I use the term `empirical science’ to include any field of disciplined study of Creation, physics and chemistry and engineering sciences but also including mathematics and history and literature.
The modern empirical sciences have advanced ahead of philosophy and very far ahead of Christian theology. Philosophers and theologians would be wise to start borrowing from those modern empirical sciences, not the particular mechanistic models but the true reasoning underlying those models. I’ll ignore for now the fact that fields such as history and sociology are in between, going through some successful and some far from successful efforts to use more advanced and abstract reasoning at least in principle similar to the powerful abstractions of those empirical sciences which had long ago adopted mathematics because of their need for quantitative analysis and then learned in the 1800s of the power of more flexible and more qualitative forms of geometric reasoning.
Yet, I’m only speaking of the arguments which, so to speak, end an analysis. It has always been true that the fundamental forms of human reasoning, including those of all modern empirical sciences, are qualitative and not quantitative. It takes a great deal of study and the development of sophisticated thinking processes to penetrate to those fundamental forms, though they are visible in the writings of Einstein in his early works and in many of the writings of the quantum theorists. They are also visible in the writings of modern theorists of evolutionary biology, in the writings of modern neuroscientists and geneticists, perhaps even in the successful (perhaps a low percentage?) and the failed experiments of modern music and painting and sculpture. As will be clear soon, the idea that quantitative fields rest on qualitative foundations comes to the fore in the thoughts of some mathematicians including educators and famous (at least in small circles) men. There are also philosophers of science and mathematics working on this and related ideas.
Let me back away for a short while and put in my own two cents worth about the need to deal, at least ultimately, with issues of being and not to treat knowledge, thoughts, and truth, and other mental `entities’ as if they were separate from being and to be imposed upon being to establish a proper order. That order, in fact, comes from being and emerges in our own mind but only as we learn to encapsulate relatively more concrete forms of being and to move on to the more abstract forms of being from which the concrete, thing-like being is shaped.
In an early and very short essay, Einstein and Bohr’s debate on the meaning of reality:, I addressed this general issue in terms of being and not knowledge, and in terms of an important debate which took place between Einstein, advocating what was really a `high’ pagan philosophy of worldly realism where things exist in themselves and can’t be inherently changed by external relationships, and Bohr who was advocating a radical view that relationships, such as those we can express in terms of the equations of quantum mechanics, are primary over stuff. Bohr’s position, as I pointed out explicitly in Quantum Mechanics and Moral Formation: Part 1, is pretty much the view advocated in the writings of St John the Evangelist and his followers and also anticipated in the Old Testament.
If we accept that relationships can be such, then we have accepted the reality of abstract being and should learn to speak in ways more appropriate to being, to reality.
I’ll move on to a few suggestive quotations on the subject of qualitative reasoning in modern mathematics and, hence, in modern scientific and engineering thought in general. For now, I’ll let the readers use their imagination to transfer these thoughts about mathematical knowledge to those aspects of being described so well by mathematics; from there we can see how this reasoning can be applied to all forms of created being, including human being.
In the article Symplectic Geometry by Alan Weinstein and published in 1981 in Bulletin of the American Mathematical Society (Volume 5, Number 1), we can read on page 1:
[G]eometry has taken a new role in mechanics through the contributions of Poincare (1854-1912) and Birkhoff (1884-1944). Now, though, the geometry is the more flexible geometry of canonical (in particular, area preserving) transformations instead of the rigid geometry of Euclid; accordingly, the conclusions of the geometrical arguments are often qualitative rather than quantitative.
In that same article, we can learn that the highly regarded 20th century mathematicians, GD Birkhoff, expressed “the disturbing secret fear that geometry may ultimately turn out to be no more than the glittering institutional trappings of analysis,” while Poinsot (1777-1859), in Weinstein’s words, “suggests that calculations are merely a tool in the service of geometrical and mechanical reasoning.” (I decided to include Birkhoff’s statement, though I can’t currently wrap my mind around it, because Weinstein indicates that Birkhoff and Poinsot were expressing the same opinion.)
