… The geometer imagines dots but thinks of points; he imagines fine threads but thinks of lines. The thinking is exact and precise, and imagination does its best to keep pace. In like manner the empirical investigator will tend to endow his images with the closest possible approximation to the laws he conceives. But while his imagination will do its best, while his perceptions will be profoundly influenced by the habits of his imagination, nonetheless the data that are available for the ideal observer make no effort towards such conformity. They go their own way with their unanalyzed multiplicity and their refractoriness to measurements that are more than approximate.
… higher viewpoints in mathematics are reached inasmuch as initial images yield insights, insights yield definitions and postulates, definitions and postulates guide symbolic operations, and symbolic operations provide a more general image in which the insights of the higher viewpoint are emergent. Now in empirical method there is a similar circle, but it follows a slightly different route. The operations that follow upon the formulation of laws are not merely symbolic. For the formulation expresses a grasp of possibility. It is a hypothesis. It provides a basis for deductions and calculations no less than mathematical premises do. But it also provides a basis for further observations and experiments. It is such observation and experimentation, directed by a hypothesis, that sooner or later turns attention to data that initially were overlooked or neglected; it is attention to such further data that forces the revision of initial viewpoints and effects the development of empirical science.
The circuit, then, of mathematical development may be named immanent: it moves from images through insights and conceptions to the production of symbolic images whence higher insights arise. But the circuit of scientific development includes action upon external things: it moves from observation and experiment to tabulations and graphs, from these to insights and formulations, from formulations to forecasts, from forecasts to operations, in which it obtains fresh evidence either for the confirmation or for the revision of existing views.
Bernard Lonergan, Insight
