I’ve made the claim in various ways that all of our knowledge and thought comes from God’s thoughts as manifested in Creation, including this universe with its concrete thing-like being. This claim implies the further claim which I would strongly support: even the structures and basic elements of our most abstract systems (say those of mathematics and logic) are drawn from empirical reality by exploration and experimentation. I am most certainly not denying that there are valid abstractions, some of which can even be labeled `absolute truths’ (at least within the context of this particular Creation), but I am saying that abstracted Darwinian selection processes are the basis of even the best of human thought.
In a classic mathematics textbook, we can read:
In applying the unifying principle of abstraction, we study concrete examples and try to isolate the basic properties upon which the interesting phenomena depend. In the final analysis, of course, the determination of the “correct” properties to be abstracted is largely an experimental process. For instance, although the limit of a sequence of real numbers is a widely used idea, experience has shown that a more basic concept is that of a limit point of a set of real numbers. [“Topology” by Hocking and Young, p 1]
Because of this process of experimentation followed by a process of analysis and contemplation—possibly leading to more experimentation, the understanding of the field of `topology’ has changed over time, though each of the understandings has held true even as a new understanding was added.
Topology was first envisioned as an abstraction of geometry; as Euclidean Geometry was the study of the movements of rigid geometric entities (abstracted from reality), so was topology a study of the possibilities of deforming some of those geometric entities into other geometric entities, say, a square into a circle. The standard video easy to find on the Internet shows a teacup being deformed `continuously’ into a donut.
And so it was that topology was partially re-envisioned as the study of `continuity’, the study of `limit points’.
More recently, topology has been again partially re-envisioned in terms of abstract algebra, abstract relationships and transformations.
The main point in this limited context is that this has all come about as a result of exploration and experimentation, not as a result of setting up pure and ideal systems of axioms; the process will involve the setting up of proposed systems of axioms which are then `played with’, tested against the `reality’ of mathematics. Systems of axioms play a major role in all fields of mathematics and should maybe play a greater role in other fields of human effort, but the useful such systems—those which hold up to hard use and are productive of further fruits—develop in surprisingly empirical ways and stand or fall by way of their usefulness and fruitfulness. Even light readings in topology (seems to be a good example for analyses of human though processes) will show that the processes of exploration and experimentation has resulted in very abstract definitions and ways of analysis—it’s hard for a neophyte to know what some of those definitions can mean and serious mathematicians will simply point out that that particular definitions of ‘topological space’ or ‘limit point’ are the well-established results of an effort to, my words, distill out some essence of topology. They work; the essence is for real and more recent distilled essences are more real than earlier distilled essences.
This is pragmatism of a sort, a pragmatism of the best sort, and it has developed in the most abstract of human fields of thought. Perhaps this is what William James and other hardheaded pragmatists were aiming at—not the denial of abstract truths and abstract ways of thought but an insight into the dependence of a flesh-and-blood creature upon its concrete world, a dependence which extends even to abstract thought. In other words, perhaps a truer pragmatism makes no statements about the nature of truth but only about the nature of human efforts to discover truths.
A directed selection process which leads to valid systems of abstract thought?