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The Computer and the Incarnation Ahriman

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Sketch of Rudolf Steiner lecturing at the East-West Conference in Vienna.

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The Computer and the Incarnation Ahriman

On-line since: 28th April, 2017

2. Methodology


“We have demonstrations about the circle, but only conjectures about the soul; the laws of nature are presented with mathematical rigor, but nobody applies a comparable diligence to research on the secrets of thinking. The source of human misery lies in the fact that man devotes more thought to everything but the highest good in life ... Whence we have the clandestine atheism planted in men, the fear of death, doubts about the nature of the soul, the weakest or at least, vacillating pronouncements about God, and the fact that many men are honest by habit or necessity rather than by virtue of their judgment.” [6]


A rather long discussion of methodology will be required if a modern reader is to be expected to make sense out of the arguments and descriptions that follow. The results I have reached are nonsensical in terms of generally accepted scientific notions. But there is a framework in which the results make good sense, and which provides a clear, logical way for arriving at results of this kind. So it is the task of this section to describe this framework, in particular focusing on the methodology I have followed. In addition, I provide an active defense of it on two points which grate on moderns: the reliance on authority (in this case, centrally that of Rudolf Steiner), and taking supersensible beings without a physical body in the ordinary sense to be fully real. Finally, I give a brief description of the logic implicit in my approach, which again is different from the forms ordinarily accepted. While we have come to realize the arbitrary character of axiomatic systems since the discovery of non-Euclidean geometries in the nineteenth century, we have not yet seen that our notion of axiomatic systems as a whole is one of several alternatives, each valid for a particular realm of experience.


2.1 Metaphysical Method

The methodology employed here is one that has been adapted from a more general metaphysical method, [7] that is, from a method designed for and suited to the treatment of questions concerning the totality of the world, which is assumed to contain a component above and beyond what is ordinarily thought of as “matter.” [8]This is not the place to state the general argument for why the world does have a non-material component, and thus why there is a need for a metaphysical method as a more inclusive replacement of the manifestly simpler materialist method. [9] Such questions are in any case generally not settled by means of arguments, however astute and cogent the arguments may be. Our powers of reasoning have been so weakened through prolonged exposure to scientism that we have learned not to trust our rational faculty whenever a truly important question is at hand, and thus sacrifice use of the facility which could most positively settle the issue. So I will simply employ the broader method, and hope that few will regret the lack of argument in its defense.

Metaphysical method starts from the recognition that every person has (whether as a result of individual striving or unconscious schooling) an attitude or stance with respect to the world and the knowledge that can be won from it. In practice, it is impossible to take a “neutral” stance to the world while one conducts an exhaustive investigation and inventory of its contents, reserving all judgment until the full results have been tabulated. So each of us necessarily adopts some stance or other usually not even on the basis of partial information, but as a result of his or her class, culture, schooling, etc. The materialist method either ignores the question of stance, or considers it to be inconsequential, imagining all the “facts” in the world to be of roughly uniform size and density, and that a determined pursuit of what it imagines to be objectively existing “facts” will put them all right in the end. Metaphysical method, on the other hand, considers the stance that humans take toward them to be of at least as much importance as the facts themselves, playing a large role in establishing or even creating the facts in the first place, and a determining role in the path a person takes through this world (i.e., the selection of the tiny portion actually experienced from the myriad of what might potentially be experienced).

The main task of metaphysical method (in this context) is to establish the “true” attitude towards a given set of facts. There is room for legitimate disagreement among those who pursue this method, just as there is among those who work in an accepted scientific field, and there are also right and wrong answers, and fruitful and fruitless avenues of investigation. Metaphysical method maintains that it is out of such stuff as “attitudes” that the destiny of souls is woven, and that this is the source of the method's importance.

Now it must be admitted that attitudes and facts are woven together, and reflect upon each other. (Modern philosophers of science have gone so far as to admit that our attitudes affect our perception of facts. [10] ) This has already been mentioned, and far from being a defect, is a motivation for taking our attitudes seriously and making a special study of them.

