On Plato, the Early Church, and Modern Science: An Eclectic Meditation

On Plato, the Early Church, and Modern Science: An Eclectic Meditation
By Jean F. Drew

God, purposing to make the universe most nearly like the every way perfect and fairest of intelligible beings, created one visible living being, containing within itself all living beings of the same natural order.

Thus does Plato (d. 347 B.C.) succinctly describe how all that exists is ultimately a single, living organism. At Timaeus20, he goes on to say:

“There exists: first, the unchanging form, uncreated and indestructible, admitting no modification and entering no combination … second, that which bears the same name as the form and resembles it … and third, space which is eternal and indestructible, which provides a position for everything that comes to be.”

And thus we find a description of the universe in which Being and Existence (Becoming) — the one God and the multiplicity of things — are bound together as a single living reality whose extension is mediated by Space (which for us moderns implies Time).

Our aim in this essay is to define these ideas and their relationships, and trace their historical development from the ancient world to the present. Taking a page from the late Eric Voegelin (1901–1985, philosopher of history specializing in the evolution of symbolization), we will follow a history-of-ideas approach to these issues. Along the way we will find that not only philosophy and cosmology, but also theology and even modern science can illuminate these seminal conceptions of Platonic thought. We must begin at the beginning, that is, with God — who is absolute Being in Plato’s speculation, of whom the cosmos itself is but the image (eikon) or reflection.

When Plato speaks of God (or when Aristotle does for that matter, as in e.g., Nicomachean Ethics), he is not referring to the Olympian gods, to Zeus, Hera, Athena, Poseidon, and the rest of the gang of “immortals.” For the Olympians are like man in that they are creatures of a creating God. Not only that, but they are a second generation of gods, the first having reigned in the antediluvian Age of Chronos; which is to say that the Olympians’ rule or law is not everlasting, but contingent. Thus they are not self-subsistent, but dependent (contingent) on a principle outside of themselves. We might say that the central difference between Plato’s God and the Olympians consists in the fact that the latter are “intracosmic” gods, and the former is “extracosmic,” that is, transcending all categories and conditions of space-time reality. In contrast, the intracosmic gods are subject to change, to contingency; and so, though they may truly be said to exist in some fashion, cannot be said to possess true Being. (More on these distinctions in a minute.)

It is clear that for Plato, God is the “Beyond” of the universe, or in other words, utterly transcendent, perfectly self-subsistent Being, the “uncaused cause” of all the multiplicity of existents in the universe. In yet other words we can say that, for Plato, the cosmos is a theophany, a manifestation or “presence” of the divine Idea — in Christian parlance, the Logos if I might draw that association — in the natural world.

As Wolfgang Smith notes, “Christian teaching is based upon the doctrine of the Logos, the Word of God, a term which in itself clearly suggests the idea of theophany. Moreover, what is implicit in the famous Prologue of St. John [“In the beginning was the Word, and the Word was with God, and the Word was God. The same was in the beginning with God. All things were made by him; and without him was not any thing made that was made. In him was life; and the life was the light of men. And the light shineth in darkness; and the darkness comprehended it not.” (John 1:1–5)] is openly affirmed by St. Paul when he declares that “the invisible things of Him from the creation of the world have been clearly seen, being understood by the things that are made, even His power and Godhead” (Rom. 1:20) … The indisputable fact is that at its deepest level Christianity perceives the cosmos as a self-revelation of God.” [Wolfgang Smith, Cosmos and Transcendence, 1984]

Being and Existence (Becoming)
Being is a concept so difficult that it comes close to eluding our grasp altogether. It is utterly beyond space and time; imperishable; entirely self-subsistent, needing nothing from outside itself in order to be complete; essential; immutable; and eternally perduring. Contrast this with the concept of existence, regarding which Plato asks “how can that which is never in the same state be anything?” And this is the clue to the profound difference between being and existence: The existing things of this world are mutable and transient.

We must in my opinion begin by distinguishing between that which always is and never becomes from that which is always becoming but never is. The one is apprehensible by intelligence with the aid of reasoning, being eternally the same, the other is the object of opinion and irrational sensation, coming to be and ceasing to be, but never fully real. In addition, everything that becomes or changes must do so owing to some cause; for nothing can come to be without a cause. [Timaeus, 3:28]

“… they come upon the scene, we know not from whence; they grow, change, and decay; and at last they disappear, to be seen no more. The physical cosmos itself, we are told, is a case in point: it, too, has made its appearance, perhaps some twenty billion years ago, and will eventually cease to exist [i.e., finally succumbing, we are told, to thermodynamic entropy or “heat death”]. What is more, even now, at this very moment, all things are passing away. ‘Dead is the man of yesterday,’ wrote Plutarch, ‘for he dies into the man of today: and the man of today is dying into the man of tomorrow.’ Indeed, ‘to be in time’ is a sure symptom of mortality. It is indicative, not of being, but of becoming, of ceaseless flux.”

All the multiplicity of existents in the universe are in a state of becoming and passing away. But Plato’s great insight is that all things in the state of becoming — that is, all existing things — are whatever they are because they are participations in Being. That is to say, “we perceive the trace of being in all that exists,” writes Smith, “and that is why we say, with reference to any particular thing, that it is.” Existence, in other words, is contingent on Being.

But we wonder: In what way is this possible? And if existents participate in being, what is that Being in which they participate?

In Exodus 3:14 Moses has experienced a theophany: While tending his flock on Mount Horeb, suddenly he hears the voice of God issuing from a burning bush: God is speaking to him! Reverentially, Moses inquires of God what is His name (meaning: what is His nature or character).

And God said unto Moses, I AM WHO AM: and He said, Thus shalt thou say unto the children of Israel, I AM hath sent me unto you.

God has told Moses: that He is Being (“I AM”). And the strong implication is that there is no “other” being: “I alone AM.” For “I” is plainly singular in form.

