Higher
Viewpoints: Part Three
Numbers, Operations, and Rules in Arithmetic and Algebra
Draft Version 1
by David Fleischacker
Copyright © 1997. All rights reserved
I. Numbers, Operations, and Rules in Arithmetic
As marks on a page, "1" and "3" mean nothing. But when they come to
represent patterns, are fruits of operations, then each represents a number.
Lonergan writes that numbers are defined by operations, and operations by rules.
What exactly does he mean? It is important to grasp the precise meaning of these
terms because Lonergan uses them to define higher viewpoints.
A. Operations in Arithmetic
Let us examine the previous work on arithmetic and
algebra. In arithmetic, we spoke of addition as a basic operation and
subtraction as its inverse. Examination of our own learning will help us to focus upon the
precise insight that results in what is called addition. Let us begin with sheep.
When I examine a sheep, I can describe its color, height, the sound it makes,
the hair that grows on its back, the shape of its eyes, or their general incomprehension
of the world. I also recognized that this sheep is a single sheep, the subject
of all these characteristics that I just mentioned. This unity, this being a
single-unified-subject is important. If I see the sheep moving to another location,
I still identify it as the same sheep. If its wool is sheared from its back,
I still say that it is the same. Something allows me to identify it as one and
the same subject that possesses these properties, and remains the same when
some of them change. Recognizing this unity presupposes a residue that remains
part of the sheep even when I grasp its unity, namely that it is an individual
(for those of you who are familiar with the empirical residue, this individuality
is the residue that remains a residue when unity-identity-whole has been grasped).
It is this individuality that becomes the basis for my recognition (again, for
those familiar with ch. 13 of INSIGHT--the principle notion of objectivity)
that this sheep is distinct from others. It is also the basis from which emerges
the notion of one in arithmetic.
I have not yet said that this sheep is numerically one sheep, because I have
reserved that particular characteristic for highlighting the fact that this
notion does not arise until one introduces the operation of addition. To focus
attention upon this, let us distinguish both unity of subject and individuality
from the number 1. As noted, unity is what makes something the subject of many
characteristics, and the same subject through a change of those characteristics
(although if too much change occurs, the unity is destroyed). Sometimes, the
term "one" is used to express this unity. The sheep is one, whole,
unified. This is a distinct meaning from the use of the term "one"
to represent a number. One as a number is always tied to other numbers. One
as unity is not. They do have something in common. As already noted, unity is
directly linked to something which is a residue which is called individuality.
Whenever experiencing a particular sheep, in its individuality of data, a unity
of that data emerges. The unity does not negate the individuality. It is individual
because we have recognized it as unified, but to recognize it as unified is
not to say why it is still an individual. The individuality is a mere fact,
given in the existence of this sheep. This individuality, as noted above, is
the condition for grasping that this sheep is not another sheep, or anything
else for that matter (for many things grasping this unity and individuality
is rather easy, but sometimes this is not always true, as when one is trying
to put the pieces together in a crime, or when studying distant stars, or in
examining atoms and sub-atomic particles). This individuality is the basis for
saying that this is not that. This is a presupposition in mathematics, which
starts from this distinctness and begins to create patterns out of it through
such operations as addition.
The distinctness is that from which arithmetic begins to create its patterns. A
distinct thing becomes one distinct thing when it is related to another
distinct thing. As children, we are first taught this creative relating by using phrases
such as "putting together." The meaning of "putting together" can be
learned in any variety of manners. It is similar to learning the meaning of the color red.
To learn the color red, any variety of objects that differ in shape, size, texture, smell,
and sound will do, as long as they are red. By pointing these things out to a child and
calling them red, he or she will eventually get the point. Without the variation, getting
the point will be extremely difficult. Try teaching a child the meaning of red without
using any variation of the object. Pointing to a red ball for instance may lead the child
to think that its name is red or its shape is red or its size is red. Only if the child
already knows the meaning of color, or is able to distinguish the meaning of red from
shape, size, texture, etc., is there any chance of grasping the meaning of the term red.
Similarly with the meaning of "putting together." At first, a child may learn
the phrase from requests of parents to put the toys together into a box, or from cleaning
a room, or from stacking blocks to form a small structure. In any case, the child is
learning to create an order from dis-order. It is this meaning of "putting
together" that becomes suitable for analogous usage in arithmetic. The parent or
teacher can say to the child that to add you need to put this distinct thing with that
distinct thing, and then we have two. This may need to be introduced through a variety of
manners. First calling one sheep 1, then introducing a second sheep as another 1, and
saying that together they make 2, then changing objects, perhaps reversing the sheep that
is first called 1, etc.. The variation that good teachers learn well is what helps the
child to get the image that is at the basis of the insight into adding, and from this
operation, the child only then understands what is meant by the term "1" and
what it means to call this sheep "1" and that adding it to a second sheep is
"2."
Notice what happens to the meaning of the term "1" when the child fails to
grasp the meaning of addition. The child could interpret it to refer to the unity of the
things, but he or she will not grasp why adding this unity to a second distinct unity
yields the number 2. The child may simply associate 2 with seeing two sheep. Or the child
may think that 2 refers to things that look similar, or things that sound similar (or feel
alike). Abstracting from these elements to what is meant by the number one and the number
two requires a great deal of variety, and a number of previous insights.
The point of this is to highlight the importance of the operation of addition for
grasping the meaning of numbers. As noted, Lonergan writes the operations define numbers,
and only by the insights that ground these definitions, does one really grasp the meaning
of a number. Only then can one call the sheep numerically one.
