Two examples from the world of mathematics are given below to clarify Albert Einstein’s famous words:
How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? … In my opinion the answer to this question is briefly this:-
As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.
Albert Einstein; Sidelights on Relativity; p28; Dover; 1983
Numbers are used in two ways:
1- To measure the distance between two points and
2- To count discrete objects
The period of a pendulum, or the time that it takes for a pendulum to travel from one end of its swing to the other side is given by the simple equation
T = a Ö L
Where T is the time it takes for a full swing of the pendulum, and L is the Length of the pendulum rod from its apex to the center of the weight. a is an arbitrary constant.
This equation is derived based upon the following Four assumptions:
1- The weight of the pendulum is concentrated at a geometrical point. That is, the weight is assumed not to occupy any volume;
2- Any tension in the rod or cable on which the weight of the pendulum is hanging is assumed to be zero;
3- Any friction at the apex where the rod is swinging is assumed to be zero.
4- The pendulum must swing a very small arc.
If these Four assumptions could exist in reality, then the above equation for a pendulum would tell us the exact period of a pendulum. Thus we see a clear example of a mathematical equation that is referring to reality, a swinging pendulum that we can clearly observe, but the equation that describes the motion of the pendulum is not certain because there exists no pendulum that complies with the above assumptions.
Someone might ask why can’t we derive an equation that describes a real pendulum with Friction, Tension and a weight that has Volume. In reality, such a task is impossible because the motion of a real pendulum is far too complicated for any mathematical system to be able to describe.
There can hardly be a more certain equation in mathematics than 1+1=2. No one would argue with that fact.
The problem arises when we try to apply it to reality!
1+1=2 says that we have in front of us two objects, each object is identical to the other, and neither object is changing in any way over time. If that was the case, then we can say that we have 2 objects.
Unfortunately no such objects exist in nature. There are no Two Identical Objects anywhere we look. Furthermore, each object that we do see is changing continuously, no matter how imperceptible it may be to us.
1+1=2 is as certain as any fact of Mathematics, but it does not refer to any reality with which we are familiar.
As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.
Albert Einstein; Sidelights on Relativity; p28; Dover; 1983;
ISBN 0-486-24511-X