The drama of a good detective story provides an illustration of the relationship between faceless facts and our weighing and weaving of them into a single web. As the story proceeds, all the facts on which the final understanding is based are mentioned, but because they make no sense in the context of the theory being built up, they are ignored by the reader and by the story's detective. As each fact is recognized for what it is, one's understanding of all the other facts shifts and alters; a wholesale alteration can occur repeatedly in the course of a single story. The master detective has to pursue two contradictory courses simultaneously, first, he has to build up as complete a theory as he can out of the facts at his disposal and pursue it as though it were certainly true, and second he has to doubt his theory with unalloyed cynicism, always looking for facts or perspectives which would cause it to collapse.

The detective's attitude is very much like that of a certain kind of scientist, the kind who can bolster a theory and rip it apart with equal facility. As one moves along the progression from detective to scientist to metaphysicist (or spiritual scientist), the range of facts to be accounted for goes from narrowly limited to as nearly all-inclusive as possible, and the focus of attention goes from being largely absorbed with the facts to being explicitly concerned with the response to the facts, or with the facts as seen in the broadest context. One starts out seeing facts as fixed and primary, and ends up seeing them as flexible and secondary.

It is possible to imagine that metaphysics is something like a psychology of scientific discovery. Actually, metaphysics has little to do with the subject matter of modem psychology, or even with the supposed “psychic” world that parapsychology attempts to penetrate. Imagine that a person starts out life a totally isolated ego, unable to make contact of any kind with an outer world. Then the person reaches out, and eventually finds a full, coherent, objective world which fills his experience. This is what the modern world understands as the normal condition of an adult human being. Now ask the question: what has reached? Where and of what is the “arm” that reached? Because we are ordinarily not aware of the reaching, and do not think of it, we picture the physical world as being immediately there, nothing more than sound or light waves (which are also a part of it) being required to bring it to us. We relegate the choice of what comes to our awareness to the psychological notion of “attention.” But in fact, the physical world which seems so immediate to us is (as a whole) as distant from us as it is possible for something to be. The gap between us and the world is filled with living substance, with a stuff which is a varied mixture of me, us, not-me, and not-us, any part of which is closer to us and more real than any part of what we think of as the physical world. It is this which maintains the distance between subject and object, and also which keeps them together. It is in this ultra-real world that our destiny is made into events and experiences. And this is the world which the metaphysician studies.

The world which the metaphysician studies is the unity or oneness in which our dual or split world has its origin. Since that world is nowhere to be found here, the researcher must position himself in the emptiness where it would be if it were here, which is in the gulf which separates subject and object, and to which our most immediate access is given by what I have characterized as “stance.” The modern scientist ordinarily absorbs himself in the object world, and takes all that his senses convey to him as being the ground of reality. The world of the subject he experiences as something present but inessential, an observer, and a source of generally unreliable commentary. The critical idealist (of which Kant is the prototype) is aware of the logical incongruities inherent in this position. He takes with full seriousness the way we confront the world from within ourselves and unavoidably impose our theories on our perceptions, constructing all sorts of notions about the source of our perceptions, but unable actually to meet or confront that source directly. The existence of realists and idealists, each caught in worlds which seem phantom-like to the other, provides a stark illustration of the subject and object worlds which are separated from each other by a seemingly unbridgeable gap. The metaphysician, affirming what is positive in each of these positions, and taking what is negative as a signpost to the interworld “emptiness” he seeks, stands resolutely in both worlds, using the contradictions between them as the force which sustains him in the emptiness, rising, until the single creative source of each reveals itself to him.

This experience is not a merely subjective experience of mystical unity or cosmic ecstasy, having no significance in the world of facts or theoretical understanding. It is like discovering a theory which has the perceptual thereness and irrefutable permanence of an observed fact, and at the same time a fact which has the transparent clarity and connectedness of a penetrating theory. It is for this reason that metaphysical method requires as a prerequisite the acquisition of the skills ordinarily valued in the subjective and objective realms, and in addition certain religious virtues which provide the actual conduit for the experiences described here.


2.2 Defense of the Methodology

A defense of metaphysical method on every point is too large a task to attempt here. I have chosen instead to consider in detail two points of apparent conflict between modern thought and metaphysical method to illustrate the kinds of defense that can be made. I have chosen a single type of argument among many as being appropriate, namely, showing that the differences are not as great as we think, so that attacks made on this method are equally appropriate to modern method.

