This is a repost of a 1997 essay written for a seminar on Insight.
Higher Viewpoints: Part One
From Arithmetic to Algebra: the transition
by David Fleischacker
Draft Version 1
Copyright © 1997. All rights reserved
(This is a reflective commentary on one facet of 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.
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)
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.
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.
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).