by Jonathan Tennenbaum

The fundamental crisis of civilization is forcing the question: Where do the ideas come from, which we find in our own heads and those of our fellow human beings, and which determine Mankind's ability or inability to survive the onrushing crisis?

The "short answer" is, that apart from the products of oligarchical manipulation, corruption, and decay, everything {positive} in our culture -- not only science and technology, but the concepts of everyday life, and language itself -- derives from nothing but the generation and assimilation of validated discoveries of principle, made by individual human minds, as measured on the metric of increase in the per-capita potential population density of the human species. The power of the oligarchy, of course, depends to a large extent on the success of its own massive efforts to cover up and distort the historical generation of culture (including science), while promoting popular belief in various varieties of empiricism and so-called "innate" or "self-evident" ideas.

It is worth stressing, that the battle against oligarchical obfuscation of the history of ideas, was a center of concern to the "American" republican circles around Schiller, Humboldt, and Gauss et al. at the beginning of last century. Moreover, exactly the concept of positive human culture as the result of integration of individual acts of resolution of fundamental paradoxes by the Platonic method of hypothesis, was key to the revolutionary work of (among others) Gauss, Weber, and Riemann on the "anti-entropic" geometry of physical space-time. Riemann discusses this explicitly in his posthumous fragments on epistemology, which provide a most useful background for comprehending his 1854 paper on "The hypotheses underlying geometry." Consider, in particular, the following passage translated from Riemann's posthumous fragment, entitled "Attempt at a theory of the fundamental concepts of mathematics and physics as the basis for the explication of Nature." (Note, that in this location, Riemann employs the terms "Begriff," concept, and "Begriffssystem," system of concepts, in a sense congruent with Lyn's use of "fundamental assumption" and "hypothesis," respectively).

"On the basis of the concepts, through which we grasp the natural world, we not only constantly supplement our observations, but, in addition, we determine certain future observations in advance as necessary, or -- in case our system of concepts is not sufficently complete -- as probable; on this basis it is determined, what is `possible' (i.e., also what is `necessary' or that whose opposite is impossible); furthermore, the degree of possibility (the `probability') of every single event so judged possible, can be mathematically determined, when the concepts are sufficiently precise.

"If an event occurs, which is necessary or probable according to the given system of concepts, then that system is thereby confirmed; and it is on the basis of this confirmation through experience, that we base our confidence in those concepts.

"But if something unexpected occurs, being impossible or improbable according to the given system of concepts, then the task arises, to enlarge the system, or, where necessary, to transform it, in such a way that the observed event ceases to be impossible or improbable according to the enlarged or improved system of concepts. The extension or improvement of the conceptual system constitutes the `explanation' of the unexpected event. Through this process, our understanding of Nature gradually becomes more comprehensive and more true, while at the same time reaching ever deeper beneath the surface of the phenomena.

"The history of the exact sciences, as far as we can follow it backwards in time, demonstrates that this, in fact, is the pathway by which our knowledge of Nature has progressed. The systems of concepts, which form the basis of our present understanding of Nature, were generated by progressive transformations of older conceptual systems; and the reasons that pushed forward the generation of new modes of explanation, can in every case be traced back to contradictions or improbabilities arising in older modes of explanation.

"Thus, the generation of new concepts, insofar as it is accessible to observation, occurs by this process.

"Herbart, on the other hand, has provided proof, that those concepts, upon which our conceptualization of the world is based, but whose origins we can neither trace back in history, nor in our own development, because they are transmitted together with language without being noticed -- all of those concepts, insofar as they are more than mere forms of connection between simple sense perceptions, can be derived from the above source; and need not be attributed to some special property of the human soul, assumed to predate all experience (as Kant claimed to do with his categories)."

Often it is most instructive, in exploring the implications of a fundamental principle such as Riemann's, to focus attention on the most deceptively simple cases -- cases of the sort fools would be likely to dismiss as being "too obvious to be worth thinking about."

Take, for example, the everyday concept of a "day." What could be more self-evident? Does Riemann actually mean to say that there is a real, creative {discovery} embedded in that idea? What would have been the paradox or paradoxes, whose resolution gave birth to the concept of "a day"? Evidently, the discovery involved predates history in the usual sense. What we might do, is try to "project" ourselves mentally back to a hypothetical, very, very distant point in time, at which the concept of "a day" did not exist, and then ask: What paradoxes {must intrinsically} confront a mind in the process of freeing itself from a naive, beast-like belief in the primacy of sense-perception? First, reflect on the following:

Could we discover anything without memory? Is a pot-headed Yahoo, who cannot remember what he saw or did 5 minutes earlier, able to make scientific discoveries? Would a Yahoo ever have been able even to discover the existence of a "day" as a recurring cycle of light and darkness? Or was the development of poetry, as a means of development of the powers of memory, crucial to the emergence of human civilization?