Once the operation of addition is grounded in an insight that can be described as
"putting together," further numbers can be defined, and one can go on to define
all of the positive integers (It would be good to spend some time creatively
reconstructing the steps in learning the meaning of addition).
B. Rules in Arithmetic
The focus has been upon the operation of addition, yet, as noted, Lonergan says that
operations are defined by rules, and so what rule defines addition? I believe the rule has
already been expressed by the phrase "putting together." Notice, when one uses
this phrase, it does not refer to what one is putting together (it does
not define a particular number), but refers to the activity, or the operation itself. It
expresses that operation, or defines it. When actually carrying out what is defined by the
expression "putting together," one is then actually defining numbers, hence one
is operating, or actuating the operation of addition. So, to distinguish operations and
rules: An operation occurs when two numbers are defined; A rule is when the operation as
such is defined.
One can go on to articulate the rules that define in arithmetic operation. To recall
from an earlier essay ("the Transition from Arithmetic to
Algebra"), multiplication is the addition of a number to itself so many times.
Taking the power of a number is to multiply a number by itself so many times. Subtraction
is removing a number from another. Division is removing a number from another so many
times. Roots are dividing a number by itself so many times. These form the basic rules of
arithmetic. Implementing these actual rules is to carry out the operations that define
numbers, and hence results in numbers. Now let us turn to algebra.
II. Numbers, Operations, and Rules in Algebra
Lonergan describes the move from arithmetic to algebra as a shift in the rules which
thereby redefine the operations, and hence the numbers. As a note, the numbers now shift
to variables, and hence the operations of algebra define variables, and the rules define
the operations. It results in these rules being "more symmetrical," "more
exact," and "more general" than those of arithmetic.(1)
In the first essay of this series ("the
Transition from Arithmetic to Algebra"), I noted that algebra begins with
problems that arise from the homogenous expansion of arithmetic. Negative numbers,
fractions, and even the meaning of addition are placed into questions that lead to the
recognition of patterns in arithmetic. One can begin with the rather simple pattern that
any number multiplied by 1 always results in the number. We can expand to the associative
and commutative properties of addition and multiplication. Multiplying a number by 1, or
the associative and commutative properties of addition and multiplication begin to tell us
more about the operations as such (In algebra books, these properties are said to be of
operations, not of numbers). So, regarding addition, A + B = B + A, it does not matter
which order one adds numbers. Similarly with multiplication, A * B = B * A. But it is not
true of powers (A^B does not equal B^A), although it is true of adding and multiplying
numbers with powers. Nor is it true of subtraction, division, and roots. Other algebraic
rules express the patterns found in arithmetic using these operations. These algebraic
rules draw attention to the operations themselves, for they begin to define them in a new
manner. These properties begin to distinguish and characterize the various operations
themselves, and allow one to grasp Lonergan's meaning when he writes, "that numbers
were generated, not merely by addition, but by any of the operations" (recall that
all the arithmetic operations can be reduced to addition).(2)
The properties of addition, subtraction, multiplication, division, powers, and roots
are defined in algebra with greater generality. In arithmetic, adding was defined as
"putting together." In algebra (elementary algebra), it is defined as being
associative and commutative, to name some of the basic properties. Not only is one
"putting together," but it does not matter which order one "puts
together" nor does it matter which numbers are "put together" first
[(A+B)+C = A + (B+C)]. Notice the difference of the rules? To move from "putting
together" to "it does not matter which order the numbers are put together"
is the move to a higher viewpoint, which is a general, symmetric, and exact account of the
nature of addition. This is true as well with the other operations (Try this comparison as
an exercise). So, to highlight his difference once again, note, that the rule in
arithmetic is to "put together," but in algebra it is that it does not matter
which order one puts together. Likewise, in arithmetic it states that multiplication is to
put together a number with itself so many times, and in algebra it states that it does not
matter which order one multiplies numbers.
A further comparison of how problems are solved will help to illustrate the difference
between these two points. In arithmetic, one might ask what is 1 + 3. The answer, placed
on the other side of the equal sign, or the line, is 4. One simply carries out the rule
and puts 1 together with 3. In algebra, one might ask what is y, when y = 2 - 3y. Then one
works to get the "y" on one side of the equation by using the rules of algebra.
I can add a number to both sides of the equation, and it does not matter which order I
place it in the equation. I can subtract, but then the order to matter. I can multiply,
and to do so I must multiply the entire side of each side in the equation by the same
multiple. So, in the case of y=2 - 3y, I might add 3y to both sides of the equation,
resulting in 3y + y = 2 - 3y + 3y, which can be reduced to 4y = 2. Then I can divide both
sides of the equation by 4, resulting in 4y/4=2/4, which can be reduced to y = ½. Notice
how the algebraic rules were applied to solve this problem. It can give you some sense of
the difference between a higher and a lower viewpoint.
The next step will be to examine, in the next essay, whether the movement from Newton
to Dalton, (from physics to chemistry) can be seen as a shift in viewpoints. In addition,
I will add an essay on the movement from Dalton to Mendeleev (higher viewpoints in
chemistry). Then, in the final essay on higher viewpoints, I will give a more precise
definition of viewpoints and the movement to a higher viewpoint as well as note other
applications of higher viewpoints.
1. Insight, 1970, p.16.
2. Insight, 17.
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