Smith draws the crucial point, “God alone IS. But how are we to understand this? ‘It seems to me,’ writes St. Gregory of Nyssa, ‘that at the time the great Moses was instructed in the theophany he came to know that none of those things which are apprehended by sense perception and contemplated by the understanding really subsist, but that the transcendent essence and cause of the universe, on which everything depends, alone subsists.’ But why? Does not the world exist? Are there not myriads of stars and galaxies and particles of dust, each existing in its own right? And yet we are told that the transcendent essence alone subsists. ‘For even if the understanding looks upon any other existing things,’ the great theologian goes on to say, ‘reason observes in absolutely none of them the self-sufficiency by which they could exist without participating in true Being. On the other hand, that which is always the same, neither increasing nor diminishing, immutable to all change whether to better or to worse (for it is far removed from the inferior and has no superior), standing in need of nothing else, alone desirable, participated in by all but not lessened by their participation — this is truly real Being.’”

Smith continues: “In the words of St. Gregory, ‘that which is always the same, neither increasing nor diminishing, immutable to all change … is truly real being.’ As concerns ‘existing things,’ on the other hand, the teaching implies that these entities are always changing, always in a state of flux, so that their very existence is in a way a process of becoming, in which however nothing is actually produced. This has been said time and again, beginning with Heraclitus and the Buddhist philosophers. And there can be little doubt that it is true: even modern physics, as we can see, points to the same conclusion. Only there is another side to the coin which is not always recognized. Existent things — the very flux itself — presuppose what Gregory and the Platonists have termed ‘a participation in Being.’ The point is that relative or contingent existences cannot stand alone. They have not an independent existence, a being of their own. ‘In Him we live, and move, and have our being,’ says St. Paul….”

I beheld these others beneath Thee, and saw that they neither altogether are, nor altogether are not. An existence they have, because they are from Thee; and yet no existence, because they are not what Thou art. For only that really is, that remains unchangeably.

Space
Space is the third essential term of the Platonic cosmology: It is the matrix in which living things and all other existents participate in Being. Plato’s creation myth — the Myth of the Demiurge in Timaeus — elucidates the Platonic conception of Space.

For Plato, the God of the Beyond is so “beyond” that, when it came time for creating the Cosmos, he didn’t even do it himself. He sent an agent: the Demiurge, a mythical being endued by God to be in divine likeness of God’s own perfect love, truth, beauty, justice, and goodness. The embodiment of divine perfections, the Demiurge wishes to create creatures just as good and beautiful as himself, according to the standard of the divine Idea — a direct analog, it seems to me, of the Logos theory of the ancient Church. Indeed, Eric Voegelin sees in the Demiurge the symbol of Incarnation [Order and History Vol. 3: Plato and Aristotle, 1957]:

“The Demiurge is the symbol of Incarnation, understood not as the result of the process but as the process itself, as the permanent tension in reality between the taxis of form or idea and the ataxia of formlessness.”

Similarly to the Christian account, the Demiurge in a certain way creates ex nihilo — that is, out of Nothing. At first glance, Plato is seen specifying, not a pre-existing “material” but a universal field of pure possibility called Chora, “Space.” Perhaps we may find in this concept a strong analogy to Isaac Newton’s concept of Absolute Space (see below).

Chora seems to indicate the idea of an eternal, universal field of pure stochastic potentiality that needs to become “activated” in order to bring actual beings into existence. In itself, it is No-thing, i.e., “nothing.” This “activation” the Demiurge may not effect by fiat: He does not, for instance, “command” to “Let there be Light!” The main tool at his disposal is Peitho, “persuasion.”

And if Chora is not so persuaded, it will remain in a state of “nothingness.” It will remain unformed, in the condition of ataxia. Of itself it is “Nothing”; by itself, it can do nothing. It cannot generate anything out of itself, not even matter in primaeval form.

And thus Plato introduces the figure of the Demiurge into his creation myth, symbolizing form or idea — the principle of (formative) taxia that draws (formless) ataxia into existence. We moderns might be tempted to describe the Demiurge as constituting an “information set” together with an “energy source,” who “persuades” the pure stochastic potentiality of formless, absolute, empty space into actualized form, and thus existence. From the cosmic standpoint, he makes unity out of multiplicity, in harmony and geometrical proportion:

“The best bond is the one that effects the closest unity between itself and the terms it is combining; and this is best done by a continued geometrical proportion.” [Timaeus, 4]

Thus the Demiurge is a kind of “divine geometer,” producing the forms (or mathematical ideas) that Chora can be persuaded to conform to, and thus come into existence.

But the Demiurge does more than just get things started: As bearer of the divine Idea — as pure love and beauty and goodness and truth — he continues always persuading Chora to generate creatures as like himself as possible (i.e., reflecting his own divine qualities at whatever generic stage), throughout all eternity. Thus creation is a continuous process in space-time. Moreover, it is the source and driver of evolution as a universal natural process.

Through the ongoing activity of the Demiurge, men and the world are constantly being informed and renewed by the divine Idea; and thus a unified cosmic whole, a “One Cosmos,” a universal order comes into being at the intersection of time and timelessness, of immanent and transcendent reality, in the medium of Space (and Time).

