Defining a circle

I have read this section of Insight many times. Each time I’ve tried to summarize, restate, or rephrase it, but have failed each time. Perhaps it’s because the content is an exposition of an example. Whatever the cause, know that much of what follows quotes whole passages of Lonergan’s text.

My previous post introduced Lonergan’s use of the desire to understand as constitutive of inquiry. This post provides a particular instance of that desire at work, as well as an exposition of the essential moments of insight into a definition.

Lonergan prompts the reader to reflect on his own knowing by attending to the definition of a circle. He constructs a scenario for the reader to attend to his own cognitive activity while striving to grasp the definition.

The clue

As every schoolboy knows, a circle is a locus of coplanar points equidistant from a center. What every schoolboy does not know is the difference between repeating that definition as a parrot might and uttering it intelligently. (1)

Imagine a cartwheel with its bulky hub, its stout spokes, its solid rim.

Ask a question. Why is it round?

Limit the question. What is wanted is the immanent reason or ground of the roundness of the wheel. Hence a correct answer will not introduce new data such as carts, carting, transportation, wheelwrights, or their tools. It will refer simply to the wheel. (2)

Lonergan takes care to specify the sense of his question. He is not concerned with the pragmatic reasons for which wheels are created nor the means by which they are built.
He is interested in the reason why we recognize any particular bulky cartwheel as an abstractly defined circle – as a locus of coplanar points equidistant from a center.

Consider a suggestion. The wheel is round because its spokes are equal. Clearly, that will not do. The spokes could be equal yet sunk unequally into the rub and rim. Again, the rim could be flat between successive spokes. (3)

He hints at a solution. Perhaps the wheel is round because its spokes are equal. However, further consideration reveals this solution as incomplete and therefore inadequate. The suggestion requires adjustment.

Still, we have a clue. Let the hub decrease to a point; let the rim and spokes thin out into lines; then, if there were infinity of spokes and all were exactly equal, the rim would have to be perfectly round; inversely, were any of the spokes unequal, the rim could not avoid bumps or dents. Hence we can say that the wheel necessarily is round inasmuch as the distance from the center of the hub to the outside of the rim is always the same. (4)

The possibility of equal spokes being sunk unequally into the hub (i.e. the limitations inherent to the concrete object) forced a shift from attending to the wheel concretely to considering it abstractly. For the definition of a circle is not imagistic, but conceptual. And images are limited.

A number of observations are now in order. The foregoing brings us close enough to the definition of the circle. But our purpose is to attain insight, not into the circle, but into the act illustrated by insight into the circle.

The first observation, then, is that points and lines cannot be imagined. One can imagine an extremely small dot. But no matter how small a dot may be, still it has magnitude. To reach a point, all magnitude must vanish, and with all magnitude there vanishes the dot as well. One can imagine an extremely find thread. But no matter how fine a thread may be, still it has breadth and depth as well as length. Remove from the image all breadth and depth, and there vanishes all length as well. (5)

Concepts

The second observation is that points and lines are concepts.

Just as imagination is the playground of our desires and our fears, so conception is the playground of our intelligence. Just as imagination can create objects never seen or heard or felt, so too conception can create objects that cannot even be imagined. How? By supposing. The imagined dot has magnitude as well as position, but the geometer says, ‘Let us suppose it has only position.’ The imagined line has breadth as well as length, but the geometer says, ‘Let us suppose it has only length.’ (6)

A few points.

First, Lonergan is framing his example so as to expose our own conscious, cognitive performance, of which we usually unaware. When we intend, we usually intend something, but only rarely do we pay attention to the intending itself. By distinguishing between imagination and conception as cognitive processes, he is drawing attention to the shifts in our natural cognitive action.

Second, the shift from imagination to conception is a shift from the concrete to the abstract. It is worth recalling Lonergan on insight as the pivot between the concrete and the abstract: “[B]y its very nature insight is the mediator, the hinge, the pivot. It is insight into the concrete world of sense and imagination. Yet what is known by insight, what insight added to sensible and imagined presentations, finds its adequate expression only in the abstract and recondite formulations…” (7)

Lonergan continues:

Still, there is method in this madness. Our images and especially our dreams seem very random affairs, yet psychologists offer to explain them. Similarly, the suppositions underlying concepts may appear very fanciful, yet they too can be explained. Why did we require the hub to decrease to a point and the spokes and rim to mere lines? Because we had a clue – the equality of the spokes – and we were pushing it for all it was worth. As long as the hub had any magnitude, the spokes could be sunk into it unequally. As long as the spokes had any thickness, the wheel could be flat at their ends. So we supposed a point without magnitude and lines without thickness, to obtain a curve that would be perfectly, necessarily round. (8)

The image

The third observation is that the image is necessary for the insight.

Points and lines cannot be imagined. But neither can necessity or impossibility be imagined. Yet in approaching the definition of the circle there occurred some apprehension of necessity and of impossibility. As we remarked, if all the radii are equal the curve must be perfectly round, and if any radii are unequal the curve cannot avoid bumps or dents.

Further, the necessity in question was not necessity in general but a necessity of roundness resulting from these equal radii. Similarly, the impossibility in question was not impossibility in the abstract but an impossibility of roundness resulting from these unequal radii. Eliminate the image of the center, the radii, the curve, and by the same stroke there vanishes all grasp of necessary or of impossible roundness.

But it is that grasp that constitutes the insight. It is the occurrence of that grasp that makes the difference between repeating the definition of a circle as a parrot might and uttering it intelligently, uttering it with the ability to make up a new definition for oneself.

