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1
CONTINUOUS VERSUS
DISCRETE
The definition of a continuous quantity
we must distinguish what is continuous from what is discrete.
A natural number is a collection of discrete units.
5 people, 8 electrons, 100 pennies. Each one is separate and indivisible -- you cannot take half of it. If you do, it will not be that unit -- that same kind of thing -- any more. Half a person is not also a person.
We count things that are discrete: One person, two, three, four, and so on.
But consider the distance between A and B. That distance is not
made up of discrete units. There is nothing to count -- it is not a number of anything. We say, instead, that it is a continuous whole. That means that as we go from A to B, the line "continues" without a break.
Now, with a discrete collection, we can take only certain parts. Of 10 people, we can take only half, a fifth, or a tenth. When we divide a collection of discrete units, such as a group of people, we eventually arrive at an indivisible one, namely one person.
But since the length AB is continuous, we could divide it into any number of equal parts. Not only could we take half of it, we could take any part we please -- a tenth, a hundredth, or a billionth -- because AB is not composed of indivisible units. And most important, any part of AB, however small, will still be a length.
The idea of a continuum, or a continuous quantity, then, is that there is no limit to the smallness of the parts into which we could divide it. We imagine a continuum to be "infinitely divisible," which is a brief way of saying that no matter into how many parts it has been divided, it could be divided still further. And each part will itself be a continuous quantity of the same kind. Each part will itself be infinitely divisible
Common boundary
An essential characteristic of a continuous quantity, such as the line AC,
is that if we divide it at any point B, the right-hand boundary B of the part AB, coincides with the left-hand boundary B of BC.
In other words, the parts of a continuous quantity will have their limits or their boundaries in common. All parts are connected.
The lines AB, B'C do not share a common boundary, a common endpoint -- they are not connected. And so there is not a continuous line that joins A and C.
But if we join BB', then what were originally two endpoints, two
boundaries, become one. That contact allows AB to "continue" into BC without a gap.
The word continuous comes from a Latin root meaning held together. What is it that holds a line together to make it whole? Again, no matter where we might divide a line, the left and right endpoints, as B above, coincide as one.
In Lesson 3 we will see how that leads to the definition of a continuous function.
Magnitudes
Whatever has size, whatever could be larger or smaller, we call a magnitude. Magnitudes are of different kinds: distance, area, time, speed. We try to measure them. For that purpose, the rational and irrational numbers, what are now called the real numbers, were originally created.
The idea of a magnitude is that it is continuous. Therefore we expect that any part of a magnitude will be a magnitude of the same kind. The idea of time, for example, as conceived in physics and presented in calculus, is that even a trillionth of a second is still an interval of time.
We sum this up in the following definition of the concept of a continuous quantity:
DEFINITION 1. We say that a quantity is continuous if there is no limit to the number of times it could be divided, and 1) no matter where it might be divided, the parts share a common boundary, and 2) each part is a quantity of the same kind.
See the Problem below.
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That is the idea of a continuous line, which is that it is potentially divisible into any number of parts or intervals,
and each interval is itself potentially divisible. A line therefore is not composed of points, which are indivisible. "Point" is a convenient word when we need it, to refer to a specific place on a line, such as where we have divided it, or the boundary of an interval. But points do not exist until we point to them! In calculus, anything more than that is unnecessary. In fact, if points were in any sense real entities, then the "two" points, B, B' above, could not merge into one. (The discussion continues in the Appendix: Is a line really composed of points?)
That, at any rate, was the meaning of the word "continuum" that prevailed from the time of Aristotle until well into 19th century, when the abstractions of modernism found their expression in mathematics as well. Certain mathematicians adopted an exact opposite meaning for that word. They began with what they called "points," and they ascribed to them a primary logical existence. They defined a "continuum," and specifically a "line," as that which is "composed of points." What were then called the real "numbers" were actually the infinity of those "points." That is logic, which does not require that words have their conventional meanings (or any meaning, for that matter). It requires only that words -- such as "points," "number," "infinity" -- obey the formal rules of a language. A logical theory therefore may be nothing more than a game, even to the point of fantasy.
Problem. Which of these is continuous and which is discrete?
To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").
Do the problem yourself first!
a) The leaves on a tree. Discrete
b) The stars in the sky. Discrete
c) The distance from here to the Moon. Continuous. Our idea of
distance, of length, is that it could have any size, however large or however small.
d) A bag of apples. Discrete
e) Applesauce. Continuous!
f) A dozen eggs. Discrete. (But if they're scrambled?)
g) 60 minutes. Continuous. Our idea of time, like our idea of
distance, is that there is no smallest unit. Any part of 60 minutes is still time.
h) Motion from one place to another. Continuous. The idea of any
quantity of motion is that there is no limit to its smallness.
i) Pearls on a necklace Discrete
j) The area of a circle.
As area, it is continuous; any part of an area is also an area. But as a form, a circle is discrete; half a circle is not also a circle.
k) The volume of a sphere.
As volume, it is continuous. As a form, a sphere is discrete.
l) A gallon of water.
Continuous. We think of volume as having any part. And any part is still a volume of water.
But:
m) Molecules of water.
Discrete. In other words, if we could keep dividing a quantity of water, then ultimately (in theory) we would come to one molecule. If we divided that, it would no longer be water!
n) A chapter in a book.
Discrete. Surely, half a chapter is not also a chapter.
o) Events.
If you think that half an event is also an event, then you will say that an event -- such as a birthday party -- is continuous. (We are not speaking of the time in which the event occurs. We are speaking of the event itself.) Otherwise, you will say that events are discrete.
p) The changing shape of a balloon as it's being inflated.
Continuous. The shape is changing continuously.
q) The evolution of biological forms; that is, from fish to man
n) (according to the theory).
What do you think? Was it like a balloon being inflated? Or was each new form discrete?
r) Words. Discrete.
s) Ideas.
If you think that the hundredth part of an idea is also an idea, (Really?), then you will say that ideas are continuous.
t) Meanings.
Discrete. Half a meaning?
u) The proof of a theorem.
Discrete. Half a proof?
v) The names of numbers.
Surely, the names of anything are discrete.
w) The universe.
Discrete. Is half a universe also a universe?
Apart from our conceptions of time, space and motion, we see that virtually everything we encounter is discrete. Even a motion picture -- where the figures on the screen appear to be in continuous motion -- is made up of individual frames, which are discrete.
Calculus, however, is the study of magnitudes; of things that are continuous.
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Copyright © 2012 Lawrence Spector
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