Let us take a string
8 units in length. Pluck
this string, it produces
a sound. Take a string 4 units in length, and pluck
it. It produces
a sound also. The frequency
of the first tone is exactly half the
frequency of the second tone.
Imagine a object moving at a constant speed that emits a constant
tone. To an listener on that object, the tone remains constant. To
a listener that stands near it, the tone will first grow higher in pitch
and then lower as it passes you.
Imagine a point. Imagine all the points that are a unit distance from
the original point. Let us give this collection of points the name
The distance an object travels when it is thrown is a function of the
initial velocity of the object and the angle at which it is thrown.
None of these concepts involve the faculties of our vision. Advanced
concepts in math and physics do not require sight. Indeed, even the
physics that we now probe, it is impossible to use sight to 'see'
these things. Indeed, if a being is so limited in its
imagination that it cannot conceive of these things without
some tangible (is a circle tangible?) evidence, it is possible to
draw them out on a metal sheet, or carve them in wood or wax so that
it can be felt with hands.
Do not prejudice yourself with the senses that you have - the logic
necessary to grasp upon foundations of math, geometry, and calculus
(and thus physics) are easy to come by.
Let us imagine a scientist, who through some disability is prevented
from using the tools that you and I use every day. It is difficult
for him to write on paper, to type on a keyboard or to use a calculator.
Yet, without these tools that we use and depend on, he has found different
ways of thinking that go far beyond the limits of the tools that we use.
His name is Stephen Hawking.