First, the question of authorities. The contrast we draw between the present practice and the previous one (which has much in common with the metaphysical method) is that “pre-scientific” papers used to establish points by referring to the revered figures who agreed with the author (a popularity contest), while now they establish points by a combination of experimental, data and strict mathematical reasoning (the test of experience and truth). There are many ways in which this contrast is misleading and self-serving.

Many important points are now established by a modern version of the popularity contest. An example is the way in which the von Neumann “disproof” of the hidden variable theory in quantum mechanics gained broad acceptance in spite of the small number of people who understood it. Von Neumann's tremendous prestige as a mathematician, coupled with his extremely long, abstruse proof, resulted in the nearly immediate acceptance of the Copenhagen interpretation of quantum mechanics, in spite of the fact that the proof, as has since been shown., has serious difficulties. [11]

More importantly, the scientific system as a whole is something we accept on authority. We do not go carefully, assuring ourselves of its validity a step at a time, as we learn to accept science. We accept it first, and may or may not actually internalize any of its explicitly held precepts. Most of us are not educated in science, and even those few of us who become proficient in some science do so after having first been thoroughly steeped in an atmosphere in which science is assumed to be authoritative. Our idea of who or what constitutes authority has changed, but our reliance on authority as such certainly has not.

It is disingenuous to claim that the ancients eschewed experience as a path of knowledge in favor of authority. They respected experience so much that they believed haying experience to be a skill in itself. They took highly skilled “experiencers” to be like fine Instruments, and put faith and trust in them just as we do in our machines. We, too, have our authorities, only they are for the most part not people; they are machines and logic.

When the ancients wished to examine physical things, they used their senses and built instruments to aid them when necessary. But they were not as concerned with the material world as we are. To answer most of the questions posed in the books we castigate as being authority-ridden, we would be unable to construct physical instruments to aid us. But the ancients knew that instruments were nonetheless needed, that the naive and untutored are unlikely to stumble across the answers to deep and subtle questions. So they subjected talented people to rigorous training, to make these people into instruments of (directly seen) knowledge, and listened with respect to what they reported. What can seem to us as undue reliance on authorities is often simply proper respect for the sort of instrument most relevant to the question at hand.

This brings us to the role of Rudolf Steiner, the primary “authority” in this case. His role is the same as the research scientist who presents his findings to a product development group of a company. The group could not have arrived at the results themselves, but they are in a position to test them, and to judge them based on their overall knowledge of the field and the previous results of the scientist. In order to do this, they have to know and trust the scientist's methods — if his previous successes have been the result only of happy coincidence, there is no reason to trust the next one. A superstitious person and a scientist might make the same prediction in a particular case, and our differing responses depend on our knowledge of the methodology which led to the result.

But the case of the spiritual investigator is different from the ordinary scientist in that his explicit work on himself has transformed what was simply attitudes towards facts (the attitudes being subjective and the facts objective) into a whole new world of facts, just as objective but more real (deeper, more inclusive, and there) than the usual set. This is the origin as facts of the supersensible beings which will be mentioned in this book, and which are treated as are any other fact which one did not discover on one's own.

Having considered the question of authority, I will now turn to the question of occult worlds and unseen beings. What we say is that people used to project their feelings on nature, and imagine an animate world hidden behind the inanimate one, while now we simply investigate the phenomena we find, and construct testable theories to account for them. But what is so much better about projecting our thoughts onto phenomena in the way we suppose the ancients to have projected their feelings? The whole point of modern science is not to rest content with the phenomena as experienced, but to “pierce through them” to the supposed “physical laws” they express, in other words, to construct a hidden or occult world which orders the manifest phenomena. The occult nature of modern physical science has long been recognized; Newton was attacked by his contemporaries not for his science, but because of what was seen as the occult nature of the gravitational force. And we are more content in our occult world than we are in the supposed primary one; theory now often precedes observation, rather than following it, and our greatest scientists put as much or more credence in good theories as they do in observations. [12]

Once we recognize the way in which modern science describes a speculative occult world which lies behind the phenomena we experience, we can consider the next level of argument. Even if modern science has a certain occult quality, the argument goes, it is an occultism which is superior to the old occultism. Because of science's strict reliance on experimental method, it obtains better results than the old occultism, and is therefore to be preferred to it.

