“Gold, is it pure or fool’s?” could well have been the question put to Archimedes by the King of Syracuse concerning the crown he was given as a votive offering. Archimedes’ ‘eureka’ is now a byword for insight - the moment of discovery, the sudden breakthrough to a solution to a perplexing puzzle, a new way of seeing the world accompanied by the rush of joy, elation and desire to share with all and sundry what one now ‘sees’. Archimedes found the answer as he lay in the baths. His solution: weigh the crown in water. In this, he had discovered the principles of displacement and specific gravity.
Insights come suddenly and unexpectedly, under the drive of inquiry into definite problems. They are an ‘inner’ event, yet cannot be commanded to come, at will. They give instant relief to the tension of inquiry, but must be ‘captured’ in some kind of expression or formulation before they disappear and are lost. Insights unify what was previously disconnected; they bring coherence to disorder, intelligibility to what was previously not understood; and they inform and give rise to concepts. Insight is the product of intelligence.
Lonergan examined the phenomenon closely, and invites us to do the same, drawing on our own experience, such as solving simple puzzles or through directed thought experiments. Here is one from Chapter 1, Lonergan’s Insight - A Study of Human Understanding, where, note, he reminds us that the purpose of the exercise is to “attain insight, not into the circle, but into the act illustrated by insight into the circle”:
“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. So, with a sidelong bow to Descartes's insistence on the importance of understanding very simple things, let us inquire into the genesis of the definition of the circle.
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.
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 hub and rim. Again, the rim could be flat between successive spokes.
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 an 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.
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 fine 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.
