Copernican heliocentric theory? If I recall correctly, we could be even more precise by observing that the earth does not exactly revolve around the sun, but both revolve around a common centre of gravity which, in the nature of the case, is very close to the latter. Our scientific descriptions always involve approximations which, to a certain extent, can be accommodated within a paradigm before the accumulation of perceived discrepancies may trigger a revolutionary shift. Even our theological formulations partake of this process of approximation as they reach out to what the Bible teaches and to what Holy Tradition reveals, and this "approximation" is also an 'approximation' in the sense of a "coming closer to" God (called "apophasis" in Eastern Orthodoxy). Thomas S. Kuhn himself has observed the similarity of his scientific "paradigm shifts" to religious conversions.

It is possible that the paradigms we think have faded away may actually still be with us in a new key, in new applications. There are many examples perhaps still to be uncovered by historians in which the ancient mystical and theological speculations concerning the heavenly and spiritual worlds became redirected to nature and embodied in the foundations of modern science. The Ptolemaic system is a case in point, if Henning F. Harmuth's suggestion is correct:

"It is generally believed that Kepler ended the dogma of the circle, but this is not so. The circle disappeared from astronomy, but it reappeared in other fields of science in disguise...the exponential function...or unit circle in the complex plane...Speaking more generally, the superposition of circles by Ptolemy and Copernicus became the Fourier series expansion in complex notation..The expression *character group of the topological group of the real numbers* does not seem to have anything to do with the circle, but its mathematical notation...reveals the truth. This character group implies the topology of the continuum for space-time, which in turn permits the use of the differential calculus for functions of space and time"(1).

While this mathematics may seem at first sight to be far-removed from the usual topics of theology, I would like to draw attention to a remarkable monograph on the theological methodology of John Calvin privately published by the Patristic and Calvin scholar, Dr. Ford Lewis Battles, not long before his death: _Calculus Fidei Some Ruminations on the Structure of the Theology of John Calvin_ (Grand Rapids: Calvin Theological Seminary, 1978). In this painstakingly-detailed study of Calvin and his Patristic antecedents, Calvin's methodology shines forth as a continuation of the pattern of his own religious conversion oriented to the limiting processes of the classical *epistrophe*, i.e., the contemplative turning around and opening up of the soul to God. Dr. Battles observed the similarity of this approximative movement of thought and the soul and the heart to the doctrine of limits informing the Calculus of Sir Isaac Newton as well as much of modern science. His details are sufficient, so it seems to me, to make the further case that the methodology of Calvin was an analogical equivalent in Theology to thinking the Differential and Integral Calculus in Mathematics.

More generally, the Differential and Integral Calculus of Leibnitz and Newton was a way of mathematizing the deep structure of contemplative mysticism, and of putting it to practical use in Science. A mathematical methodology came to replace the differentiative and integrative functions of Gnosticism, Hermeticism, Cambridge Platonism, and Christian Hesychasm. The topological circle in the foundations remained unbroken, and "Ptolemy" lived on! Actually, Ptolemaic cycles must then have returned to modern Astronomy and Science, according to this thesis, in Time-Series (Spectral) Analysis and its "frequency domains", and in the convolutions and deconvolutions of the mathematics of optical image-enhancement and pattern-recognition techniques (corresponding to the enhancement of perception reported in states of contemplation). There are some strong isomorphisms here also to Cybernetics and the Dialectics of Hegel expressed in terms of his teleological limit thinking and "circles of circles", and understood as a Theory of Time and as a "circular epistemology"(2, 3, 4, 5).

Getting back to Thomas S. Kuhn and his Social Historiography of Scientific Revolutions, I think we can safely say that here too there are some very interesting contemplative and mathematical isomorphisms, to use again that felicitous word which Rob...introduced in his post on the mathematics of Bernard Lonergan. Kuhnian Revolutions, of course, can be somewhat helpfully understood in terms of Hegelian dialectics as an ongoing process of theses, antitheses, and syntheses. His Phenomenology is not without Hegelian links, and his Sociology of Paradigms is not without the touch of 19th-century Russian Romantic "sociology of knowledge" via the Russian/French historian of science, Alexandre Koyré. This is to imply that when we really begin to probe the depths, Kuhnian theory, too, discloses its clear resemblance, both in structure and dynamics, to traditional contemplative mysticism and to the workings of the Differential and Integral Calculus. Pushing problem solving to the utmost limits of the paradigm, we eventually find that the paradigm itself must be differentiated, and conventional constants stripped away, in order to reintegrate between the new, more scientifically-productive sets of limits.

With reference to Sir Isaac Newton's mathematical Calculus and to a similar apophatic (limit-approximating) pattern discerned in the theological methodology of John Calvin, Dr. Ford Lewis Battles wrote that "The theory of limits, on which modern mathematics, and as a corollary, all modern science and technology depend, is actually a secularized form of the Scriptural view of reality. Mathematics and Scripture both view the human grasp of reality as increasingly approximated to, but never identified with, the human symbols we use to represent reality"(6).