Compare the Platonic creation myth with the philosophy of Dionysius the [Pseudo-]Areopagite, said to be the Greek converted by St. Paul in Acts, 17:34. For Dionyius, the “names of God” — the divine qualities — are goodness, being, life, wisdom, power, and justice. Joseph Stiglmayr writes [Cath. Encycl. at the entry for Dionysius the Pseudo-Areopagite], that for Dionysius, God is

“… the One Being (to hen), transcending all quality and predication, all affirmation and negation, and all intellectual conception, [Who] by the very force of His love and goodness gives to beings outside Himself their countless gradations, unites them in the closest bonds (proodos), keeps each by His care and direction in its appointed sphere, and draws them again in an ascending order to Himself (epistrophe) … all created things [proceed] from God by the exuberance of being in the Godhead (to hyperpleres), its outpouring and overflowing … and as a flashing forth from the sun of the Deity. Exactly according to their physical nature created things absorb more or less the radiated light, which, however, grows weaker the farther it descends. As the mighty root sends forth a multitude of plants which it sustains and controls, so created things owe their origin and conservation to the All-Ruling Deity…. Patterned upon the original of Divine love, righteousness, and peace, is the harmony that pervades the universe…. All things tend to God, and in Him are merged and completed, just as the circle returns into itself, as the radii are joined at the centre, or as the numbers are contained in unity.”

The Platonic resonances seem unmistakeable in these lines. It appears that both Platonic speculation and the Logos doctrine of the ancient Church as articulated by Dionysius are in agreement that Creator must be “beyond” Creation in order to resonate with it — which resonance is what makes the universe to be alive — i.e., a living universe.

C. A. Dubrey points out [Cath. Encycl. at the entry “Teleology”], that the theology of St. Thomas Aquinas makes it clear that, “Intrinsic finality [we are to think of this as a blend or merger of efficient and final causes in the Aristotelian sense] consists in the fact that every being has within itself a natural tendency whereby its activity is directed towards the perfection of its own nature…. St. Thomas does not hesitate to speak of ‘natural appetite,’ ‘natural inclination,’ and even ‘intention of nature,’ [we moderns might be tempted to add ‘instinct’ to this list] to mean that every being has within itself a directive principle of activity. Accordingly, God does not direct creatures to their ends from outside, but through their own nature…. The Divine plan of creation is carried out by the various beings themselves acting in conformity with their nature.

When, however, this finality is called immanent, this expression must not be understood in a pantheistic sense, as if the intelligence which the world manifests were to be identified with the world itself, but in the sense that the immediate principle of finality is immanent in every being…. Thus the unconscious finality in the world leads to the conclusion that there must be an intelligent cause of the world.” [Emphasis added.]

Aquinas’ insight, and also Plato’s, evokes a reconsideration of Isaac Newton’s concept of Absolute Space. Possibly this may be understood in the following terms. First, Absolute Space is “empty” space. Second, it is not a property of God, but an effect of His Presence; i.e., we advert to theophany again. The question then arises, in what “where” or “when” does this theophany take place? Perhaps Newton’s answer would be: In the beginning, and continuously thereafter. Second, it has been suggested that Newton intends us to understand Absolute Space as the sensorium Dei: “God constitutes space and time through his eternity and omnipresence” [ existendo semper et ubique, durationem et spatium consitutit: Philosophiae Naturalis Principia Mathematica, 3d ed., 1726]. Wolfhart Pannenberg writes,

“Now there are a number of good reasons — suggested by both philosophical and scientific thought — to consider time and space as inseparable. Einstein’s field concept comprises space, time, and energy. It takes the form of a geometrical description, and this seems to amount to a spatialization of time. The totality of space, time, and energy or force are all properties of a cosmic field.

“Long before our own age a theological interpretation of this subject matter had been proposed, and it was Isaac Newton who offered this proposal. It too referred everything to space or, more precisely, to the correlation of force as in the case of a force like gravitation acting at a distance. Newton’s well-known conception of space as sensory of God (sensorium Dei) did not intend to ascribe to God an organ of sense perception, the like of which God does not need, according to Newton, because of divine omnipresence. Rather, Newton took space as the medium of God’s creative presence at the finite place of his creatures in creating them.” [Wolfhart Pannenberg, Toward a Theology of Nature, 1993]

Thus the infinite takes priority over every finite experience, including intellectual experience — a position decisively argued by Descartes, as Pannenberg avers, “in his thesis that the idea of God is a prior condition in the human mind for the possibility of any other idea, even that of the ego itself.”

The Influence of Platonic Speculation on the Early History of the Church
D. Edmund Joaquin, an insightful and gracious Christian friend, writes, “We understand that the universe is created and sustained by the Word [the Logos], and not only that, but by the Word sounding. God sustains the universe consciously and actively. He has not gone away and left us. In fact, He reveals Himself to us, and His final revelation is in the person of Christ [the Logos]. Christ is not an abstract aspect of God, like wisdom. He is God. He is God incarnating in the world that He himself has made.”

Joaquin further observes that “[the Gospel of] John is written to the Greeks and put into words that they could understand.” It seems there’s a mystery buried in here somewhere. Consider: Socrates was the teacher of Plato, who was the teacher of Aristotle, who was the teacher of Alexander — and Alexander spread Greek culture throughout Eurasia, the Middle East, and the Indian subcontinent. Add to this the fact that the great evangelist, St. Paul, had some difficulty converting the Jews to the Christian faith; but he converted the Greeks in droves. Not only St. John, but also St. Paul speaks in terms the Greek mind could readily grasp, as when he says God is He “in Whom we live and move and have our being.” These historical connections do not appear to be accidental, coincidental, nor incidental to the spread of the early Christian Church.

According to The Catholic Encyclopedia, the Greeks strongly responded to Christianity for its moral beauty as well as its truth. A case in point is St. Justin Martyr. He was a man of Greek culture, born in Palestinian Syria about the year 100 A.D, who converted to the faith around 130 A.D. Justin became one of Christianity’s earliest and most powerful apologists, and ended up condemned by the Roman authority for refusing to sacrifice to the pagan gods, for which offense he was summarily executed by the Imperium, along with several other of his “refusnik” co-religionists. The official record of their martyrdom is extant:

“The Prefect Rusticus says: Approach and sacrifice, all of you, to the gods. Justin says: No one in his right mind gives up piety for impiety. The Prefect Rusticus says: If you do not obey, you will be tortured without mercy. Justin replies: That is our desire, to be tortured for Our Lord Jesus, and so to be saved, for that will give us salvation and firm confidence at the more terrible universal tribunal of Our Lord and Saviour. And all the martyrs said: Do as you wish; for we are Christians, and we do not sacrifice to idols. The Prefect Rusticus read the sentence: Those who do not wish to sacrifice to the gods and to obey the emperor will be scourged and beheaded according to the laws. The holy martyrs glorifying God betook themselves to the customary place, where they were beheaded and consummated their martyrdom confessing their Saviour.”