Certainly we have been able to make many measurements and predictions more accurately than the ancients — but this is in spite of our “occult” method rather than because of it. We have won many battles over accurate measurements not because of superior weapons, but because of having more of them; we have won by means of the scale of our war machine, not its efficiency or appropriateness. The history of astronomy, usually taken to exemplify the triumph of modern science, can be used to show precisely the opposite: the misrepresentation of its history incidentally provides evidence of the disingenuousness and lack of self-consciousness of modem “occultism.”

The picture we have is that Ptolemy constructed his arbitrary system of planetary spheres in a primitive attempt at celestial mechanics, and had to introduce all sorts of “fixes” just to make it work at all. [13] Then along came Copernicus, who advanced the heliocentric view against a millennium of tradition, because the emerging scientific mood demanded a theory which fit the facts better than Ptolemy's. Finally, Kepler saw that orbits were ellipses, and the modem age was launched.

The real story is more interesting. It starts with the Babylonians, who accumulated many centuries of planetary observations, and who by the third century before Christ determined things like the period of the Sun to an accuracy not surpassed by modem astronomy until the nineteenth century, using a purely empirical theory. [14] Then, using Babylonian observations, Ptolemy (c. 100 - c. 178) constructed his theory to “save the appearances.” [15] That he needed such a theory shows that modernism was already at work in him, but he did not reify his concepts, nor did he introduce elaborate ideal notions into them. Although he maintained that all motion in the heavens is spherical, he introduced the equant into his constructions, which made his circles mathematically equivalent to ellipses.

Copernicus appeared on the scene in the sixteenth century. He admitted that he rarely made observations, and stated a prime motivation to be establishing the planetary orbits as perfect circles. [16] Therefore, putting theory ahead of observation he threw out Ptolemy's ellipse construction, and talked about how the Sun is “really” the center of our system. Actually, of course, one can treat either the earth or the Sun as the center of our system — it is only a question of where one would prefer (as a matter of convenience) the center of the coordinate system to be. Copernicus, however, thought this matter of coordinate systems — which does not affect the phenomena one way or the other — to be crucially important. In so doing, he is properly thought to stand at the beginning of our age, because he took what is not and cannot be seen and which does not alter phenomena to be more important than the phenomena themselves. The significance of Kepler is that he worked within the new “occult” realm and showed how more elaborate ideas may be used within it; he improved the efficiency of the method without altering its quality. [17]

History shows that, to the extent they cared about what we care about, the ancients obtained unqualifiedly admirable results, and that they did so without postulating elaborate worlds which stand unseen behind the phenomena; we are the occultists, not they. And if we use “results” as the measurement of virtue, our method does not stand up as well as theirs, since we have a commitment to the direct connection between the occult world and the world of phenomena which the ancients were not hobbled by. Perhaps that is why they were able to obtain results more accurate than the accuracy of their observational tools with such an economy of means.

There are important differences between the occult world postulated by modem science and the one observed by some of the ancients and a few modern spiritual scientists. In particular, my occult world is populated by living beings. But an occult world of living beings is not intrinsically more difficult to justify than an occult world of mathematically expressed “laws,” or other mathematical quasi-objects such as “atoms” or “electrons,” once some occult world has been admitted to exist. Of course, we are not used to having our occult world populated by living beings; we find it strange and uncomfortable — but what of that? The only scientific question is: can we know that world (to the extent that an occult world is knowable), can we show by its use that we can account for phenomena which otherwise leave us perplexed? Once we arrive at this question, it is possible that the ground is emotionally and intellectually cleared for the new thoughts to be advanced here, and we may proceed.


2.3 Material and Spiritual Logic

Since this is a book about computers and not methodology itself, the present discussion of methodology must soon come to an end. But because something called “Ahriman” will be brought into a definite association with the machines, one more set of thoughts must still be conveyed, thoughts about the logic which permeates our thinking.

The laws of thought and logic which have developed in the west starting with the Greeks are adequate for treating the nature of the computer in a clear way. But it is impossible to treat adequately the notion of “Ahriman” and remain within the bounds of ordinary logic; one is compelled to choose between being truthful but unclear and illogical, and clear and consistent but misleading. The source of the problem is that, with a few notable exceptions, our logic has come to us through immersion in the material aspect of the world. It is not ail-inclusively about thought, but is more narrowly about thought-of-matter. As a result, we cannot think clearly (in the ordinary sense) about something such as Ahriman which does not have a simply material existence.

