An Imaginary World
Many years ago the philosopher Abu Hamid al-Ghazali
(1058-1111) said that mathematicians suffered from pride. Their proofs led them
to believe that they had gone beyond religion. Nowadays many mathematicians believe
that they have discovered the ‘mind of God’. This arrogance comes about because
they believe that mathematics is the law by which the universe works. The physical
universe is translated into a mathematical model, and by understanding this model
they believe they can translate this back into the real world and deduce how it
works. Such physical entities as black holes were discovered in this way. I've
mentioned before (“The Conjuring Trick”) the traps that such a belief can lead
to. Many (though not all) of these mathematicians are sceptical about things such
as psychic phenomena as they claim they don't fit in with their mathematical models.
The belief in a mathematical model of the universe has to be complete. You can’t
pick and choose which entities you are going to accept and which you are not.
If the model predicts something than it has an existence, or at least a theoretical
existence. A calculation may produce, for example, negative distance. Negative
distance doesn't exist in actuality but the concept of negative distance lets
us perform calculations that involve moving backwards. Even the concept of negative
time is essential to understand the idea of yesterday.
However, there is a part of the mathematical model that represents an alternative
universe but is usually ignored by mathematicians, and that is the concept of
imaginary numbers. Those who have an understanding of maths may skip the next
few paragraphs. For those who don't I will explain the concept very simply.
In maths a squared number is a number multiplied by itself. 2 squared is 2 multiplied by 2 which is 4. 3 squared is 9. A square root of the number is the number that when squared equals it. So the square root of 4 is 2, the square root of 9 is 3; the square root of 64 is 8.
When we multiply a negative number by a positive number the result is negative. So -3 multiplied by 2 is -6. A negative number multiplied by another negative number is positive. -3 multiplied by -2 is 6. This makes sense when you look at real-life examples. If we go backwards at twice the speed we would cover twice the negative distance. If we go backwards and then move backwards from that, we move forward. So it follows, then, that a negative number squared is positive. -3 squared is 9. It is worth noting that 1 squared is 1, and -1 squared is also 1. So the square root of 1 is also 1.
What, then, is the square root of a negative number? In other words, what number, when multiplied by itself, produces a negative result? The answer is no number that we normally come across. They are called imaginary numbers and are used in some maths equations – simply because it is the only way they can be solved - but generally are otherwise avoided.
When I said earlier that the square root of 1 is 1, that is not strictly true. 1 actually has two square roots: 1 and -1.
Einstein's formula, e = m c squared, calculates the amount of energy (e – in Joules)
that is produced when a certain amount of matter (m – in Kilograms) is converted
completely. The mass is multiplied by c (the speed of light in metres per second)
squared, a huge number. We can transpose this formula to yield the speed of light:
c = square root of (e divided by m). But this gives two answers for the speed
of light. A real one (100,000,000 metres per second) and an imaginary one (the
same speed but in the imaginary plane).
Mathematicians usually ignore this, but these imaginary number represent the keys to an alternative universe that exists, as were, at 90 degrees to our own. We rarely see it but it is there, and without it the universe could not exist.
