Let’s do a Plato-update. Mathematics is the key, as it was for that heavyweight wrestler (according to some rumors which even hint at championships). Very roughly speaking, philosophy to Plato was a generalization of the thinking done in geometry and arithmetic. It may not be true that Plato’s school (the Platonic Academy Academy) had a sign above the entrance which said: “Let None But Geometers Enter Here,” but that seems to be the spirit of Plato’s way of thinking—at least after he’d absorbed some of (much of?) the seemingly qualitative (more informal?) thoughts of Socrates.
Plato, his predecessors and his successors, used mathematics as a form of training and also as a source for thinking tools capable of handling abstract realms of being—as I would put it. Those creative borrowers, call them clever burglars if you wish, were able to produce higher level understandings of various aspects or realms of created being.
This business of speaking of moral and even divine matters by using the relationships, geometrically considered, of material things was natural and even necessary for all those trying to discern the abstract relationships in all of this stuff in flux. It certainly seemed natural to the ancient Judeans to press their argument of Jerusalem’s close relationship to God by emphasizing it’s location, on a hill, a high hill by the standards of the immediate region. Were they saying it was closer to the Heavens? I don’t know but they were clearly saying that Jerusalem was above the cities of Canaan and even above Babylon and that meant…it was holy; the city of David, God’s beloved, looked down upon the other (known) cities of man. To Greeks and Hebrews alike, good men followed straight paths and men, like camels, had trouble passing through the eye of a needle (the name for narrow gates through deep portions of the wall of a city, used to deny access to bands of undesirable visitors). Camels are pretty big animals and those gates weren’t much bigger; it could be a very tight fit for camels carrying any sort of load. The gates were narrow but sheerly impassable to those who thought to bring in large loads of goods or to those trying to enter in large groups rapidly and perhaps for nefarious reasons. Again, burden yourself with riches and your entry into the City of God becomes more difficult if not impossible for men to accomplish. Try to invade the City of God for man-centered purposes and the entry was also difficult.
We live in a world where the most obvious forms of being are physical things and physical relationships, many of which things and relationships are at least partially described by mathematics of various sorts, ranging from the simple and (maybe) non-theoretical geometries and trigonometries of ancient pyramid- and temple-builders. The very development of (again-maybe) purely empirical technologies combined with the preference of many ancient Greeks for abstract understandings. By the time that Ptolemy, the Greco-Egyptian scientist, built—among other serious intellectual edifices—an extraordinarily complex and very accurate model of the movements of the known large bodies of the solar system. This, like all higher-level scientific accomplishments through the ages, was based upon a great respect, almost idolatrous at times, for order of a sort allied to human reasoning—most especially in its abstract forms, metaphysical and mathematical. (The idolatry shows up mostly when our forms of reasoning prove inadequate but we refuse to pay proper attention and respect to empirical reality and the abstract realms to which they point, thus refusing to pay proper attention and respect to God in His role as Creator.)
I’ve provided a metaphysical understanding of this world and the greater Creation in which this world is embedded. In the tradition of all major schools of human thought, Christian and Jewish and Islamic and pagan and others, I’ve made a claim that we can understand each and every thing as well as the greater realms of Creation by keeping that understanding of Creation in mind and constantly using our understanding of the small to better our understanding of the large and vice-versa. My metaphysical understanding of Creation was based upon a reconstruction of something of a Thomistic system—radically existentialist among other traits, allowing for a more upfront recognition of the necessary respect for empirical knowledge—that of Creation. This respect requires that we consider: modern theories of spacetime, of biological evolution and development, of quantum physical theories of matter, and modern abstract mathematics which have sometimes spoken of facts and truths in empirical reality before we noticed them. My reconstruction, arguably a new Christian metaphysics as much as that of Aquinas was new relative to that of Anselm and earlier fathers, is somewhat superficial right now, but is presented as very much an open work in progress; if I’m heading in the correct direction, there is plenty of work for many generations of thinkers and teachers and popularizers.
I’ve not even scratched much of the surface of modern empirical knowledge, though a little bit more than I described above because I have made some serious consideration of the knowledge in modern fiction and in modern history and other `softer’ sciences.
But there is something missing in my work, despite my efforts to present my sketches of a metaphysical system in a way that shows all realms of created being, from our concrete realm back through abstract realms of created being and right to the truths and divine thoughts manifested as the raw stuff of Creation. That something which is missing is perhaps a necessary clarity rather than substantial claims to knowledge.
Let’s assume, as a matter of necessity of the common sense variety, that Creation is made up of all of its pieces: creatures abstract and concrete as well as specific manifestations of divinity beyond or different from creatures—I write with a necessary vagueness for now. In what sense is my understanding of Creation, call it my worldview, made up of all the understandings I draw upon: Biblical revelation as well as a small body of other revelations along with human knowledge of concrete things and abstract things and all their various relationships?
We don’t want things to fit together too tightly. That would conflict with our feelings and well-reasoned thoughts about freedom—not any sort of absolute freedom and not under the control of an agent called `free-will’ but some sort of freedom we can’t describe but might be describable with appropriate qualitative abstractions of ideas from modern mathematics. In particular, we must protect the experienced constrained freedom of individual entities within larger entities (for individual human beings, this means communities as well as other corporate entities of a mostly physical nature).
Before coming to a conclusion, I’ll point out the obvious—at least it’s obvious to those with knowledge of history, including the history of human thought. What I’m saying has consequences for all fields of human thought and endeavor. For example, while there are no implied political policies of a specific sort in my work, there is a strong implication that true politics for human beings as they exist in this world as individuals and communities exclude most modern politics of a left-wing and right-wing sort, as well as any mushy politics in the so-called center. In fact, most `idealistic’ systems of thought about politics and even human nature are disallowed in favor of an ongoing analysis of revelation and empirical knowledge.
The conclusion of this meandering essay is basically the theme of my latest major effort: modern mathematics has tools, qualitative and conceptual as well quantitative, for dealing with parts and wholes—without dissolving the parts into the wholes or considering the wholes as merely nominal conglomerates. (This book is tentatively titled The Shape of Reality and is at least six months away from completion.)
This is philosophy and theology done right—to make greater sense of it all while not losing the sense of the smaller realms or particular things. Mathematics has provided, somewhat, the tools for this essential task in an age facing an embarrassing treasure of mostly undigested knowledge of the concrete and abstract realms of Creation. I qualify by `somewhat’ only because highly technical tools of modern, abstract mathematics (topology as a qualitative or abstracted geometry, abstract algebra as qualitative or abstracted studies of structure, etc) need to be understood and then appropriated (stolen?) for more general use by philosophers and theologians, and then by historians and mind-scientists and so on. A lot of work lies ahead of us if my views correspond at all to reality.