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Mathematics and Reality |
“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:-
Numbers are used in two ways:
1-
To measure
the distance between two points and
2-
To count
discrete objects
An example of the fact that when mathematics refers to reality, it is not certain.
The period of a pendulum, or the time that it takes for a bob to travel from one end of its swing to the other side is given by the simple equation
T = (a) sqrt(L)
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 bob is assumed not to occupy any volume;
2- Any tension in the rod or cable on which the weight of the bob 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 an 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 bob 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.
[…] in evaluating the various systems of modern physics which all employ a mathematical approach to nature, it is important to
underscore in bold type that the most salient fact about our universe, the motions, resists any reduction to mathematical categories.
The second example will show the fact that when a mathematical equation is certain, it does not refer to reality.
Vincent Edward Smith; On the Mathematical Approach to Nature; in
Progress in Philosophy; 1955; p115
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.
If we are to add at all, we must add
unlikes,
in violation
of all mathematical regulations.
Billy E. Goetz; President of MIT 1958; The Usefulness of the Impossible; 1963; p189
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; (1879-1955); Sidelights on Relativity; 1921/1983; p28
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