So as a final methodological subject, I must indicate briefly the nature of the logic which underlies the main content of this book, and which also underlies other internally transparent expositions of spiritual realities. For simplicity's sake, I will call all the ordinary logics from the syllogism through the predicate calculus “material logic,” and the family of logics whose general characteristics I will outline here “spiritual logic.”

Material logic has appeared in many different forms, and has undergone significant transformations during its history. Even relatively small differences of notation have at times had a major impact. But there are certain characteristics which all material logics share. All are based on a relatively small set of statements called axioms or postulates. Axioms are the basis of any logical system because they appear first, asserted by the constructor of the system for his own meta-logical reasons. Axioms are extremely simple statements, so simple that their truth is self-evident. At one time axioms were held to be universally true, but now it is generally accepted that they are arbitrary, that they form a system's basis not because of their necessary truth, but because a system must be based on something.

A famous axiom in geometry is “parallel lines in a plane never meet.” [18] In symbolic logic, a typical axiom is “a or not-a,” which states that a proposition must either be true or its negation must be true.

Two distinct sorts of objects which are found within logical systems may be called operands and operators. [19] Operands are the passive objects of the system. In geometry, they are things like points and lines, while in propositional logic they might consist simply of “true” and “false.” Operators are the dynamic actors of the system, typically serving to relate operands to each other. In arithmetic, “plus” is an operator, and in propositional logic, “and” is a typical operator. Statements in the logical system consist of lawful sequences of operands and operators.


Text Box: original transformation
 s = t s + c = t + c


Logical systems must also have transformation rules, which turn one true statement into another true statement. These are perhaps most familiar to us in algebra, in which a simple transformation rule might be


This rule expresses the thought that if one starts with any equation “s=t” and adds a constant “c” to each side of the equation, the truth-value of the equation is unaltered. When applied to the axioms, the transformation rules allow one to produce an indefinitely large number of true statements, the most significant of which are termed theorems. Theorems are simply compound statements which have been spun out of the axioms by means of the application of the transformation rules in a particular order.

Just as axioms are the basis of a logical system, the theorems are in a sense the goal of it. Theorems are often complicated enough so that their truth is not self-evident to most of us, but can nonetheless be shown to be as true as the axioms. In the appropriate circumstances, showing a statement to be necessarily true or false (proving or disproving a conjectured theorem) can be challenging and useful.

To make the notion of a logical system clear, consider a system which we may call “the odd numbers.” [20]

The system's definition has three parts. The first is the alphabet, which is the set of signs which may appear in statements. In this case, the alphabet has only one sign, “1”. The second is the set of axioms, in this case the single axiom “1”. Finally, there is a single production rule as illustrated,


Text Box: original transformation
 x x11


where “x” is understood to stand for any statement (axiom or theorem). The rule states that any true statement remains true after “II” is appended to it. Successive application of the production rule to the axiom results in the theorems “111”, “11111”, “1111111”, etc. If one interprets each series of ones as representing a number (in the unary number system), it is evident that our logical system encompasses the odd numbers 1, 3, 5, 7, etc. While this system is trivial, more complicated systems are different only in having larger alphabets, more axioms, [21] and more production rules.

The power of material logic derives from the fact that it is purely formal. It is nothing but a set of rules which tell how to transform strings of signs into other strings of signs. The signs have no meaning of their own. They are not symbols or even signs of anything: it is not necessary to admit any commonality between four objects and the sign “4” in order to have a logically sound system of arithmetic. In fact, it is a miracle of felicity that correspondences nonetheless exist, that accurate maps of much of the world can be made out of a totally vacuous system. To a logician, though, perhaps a greater delight is the way that theorems of such power, beauty, and subtlety can be built up out of a small pile of trivialities.