Jules Lebreton writes (at the entry for St. Justin Martyr in Cath. Encycl.) “Justin tries to trace a real bond between philosophy and Christianity: according to him, both one and the other have a part in the Logos, partially disseminated among men and wholly manifest in Jesus Christ.”

Yet for all their apparent similarities and resemblances in many respects, there is a profound difference between Platonic insight and the Christian one: and this pertains to the relations between God and man.

Both Plato and Justin proclaim the transcendent God. Yet for Plato, God is so “beyond” as to be almost impossible of human grasp. Yet Plato felt the “divine pulls” in his own nature. These Plato thought could be accounted for and articulated by an act of pure unaided intellect, that is by nous, in a state of intense contemplation.

Contrast this position with Justin Martyr’s, who insisted that human wisdom was impossible without the testimony of the Prophets (whom God himself had informed and instructed) and the action of the Holy Spirit. For Plato, man’s relations with God consist of operations of the mind. For Justin, they are operations of the heart, of the Spirit. For Justin, God is not a mental abstraction: He is real Personality with whom one can have direct personal relations, in the Spirit.

A later writer, John Scotus Eriugina (ninth century) elaborates the Justinian position, in the process noting that there is a “downward tendency” of the soul towards the conditions of animal existence, and that this has only one remedy: Divine grace, the free gift of the Holy Spirit. “By means of this heavenly gift,” writes William Turner [at the entry for Scotus in the Catholic Encyclopedia], “man is enabled to rise superior to the needs of the sensuous body, to place the demands of reason above those of bodily appetite, and from reason to ascend through contemplation to ideas, and thence by intuition to God Himself.”

The pull of animal nature is an idea we also find in Plato, and also the countervailing pull from the divine Beyond. Man lives in the metaxy, in the “in-between reality” constituted by the two. Man’s task is to resolve this tension, and establish the proper balance that expresses the highest and best development of his human nature. But man must do this entirely by himself by means of nous or reason. There is no spiritual help “extra” to the human psyche available to facilitate this process.

“…admits that the soul can naturally comprehend what God is, just as it understands that virtue is beautiful … but he denies that the soul without the assistance of the Holy Ghost [Spirit] can see God or contemplate him directly through ecstasy, as the Platonic philosophers contended. And yet this knowledge of God is necessary for us: ‘We cannot know God as we know music, arithmetic, or astronomy’; it is necessary for us to know God not with an abstract knowledge but as we know any person with whom we have relations. The problem which it seems impossible to solve is settled by revelation; God has spoken directly to the Prophets, who in their turn have made Him known to us…. It is the first time in Christian theology that we find so concise an explanation of the difference that separates Christian revelation from human speculation.” [Emphasis added]

Natural Law, Contingency, and the Scientific Method
The Platonic model encourages us to recognize that the universe is zoon empsychon ennoun, a living creature endowed with soul and intelligence. The myth of the Demiurge describes the world process as a type of incarnation, a dynamic relation of absolute being and contingent becoming evolving in space and time in a manner expressing a perduring taxia–ataxia relation. The Cosmos itself — the totality of all existing things — like its constituents, for example man and even the stars, is an eikon of being-in-becoming, a reflection or image of the divine Idea. Time itself is but a “moving image of eternity.” The life of the cosmos is wholly dependent, contingent on the Idea from which it manifests.

It is a lawful, orderly universe, yet one in which new occurrences are always arising. These new events are coming from, as it were, a “sea of contingency” analogous to Plato’s conception of Space, that is Chora — the infinite field of unformed, pure potentiality.

The immediately foregoing ideas, of course, are not scientific ones strictly speaking. Still, there are elements here that perhaps science would do well to consider, in order to maintain the integrity of its own method. For one thing, it seems science itself, in its disclosure of the regularities of nature, seems to have an in-built tendency to overlook contingency. We may define an event as contingent if a description of it is neither self-evident nor necessary, “if it could have happened differently,” as Ted Peters puts it in his Preface to Pannenberg’s Towards a Theology of Nature.

C. A. Dubray writes [“Teleology,” Cath. Encycl.], “The fact that the world is governed by laws, far from giving any support to the mechanistic conception, is rather opposed to it. A law is not a cause, but the expression of the constant manner in which causes produce their effects.” In other words, natural laws are expressions of observable regularities that occur in the world of existent phenomena in ordinary space-time reality. Thus, the laws themselves have no force as “causes”: they are descriptions.

Yet the focus on regularity inevitably masks the particularity and contingency of unique events. As Ted Peters notes, it is here that “we run into a problem of focus in the scientific community, because virtually all the theoretical attention is given to the regularity of nature’s laws, while the contingency of natural events slips into the nearly invisible background.” Peters continues:

“What researchers concentrate on are the uniformities that can be expressed in timeless equations. A dictionary of equations describing these uniformities allegedly constitutes scientific knowledge…. A closer examination, however, reveals that the applicability of these equations to concrete cases of natural processes requires certain initial and marginal conditions, conditions that in every case are contingent. Only when contingent conditions permit can we expect a natural law to operate as expected.”

