Differential Geometry and Moral Narratives

Pilgrims travel paths, journeying from one location on earth to another, sometimes those places are fictional but usually quite concrete. To be sure, Dante’s pilgrim found (as some of the more recent translations of The Inferno attest) that the path could wander away from him, a strange event from a fully concrete view of paths. It requires sharp vision and alertness to keep track of that path which seems to often test us in so many ways. That Christian poem, or other such Christian works as Pilgrim’s Progress, follow in an ancient tradition scouted out by Homer in the Odyssey and likely many preceding verbal works that were not recorded in a more lasting form. C.S. Lewis followed a less popular variant of pilgrim literature when he wrote Pilgrim’s Regress as a highly metaphorical work in which the pilgrim moved mostly, as I recall, from one site to another where took place conversations intended to show the problems in modern philosophies.

Our moral and spiritual and intellectual journeys bear close comparison to physical journeys and there will always be room for concrete journeys in our various tales of our lives, individual or collective. But there’s more to be said for modern man because there has been progress of an sort in complexity and richness of human life if not necessarily in quality of life. Moreover, there’s at least one way in which the journey across roads or over oceans was always a distortion and oversimplification of the human experience:

There’s not so strong a boundary between inside and outside for any creature in this universe though there was an illusion of such during the period that the human mind was evolving through a stage where our individual existences were coming into somewhat sharper focus.

But there’s more:

The number of human beings in our lives, the number of things, the level of abstract relationships in our lives (such as our relationships to far-away, centralized governments), the sheer richness of our relationships in total, demands better ways of examining and analyzing our moral and social lives.

We need metaphors as rich as our moral lives. Seemingly by accident, I’ve constructed the possibility of such metaphors by my efforts to view Creation from a Christian position while accounting for modern empirical knowledge, including the most theoretical of physics and mathematics. But it’s no accident because my major concerns all along have been with the nature of created being and the nature of the human mind.

Modern empirical knowledge has increased by extraordinary amounts in recent centuries. In fact, much of it is really data that hasn’t yet been digested enough to call it knowledge. Yet, we’ve greatly enriched our understanding of being, at least so far as atoms and stars are concerned. Still more importantly, we’ve come to have some understanding of the universe as an entity in its own right and not just a collection of all that we’ve observed so far.

We certainly can’t claim to understand even the aspects of created reality covered by our most successful theories: general relativity and quantum physics — especially quantum electrodynamics. Yet, our understanding of certain aspects of being in this universe, and maybe a bit beyond, has been greatly enriched. It’s been enriched far beyond the understandings of created being which are a part of our philosophical and theological traditions, including our moral traditions. The concepts of created being which underly the study of black-holes is far richer than the corresponding concepts which underly the understanding that nearly all philosophers and theologians have for their study of human nature.

So, let’s get to work…

Under a Thomistic existentialist philosophy, the human mind is seen as forming in active response to the environments of the human being or a multitude of environments or even the entire universe in some meaningful sense. Our richer understanding of Creation is reflected in an increased richness and complexity of the human mind, at least a mind which is open to this new knowledge while also having a worldview which allows that mind to make sense of those mountains of knowledge.

The path is not just beneath our feet, it’s in our heads. Our minds partake of the slope and surface material of that path. Moreover, we’re constantly being jostled about or just generally pushed left or right by the masses of our fellow-travelers — the traditional journeys don’t respect the social nature of human beings and portray us as more or less pure individuals when it comes to our moral natures. To be sure, we often spend parts of our journeys in some sort of lonely desert, but we’ll then return to more heavily populated regions.

The journeys of a self-aware moral creature with a complex mind lie well beyond the reach of our current ways of speaking, beyond metaphors drawn from simple paths across the surface of the earth. In this case, a need for complexity is also linked to a need for greater abstraction. I’m going to use differential geometry as one likely source for useful models for advancing our understanding of our increasingly complex and abstract selves, a change brought about by various factors including the increasing size of the human population but mostly the increasingly complex and abstract nature of our social and political relationships. Differential geometric models are very abstract, but so is the world of a human being living in a complex society and developing a complex mind. Differential geometric models allow a much richer description of possible future paths at each point on a journey. Along with tensors, they also allow ways of speaking about the distortion in a space, say a moral space, caused by the presence or the movements of other creatures. Tensors are typically used along with these models (or separately for some reasons) partly because they allow a concise expression of complicated and complex physical actions, such as stretching or twisting or both at the same time.