In material logic, the idea of higher order logics is already present. For example, one speaks of primary logic or simple propositional logic, and then of general logic which includes quantifiers such as “some” and “every.” This way of building a logic brings into play a (symbolically speaking) vertical element, but the vertical element is unfortunately of a false kind. This point may be grasped by a comparison to the plant and animal kingdoms, in which animals are a genuinely higher order of being than plants; they add a qualitative element (with appropriate physical expression) not present in plants. If the animal kingdom were higher order in the sense that term may be understood in material logic, animals would be constructed out of the same principles and materials as plants, the only difference being that they would somehow feed on other plants instead of or in addition to conducting photosynthesis. Indeed, parasitic plants such as mistletoe and insect eating plants such as the Venus's-fly trap are counterfeit animals, higher-order plants, in this sense.

Truly higher-order logics are, however, possible. I know of two levels, and more may exist. The level immediately above material logic is a logic of metamorphosis and transformation in a world ruled by dynamic polarities. Goethe sensed this logic while he did his botanical studies, and Hegel developed it under the rubric of dialectics. I have described this logic in a preliminary way and demonstrated its application in detail elsewhere. While material logic is appropriate to the mineral world, this first higher order logic permeates wherever living being unfolds itself organically,

It is the second order logic which is of interest to us here, and which I have termed spiritual logic. So far as I know, a formalism in which statements in this logic may be expressed has never been devised, nor will I present one here. Nonetheless, it is possible to see that spiritual logic permeates spiritual realities in a way appropriate to their nature, and that people who have investigated these realities in an exact way have intuitively made their descriptions conform to spiritual logic, whether or not they were consciously aware of the fact. In what follows, I will attempt to characterize and describe spiritual logic, but not fully define it.

Spiritual logic is related to material logic by a series of inversions, reversals, and coalescings involving its central elements and characteristics. The most obvious inversion involves the vacuousness that characterizes material logic as a whole. Its signs and strings are empty, arbitrary, and trivially obvious. The alphabet is nothing but a set of place holders. The theorems, however clever, are mechanically derivable from the axioms via the production rules. In spiritual logic, it is appropriate to say that one finds not signs but symbols. All the statements, both axioms and theorems, indicate sources of meaning, being, and quality. Of course, the marks that one might make on a piece of paper superficially appear similar to those of material logic. The point is that material logic may be fully represented by marks on paper, while spiritual logic may only be appropriately indicated by correctly formed graphic symbols.

Material logic is more trivial in its axioms and more sophisticated in its theorems, some of which take a stroke of genius to discover. Its axioms are so obvious that beginning students are often confused by them: who in his right mind would trouble to state the principle of identity, that any variable or constant is equal to itself? In spiritual logic, this relation is reversed — the axioms are full, necessary, and the part of the system which is the most profound and difficult to understand, while the theorems are (relatively speaking) easier to grasp and arbitrary. The point in material logic is to discover and elucidate theorems; in spiritual logic, one stumbles across theorems more easily, and the point is to elucidate the axioms, the profound sources of the system.

Another aspect of the reversal of the relation between axioms and theorems concerns unity and compounded-ness. In material logic, axioms are simple statements, while theorems are almost always compound. In spiritual logic, the axioms are still in some sense unities, but they are so in a complex, multi-faceted way, while by the time one gets out to the consequents, the theorems, the complexity is at least greatly reduced. The theorems most distant from the axioms are simple irreducibles, such as individual percepts experienced by humans.

In most instances of material logic (though not in the example given above — think instead of algebra), there is a clear distinction between passive operands and active operators. A similar distinction holds between passive logic systems and active abstract (e.g. Turing) machines, even though one can completely model one in terms of the other. [22] Again, one distinguishes between active production rules and passive theorems. Although such dualities pervade material logic, it is as a whole passive (substance-like) in relation to spiritual logic, which as a whole is active (essence-like). A facet of this relation appeared in the discussion of the fullness and emptiness of the logics above.

As one moves from passive (as a whole) material logic to active spiritual logic, the duality active/passive recedes into the background, so that for example the clear distinction between operands (such as variables in an equation) and operators (such as “+”) disappears. The symbols of spiritual logic partake of the natures of substance and of essence at the same time.

Spiritual logic is not an alternative to material logic, because it does not supersede material logic in that logic's proper realm of application. One must still be able to ferret out and eliminate ordinary-logic contradictions. However, this admirable practice universally applied, effectively makes one unable to think about phenomena which have a primarily spiritual basis. Hence spiritual logic, which provides a basis for thinking clearly about spiritual realities, and which is implicit in (and thus necessary for rationally comprehending) existing expositions of spiritual phenomena.