To the extent that the scientific method of inquiry is premised on an “If/Then” logical construction — which seems ever to be the case — the method itself is an exercise in contingency, yet nonetheless one in which “Determinacy gets thematized, whereas contingency gets ignored.” Arguably this is a serious bias having epistemological implications; for e.g., “if the laws of classical dynamics are in principle temporally reversible, the actual course of natural events from which those laws have been abstracted is not. The reality of nature is first and foremost a historical reality.”

Pannenberg suggests a corrective for this “bias,” acknowledging: “That modern science so easily lends itself to abuse cannot be prevented in principle. It is one of the risks involved in the abstract study of regularities that either are inherent in nature itself or can be imposed on natural processes [e.g., as in ideological, technical, or engineering solutions]. This risk cannot be met on the level of scientific description itself but must be met first on the level of philosophical reflection on the work of science. It is on this level that the abstract form of scientific description must be considered with special attention to what it is “abstracted from” and what is methodically disregarded in the abstract formulas of science.”

And so contingent conditions — i.e, initial and boundary conditions — must be restored to their proper place in our deliberations, for they “are required for any formula of natural law to be applied. They are contingent at least in that they cannot be derived from the particular formula of law under consideration.… The mathematical formula of a natural law may be valid without regard to time. The physical regularity that is described by such a formula is not independent of time and temporal sequence. But it is only that physical regularity which makes the mathematical formula a law of nature. This suggests that the laws of nature are not eternal or atemporal because the fields of their application, the regularities of natural processes, originate in the course of time. Thus it also becomes understandable that new patterns of regularity emerging in the sequence of time constitute a field of application for a new set of natural laws….”

We may recognize that the total process of natural events presents itself to observation as a mesh of contingency and regularities. It is the task of science to pursue thematically the aspect of regularity. But, asks Pannenberg, can science “ever succeed in bringing into view the entirety of nature as determined in all details by a number of laws that are in any case not infinitely complex? This would mean at the same time that a stage of research is conceivable from which nothing more could be discovered. Many natural scientists have had this nightmare because of the successes of their own research. Fortunately it probably is not a truthful dream.”

For, says Pannenberg, “laws always uncover what is necessary superimposed on what is contingent. Given the undeniable contingency of occurrences in natural events, can we recognize in their special character as occurrences … [that] regularity as their own element in such a way that the presence of regularity can be thought together with the contingency of occurrences, not only under abstraction from the contingency of occurrences?” [Emphasis added]

Which is why Pannenberg advocates an opening up of new viewpoints in scientific research, “not because physical hypotheses or insights can be derived from them but because they open up and enlarge the intellectual space on which the formation of physical hypotheses depends…. In physics also, horizons of questioning have to be opened up first of all in order that hypotheses that arise in them can be examined by experiment and classified theoretically.”

Perhaps we need a greater appreciation of the “fitness” of the scientific method to engage the truly great questions of life, which ever seem to involve the relations of law and contingency. Leibniz propounds two great questions of perennial interest to the human mind: (1) Why are things the way they are and not some other way? (2) Why does anything exist at all?

Such questions, scientists will readily tell you, are beyond the purview of the scientific method. But does that mean such questions have no force or meaning such that they should not be asked at all?

Perhaps the incapability of the scientific method to answer such questions owes to the fact that all the great physical laws are acknowledged to be time-reversible; but we know that existence in space and time is not a time-reversible process. As Pannenberg states, it is a historical process. We might even say it is an evolutionary process.

Which suggests an analogy that might enlighten these questions, sharpen their meanings, and suggest additional questions: an analogy to direct human experience. Pannenberg writes of human beings, who do seem to live in a “time-irreversible,” that is “historical” process:

“Human beings never live only in the now. Rather, they experience their present as heirs of the past and as its active change. They anticipate the future in fear, hope, and planning; and in the light of such anticipation of the future they return to their present and the heritage of their past. The fact that we know of historical continuity is at least also conditioned by this peculiarity of human experience with time. If there is a new event, then it modifies the context of our consciousness of time which is already found present. It throws light back on earlier occurrences which have become a part of our experience already. In the same way, ideas that occur to us throw light on our previous expectations and plans in justifying, fulfilling, modifying, or disappointing and thwarting them. Thus the contingent event always enters already into a context of experience or tradition…. The future, beginning in the present happenings, is thus the origin of the perspective in which the past occurrences are put by every new experience.”

Worldviews and Paradigm Shifts
It is perhaps a truism that we tend to find what we’re looking for by screening out any and all potential elements which do not fit the pattern of our expectation. Arguably, the scientific method may be said inherently to suffer exposure to potential danger from this side, as suggested in the above remarks. Indeed, Schröedinger’s theory of wavefunction seems to predict this. Consider these remarks from Stephen M. Barr [Modern Physics and Ancient Faith, 2003]:

“In quantum theory, as traditionally formulated, there are ‘systems’ and ‘observers.’ Or rather, in any particular case, there is the system and the observer. The observer makes measurements of the system. As long as the system is undisturbed by external influences (that is, as long as it is ‘isolated’), its wavefunction — which is to say its probability amplitudes — will evolve in time by the Schröedinger equation…. However, when a measurement is made of the system the observer must obtain a definite outcome. Suddenly, the probability for the outcome that is actually obtained is no longer what the mathematics said it was just before the measurement, but jumps to 100 percent. And the probabilities for all the alternative outcomes, the ones that did not occur, fall to 0 percent.”

Thus we might say that the “reality” we humans experience ever involves “a moving goal-post.” And as the mover of this goal-post, the human agent is most indispensably involved in this process.

Faced with such “indeterminacy” regarding the foundations of experience, it is not surprising that people usually have recourse to mediating worldviews, or organized frames of ideational reality that constitute the conceptual space in which active experience is engaged and accordingly analyzed and interpreted. Certainly Plato has offered such a model. And so has Nobel laureate Jacques Monod [in Chance and Necessity, 1971]:

“Chance alone is the source of every innovation, of all creation in the biosphere. Pure chance, absolutely free but blind, is at the very root of the stupendous edifice of evolution. The central concept of biology … is today the sole conceivable hypothesis, the only one compatible with observed and tested fact. All forms of life are the product of chance….”

