This post completes a series of posts in on the nature and moments of insight.
Nominal and explanatory definition
Lonergan’s sixth observation concerns different kinds of definition. To begin with, he has nominal and explanatory definition in mind.
As Euclid defined a straight line as a line lying evenly between its extremes, so he might have defined a circle as a perfectly round plan curve. As the former definition, so also the latter would serve to determine unequivocally the proper use of the names ‘straight line,’ ‘circle.’ But in fact Euclid’s definition of the circle does more than reveal the proper use of the name ‘circle.’ It includes the affirmation that in any circle all radii are exactly equal; and were that affirmation not included in the definition, then it would have had to be added as a postulate. (1)
Nominal definition, then, is an insight into the correct usage of language. To this end, defining a circle as a perfectly round plan curve would suffice.
Explanatory definition supposes a further insight into the objects to which language refers. In this case affirming that all radii in a circle are equal goes beyond the mere usage of language.
Primitive terms
The seventh observation concerns primitive terms.
Every definition presupposes other terms. If these can be defined, their definitions will presuppose still other terms. But one cannot regress to infinity. Hence, either definition is based on undefined terms or else terms are defined in a circle so that each virtually defines itself. (2)
This is a traditional concern about definition. However, Lonergan’s discussion of the nature and moments of insight outlined in my previous posts avails us of more resources than are typically brought to bear on the matter.
Many philosophers of language fall into the regress above because they only employ other linguistic elements to fix the meaning of a term. Alternatively, Lonergan’s cognitional analysis of insight admit experiences, images, questions, and other insights into consideration.
Let us say, then, that for every basic insight there is a circle of terms and relations, such that the terms fix the relations, the relations fix the terms, and the insight fixes both. If one grasps the necessary and sufficient conditions for the perfect roundness of this imagined plane curve, then one grasps not only the circle but also the point, the line, the circumference, the radii, the plan, and equality. All the concepts tumble out together, because all are needed to express adequately a single insight. All are coherent, for coherence basically means that all hang together from a single insight. (3)
Implicit definition
A final observation introduces Hilbert’s notion of implicit definition.
Implicit definition consists in explanatory definition without nominal definition. Again, he employs another familiar example.
Thus, the meaning of both point and straight line is fixed by the relation that two and only two points determine a straight line.
[…]
It consists in explanatory definition, for the relation that two points determine a straight line is a postulational element such as the equality of all radii in a circle. It omits nominal definition, for one cannot restrict Hilbert’s point to the Euclidean meaning of position without magnitude. An ordered pair of numbers satisfies Hilbert’s implicit definition of a point, for two such pairs determine a straight line. Similarly, a first-degree equation satisfies Hilbert’s implicit definition of a straight line, for such an equation is determined by two ordered pairs of numbers. (4)
Larger implications
In chapter one, Lonergan explains how insight functions in mathematical knowing and, as such, all of the preceding examples and definitions are taken from mathematics.
Clearly though, not all definitions are implicit definitions. Still, the sort of generality operative in implicit definitions is appropriate to mathematical and scientific objects. It is in part because these objects can be defined implicitly that Lonergan considers mathematics and science to constitute a separate realm of meaning, or discourse, from our everyday, commonsense one.
The significance of implicit definition is its complete generality. The omission of nominal definitions is the omission of a restriction to the objects which, in the first instance, one happens to be thinking about. The exclusive use of explanatory or postulational elements concentrates attention upon the set of relationships in which the whole scientific significance is contained. (5)
From here, Lonergan gives an explicit treatment of the insights into mathematics that allowed successive branches of mathematics to develop.
Notes
- Bernard Lonergan, Insight: A Study of Human Understanding, p. 35.
- ibid., p. 36.
- ibid., p. 36.
- ibid., p. 37.
- ibid., p. 37.
Dissertating Outloud 