Lonergan has shown us five features of an insight.
- The release of the tension of inquiry
- Its sudden and unexpected character
- Its resulting from inner rather than outer conditions
- That it pivots between the concrete and abstract
- That each new insight eventually becomes habitual
Additionally, he has shown us how the various elements of an insight relate, resulting in a mathematical definition. Clues, concepts, images, and questions all serve as primitive terms of the definition.
And further, he has distinguished between nominal, explanatory, and implicit definitions, and it is implicit definitions with which he is concerned in his investigation of mathematical and scientific insights because of their complete generality.
Hence, we’ve understood a single insight, but what of clusters and series of insights that are related and which fall within the same discipline?
The next significant step to be taken in working out the nature of insight is to analyze development. Single insights occur either in isolation or in related fields. In the latter case, they combine, cluster, coalesce, into the mastery of a subject; they ground a set of definitions, postulates, deductions; they admit applications to enormous ranges of instances. But the matter does not end there. Still further insights arise. The shortcomings of the previous position become recognized. New definitions and postulates are devised. A new and larger field of deductions is set up. Broader and more accurate applications become possible. Such a complex shift in the whole structure of insights, definitions, postulates, deductions, and applications may be referred to very briefly as the emergence of a higher viewpoint. Our question is, Just what happens? (1)
In order to chart a progression of this sort, Lonergan maps kind of insights required to transition from arithmetic to elementary algebra.
Positive integers
Lonergan sets about to define the positive integers.
Let us suppose an indefinite multitude of instances of ‘one.’ They may be anything anyone pleases, from sheep to instances of the act of counting or ordering.
Further, let us suppose as too familiar to be defined the notions of ‘one,’ ‘plus,’ and ‘equals.’
Then, there is an infinite series of definitions for the infinite series of positive integers, and it may be indicated symbolically by the following:
1 + 1 = 2
2 + 1 = 3
3 + 1 = 4
etc., etc., etc. … (2)
You no doubt look at the sequence of expressions above and immediately understand the principle at work, as well as the meaning of the “etc., etc., etc. …” However, if we’re to be more specific, we explain that the etc., etc., etc. … means that we’ve had sufficient insight into the principle at work to generalize and apply it beyond the particular cases before us above.
Addition table
Next, Lonergan seeks additional precision for the notion of equality.
Let us say that when equals are added to equals, the results are equal; that one is equal to one; and that, therefore, an infinite series of addition tables can be constructed.
The table for adding two is constructed by adding one to each side of the equations that define the positive integers. Thus,
From the table 2 + 1 = 3
Adding one 2 + 1 + 1 = 3 + 1
Hence from the table 2 + 2 = 4
…
Thus, from the definitions of the positive integers and the postulate about adding equals to equals, there follows an indefinitely great deductive expansion. (3)
The homogeneous expansion
“The homogeneous expansion constitutes a vast extension of the initial deductive expansion. It consists in introducing new operations. Its characteristic is that the new operations involve no modification of the old.” (4)
To addition is added the notions of multiplication, powers, subtraction, and division, all of which either build upon the principles at work in addition or simple invert them.
Thus, multiplication is to mean adding a number to itself so many times, so that five by three will mean the addition of three fives. Similarly, powers are to mean that a number is multiplied by itself so many times, so that five to the third will mean five multiplied by five with the result multiplied again by five. On the other hand, subtraction, division, and roots will mean the inverse operations that bring one back to the starting point. (5)
The need of a higher viewpoint
However, when granted the free reign of full generality, even the rules and operations of addition, multiplication, subtraction, and division can carry us beyond our mere starting point. These very rules and operations give rise to complications that cannot addressed by the rules and operations themselves.
A fourth step will be discovery of the need of a higher viewpoint. This arises when the inverse operations are allowed full generality, when they are not restricted to bringing one back to one’s starting point. Then, subtraction reveals the possibility of negative numbers, division reveals the possibility of fractions, roots reveal the possibility of surds. Further, there arise questions about the meaning of the operations. What is multiplication when one multiplies negative numbers or fractions or surds? What is subtraction when one subtracts a negative number? Etc., etc., etc. (6)
Formulation of the higher viewpoint
In order to formulate the higher viewpoint, Lonergan first seeks greater precision than we have thus far achieved by distinguishing between 1) rules, 2) operations, and 3) numbers.
Numbers are defined implicitly by the operations, “so that the results of any operation will be a number, and any number can be the result of an operation.” (7)
Operations are also defined implicitly by the rules, “so that what is done is accord with rules is an operation.” (8)
In pursuit of the higher viewpoint, the rules must be obtained which fix the operations which in turn fix the numbers. This requires an insight into 1) the operations which occur according to the current rules and 2) the formulation of new rules.
