One hears of talk about how even math is culturally based. There is an element of truth in that, but there is something true as well that makes it trans-cultural. The answer as to its relation to culture begins by answering a question. What is math?
Lonergan’s answer to that question is very profound, and once understood, it opens up connections between math and the natural sciences, as well as human thinking and questions about epistemology, as well as reality. So, let us begin to answer that first question.
Lonergan in Insight, the first chapter, develops a number of points using examples taken from math. He is not yet moving to an account of math, but really is drawing the reader to think about what it means to understand something. And math, oddly, is one of the easier places to start getting insights into insights. So, he talks about Archimedes and his thinking about how to figure out how pure was the gold crown, the answer being a combination of weight and water displacement. Purer gold has a certain weight per volume. Another example is that of the circle. One starts by playing with the image of a cartwheel and ends up getting an insight, then goes further to define the insight in terms of lines and points. In both cases, insight springs up in images, but it starts with wonder and a clue. And in both, the insight is not the end, but rather, one goes on to definitions. Lonergan then moves on to the expansion of a viewpoint in arithmetic, starting with positive integers through the operation of addition, but then by reversing the operation in subtraction, one moves back to one’s starting point, and can even go further. He then exemplifies the meaning of a homogeneous expansion of a viewpoint by adding multiplication and division, then powers and roots. Together, these six operations are the main driving tools for both building and being able to move around in the horizon of arithmetic. He then goes on to explore what he calls higher viewpoints by turning to algebra, which discover patterns in the doing of arithmetic, such as A + B = B + A. Various properties and laws, even functions arise out of this higher viewpoint, and one could go a step further, to the next higher level in math, namely calculus.
On top of this, Lonergan then adds inverse insights, where one expects certain answers to questions about the square root of 2 or other irrational numbers where one expects an answer, but discovers that there is no such answer. Finally, he ends the chapter with the empirical residue, and this perhaps introduces one of the most interesting elements that later will become key for understanding how math is more than logic. That residue includes such things as particular places and times and the continuum, both of which have many connections to math.
The next chapter starts by comparing math to science, and he will begin to add statistics into the picture as well.