Faith itself is approximative in this manner, reaching out to that which lies beyond, stetching forth according to the pattern that long ago was called *epectasis* or the *intentio animi*. Nowadays, so it seems to me, faith-functioning in this sense has not uncommonly been secularized and visually reduced to Phenomenological *intentionality* stemming at least in part from Husserl's inclination to Rhineland mysticism. Dr. Battles places the Calculus Fidei of Calvin in the context of the latter's not-so-well-known contemplative mysticism and its links with Patristic Orthodoxy.

Once, when I quoted Dr. Ford Lewis Battles to the effect that "The theory of limits, on which modern mathematics, and as a corollary, all modern science and technology depend, is actually a secularized form of the Scriptural view of reality", one respondent replied, "Utter nonsense", asserted no connection between the mathematical notion of limit and Scripture or John Calvin, invoked Non-standard Analysis as a counter-example, and suggested "Foundations" (if anywhere) as Mathematics's preferred basis. All well and good, perhaps, if we are willing as Christians to remain within the confines of politically-correct, pure mathematical theorizing, where even the infinitesimals, long so practically useful to many mathematicians in secret, took generations and generations and then some to surmount mathematical prejudices and to appear in the light of the present day in their new, Non-Standard format (crossing paths with Leibnitz and monads and Alchemy along the way). Surely as Christians, however, we will recognize that the true foundation of Mathematics is nowhere else than in the Word of God Incarnate. (Colossians 1:16-17; 2:3). Faith is the eye of reason, mathematical reason included.

Does this sound strange? It is certainly in keeping with my own theological formation in High Calvinism (more supralapsarian than the 5-pointer kind) and Hesychast Patristic Orthodoxy, still close today to the ancient tradition which held Mathematics itself to be contemplatively a way to God (in Christ, now we know, via the "ladder of creation"). There is no problem here for the Mystical Theology of the Eastern Church, nor with Calvin's *meditatio coelestis (or *futurae*) vitae*, I am sure, even though Calvin himself did not develop the formal mathematics of it. In all fairness to Dr. Ford Lewis Battles at this point, we do need to take note that it was not of individual "theories" that he really spoke, but of "theory" in the earlier, contemplative sense of "theoria".

As far as individual theories go, Non-standard Analysis is relatively-speaking simply a very recent newcomer. Not even a well-developed theory of Mathematical Limits was in place at the time of Newton and Leibnitz. Nor was the Theory of Character Groups in place, by which Henning F. Harmuth has suggested that we can link the real numbers and the Calculus with the ancient cyclical-dynamic views of the cosmos (i.e., via the unit- circular form in the complex plane of the notation for the character group of the topological group of the real numbers). The ancient tradition of "theoria', on the other hand, has been in place for millenia, and to this in its specifically Calvinistic format, with its antecedents in Patristic Orthodoxy, Dr. Battles wished to draw our attention and our faith.

The specific form of the limit methodology which Dr. Battles discerned as operative in the "antithetical structure" of Calvin's _Institutes_, and recorded in meticulous detail, was very much other than the dichotomous visual logic of Peter Ramus, and passed far beyond the Aristotelian principle of *reciprocal excess and defect* and the *true-false* principle which had most recently been used by Erasmus, Luther and Zwingli, to actually take on the pattern of a mystical *double movement* of *descent* and *return/ascent* both Biblical and mathematical as illustrated with reference to Isaac Newton, and replicating the dynamic structure of Calvin's own conversion to Christ. Calvin's theological consciousness operated with multiple limiting ratios of antitheses and differences.

In the Byzantine world, the "Calculus Fidei" in this sense had already been stabilized many centuries before Calvin as the *lived experience* in contemplation of the *Apophatic Theology*. There is currently in Eastern Orthodox theology some recognition that *limit thinking/faith-functioning/ intentionality/intentio animi/epektasis* combined with *conversion* (the unitary *double [or *circular*] movement* of *descent* and *return/ascent*) is in fact an "embodied calculus", a mathematics *lived out* (participation, mimesis). We can think of the mathematical differentiation and integration as an analogue of the mystical stripping away (of constants, ideas, images, symbols, idols, temporal means of support) and (re)integration. "Apophasis" conveys the notion of "detachment from" and "faith approximation to", while the Non-Standard infinitesimal imagery points to the mystical principle of *kenosis*, of humility, of becoming small, of finding the greatest in the least.

"Just as in the scientific world approximation to the infinitely small started the conquest of infinite space, so approach to the divine mysteries lies through humility and the kind of *kenosis* that we see in Christ, Who `made himself of no reputation'"(7). As Fr. Alexander Webster points out, this "Orthodox experience of the mystical participation of the human person in the life of God is neither a static contemplation of the "essence" of God nor a gradual merging of the many into the One, but rather an eternal dynamic progress that approximates the divine in a manner analogous perhaps to the mathematical concept of limit in calculus and analytical geometry"(8).