It follows that the image is necessary for insight. Inversely, it follows that the insight is the act of catching on to a connection between imagined equal radii and, on the other hand, a curve that is bound to look perfectly round. (9)

That points and lines cannot be imagined, but are concepts, is crucial, both here and for subsequent arguments in Insight. For example, Lonergan argues that failing to recognize the abstract nature of scientific laws results in the corresponding failure to distinguish classical laws from statistical laws, which in turn results in misunderstanding the relationship between scientific laws and the concrete events the former explains.

Hopefully it will become apparent how understanding insight – the pivot between the concrete and the abstract – will help to explicate the relationship between abstract, scientific laws and the concrete things to which those laws apply.

The question

Lonergan’s fourth observation concerns the question itself – Why is the wheel round? – which is the expression of the desire to understand.

Though he draws attention to it last, the desire to understand is the necessary condition for the existence of all inquiry.

The fourth observation adverts to the question.

There is the question as expressed in words. Why is the wheel round?

Behind the words there may be conceptual acts of meaning, such as ‘wheel,’ ’round,’ etc.

Behind these concepts there may be insights in which one grasps how to use such words as ‘wheel,’ ’round,’ etc.

But what we are trying to get at is something different. Where does the ‘Why?’ come from? What does it reveal or represent? Already we had occasion to speak of the psychological tension that had its release in the joy of discovery. It is that tension, that drive, that desire to understand, that constitutes the primordial ‘Why?’ Name it what you please – alertness of mind, intellectual curiosity, the spirit of inquiry, active intelligence the drive to know. Under any name, it remains the same, and is, I trust, very familiar to you.

This primordial drive, then, is the pure question. It is prior to any insights, any concepts, any words; for insights, concepts, words have to do with answers, and before we look for answers we want them; such wanting is the pure question.

On the other hand, though the pure question is prior to insights, concepts, and words, it presupposes experiences and images. Just as insight is into the concretely given or imagined, so the pure question is about the concretely given or imagined. It is the wonder which Aristotle claimed to be the beginning of all science and philosophy. But no one just wonders. We wonder about something. (10)

Genesis (of a definition)

Here, Lonergan outlines the sequential moments of a definition, all of which have already been introduced. He identifies four moments:

• wonder (i.e. the desire to understand)
• the clue
• cooperation of imagination with intelligence, and
• insight into the definition

When an animal has nothing to do it goes to sleep. When a man has nothing to do he may ask questions. The first moment [then] is an awakening to one’s intelligence. It is release from the dominance of biological drive and from the routines of everyday living. It is the effective emergence of wonder, of the desire to understand.

The second moment is the hint, the suggestion, the clue. Insight has begun. We have got hold of something. There is a chance that we are on the right track. Let’s see.

The third moment is the process. Imagination has been released from other cares. It is free to cooperate with intellectual effort, and its cooperation consists in endeavoring to run parallel to intelligent suppositions, while at the same time restraining supposition within some limits of approximation to the imaginable field.

The fourth moment is achievement. By their cooperation, by successive adjustments, question and insight, image and concept present a solid front. The answer is a patterned set of concepts. The image strains to approximate to the concepts. The concepts, by added conceptual determinations, can express their differences form the merely approximate image. The pivot between images and concepts is the insight. And setting the standard which insights, images, and concepts must meet is the question, the desire to know, that could have kept the process in motion by further queries had its requirements not been satisfied. (11)

As the argument develops

Again, Lonergan structured his argument in Insight pedagogically. Correctly, I think, he understood the need for individuals to understand that they themselves are knowers before they could move on to answer larger questions. As such, he assumed nothing about the opinions or positions of his readers, but merely invited them to examine their own cognitive performance, along with that of other whom we assume to be knowers, such as mathematicians and scientists.

Therefore, Insight (or at least the first half) is not a series of statements, claims, or true propositions. It is a guided activity. One which proceeds slowly and refines its process and claims as it proceeds.

But the value of even the earliest pages of Insight is undeniable. Jumping ahead to chapter 3, “The Canons of Empirical Method,” one can already see how what Lonergan is explaining here with the circle shows up in his robust treatment of scientific method and understanding:

The basic distinction is between abstract systems and particular cases. Both are objects of insight. But the particular case is the typical instance, presented by sense or imagination and understood by insight into the presentation. In contrast, the abstract system is neither sensible nor imaginable; it is a conceptual object constituted by terms and relations that, at least in the last resort, are defined implicitly.

[…]

Now let us suppose full knowledge of all classical principles and laws. Then we suppose full knowledge of abstract system: for principles and laws are relations; such relations necessarily involve the terms that they define implicitly; and abstract system consists in terms implicitly defined by the relations expressed in verified principles and laws.

However, if this full knowledge of abstract system is to be applied to the concrete universe, there will be needed a manifold of insights into particular cases. For, as was noted above, abstract system is applied to concrete situations only inasmuch as insight into the situations selects the relevant laws, determines the mode of their combination, and substitutes numerical values for the variables and undetermined constants of the laws. (12)

Determining how it is that we arrive at the definition of a circle is incomplete. Next, Lonergan explains different kinds of definitions, namely nominal, explanatory, and implicit.

Notes

1. Bernard Lonergan, Insight: A Study on Human Understanding, p. 31.
2. ibid., p. 31.
3. ibid., p. 31.
4. ibid., p. 31 – 2.
5. ibid., p. 32.
6. ibid., p. 32.
7. ibid., p. 30.
8. ibid., p. 32.
9. ibid., p. 33.
10. ibid., p. 33 – 4.
11. ibid., p. 34 – 5.
12. ibid., p.109 – 10.