The American physicist John Wheeler once summarized general relativity by telling us that matter tells space how to shape itself and space then tells matter how to move. Maybe we can play around with this metaphor:

Human beings tell moral space how to shape itself and moral space then tells human beings how to move through life — how to act.

Now we’re talking.

I can relate to moral metaphors that speak not so much of a simple path beneath my feet as a path that merges into me, twisting and stretching me even to the depths of my being. This ‘path’ is all that I move through or communicate with and all of my own bodily substance.

I can relate to a human world in which, for good or bad, we shape our moral space, the space in which we live our moral lives, including both our lives in public spheres and private spheres. ‘We’ shape our moral space. Few there are strong enough, brave enough, to move onto a path far from those they’ve traveled to that point in their lives, far from the path they were raised to travel. But it would be easier for all of us to see the more abstract possibilities and demands of our increasingly complex moral lives if we had better ways to speak and think about those lives. An obvious example of human beings who were, so to speak, in over their heads is the bulk of ‘nice’ Germans from the 1930s and 1940s who continued to live local lives that were decent and morally well-ordered even as they failed to understand or to respond to the more abstract demands upon them as citizens of a nation which had fallen into evil ways because of actions of a centralized and somewhat abstract national government. It’s certainly true that some did see and understand their moral responsibilities, but even those had problems, so far as I know, bringing matters into focus. They had no clear ways of communicating the horrors which were developing throughout Europe.

I can relate to a human nature which is stretched and compressed and twisted as it travels along those paths, paths which can shoot away in unexpected directions without warning and sometimes without giving good indication of what’s happening until we’re well along a path we falsely thought to understand. Again, those nice, middle-class Germans were doing all they’d been raised to do, hold good jobs and pay their bills and take care of their own children, but the path leading to regions of moral order had veered well away from those simple paths they’d traveled, not because they’d been raised wrong in ways that could have been forecast but because the world had become complex in a way that modern middle-class thought doesn’t even perceive. The world had gotten more complex than could be handled by modern middle-class ways of moral thought and behavior.

I can relate to a human life and to a human community life in which the path veers away from us as it did to Dante’s pilgrim. We can find out late in the game that the ‘right’ path we’d followed has led us or our children into a state of moral corruption.

Like it or not, human beings don’t live in a world which is ‘only’ concrete — as I’ve tried to communicate in my various writings on this blog as well as in my first published book: To See a World in a Grain of Sand. We live in a world in which the abstract is with us always. How else could mathematicians and other thinkers know of such strange entities as transfinite numbers? In my way of thinking and speaking, God shapes the concrete aspects of our world from a more abstract foundation which is the manifestation of the truths He chose for Creation. I call that abstract source of created being the ‘Primordial Universe’. We can, so to speak, coax the abstract aspects of our universe, and perhaps those of Creation in general, out of what seems to be only concrete by way of proper tools of thought, such as differential geometry and tensor analysis and metaphors and analogies and so forth.

I say ‘proper tools’ because they are not merely useful fictions. Good metaphors draw upon more abstract levels of reality where, for example, the moral paths of a mind-ed creature can be more complex than any possibilities seen directly in his concrete world and the visible events of his life. Differential geometry and tensor analysis and other abstract tools developed by mathematicians and other scientists point at certain aspects of being as manifested in the Primordial Universe, aspects of being which we share with stars and atoms. As creatures whose minds can be shaped by responses to this universe, we share in the full being of this universe in a special way. In order to understand ourselves, we need to use our richer and more complete understanding of all aspects of this universe, including very much so, the fundamental aspects which can be described by the mathematical tools of modern theoretical sciences, but we clearly need to also learn how to integrate into our moral understandings all that knowledge which comes from modern historical studies, from evolutionary biology, from literary explorations of reality, and so forth.

These are interesting and exciting times for those who would like to help build the foundations of the next phase of human civilization.