Alice laughed: “There’s no use trying,” she said; “one can’t believe impossible things.”
“I daresay you haven’t had much practice,” said the Queen. “When I was younger, I always did it for half an hour a day. Why, sometimes I’ve believed as many as six impossible things before breakfast.”
Today’s VERITAS is inspired by a brilliant essay by the eminent Physicist Eugene Wigner : “The unreasonable effectiveness of mathematics in the natural sciences”. In this essay Wigner expresses surprise at how successful a tool created by the human brain(Mathematics) can be at explaining nature(Physics).
How come our brains are able to understand so much. How come the mathematics invented in a year x becomes useful in explaining physical phenomenon in the year x+50? Isnt it a wonder.
In his essay Wigner tells the story of two friends who land into different fields: one becomes a statician and the other a shopkeeper.
After some years they meet again and the statistician shows the shopkeeper his latest research paper on population statistics. The shopkeeper does not understand much but finds some funny looking formulae in the paper. He turns to the statician and asks him ” what is this?”.
The statistician says ” This is pi”. The shopkeeper says “what is pi?”.
The statistician says ” it is the ratio of the circumference of a circle to its diameter”. The shopkeeper is shocked ” SURELY THE CIRCUMFERENCE OF A CIRCLE HAS NOTHING TO DO WITH POPULATION!!”.
Is it now strange how our small set of mathematical contants find themselves everywhere.
In VERITAS today we ask ” If imaginary numbers are not “real” then why are they used everywhere(electrical engineering, Physics etc)?”
A complex number(or an imaginary number) is the square root of -1. They were invented by Mathematicians for their own enjoyment(maths). The Maths guys had invented them because they wanted to solve all algebric equations including the one : x^2 + 1 = 0
Okay so these are not real numbers. So why do we use them?
They are useful because of a most beautiful formula(Feynman calls it a jewel):
exp(ix) = cos(x) + i sin(x)
where i is the square root of -1.
What this means is that whereever you see a sin or a cos you can replace it with exp(ix) and do all your calculations and in the end take the real part of the result.
But why would anyone want to do that?
We do this because calculations with exp(x) etc are very easy compared to the calculations with sin , cos and such. for example the differential of exp(ax) is a*exp(ax): so differentiation is as simple as multiplication. Similarly exp(x) * exp(y) = exp(x+y) : multiplication is as easy as addition.
So in any equation we can change the sin and the cos to the exp(ix) and do all calculations and in the end take the real part as the answer.
So what I am saying here is that complex numbers make Physics simple but they are not necessary. Even if complex numbers were not invented things would still have been okay(**well almost! see later :-)**)
Now when I first read about negative numbers I felt uneasy. How can there be less numbers than 0? Later we understood that negative numbers mean that you have made loss or that you have less than an arbitrary assigned 0.
So in case of negative numbers we assigned a number not to a quantity but to a concept( the concept of loss etc). You cannot see a negative number in terms of set of pencils etc.
Similarly imaginary number is a number assigned to a concept. The concept that there is a “something” whose square is -1. By itself it may have no meaning. But the concept is useful when it is a part of a larger concept( a larger equation perhaps).
The human brain has gone beyond the number of fingers in our hands. Now we can think in terms of more abstract things and that has helped us understand nature.
Now I will tell you something that makes me uneasy. I said earlier that complex numbers are used only to simplify the equations of Physics. And that they have no “real” meaning. But the equations of Quantum Mechanics(the study of subatomic particles) NEEDS the concept of imaginary numbers! Without complex numbers Quantum mechanics cannot be formulated! I find this very very crazy! Is God making a fool of us by making us play with the toys that we invent.
And I ask myself and the rest of the readers of VERITAS the question: “WHY IS MATHEMATICS SO SUCCESSFUL??”
Alice: If I had a world of my own, everything would be nonsense. Nothing would be what it is, because everything would be what it isn’t. And contrariwise, what it is, it wouldn’t be, and what it wouldn’t be, it would
Go, wondrous creature! mount where Science guides:
Go, measure earth, weigh air, and state the tides:
Instruct the planets in what orbs to run,
Correct old time and regulate the Sun;