Needless to say, these two models are polar opposite conceptualizations. Yet having received each on “good authority,” which do we choose?

Such are not idle considerations; for as James Hannam points out [“The Development of Scientific and Religious Ideas,” 2003], “grand theories … often suffer death by detail where it is found that up close the situation is too complicated for the theory to handle…. [Yet] in the end, after it has changed the course of the river of enquiry, the theory can end up as a mortlake cut off from the general flow….”

Hannam cites historian Thomas Kuhn, who documents an historical process he terms “paradigm shift,” describing a situation in which the findings of authoritative science move “out of science and into practically every other field of human endeavor.” Once a given, albeit partial or even defective theory becomes “dominant,” writes Hannam, “far from being thrown out, a falsified theory is enhanced to deal with new information until such time as it finally collapses under the weight of anomalous results. Then, after a chaotic period, a new theory emerges that can deal with the anomalies and normal service resumes…. A paradigm refers to but one field, say classical mechanics or health policy whereas the ideology/worldview is the general background that underpins all the paradigms.”

The worldview (or ideology, if you prefer), for better or worse, implicitly shapes the background knowledge of thinking agents to which new experiences constantly are being conformed. Hannam says that worldview “is often so deeply embedded in the psyche that it is very rarely considered explicitly except by specialists,” but that nonetheless, “the worldview is seen as [a] self-confirming fact of life and hence it is not strictly rational…. The existence of a dominant worldview does not mean that a particular individual is unable to think outside the box but rather that his ideas are unlikely to fall on fertile ground. Unless new ideas can be stated in a language that makes them comprehensible to his peers, his intention in writing will not be met.”

Which is the not-too-subtle way to put the fact that every man has a worldview, without exception, whether articulate or inarticulate; and that somehow, for the “intention of writing to be met” — that is, for accurate and meaningful (i.e., successful) communication of ideas to take place — some deeper, common ground of shared truth must first be accessed, for the purpose of providing a more capacious intellectual space in which the human pursuit of knowledge and wisdom might unfold or evolve from its present point of attainment.

But where today in our modern world is such a common ground or field to be found? Hannam proposes the examination of the history of ideas as a possibly useful method in the search for common ground. He writes,

“To examine the history of ideas the only fair way to proceed would seem to place before ourselves the evidence and authority that the historical agents had before them and assume they acted rationally on that basis. Otherwise, there is no hope of ever tracing intellectual development because ‘cause and effect’ assumes some sort of logical causality that is impossible with non-rational agents. The best that could be hoped for would be a catalog of mental positions, with no way to say how one led to another except by being pushed by blind exterior forces. This might be precisely what determinists are advocating but they would have to give up any hope of finding causes and restrict themselves to explanations.”

Perhaps we moderns would do well to reconsider the common assumption that people living before our own time were somehow inferior in knowledge, experience, and observational powers as compared with our own status as enlightened individuals. Arguably, the ancient world produced some of the most powerful thinkers in the history of mankind, formulating ideas that were, in the words of Hannam, “the fruits of unfettered metaphysical speculation that inevitably hits on the right answer occasionally.”

Democritus, for example, proposed a theory predicting the atom as the ultimate constituent of matter, more than two-thousand years before the technical means existed to isolate atoms experimentally or, as Hannam notes, any “useful applications for them” could be found. Then it was discovered that the atom itself is an ordered constellation of even finer parts. There seems to be an historical progression of ideas here, the new building up on a framework originally laid up in the past, modifying it, improving on it in light of new insights and technical capabilities.

Hannam gives another example of more recent vintage: “Copernicus needed Nicole Oresme’s solution as to why we do not feel the movement of the Earth even though in Oresme’s time it was just a curiosity as no one thought the Earth actually was moving … each new idea, once accepted, shifts the boundaries of the worldview and makes it possible for further new ideas to be accepted into the pale.”

We can extend the examples even further. Reimann constructed a geometry, apparently because his mind could grasp the logic and beauty it revealed for its own sake. But at the time, it had no apparent “external referent” in the field of nature. It was a beautiful and glorious abstraction — until Einstein came along, and picked it up “off the shelf” as it were, to become the very language of relativity theory.

Thus it might be said that the evolution or “progress” of science depends on successive enlargements of the conceptual space it requires to do its work. In other words, science inherently is a participation in the historicity of the world.

Whatever our personal worldview, perhaps it would be well to recall that science is an historical process. Perhaps this understanding could open up additional, needed conceptual space that science itself requires in order to advance.

The Real Number System

The real number system evolved over time by expanding the notion of what we mean by the word “number.” At first, “number” meant something you could count, like how many sheep a farmer owns. These are called the natural numbers, or sometimes the counting numbers.

Natural Numbers

or “Counting Numbers”

1, 2, 3, 4, 5, . . .

The use of three dots at the end of the list is a common mathematical notation to indicate that the list keeps going forever.

At some point, the idea of “zero” came to be considered as a number. If the farmer does not have any sheep, then the number of sheep that the farmer owns is zero. We call the set of natural numbers plus the number zero the whole numbers.

Whole Numbers

Natural Numbers together with “zero”

0, 1, 2, 3, 4, 5, . . .

About the Number Zero

What is zero? Is it a number? How can the number of nothing be a number? Is zero nothing, or is it something?

Well, before this starts to sound like a Zen koan, let’s look at how we use the numeral “0.” Arab and Indian scholars were the first to use zero to develop the place-value number system that we use today. When we write a number, we use only the ten numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. These numerals can stand for ones, tens, hundreds, or whatever depending on their position in the number. In order for this to work, we have to have a way to mark an empty place in a number, or the place values won’t come out right. This is what the numeral “0” does. Think of it as an empty container, signifying that that place is empty. For example, the number 302 has 3 hundreds, no tens, and 2 ones.

