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Geometry, Modern: Theological Aspects

Geometry, Modern: Theological Aspects


The discovery of mathematics in deep Antiquity, together with its essential pair, geometry, was an important factor shaping rationalistic tendencies of the European spirit. From Plato's belief that "God geometrizes" through Einstein's conviction that the goal of science is nothing else but "to discover the mind of God," interaction between geometry and theology continued with change a changing rate and intensity.


Middle Ages through the nineteenth century

During the Middle Ages theology formed the natural environment for the sciences. For instance, the shift in theology from the understanding of God's presence in the world in terms of "his power" to the understanding of his omnipresence in terms of "all places" fostered the gradual emergence of the modern idea of space extending to infinity. This process culminated with the French philosopher and mathematician René Descartes (15961650), who identified matter with only one of its attributes, extension: body is nothing but an extended thing. Descartes was doubtless inspired by his monumental discovery of analytic geometrythe first really important discovery in geometry after Euclid and Apollonius. In Descartes's view, science, which should be done "in a geometric manner" (more geometrico ), is concerned with extended bodies, thus leaving to philosophy the realm of consciousness.

In the seventeenth century a kind of fusion occurred between science and theology (called physico-theology ) to an extent unheard before. This is clearly seen, for instance, in the writings of Isaac Newton (16421727). In creating his concept of absolute space Newton was a direct successor of former disputes on God's omnipresence. Newtonian absolute space, which "in its own nature, without relation to anything external, remains always similar and immovable" (Principia ; 1687), has three attributes: homogeneity, immobility, and infinity, which qualify it as both the universal arena for physical processes and the "sense organ" of God (sensorium Dei ). The enormous successes of Newton's physics overshadowed his theology and only the former function of the Newtonian space continued to exercise its influence on subsequent generations of thinkers.

Newton's absolute space as an arena for physical processes constituted an inherent element of the mechanistic worldview, and it came as a shock when it turned out that Euclidean space is not the only possibility. The dispute concerning Euclid's "fifth postulate" lasted from antiquity. The question was whether the fifth postulate has to be accepted as an independent assumption or could be deduced from other postulates. Many proofs of the fifth postulate produced during the centuries invariably turned out to fail. Around 1830, three mathematiciansNikolai Ivanovich Lobachevsky (17931856), Janos Bólyay (18021860), and Carl Friedrich Gauss (17771855)demonstrated independently but almost simultaneously that one can obtain a new geometry, a geometry that is absolutely consistent from a logical point of view, based on the negation of Euclid's fifth postulate. This shows that Euclid was right: The fifth postulate is an independent assumption and cannot be derived from other postulates. This long expected conclusion was overshadowed, however, by the fact that a new non-Euclidean geometry was possible. Soon it became manifest that by playing with axioms an infinite number of geometries could be created. In fact, in the second half of the nineteenth century many new geometric systems were created and extensively explored. The philosophical significance of this mathematical revolution was comparable to that of Copernicus (14731543): Humans are not only creatures from the outskirts of the universe, but even the universe, at least conceptually, is not unique; it is a member of an infinite family of geometric universes.

German mathematician Georg Friedrich Bernhard Riemann (18261866) in his 1854 inaugural lecture created a broad conceptual setting for modern geometry, which admitted more than three spatial dimensions. He also foresaw its physical applications: The world, with all its physical fields, could be but a system of fluctuating geometries.


Relativity

At the end of the nineteenth century, peoples' imaginations were fed with multidimensional geometric pictures. Some philosophers started speculating on "other dimensions" as living places for spirits, and the popular writer Edwin A. Abbot published a book in 1884 entitled Flatland, the principal aim of which was criticism of Victorian England, but which in fact inspired both philosophers and scientists to deal with new geometric spaces.

With the advent of the special and general theories of relativity the concept of space-time entered the imaginary requisites of popular and philosophical literature and became a powerful tool of scientific investigation. From then on, geometry would not only deal with the problem of space but also with at least some aspects of the time problem. Consider only two such problems that have repercussions in theological matters. The first problem concerns the nature of time flow and its relationship to eternity. The theory of relativity favors, but does not require, a picture of space-time as existing in one totality with the idea of the flowing time being only a "projection" of human psychological experience onto the world. Such a picture is consonant with the traditional idea of God's eternity (going back to Augustine of Hippo [354430 c.e.] and Boethius [c. 480c. 526 c.e.]) as existence outside time rather than existence in time flowing from minus infinity to plus infinity. The second problem concerns the interpretation of the initial singularity appearing in some solutions of Einstein's equations describing the evolution of cosmological models. The question whether such a singularity (for instance the one corresponding to the Big Bang in the standard cosmological model) could be identified with God's act of creation was once heatedly discussed. The prevailing view at the start of the twenty-first century is that such interpretations should be postponed (if they are methodologically legitimate) until a trustworthy quantum cosmology becomes available

Rapid progress in relativity theory, especially during the second half of the twentieth century, greatly contributed to the development of geometry. New physical problems required the sharpening of known geometric methods and the invention of new ones. In fact, the necessity to consider more and more abstract spaces gradually led to the broadening of the notion of geometry itself. The process of the geometrization of physics has changed both physics and geometry.


Noncommutative geometry

It seems that a long dialogue between science and religion has made people more cautious about drawing theological conclusions from scientific premises, but there is still one lesson the theologian can learn from this process. The degree of generalization of spatial and temporal concepts one meets in geometry and its applications to physics is a good warning against anthropomorphisms in theological language.

One notable achievement in geometry at the end of the twentieth century is the creation and rapid progress in the so-called noncommutative geometry, which has some roots in the mathematical formalism of quantum mechanics. One of its aims is to deal with spaces that are intractable with the help of the usual geometric methods. Noncommutative spaces are, in general, purely global entities; no local concepts have, in general, any meaning. For example, the concept of point, as a typically local concept, has no meaning in many noncommutative spaces. The number of attempts to apply noncommutative geometry to physics, for instance to create a fundamental physical theory, is constantly increasing. Some such attempts can have a profound philosophical meaning. For example, it is possible to create a model of the fundamental physical level in which there is no space and no time in their usual senses (space consisting of points and time consisting of instants, which are local concepts) and yet, in spite of this, an authentic dynamics (i.e., equations modeling behavior of physical systems under the action of forces) can be defined in them. Even if such models will turn out to be false, they demonstrate, by being logically consistent, that time (in the usual sense as transient succession of events) is not the necessary condition for an authentic activity. This seems to falsify the claim of some theologians that the idea of an active agent existing outside the flow of time is contradictory in itself.

Conclusion

To conclude, it could be said that although in the past there were many direct influences coming from geometry to theology, it seems unlikely that this will happen in the future. However, one could expect an indirect influence. Modern geometric methods and their application to physics and other natural sciences doubtless shape people's sense of rationality, and this feeling for the rational will continue to be a powerful source of theological inspirations.


See also Big Bang Theory; Einstein, Albert; Mathematics; Newton, Isaac; Physics, Quantum; Relativity, General Theory of; Singularity; Space and Time


Bibliography

boyer, carl b. a history of mathematics. new york: wiley, 1968.

connes, alain. noncommutative geometry. new york: academic press, 1994.

davies, paul c. w. the mind of god: the scientific basis for a rational world. new york: simon and schuster, 1992.

funkenstein, amos. theology and the scientific imagination from the middle ages to the seventeenth century. princeton, n.j.: princeton university press, 1986.

kline, morris. mathematics in western culture. london: penguin books. 1977.

torrance, thomas f. space, time, and incarnation. oxford: oxford university press, 1969.

michael heller

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