler/ , this problem can be visualised by
considering the antilogarithm of an imaginary number as a SPIRAL
IN COMPLEX SPACE. What is the lowest point? One for every turn
of the spiral. Therefore multiple solutions.
I then continued the research by delving deeply into Euler's
GAMMA FUNCTION. Here, for numbers below zero, the result negates
in unit steps. I called this the NEGATION FUNCTION, which is
also a spiral in complex space.
I posed the riddle, how do you create a diagram of the Gamma
function when it spirals in one place and not in the other?
I solved that riddle with a surprising answer. NATURE IN THIS
CONTEXT REJECTS THE IMAGINARY. There is no solution to the Gamma
Function if you use complex maths. There is no complex space.
That is to say, the square root of -1 has two answers, but
Nature insists that you must take them BOTH AT THE SAME TIME.
This is quite different to the roots of real numbers. The roots
of 4 are 2 *OR* -2.
The roots of -1 are i1 *AND* -i1. Take BOTH or NONE, in the case
of the Gamma Function. Otherwise, the riddle is insoluble.
But i1 makes the spiral go left whilst -i1 makes the spiral go
right. Making the spiral go equally left and right will STOP it
from spiralling. So both is the equivalent of none. Nature has
REJECTED the imaginary.
At this point I wrote (http://www.wehner.org/euler/solution.htm
) that "If it can be shown for other functions unrelated to the
Gamma function, and having a negation function, that Nature
rejects the imaginary axis in this way, then it can be said that
Nature rejects complex mathematics as a human product. "
I had not expected that Nature would then reveal to me a
function that rejects the imaginary axis without having a
negation function. It is the INVERSE function to the
exponential. That is to say, it is the LOGARITHM.
>From the first page above, you can visualise Euler's equation
for the exponential (exp(i X) = cos(X) + i sin(X)) as a spiral.
You can see that exp(i Pi) gives -1. So does exp (i 3Pi). So
does exp (i 5Pi). So do ALL the odd imaginary multiples of Pi.
So do all the negative values of those odd imaginary multiples.
So the log of -1 is ALL of those positive and negative odd
imaginary multiples of Pi. It does not really matter whether
they are AND or OR. It only matters that they are there.
I then considered the logarithm of zero. This is known to be
MINUS INFINITY.
But what of MINUS ZERO? This may seem like a silly question, but
it is not. We have considered the log of any negative number to
be the log of the positive equivalent plus one or more odd
imaginary multiples of Pi. So the log of minus zero will be
minus infinity just like the log of zero - but with a DIFFERENCE.
For all the negative numbers, the logarithm is as for the
positive but TOGETHER WITH THE SERRIED RANKS OF ODD IMAGINARY
PIs. As the numbers rise, they get closer and closer towards
zero, and the real part of the logarithm goes downwards towards
minus infinity.
Then they hit zero. Those serried ranks of imaginary numbers
STOP. That's it! They VANISH.
One would call this a SINGULARITY, but it is more than that.
There are INFINITE odd imaginary multiples of Pi, and they ALL
stop together at the point X=0.
So Nature again is REFUSING the imaginary.
Natural laws tend to flow smoothly throughout the number
continuum. For an entire family of numbers - all odd imaginary
multiples of Pi - and therefore by inference a whole SET - the
set of all imaginary - to stop is quite unnatural.
So the imaginary numbers are the imaginings of Humankind. Nature
does not imagine.
Charles Douglas Wehner