What is the relationship between the ‘real’ world and the mathematical world?
If you studied physics at school you may have performed experiments with lights boxes and weights and other such things to understand how the world works. If you went on to study physics at university you would probably find that most of the time you are no doing experiments on real things but doing a lot of applied mathematics. How is it that by performing mathematical problems we can predict outcomes in the real world?
Imagine that you were walking through the hallowed corridors of a college at Cambridge University sometime in the last century and an excited person comes running up to you and says that he’s discovered something amazing about black holes. He drags you into his room and shows you sheets of paper with calculations. There, he says, I’ve proved that black holes emit radiation. Now you don’t understand the maths. How do you know he’s proved what he says he’s proved and not that ice cream will melt at room temperature? Done correctly he has represented a real-world entity with a mathematical representation and then manipulated the representation to produce an answer. The key is in the transformation of the real world into the mathematical world.
John and Kate have apples. We know who owns what apples because each has box next to them containing the apples. If John has five apples and Kate gives him two apples we know that John will have seven apples (assuming that John doesn’t eat any of the apples!). We can do the calculations and when we go back from the mathematical world to the real world and count the apples in John’s box we find that our calculations were correct.
If John has five apples and he gives Kate seven apples, how many apples does John have? One answer is that he can’t do it. Another answer is that John owes Kate two apples. Suppose John gives Kate two real apples and an IOW for two apples. Kate now has a piece of paper which represents negative two apples. The paper only has any meaning because John and Kate have given it this meaning.
Alternatively john and Kate could simply remember that John owes Kate two apples. Owing is a state of mind so in our conversion from the mathematical world to the real world we’ve done something very strange: instead of converting to real, solid apples we have changed something in our minds. We could say that we have mathematically proved karma — the concept that there is a mathematical sum (positive and negative) that is taken of your life. Not just your activities but also your intentions.
Numbers could be considered as being on a line which stretches to infinity in each direction. At zero the numbers change from positive to negative.
It became clear that there were inconsistencies in the number theory and mathematicians had to fill in gaps in our concept of numbers in order to retain consistency. For instance, it quickly became obvious that if we have whole numbers (what are called integers), we also have to have fractions and it also became obvious that we have allow for so-called irrational numbers, such as pi.
An irrational number is a number that can’t be expressed as a ratio. Examples of rational numbers are:
½ = 0.5
¼ = 0.25
√4 = 2
√9 = 3
Examples of irrational numbers are:
√2 ~ 1.414213562373095…
√7 ~ 2.645751311064591…
π ~ 3.141592653589793…
Pi is the relationship between the diameter and the circumference of a circle and cannot be described with compete accuracy to any number of decimal places.
However to maintain a consistent number system, mathematically there is also a concept of imaginary numbers. The square root of a number (shown as √) is the number that, when multiplied by itself, equals the number squared. In fact, each square root of a positive number has two square roots: the positive number (√4 = 2) and the negative number (√4 = -2), So for example:
2 X 2 = 4, 2^2 = 4, √4 = 2 or -2
3 X 3 = 9, 3^2 = 9, √9 = 3 or -3
10 X 10 = 100, 10^2 = 100, √100 = 10 or -10
1 X 1 = 1, 1^2 = 1, √1 = 1 or -1
-1 X -1 = 1, -1^2 = 1, √-1 = ?
The number that when multiplied by itself equals -1 is called i1 (i for imaginary). Graphically it is usually depicted as shown in the diagram. We can see that there is a complete duplicate of our normal numbers but in an imaginary plane. So every object has its mathematical counterpart in the imaginary number plane. We could consider that this is a world that exists at right angles, as it were, to our world.
We have said, even if just as an exercise, that there is a relation between the mathematical world and the real world. Of course the dispute would be that even though we can perform these mathematical exercises, how do we translate the result back into our real, non-mathematical, world?
The why and the how
Consider the following scenario:
Imagine that you bump into an old friend who you haven’t seen for some years. You ask them how things are and they tell you that they have recently become divorced. You ask them “how did you get divorced?”. Assuming they answer the question you asked, they would reply with an account of the proceedings: they filled out some forms, went in front of a magistrate, etc. It would be a factual account: times and places that are verifiable. If you asked either partner you would probably get a similar answer. If you asked their friends and children, insofar as they knew, you would get a similar account as well. This kind of response is left-brained.
Now suppose you asked them ‘Why did you get divorced?”. You would get a very different answer. The husband and wife may give totally different answers, the friends may give different answers still, as would the children. The answers would not be factual or probably verifiable. They would be opinions and feelings. This kind of response is right-brained.
The two ways of seeing the same event are like two universes that exist in right angles to each other. Meeting at zero but with no other point of reference.
These two ways of seeing permeates every facet of our existence. Many people go through life completely unaware of this parallel universe that runs alongside our own.
In this example one interpretation is to consider the ‘real’ numbers to be the ‘how’, and the imaginary numbers to be the ‘why’. Science is very good at the ‘how’ but the ‘why’ is outside of science — it is metaphysics, literally outside of or beyond physics.
By Philip Braham on July 24, 2018