Just as our mathematical models are mere approximations to a greater reality intended (i.e., in intentionality), so also our Christian theological symbols, concepts, percepts, ideas, must negatively and apophatically be left behind ("detachment from", Husserl's equivalent was the "bracketing") in the positive face of an overwhelming Reality. This, I understand, is why Randal...wrote...on this same topic, "And this is why I'm Catholic...Catholic theology...regards the Eucharist as THE reality, THE physical presence of God among us...Thus the Catholic faith, although permeated with symbols perhaps analogously with arithmetic, is much more than symbols alone." Perhaps this, to some degree, it is this at what our other respondent was getting also, when he said (if indeed this was his intention) that the inadequacy of symbols is not what theology is about. Most certainly, this (Real) Presence is the central focus of the Mystical Theology of the Eastern Church also, and of High Calvinism as well (in spite of different theological explanations).

"This knowledge of God only comes when love for him begins to take on a personal character; when on the pathway of life for the first time you have _*met*_ Him; when the Lord has become a Personal Presence by the side of your own self; when God and you have entered into a conscious, vital, personal particular relationship -- He your father, you his child...First from afar, then ever closer, until at length all distance falls away, and the meeting takes place -- a moment of such blessedness as can never be expressed in words.. "Then, and only then, comes the "nearness"...He also who has not entered into this secret, may say with others, "it is good for me to be near unto God (Ps. 73,27), but as yet he does not grasp it...He thinks it means a pious frame of mind, but feels no slightest burning of a spark of this mystical, most most intimate and personal love in his own heart...To be "near" is to be so close to God that your eye sees, your heart is aware of, and your ear hears him, and every cause of separation has been removed; near in one of two ways: either you feel yourself, as it were, drawn up into heaven, or that God has come down from heaven to you, and seeks you out in your loneliness, in that which constitutes your particular cross, or in the joy that falls to your lot"(9).

By the way, while I do not think it possible that Husserl would have ever really been successful in his expressed intention of developing a method by which those without God could bootstrap their way to Him, still I do very much appreciate the way in which his bracketing, intentionality, and the phenomenological reductions are of value in elucidating the structure of contemplative mysticism. I can appreciate his desire in this task not to be so heavenly minded that he was no earthly good. If his early mathematical training did happen to play a role in his continuation of the philosophical tradition of secularizing faith-functioning as intentionality, I certainly hope I do not appear to be "blaming" mathematics. His mathematics would have more likely made for further clarification. Think "faith-functioning" (a human function which can be directed either to the true God or to an idol) when you hear or read or think "intentionality" and perhaps reach some insights into your own faith that you never knew before. In this connection, see Evan M. Zuesse, _The Role of Intentionality in the Phenomenology of Religion_, Journal of the American Academy of Religion 53(1):51-73 (March 1985)!

Note:

"...According to some modern scholars, the reception of Plato's philosophy [from the Byzantine East] did more to widen the intellectual horizon of the West during the Renaissance than almost any other single factor. Certain authorities, however, take a narrower view. They believe that the most significant contribution of Platonic philosophy consisted, rather, in an emphasis on a mathematical type of thinking derived from certain Pythagorean materials incorporated in Plato. It was this mathematical emphasis, in contrast to the medieval Western Aristotelian stress on logic that paved the way for the advent of modern Western science, especially acceptance of the Copernican theory"(10).

References

(1) Henning F. Harmuth, _Sequency Theory Foundations and Applications_, New York/San Francisco/London: Academic Press, 1977, p.2.

(2) Bradford P. Keeney, _Aesthetics of Change_, New York/London: The Guilford Press, 1983. Circular / cybernetic epistemology.

(3) Georges Poulet, _The Metamorphoses of the Circle_, Baltimore: The Johns Hopkins Press, 1966 [trans. from the French].

(4) Tom Rockmore, _Hegel on Epistemological Circularity and Certainty_, International Philosophical Quarterly 21:235-248(1981).

(5) _Circularité des *Exercices* et circularité: du savoir absolu. D'Ignace à Hegel par Hölderlin_, pp.164-177 of Gaston Fessard, _La Dialectique des Exercices Spirituels de Saint Ignace de Loyola_, Paris: Aubier-Éditions Montaigne, 1956.

(6) Ford Lewis Battles, Calculus Fidei Some Ruminations on the Structure of the Theology of John Calvin_, Grand Rapids: Calvin Theological Seminary, 1978 [Privately published], p.44).

(7) Archimandrite Sophrony, _Wisdom from Mount Athos The Writings of Staretz Silouan 1866-1938_, London & Oxford: Mowbray's, 1974 [trans. from the Russian by Rosemary Edmonds], p.7.

(8) Alexander F.C. Webster, _Orthodox Mystical Tradition and the Comparative Study of Religion: An Experimental Synthesis_, Journal of Ecumenical Studies 23:(4):621-649 (Fall,1986), p.631.

(9) Abraham Kuyper, _To be Near Unto God_, Grand Rapids: Baker Book House, 1979 [trans. J.H. De Vries, 1925]. See pages 13,15,16,22,23).

(10) Deno John Geanakoplos, _Interaction of the "Sibling" Byzantine and Western Cultures in the Middle Ages and Italian Renaissance (330-1600)_, New Haven and London: Yale University Press, 1976, p. 64.

April, 1996; 2005

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Index