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Higher
Viewpoints: Part One
From
Arithmetic to Algebra: the transition
by
David Fleischacker
Draft
Version 1
Copyright
© 1997. All rights reserved
(Before
you read this, please either read, or reread, sections 1 - 3 of
chapter one in Insight.)
I.
The Viewpoint of Arithmetic:
In
Insight, Lonergan builds to the notion of a higher viewpoint
after he has developed an understanding of clues, insight, concepts,
questions, images, and definitions. A viewpoint is not merely
a definition, but a set of systematically related definitions
(and of the operations that underpin both the definitions and
their systematic relations). It is not a single definition. Defining
a circle, for example, is not a viewpoint, but it does arise out
of a geometrical viewpoint, and contributes to it. The same is
true of the distributive or commutative properties of algebra,
or the power rule of calculus. They do not constitute an entire
viewpoint, but they are components.
Lonergan
illustrates lower and higher viewpoints with arithmetic and algebra.
A mathematical viewpoint is constituted by rules, operations,
and symbols (or numbers). The rules implicitly define the operations,
and the operations implicitly define the symbols. What does he
mean by this?
A.
The Deductive Expansion of Arithmetic (the first horizontal development
in mathematics):
Lonergan
begins with arithmetic, more specifically with addition. One may
count sheep or goats or troops in an army or persons inhabiting
a town. The counting involves the operation of addition-- one
plus one plus one, and so forth. It is an activity relating quantities
and defining them in terms of each other. The basic unit of this
quantity can be symbolized, let us say with a "1" or
"I". Other symbols can be used to represent what one
is doing when adding, such as "+" or "plus."
Any
number of symbols can be invented to represent operations (addition,
subtraction, etc..) and numbers (some of which, Lonergan notes,
are better conducive to the future development of mathematics
than others because of their potential for leading to further
insights). In order to simplify the ongoing definitions of numbers
most cultures that developed mathematics introduced repeating
schemes. Some introduced repetitions based on 30 or 60 (think
of our clocks and watches). Our present system is based on repetitions
of 10, so we developed a symbol for zero through nine, and then,
once ten is reached, we add a place to the left indicating the
number of "tens." Then once the tens reaches beyond
the ninth position, we add the hundreds, then thousands, and so
on (Computers, you may have heard, are based on a binary, with
ones and zeros).
From
adding numbers we can develop, as Lonergan notes, a definition
of the positive integers.
So,
1
+ 1 = 2
2 + 1 = 3
3 + 1 = 4
etc., etc., etc..
Once
the insight is gained, or in other words, when one understands
what is meant by "etc., etc., etc." then one can continue
to indefinitely define any positive number. From this, one can
create an entire deductive expansion of a viewpoint or horizon
in arithmetic, and continue indefinitely to define the whole range
of positive integers. One can also construct mathematical tables
using 2s, 3s, 4s, etc..
2
+ 2 = 4
4 + 2 = 6
6 + 2 = 8
Etc., etc., etc.. ("2" is added in a repeating fashion)
3
+ 3 = 6
6 + 3 = 9
9 + 3 = 12
Etc., etc., etc.. ("3" is added in a repeating fashion)
4
+ 4 = 8
8 + 4 = 12
12 + 4 = 16
Etc., etc., etc.. ("4" is added in a repeating fashion)
Then,
ETC.,
ETC., ETC. (For the entire process above)
Notice
how all of the numbers are defined in terms of the operation of
addition. It is the basic insight that grounds this operation
which allows for the construction of an entire deductive expansion
which creates a "world" or viewpoint, even if rather
limited. It is a first, horizontal development of a horizon in
mathematics.
B.
The Homogeneous Expansion (the second horizontal development in
mathematics)
One
can become more creative, and begin to add a number to itself
say three or four or five times.
So,
1
+ 1 + 1 = 3
1 + 1 + 1 + 1 = 4
2 + 2 + 2 = 6
3 + 3 + 3 + 3 = 12
Etc., etc., etc..
And,
instead of writing this with three 1's or four 1's or three 2's
or four 3's, shorthand notation can be developed.
1
x 3 = 3
1 x 4 = 4
2 x 3 = 6
3 x 4 = 12
So,
one introduces a different symbol, namely an "x" to
indicate the number of times one is added to itself. Notice how
this new symbol is still defined in terms of the old operation
of addition. It means "adding" a number to itself so
many times.
Creativity
does not have to stop. If we can add one to another, then what
happens if we take something away. We had three sheep, sold one,
and now are left with two. This is the opposite of addition, and
we can give it the symbolization of "-"(1)
and call it subtraction. Again, one can develop charts of subtraction,
just as with addition and multiplication. Likewise, just as one
can reverse addition by removing something, so one can reverse
multiplication by removing a number so many number of times. So,
just as one can add 2 to itself four times to get 8, so one
can remove 2 from 8 four times. This of course, is division.
Creatively
constructing a viewpoint by introducing new symbols such as subtraction,
multiplication and division, is what Lonergan calls a homogenous
expansion. One has introduced new symbols relating numbers, but
notice how everything is still thought of in terms of addition.
