2 Using Everyday Words
Our next aim is to show how everyday words should be used in mathematics to
describe numbers and quantities - their use here is close to their everyday
meanings. For example, we can say if a number or quantity is known or not,
changing or not, constant or not, increasing, decreasing, shrinking, growing,
confidential or embarrassing, top-secret or simply forgotten. Everyday words
give the descriptive vocabulary of mathematics. Describing and talking about
quantities and numbers is a part of mathematics after arithmetic. More examples
follow.
2.1 Airplanes or Jets
We can speak about the height of an airplane above the ground. We can speak
about it without measuring it and without knowing it exactly. The height will be
zero when the airplane is on the ground. This height increases as the plane
takes off. The height will then remain almost unchanged and nearly constant when
the plane has reached its maximum height or cruising altitude. Then at the end
of the trip, the height of the plane will decrease (get smaller) until the
plane, we hope, gently lands.
2.2 People
We can also speak about the number of people in a room. When nobody enters or
leaves, this number remains constant. When somebody enters or leaves, this
number varies. This number or count is usually a whole number or zero. When
someone is just leaving and partly in and partly out of this room, we cannot
count or we have to allow fractions.
When we speak about the number of people in a room do we mean completely in,
do we include fractions, or do we just say the count cannot be done at those
moments when someone is partly in or out, moving or not? This number or count
needs to be clearly defined. Words are needed to say precisely how it is
computed, otherwise ambiguity results.
2.3 Height
When a building is being constructed, its height is increasing. The
construction and the increase in height of the building may take place over one
or two years. While the building is used, say seventy years, its height may be
constant - unchanging. At the end of the building's useful life, the building is
left to fall down or it is demolished - torn down. Here over a long or short
time, the height decreases.
The height of the building varies. This height is therefore a variable during
the construction and the demolition (collapse or falling down) of the building.
The height is usually a constant, unchanging and invariable quantity during the
seventy or so years that the building is used.
The height of the building may or may not be known to us during the lifetime
of the building. Yet we can still refer to the height of the building, and to
its other dimensions, even if we have not measured these quantities and even if
they are unknown to some or all of us.
Here are some more questions, just for fun. What do we mean by the height of
the building? Before the building is built, can we talk about its height? Can
the height be taken to be zero? When the building is being built, is the height
of the building equal to the height of its walls as they are being put up? If
the building has a basement or a foundation, do we say the height of the
building is negative or is it undefined while the basement is being dug, or the
foundations being built? When the building is being demolished, does it have a
height? What is it?
What do we mean by height? Better yet, we can speak of the height of a
building whenever we can say what it represents (means) and/or how we might
measure it. This permits us to speak of the current height, the planned or
intended height, the past height, the future height. Is the height of a
demolished building zero, or undefined? Is the planned height of a building
equal to its actual height before construction, during construction, during its
use or during demolition? A definition or identification of the height we want
to speak about, is needed.