Let me explain. From the image of a cartwheel we proceeded by insight to the definition of the circle. But while the cartwheel was imagined, the circle consists of a point and a line, neither of which can be imagined. Between the cartwheel and the circle there is an approximation, but only an approximation. Now the transition from arithmetic to elementary algebra is the same sort of thing. For an image of the cartwheel one substitutes the image of what may be named ‘doing arithmetic’; it is a large, dynamic, virtual image that includes writing down, adding, multiplying, subtracting, and dividing numbers in accord with the precepts of the homogeneous expansion. Not all of this image will be present at once, but any part of it can be present, and when one is on alert any part that happens to be relevant will pop into view. In this large and virtual image, then, there is to be grasped a new set of rules governing operations. The new rules will not be exactly the same as the old rules. They will be more symmetrical. They will be more exact. They will be more general. In brief, they will differ from the old much as the highly exact and symmetrical circle differs from the cartwheel. (9)
Successive higher viewpoints
Recall that Lonergan is taking us through the mathematical transition one makes in moving beyond mere arithmetic. However, this is not a historical analysis of the development, nor even a logical one. As ever, Lonergan is concerned to have insight into the nature of insight, and in this case, into mathematical insight.
What is it that we must understand in order to properly mark the usage and meaning of the rules that govern the operations in each correspondingly higher form of mathematics?
At each stage of the process there exists a set of rules that govern operations which results from numbers. To each stage there corresponds a symbolic image of doing arithmetic, doing algebra, doing calculus. In each successive image there is the potentiality of grasping by insight a higher set of rules that will govern the operations and by them elicit the numbers or symbols of the next stage. Only insofar as a man makes his slow progress up that escalator does he become a technically competent mathematician. Without it he may acquire a rough idea of what mathematics is about, but he will never be a master, perfectly aware of the precise meaning and the exact implications of every symbol and operation. (10)
The significance of symbolism
While there is no doubt that each instance of mathematical symbolism is conventionally chosen, still the symbolism chosen is always more or less adequate to the task at hand.
Take the square root of 1794.
Now take the square root of MDCCLXIV.
Similarly, “the development of the calculus is easily designated in using Leibniz’s symbol dy/dx for the differential coefficient; Newton’s symbol, on the other hand, can be used only in a few cases, and what is worse, it does not suggest the theorems that can be established.” (11)
Why is this so? It is because mathematical operations are not merely the logical expansion of conceptual premises. Image and question, insight and concepts all combine. The function of the symbolism is to supply the relevant image, and the symbolism is apt inasmuch as its immanent patterns as well as the dynamic patterns of its manipulation run parallel to the rules and operations that have been grasped by insight and formulated in concepts. (12)
Lonergan gives five reasons why the selection of the appropriate symbolism – which really is a kind of image – is central for having further insights and discovering solutions during inquiry.
First, symbolism takes over a notable part of the solutions to problems. The symbolism itself, once learned and habitual, dictates how one moves from problem to solution.
Second, the symbolism constitutes a heuristic technique. Symbolism allows one to label the unknown, assign it a symbol, situate it within a precise set of operations; the symbolism allows that which is to be discovered to be present throughout the investigation.
Third, the symbolism itself offers clues, hints, and suggestions. Because an aptly chosen symbolism provides a heuristic structure for the inquiry, it suggests how to find the hints and clues required in order to proceed to the anticipated solution.
Fourth, there is highly significant notion of invariance.
An apt symbolism will endow the pattern of a mathematical expression with the totality of its meaning… The mathematical meaning of an expression resides in the distinction between constants and variables, and in the signs or collocations that dictate operations of combining, multiplying, summing, differentiating, integrating, and so forth. It follows that, as long as the symbolic pattern of mathematical expression is unchanged, its mathematical meaning is unchanged. Further, it follows that if a symbolic pattern is unchanged by any substitutions of a determinate group, then the mathematical meaning of the pattern is independent of the meaning of the substitutions. (13)
Fifth, the symbolism appropriate to any stage of mathematical development provides the image in which may be grasped by insight the rules for the next stage.
Related, mathematical insights
I hope it is clear, then, that the symbolism and operations with construct the rules of any specific form of mathematics. Further, it is by understanding the complexity of the operations and the their limitations that one can understand the precise nature of the operative rules of, say, arithmetic. Finally, this precise understanding of the current, operative rules suggests that operations are not possible according to those rules and therefore, suggests potential new directions in which to develop new rules.
In providing us this series of examples, whether of cartwheels and circles, or arithmetic rules and operations, Lonergan has spelled out the functioning of insight in its precise and static functioning. As we move from insight in mathematics to insight in science, these become much more complex, even messy.
Notes
- Bernard Lonergan, Insight: A Study of Human Understanding, p. 37 – 8.
- ibid., p. 38.
- ibid., p. 39.
- ibid., p. 40.
- ibid., p. 40.
- ibid., p. 40.
- ibid., p. 40.
- ibid., p. 41.
- ibid., p. 41.
- ibid., p. 42.
- ibid., p. 42.
- ibid., p. 42.
- ibid., p. 43.
Dissertating Outloud 