In a book of writings by Soviet mathematicians, Mathematics: Its Content, Methods, and Meaning, Volume 2, the truly great Russian mathematician, A N Kolmogorov, has this to say on Page 258 of the article, The Theory of Probability:
The proponents of mechanistic materialism assumed that such a formulation [of systems describable in terms of relatively simple differential equations, such as gravitational fall] is an exact and direct expression of the deterministic character of the actual phenomena, of the physical principles of causation. According to Laplace, the state of the world at a given instant is defined by an infinite number of parameters, subject to an infinite number of differential equations. If some “universal mind” could write down all these equations and integrate them, it could then predict with complete exactness, according to Laplace, the entire evolution of the world in the infinite future.
But in fact this quantitative mathematical infinity is extremely coarse in comparison with the qualitatively inexhaustible character of the real world. Neither the introduction of an infinite number of parameters nor the description of the state of continuous media by functions of a point in space is adequate to represent the infinite complexity of actual events.
It’s interesting and enlightening and refreshing to see determinism shot down, not by probabilistic arguments but by arguments pointing to the “qualitatively inexhaustible character of the real world.”
In the article, Non-Euclidean Geometries by another highly-regarded Russian mathematicians, A D Alexandrov, we can read on page 155:
The real significance of this point of view is that it makes it possible to use the concepts and methods of abstract geometry for the investigation of diverse phenomena. The realm of applicability of geometric concepts and methods is extended immensely in this way. As a result of the concept of space the term `space’ assumes two meanings in science. On the one hand it is the ordinary real space (the universal form of existence of matter), on the other hand it is the `abstract space,’ a collection of homogeneous objects (events, states, etc.) in which spacelike relationships hold.
On page 158, we can read a description of `space’ from an abstract viewpoint:
By a `space’ we understand in mathematics quite generally an arbitrary collection of homogeneous objects (events, states, functions, figures, values of variables, etc.) between which there are relationships similar to the usual spatial relations (continuity, distance, etc.). Moreover, in regarding a given collection of objects as a space we abstract from properties of these objects except those that are determined by these spacelike relationships in question. These relations determine what we can call the structure or `geometry’ of the space. The objects themselves play the role of `points’ of such a space; `figures’ are sets of its `points’.
Abstract spaces might well be a `graphing’ of abstract forms of being in so far as they show up in properties or behaviors of concrete being, including human being.
Finally, we can read on page 1 of the book, Topology and Geometry for Physicists (Dover Publications, 2011), by Charles Nash and Siddhartha Sen:
[T]opology produces theorems that are usually qualitative in nature—they may assert, for example, the existence or non-existence of an object. They will not in general, provide the means for its construction.
Some of my readers may have better knowledge and skills in these fields than do I, but, for the others: qualitative, topological reasoning is very important in modern physics, even in fields where the goal is to ultimately produce a quantitative result.
We have hints of a greater unity, one in which knowledge is not `of’ being, but rather an encapsulation of being in a way that makes it a sharing in being. Created being is manifested thoughts of God so that Creation is certain acts, acts-of-being, of the mind of God, acts-of-being become objects in which God is ever-present because that divine presence is God’s thought. God is unified in such a way that thoughts and acts and feelings aren’t really separate, as will also be true of that completed and perfected human being, the Body of Christ.
By exercising the human ability to encapsulate large chunks of the world and even of those realms of Creation invisible to the sense organs of our thing-like being, we share in those thoughts and acts and feelings of God in His freely-chosen role as the Creator. Being able to quantify, say, the effects of gravity under relativistic or non-relativistic conditions is a part of that sharing but it is more complete and more perfect when it becomes that greater understanding in a way similar to the greater, geometric understanding of spacetime which is the General Theory of Relativity, an understanding which can be made more specific so that it is the quantitative description of a particular universe or relativistic object such as a black-hole. Yet, the more general understanding, the more abstract description, doesn’t go away even when you tighten the constraints on the field equation of General Relativity so that a specific universe/object is described and the equations can be solved.
More importantly for my purposes, if we stay at the very abstract level for a while, we can begin to better understand the being of that abstract level, being from which this concrete, thing-like world is shaped. Using abstract space and geometry, I can propose that we might be able to understand the cohesion of societies which have expanded far beyond kinship and proxy-kinship relationships in terms of a bending of the state space which brings citizens together.
There’s much work to be done and all I can say at this point is that we won’t come to better, even more exact, understandings of human communities by way of straightforward adoptions of quantitative methods from physics nor by way of more clever use of our current stock of words and concepts regarding the concrete, thing-like, directly perceptible properties of those communities.