So is zero a number? Well, that is a matter of definition, but in mathematics we tend to call it a duck if it acts like a duck, or at least if it’s behavior is mostly duck-like. The number zero obeys most of the same rules of arithmetic that ordinary numbers do, so we call it a number. It is a rather special number, though, because it doesn’t quite obey all the same laws as other numbers—you can’t divide by zero, for example.

Note for math purists: In the strict axiomatic field development of the real numbers, both 0 and 1 are singled out for special treatment. Zero is the additive identity, because adding zero to a number does not change the number. Similarly, 1 is the multiplicative identity because multiplying a number by 1 does not change it.

Even more abstract than zero is the idea of negative numbers. If, in addition to not having any sheep, the farmer owes someone 3 sheep, you could say that the number of sheep that the farmer owns is negative 3. It took longer for the idea of negative numbers to be accepted, but eventually they came to be seen as something we could call “numbers.” The expanded set of numbers that we get by including negative versions of the counting numbers is called the integers.

Integers

Whole numbers plus negatives

. . . –4, –3, –2, –1, 0, 1, 2, 3, 4, . . .

About Negative Numbers

How can you have less than zero? Well, do you have a checking account? Having less than zero means that you have to add some to it just to get it up to zero. And if you take more out of it, it will be even further less than zero, meaning that you will have to add even more just to get it up to zero.

The strict mathematical definition goes something like this:

For every real number n, there exists its opposite, denoted – n, such that the sum of n and – n is zero, or

n + (– n) = 0

Note that the negative sign in front of a number is part of the symbol for that number: The symbol “–3” is one object—it stands for “negative three,” the name of the number that is three units less than zero.

The number zero is its own opposite, and zero is considered to be neither negative nor positive.

Read the discussion of subtraction for more about the meanings of the symbol “–.”

The next generalization that we can make is to include the idea of fractions. While it is unlikely that a farmer owns a fractional number of sheep, many other things in real life are measured in fractions, like a half-cup of sugar. If we add fractions to the set of integers, we get the set of rational numbers.

Rational Numbers

All numbers of the form , where a and b are integers (but b cannot be zero)

Rational numbers include what we usually call fractions

Notice that the word “rational” contains the word “ratio,” which should remind you of fractions.

The bottom of the fraction is called the denominator. Think of it as the denomination—it tells you what size fraction we are talking about: fourths, fifths, etc.

The top of the fraction is called the numerator. It tells you how many fourths, fifths, or whatever.

RESTRICTION: The denominator cannot be zero! (But the numerator can)

If the numerator is zero, then the whole fraction is just equal to zero. If I have zero thirds or zero fourths, than I don’t have anything. However, it makes no sense at all to talk about a fraction measured in “zeroths.”

Fractions can be numbers smaller than 1, like 1/2 or 3/4 (called proper fractions), or they can be numbers bigger than 1 (called improper fractions), like two-and-a-half, which we could also write as 5/2

All integers can also be thought of as rational numbers, with a denominator of 1:

This means that all the previous sets of numbers (natural numbers, whole numbers, and integers) are subsets of the rational numbers.

Now it might seem as though the set of rational numbers would cover every possible case, but that is not so. There are numbers that cannot be expressed as a fraction, and these numbers are called irrational because they are not rational.

Irrational Numbers

Cannot be expressed as a ratio of integers.
As decimals they never repeat or terminate (rationals always do one or the other)

Examples:

	Rational (terminates)
	Rational (repeats)
	Rational (repeats)
	Rational (repeats)
	Irrational (never repeats or terminates)
	Irrational (never repeats or terminates)

More on Irrational Numbers

It might seem that the rational numbers would cover any possible number. After all, if I measure a length with a ruler, it is going to come out to some fraction—maybe 2 and 3/4 inches. Suppose I then measure it with more precision. I will get something like 2 and 5/8 inches, or maybe 2 and 23/32 inches. It seems that however close I look it is going to be some fraction. However, this is not always the case.

Imagine a line segment exactly one unit long:

Now draw another line one unit long, perpendicular to the first one, like this:

Now draw the diagonal connecting the two ends:

Congratulations! You have just drawn a length that cannot be measured by any rational number. According to the Pythagorean Theorem, the length of this diagonal is the square root of 2; that is, the number which when multiplied by itself gives 2.

According to my calculator,

But my calculator only stops at eleven decimal places because it can hold no more. This number actually goes on forever past the decimal point, without the pattern ever terminating or repeating.

This is because if the pattern ever stopped or repeated, you could write the number as a fraction—and it can be proven that the square root of 2 can never be written as

for any choice of integers for a and b. The proof of this was considered quite shocking when it was first demonstrated by the followers of Pythagoras 26 centuries ago.

The Real Numbers

Rationals + Irrationals
All points on the number line
Or all possible distances on the number line

When we put the irrational numbers together with the rational numbers, we finally have the complete set of real numbers. Any number that represents an amount of something, such as a weight, a volume, or the distance between two points, will always be a real number. The following diagram illustrates the relationships of the sets that make up the real numbers.

An Ordered Set

The real numbers have the property that they are ordered, which means that given any two different numbers we can always say that one is greater or less than the other. A more formal way of saying this is:

For any two real numbers a and b, one and only one of the following three statements is true:

1. a is less than b, (expressed as a < b)

2. a is equal to b, (expressed as a = b)

3. a is greater than b, (expressed as a > b)

The Number Line

The ordered nature of the real numbers lets us arrange them along a line (imagine that the line is made up of an infinite number of points all packed so closely together that they form a solid line). The points are ordered so that points to the right are greater than points to the left:

Every real number corresponds to a distance on the number line, starting at the center (zero).
Negative numbers represent distances to the left of zero, and positive numbers are distances to the right.
The arrows on the end indicate that it keeps going forever in both directions.