[6] Leibniz: “On True Method in Philosophy and Theology”, 1686, from Selections, P. Wiener, Ed., New York, 1951.

[7] The most succinct summary of metaphysical method outside the work of Rudolf Steiner is given in Rene Guenon: “Oriental Metaphysics”, Tomorrow, vol. 12, no. 1; also in Jacob Needleman: The Sword of Gnosis, Baltimore, 1974.

[8] This does not necessarily involve adopting a dualistic position, which is an artifact of the approach to a materialistic world conception. The “supersensible component” referred to rests on a conceptual distinction between spirit and matter necessitated by the modem context, but which should not be taken to imply an ontological distinction between them operative at all levels.

[9] See Guenon: The Reign of Quantity, Baltimore, 1972, and Symbolism of the Cross, London, 1945, for argumentation along these lines.

[10] See for example, N. R. Hansen: Patterns of Discovery, Cambridge, 1958, especially chapter one.

[11] This example is explained in detail in Feyerabend: Science in & Free Society, London, 1978, p. go

[12] Einstein regarded negative results of a test of special relativity as “improbable because their basic assumption ... are not suggested by theoretical systems which encompass wider complexes of phenomena.” With regard to a test of the general theory of relativity, he said, “It is really strange that human beings are normally deaf to the strongest arguments while they are always inclined to overestimate measuring accuracies.” References and further discussion of this point are given in Feyerabend: Against Method, London 1975

[13] We are so convinced of this that one of these, the “epicycle,” has come to mean any ad hoc mechanism which elaborately extends a theory without deepening it.

[14] “It is now recognized that [Oppolzer's 1887] value for the motion of the sun from the node was 0.7” too small per annum; (the fourth century B.C. Babylonian} Kidinnu was actually nearer the truth with an error of 0.5 “too great.” Quoted in Toulmin and Goodfield: The Fabric of the Heavens, New York, 1961, p. 39.

[15] This is the phrase used by Ptolemy to express what he wished to achieve in his mathematical theory. Owen Barfield, in his Saving the Appearances, New York, 1965, goes far in making clear the implications of this phrase in subject-object relations, and in the evolution of consciousness.

[16] “The movement of the celestial bodies is regular, circular, and everlasting — or else compounded of circular movements.” heading of Book I, section 4,  On the Revolutions of the Heavenly Spheres

[17] Mathematically speaking, Kepler's celebrated “discovery” of the elliptical nature of the orbit of Mars would be more justly termed a “recovery,” since it did no more than bring astronomical theory back to where It had been in Ptolemy's time. There is an exact equivalence between the three Ptolemaic points 1) the earth, 2) the center of the eccentric, and 3) the center of the equant, and the three Keplerian points 1) the sun at one focus of the ellipse, 2) the center of the ellipse, and 3) the other focus of the ellipse.

[18] “That, if a straight line failing on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that Bide on which are the angles less than the two right angles.” Euclid: The Elements , Book I, Postulate 5. Euclid based his system on what may be translated as 23 definitions, 5 postulates, and 5 common notions. The differences between these and the changes they hare undergone, while of great importance within the history of material logic, are unimportant in the present attempt to characterize material logic as a whole.

[19] This distinction is central to formulations of logic for practical purposes such b computer languages, and thus pervades the thinking of those who work with it. In the purely theoretical forms of logic devised for the purpose of proving theorems about the boundaries and powers of logical systems as such, this distinction is typically unimportant.

[20] The form taken by the example is that of a Post production system. As Poet showed (“Formal reductions of the general combinatorial decision problem*, Am. Journal of Math, €5, pp. 197-268, and described in Minsky), any formal system, including Turing machines, may be reduced to the canonical form of a production system of the type illustrated here. More examples may be found in Hofstadter: Gödel, Escher, Bach, New York, 1980. For an intermediate presentation, see Mínsky: Computation: Finite and Infinite Machines, Englewood Cliffs, 1967.

[21] Unless the system is in Post “normal” form, in which case there is always a single axiom.

[22] This distinction is more than just one of notation. Even since the isomorphism of all such systems has been definitely demonstrated, logicians still think in terms of modeling a Turing machine with a Post production system, rather than simply translating between notations.

Last Modified: 28-May-2023
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