Geometry and God: How mathematical thought can illuminate metaphysics

Posted on 15 December 1993

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The great power of mathematics rests in the rigor with which it demands clarity of definition. And although ambiguity is present in mathematical discourse, it is made explicit and therefore disabled. Many an argument immediately dissolves once the two sides realize that they were starting from different premises or definitions. So we would do well to exercise this kind of discipline in situations where the danger of ambiguity and misunderstanding is great.

I wonder if the following analogy might serve to clarify our understanding of this issue. As was known since Euclid, geometry can be formulated as a system of propositions that can be logically deduced from a certain set of axioms and definitions. Modern mathematicians have realized that one can modify these axioms slightly to obtain non-Euclidean geometries, which are just as equally true in the strict mathematical sense as Euclidean geometry. Nevertheless, many people dismissed these “imaginary” geometries as mere fanciful speculation since only Euclidean geometry was physically real. Or so they thought. Since Einstein, we now know that, in fact, non-Euclidean geometry is more accurately called the “real” geometry since physical space is curved and not flat. But this curvature is a very subtle phenomenon that can only be detected under the most extraordinary and precise experimental conditions. In other words, although Euclidean geometry is not entirely accurate, it is very close to being accurate. Moreover, this fact itself is “explained” by non-Euclidean geometry since curved space on a very small scale is very nearly flat.

The moral to the story is that (1) we fool ourselves if we think our conceptual systems have unlimited validity, and (2) even though a choice of axioms or definitions may have limited validity and ultimately be superseded by a more comprehensive system, it may nonetheless be quite sufficient for many purposes. In other words, any system is perfectly true in the sense that it perfectly represents exactly that aspect of reality which is delimited by the definitions and axioms at its foundation. Falsity and confusion is introduced when we superimpose upon such conceptual systems the baseless assumption that they have unlimited validity.

Another insight that can be gained from the study of this example from mathematics is the fact that certain conceptual systems or views can be seen as subsets of others. Non- Euclidean geometry in a certain sense contains Euclidean geometry since curved space is nearly flat on small scales. In this case it is not a question of choosing sides and having an argument over which geometry is right. We simply choose the one that best serves to communicate what we wish to express in the clearest possible way. If I want to give directions to my home, I will not use non-Euclidean geometry in the case of a friend across town. But if my friend is sailing across the ocean, I may well need to take into account the curvature of the earth if she is to find me.

Now can we perhaps clarify our understanding of the issue of defining God by analogy to this example from mathematics? In other words, perhaps there are, as it were, different views of God that are each used to express different aspects of reality for different purposes and in different circumstances. No one view is itself absolutely valid. Yet each is perfectly valid as far as it goes. Moreover, it may be the case that these different views are nested within each other as Euclidean geometry is nested in non-Euclidean geometry. In this case, there is a possibility of seeing them all in harmony.

Strictly speaking, the non-Euclidean axioms are not consistent with the Euclidean axioms. If space is curved, it is not at the same time flat. So non-Euclidean geometry denies flat space. But this is on one level. As we consider smaller and smaller portions of space, the curvature is less and less significant, and we can treat space as if it were flat. In this sense the non-Euclidean axioms and the Euclidean axioms are compatible. But the non-Euclidean ones are the “real” ones and the Euclidean ones are the “as if” ones. They don’t have the same status. Because we maintain this distinction, we are able to see these different incompatible systems in harmony.

Are there cases where one system supersedes another in such a way that the earlier one is seen to be entirely unacceptable, even on an “as if” or limited basis? It seems to me that if the superseded system had any truth to it at all, then that truth should be able to be seen within the context of the larger truth. Either that or the new system does not supersede it, but rather complements it. To take an example from physics, both the theory of general relativity and quantum field theory are systems that supersede classical mechanics and yet are each compatible with it in the above “as if” sense. Yet relativity and quantum mechanics are not compatible with each other. The great “holy grail” of physics is to find a larger system which contains both general relativity and quantum field theory. Even though these theories are inconsistent, they are valid in their own domains. Thus they are complementary. Neither one contains the other.

Using this analogy, just as a grand unified theory of physics would show how these complementary theories are compatible, a more comprehensive understanding of religious concepts would show how many complementary concepts are compatible. Even if we don’t yet see how apparently contradictory views can be harmonized, we must have faith that this harmony is possible if we are to find a unified vision. And this faith is the driving spiritual force behind both scientific and religious breakthroughs.

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Posted in: Philosophy, Science