Absolute Value

When we want to talk about how “large” a number is without regard as to whether it is positive or negative, we use the absolute value function. The absolute value of a number is the distance from that number to the origin (zero) on the number line. That distance is always given as a non-negative number.

In short:

If a number is positive (or zero), the absolute value function does nothing to it:
If a number is negative, the absolute value function makes it positive:

WARNING: If there is arithmetic to do inside the absolute value sign, you must do it before taking the absolute value—the absolute value function acts on the result of whatever is inside it. For example, a common error is

(WRONG)

The correct result is

I think the same analogy would describe how different people (beliefs) view the concept of God. Some view God as having only a single aspect. Others, who have moved from their position, view God as a Triunity -- each representing the whole.
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historically the arian position, while dominant is arabia--is an anomaly in the west.
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I'm not certain of your meaning, ckilmer. Can you re-phrase? Thank you.
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I'm responding to the "who have moved from their position" part of your statement.

I'm not sure what that's about. So I just stated that historically trinitarianism has been the dominant theology in the west. It has only been since the enlightenment that monism has been more prevalent--as practiced by unitarians (-1750)& jehovah witnesses +-1900)officially and liberal protestants (+-1935) unoffically in the USA and most protestant churches in Europe since the mid 1800's. The caveat for European churches is that they have very rapidly gone from monism to extinction. In Berlin alone some 70 churches are up for sale this year because no one attends. Monism is what happens when Jesus is considered to be no longer equal to God the Father. That is he is considered a great teacher/philosopher/prophet and very very good. The Moslems think of Jesus as being a wise man/teacher/prophet/very very very good--but not God. They take the assertion that Jesus is God in the same way that the Jews of Jesus time took Jesus words--as blasphemy.

There is no proof. Rather I have heard it said that the form of christianity that Mohammed encountered was arianism. Arianism, Named after Arias of Alexandria Egypt-- was the third century heresy against which the council of Nicea ruled in +-325. I have learned in recent years that-- during the civil wars of the roman empire of the period--the Arians and the neoplatonists were fellow travelers. Similiarly, you'll find that starting from the late 1700's the neoplatonists in the philosophy depts in the west and the purveyors of "higher criticism" in the seminaries were also fellow travelors.

Interestingly, I heard a presentation last sunday by the second of a team of free church affiliated--Calvinist German Seminarians who want to set up calvinist church in downtown Berlin. It would be the first in more than 200 years. He has been training at Redeemer Presbyterian Church in NYC. Redeemer is the first Calvinist (conservative/trinitarian) Presbyterian Church to be set up in any downtown area in the USA in many decades. Many liberal presbyterian downtown churches in NYC are virtually abandoned but they are not as far along as the liberal churches in Berlin. The German church planter I talked with was Christian Nowatzky. His historical take was much similiar to the one I presented above.

here's the url for that group.
http://www.redeemer2.com/themovement/issues/2004/august/berlinprojekt.html

My impression is the guys have the right stuff. Its my prayer that it can be said of the Germans in the next decade or so "who have moved from their position" ...

Back 350 years ago when Liebniz helped to invent the calculus, mathematicians were not resticted to the real number system. They also used numbers called infinitesimals which are so small that you can think of them as zero. To find the slope of the spaghetti curve, Leibniz shrank the vertical and horizontal sides left over from our original triangle until those sides became infinitesimals. So you can think of this as dividing zero by zero, i.e. 0/0. The amazing result was an equation that gave him a finite slope that varied for every point on the curved spaghetti. Wow!!!

About 100 years ago mathematicians became very uneasy about infinitesimals and dividing 0 by 0, so they threw out infinitesimals, and redefined calculus in terms of the concept of limits and real numbers only.

Now, when I first took calculus, I was lost. I didn't understand limits at all and found them confusing and cumbersome. I started studying the history of the development of calculus. Since Leibniz used infinitesimals, I began to think of calculus in terms of infinitesimals. A light bulb went on, and I began to solve calculus problems with ease. But I didn't dare to tell my instructor that I was secretly thinking about dividing infinitesimals instead of taking limits of real numbers.
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The length of the diagonal of a right triangle is an irrational number. Therefor even if the vertical and horizontal of the right trangle were set at zero the hypotonuse would still be an irrational number.

Are you saying that Leibnitz's equation produced a sequence of irrational infintessimals? This raises two questions. does this mean that the space between each irrational infintessimal is --irreducable. If so then perhaps the answer to the old question "how many angels can you get on the head of a pin" ... is--a finite number of irrational infintessimals. This would also answer why it is that if you tried to get from one to two by going half the distance and then half the distance again forever---that you would never get from one to two.

The reason is that the distance between 1 and 2 is a straight line.

If not then none of the above conclusions hold. But perhaps you could comment.

second question: why is a sequence of irrational infitessimals that describe a finite slope--so darn exciting. Is it because theres so many angels in one spot/curve or because there's no space between them.

never mind. I'm getting jazzed thinking about it too. And I don't even understand math. Now I think I understand how you were motivating betty boop to expand on her thinking.

so now that we know that space is not nothing...what is space.

I believe that liebnitz was Newton's competitor/contemporary. Newton was such a towering figure in the sciences that his religious writings are generally overlooked. He was a thorough going Arian--in the sense that he believed that Jesus was fully man but not really God. Because 200 years before Newton--the reformation coincided with the overturning of the Ptolemaic cosmology
in the early 1500's--I wonder whether the math and science that Newton produced didn't also the overthrow reformation theology. Or whether it was the man himself who set things in motion.

If the the clue to Newton's theology could be found in Newton's calculus--what number or calculus would you say it was. I don't know anything about Liebnitz beyond what I've learned in this thread. Perhaps his numbers have the clue.

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