Subtraction is the reverse of addition. Multiplication is the
addition of a number to itself so many number of times. Division
is the reverse of that. And if one wishes to add powers and roots,
they likewise can be defined in terms of addition. A power is
the multiplication of number by itself so many number of times.
Thus to define this in terms of addition, let the number that
is powered equal y, and the power equal z. Thus,
the answer is defined as the number, y, which has been added
to itself y number of times, thus forming a group, a group which
in turn is added to itself again y number of times, and repeating
this formation of groups z-1 number of times. This
can be illustrated with 3 to the power of 4. 3 to the
power of 4 is the same as 3 x 3 x 3 x 3. The first group
arises by converting 3 x 3 into 3 + 3 + 3. This group is then
added to itself 3 times in order to get the equivalent of 3 x
3 x 3. This results in a second group that can be written
as [(3 + 3 + 3) + (3 + 3 + 3) + (3 + 3 + 3)]. Finally, take
this second group and add it three times to itself. The
final answer comprises this third group, which can be written
as
[(3+3+3)
+ (3+3+3) + (3+3+3)] + [(3+3+3) + (3+3+3) + (3+3+3)] + [(3+3+3)
+ (3+3+3) + (3+3+3)] = 81
A
"root" is the reverse of this procedure. So, the 4th
root of 81 requires breaking down the 81 into three groups, where
the basic group, which when discussing powers was called the first
group, is comprised of a number that has been added to itself
its own number of times. This number is the answer. So,
even powers and roots can be thought of in terms of addition.
II.
Algebra: The Higher Viewpoint ( a vertical expansion in mathematics)
The
homogeneous expansion of arithmetic has not introduced any new
rules. One can define each of the new operations in terms of addition
(or the reverse of addition). New rules are only introduced when
one starts "observing" patterns in arithmetic, and doing
this initiates algebra (Lonergan notes that the image which leads
to algebra is the doing of arithmetic). What does this mean? Lonergan
notes that this "turn of question" that lead to the
discovery of patterns in arithmetic occurred because of questions
such as;
What
happens when one subtracts more than one had?
Or what happens when division leads to fractions?
Or roots to surds?
Each
of these refers to various problems that emerge in the homogenous
expansion. Their answer lies in grasping patterns. Questions emerge
which ask, what, in general, happens when one subtracts numbers,
or adds numbers, or divides numbers, or adds powered numbers,
etc.? Today, these initial patterns are given such names as commutative,
distributive, and associative properties. Let us illustrate these
laws.
A
simple pattern is adding a number to zero.
1
+ 0 = 1
2 + 0 = 2
3 + 0 = 3
4 + 0 = 4
5 + 0 = 5
6 + 0 = 6
etc., etc., etc..
The
etc., etc., etc., again is introduced to gain the insight. A number
added to zero results in an answer that is that number. This can
be symbolized by creating a symbol that represents a number (or
in other words, a variable). Let that symbol be "A."
Below is the formulation of this pattern,
A
+ 0 = A
Another
example is the multiplication of a number by 1.
1
x 1 = 1
2 x 1 = 2
3 x 1 = 3
4 x 1 = 4
5 x 1 = 5
etc., etc., etc..
If
one recognizes the pattern, then one notices that a number multiplied
by one, gives the number. Hence, this insight can be symbolized.
A
x 1 = A
The
same is true with the various laws or properties (as they are
actually called) mentioned earlier. The commutative property of
addition states that A + B = B + A. The commutative property of
multiplication states that A x B = B x A. The associative property
states that (A + B) + C = A + (B + C). The parenthesis means add
these numbers first. The associative property of multiplication
states that (A x B) x C = A x (B x C). As an exercise right now,
try expressing these patterns using actual numbers and the "etc.,
etc., etc." as I did above.
You
can practice this further by opening any algebraic text, and examining
the numerous rules about addition, multiplication, division, powers,
roots, addition of powers, multiplication of powers, multiplication
of roots, and the inverses of each of these formulas (subtraction
of powers and roots, division of powers and roots, etc..)
Notice
how one is understanding the operations in a manner beyond that
of addition. One begins to grasp, for example, that multiplying
two negatives leads to a positive, that dividing a negative into
a negative also leads to a positive and many other characteristics.
These recognized patterns then begin to form new rules, which
constitute the higher viewpoint called algebra. These rules guide
one in solving problems, since they implicitly define how one
is to carry out operations and define the new symbols of A's,
B's, and C's, which represent variable numbers (this will be discussed
with more precision and examples in a later commentary). The rules
constitute a vertical expansion of the mathematical horizon.
Like
arithmetic, algebra also has a deductive and homogeneous expansion,
or, at least something analogous. This is for a later section.
David
Fleischacker
Copyright
© 1997. All rights reserved
1.
For a history of mathematics that discusses these symbols,
see a book that is frequently recommended in Lonergan circles,
Carl Boyer, A History of Mathematics (New York: John Wiley
& Sons, Inc., 1991).
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