Appendix 1

CONTINUOUS VERSUS DISCRETE

WE MEASURE things that are continuous. Therefore we need fractions, which are the divisions of number 1 into parts. We count things that are discrete. What exactly is the difference?

Half a chair is not also a chair; half a tree is not also a tree; and half an atom is surely not also an atom. A chair, a tree and an atom are examples of a discrete unit. A discrete unit is indivisible, in the sense that if it is divided, then what results will not be that unit, that thing, any more -- half a person is not also a person

What is more, a collection of discrete units will have only certain parts.  Ten people can be divided only in half, fifths, and tenths. You cannot take a third of them.

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.

Since the length AB is continuous, 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.

This distinction between what is continuous and what is discrete makes for two aspects of number; namely number as discrete units -- the natural numbers -- and number as the measure of things that are

continuous.  That gives rise to the "fractions." We do not need fractions for counting.  We need them for measuring; for assigning a number to whatever is continuous.

Problem 1.

a)  Into which parts could 6 pencils be divided?

Halves, thirds, and sixths.

b) Into which parts could 6 meters be divided?

Any parts. 6 meters, which is a length, are continuous.

Problem 2.   Which of these is continuous and which is discrete?

a)  A stack of coins   Discrete

b)  The distance from here to the Moon.

Continuous. We can imagine half of that distance, or a third, or a fourth, and so on.

c)  A bag of apples.   Discrete

d)  Applesauce.   Continuous!

e)  A dozen eggs.   Discrete

f)  60 minutes.

Continuous. Our idea of time, like our idea of distance, is that there is no smallest unit.

g)  Pearls on a necklace.   Discrete

h)  The area of a circle.

As area, it is continuous; half an area is also an area.  But as a form, a circle is discrete; half a circle is not also a circle.

i)  The volume of a sphere.

As volume, it is continuous. As a form, a sphere is discrete.

j)  A gallon of water.

Continuous. We imagine that we could take any part.

But

k)  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 not be water any more!

l)  The acceleration of a car as it goes from 0 to 60 mph.

Continuous. The speed is changing continuously.

m)  The changing shape of a balloon as it's being inflated.

Continuous. The shape is changing continuously.

n)  The evolution of biological forms; that is, from fish to man (according
n)  to the theory).

What do you think? Was it like a balloon being inflated? Or was each new form discrete?

o)  Sentences.

Discrete. Half a sentence is surely not also a sentence.

p)  Thoughts.   Discrete. (Half a thought?)

q)  The names of numbers.

Surely, the names of anything are discrete. Half a name makes no sense.

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