Pre-Socratic Greek tradition often spoke of the origin of the Universe in terms of the creation of Order (Cosmos) out of Chaos. Does this not exactly describe the subjective process by which a human mind frees itself from the blind impulses of "animal instinct" and "sense certainty"? The world of the existentialist Yahoo, or a newly-born infant, is a kind of Chaos, a "kaleidoscope of feelings" replacing each other in more or less rapid succession. Mankind could not survive, were it not possible to awaken a power of {creative discovery} in the infant or the supposedly infant-like, primitive man -- a mental function energized by the most powerful form of human emotion, {Agape}. It is that agapic power, inseparable from the faculty of {memory} as understood by the Renaissance, which conquers the Chaos of bestiality and creates the Cosmos of human development as an ordering of successive acts of discovery.

Next, consider the elementary paradox of change, as it is addressed by the simplest of astronomical discoveries. The following exploration is hypothetical, but necessarily touches upon a discovery actually made (and in fact, made repeatedly in various forms) in human history.

You are a prehistoric human being, living perhaps 500,000 years ago. On a beautiful clear night, you seek a place to lie down under the open sky. Gaze up, from there, at the magnificent canopy of the heavens! The myriad stars shine down on you in majestical silence, like little lights afixed to a lofty dome. Here is peace, here is rest! You close your eyes and relax.

You awaken later that night. As your eyes once more open to the sky, you are struck with a sudden sense of strangeness. Something is different! Something has happened! The stars seem to have changed. Looking around, you recognize a group of bright stars, whose form you remember having remarked before you took your nap. That group of stars is no longer where it was before; the stars have changed position!

Changed? How is that possible? You stare intently at the stars. Not the slightest motion is perceptible; only a gentle twinkling while they remain, seemingly immovable, in their places.

A paradox! On the one hand, your faculty of sense perception, swears to you that the stars are fixed and motionless. On the other hand, you remember that the same faculty has earlier testified, no less insistently, of an arrangement of stars in the sky, which is different from the one it now reports!

Intrigued, you repeat the experiment, but with a variation: you ask a friend to keep watching the stars, without interruption, during the time your eyes are closed. The experiment is performed. Once again, you find an undeniable change in the positions of the stars, when you look at the sky again after a nap. Your friend, however, swears he never saw the stars move!

The paradox strikes deep into your mind. Whatever follows, will depend on how you respond to the paradox. However you respond -- or even if you do {not} respond -- that response will reflect some sort of {hypothesis}, an hypothesis generated nowhere but inside your own mind.

Shall you merely conclude that your eyes (or those of your friend) have lied to you in some arbitrary fashion? Or that the Universe itself is maliciously arbitrary? If so, then how would human existence be possible?

Or is there another way out? Perhaps we should not {completely} reject the evidence of our senses. Perhaps it were better to assume, that although our sense perceptions in themselves do not represent reality, still there must be some implicitly discoverable, lawful relationship between sense perception and reality. This is the pathway of science.

Choosing that pathway, the paradox moves us to hypothesize the existence of something, which our senses -- in virtue of some lawful limitation of the same -- cannot grasp: to hypothesize a concept of a {process of change}, which in itself is {invisible} to the senses, but yet efficiently accounts for the observed (or rather, remembered) {difference} in positions! That adduced concept, of an invisible -- but efficient! -- process of change, is an object of a different sort than a sense perception (including the paradoxical entity we commonly identify as the "perception of motion"). It is not sufficient to account for that new concept, by merely saying: "the stars move too slowly for our eyes to see." The point is, that the paradox just presented, evokes the potential of a {new quality of relationship of our mind to the Universe}.

A change in the substance of our mind! Prior to the explosion of the paradox, you looked at the Universe (the starried heavens) as an object of sense perception. Now, you are looking at the Universe from the standpoint of a process of discovery, which stands in ironical contrast to naive belief in sense perception. To the reflecting mind, that {difference in mental attitude}, from before to now, provokes the hypothesis of {higher species of change} -- a process of improvement of human cognitive powers, which is invisible to our senses, but real and earthshakingly powerful nonetheless.

Turning once more to our nightly observations, what shall be our next step? Does our power of discovery give us the capability to hypothesize, not only the existence, but also the {form} of the process of change of position of the stars? How would we discover the coherence between the paradox of the stars' motion, and a similar paradox, posed by the behavior of the Sun? And how could we do that, using nothing more than the means which were available to prehistoric Man?

Lest the reader find the above discussion "too trivial" to be important, consider the following. Nearly everyone today is faced (or will be soon) with a congruent form of paradox: On the one hand, most people would claim that their most deeply held "values and beliefs," being absolutely self-evident (to them!) are fixed and unchangeable. On the other hand, comparing those people's "deeply held personal values" of today, with the corresponding values held "self-evidently" by those same people 30 years ago, we find almost nothing in common! If mankind is to survive, an increasing ration of leading and ordinary citizens must be brought to discover, as an "enemy image," the process by which the oligarchy was able to induce that radical, downward "paradigm shift" in their own minds.

by Jonathan Tennenbaum

Have you ever stopped to consider, how a human being, a "mere infinitesimal" on the scale of the world as a whole, could actually come to know the vast dimensions of the solar system, or to measure astronomical cycles hundreds or thousands of times longer than the brief span of his or her individual life? The existence of such powers of cognition, by which the "infinitesimal" can know the macrocosm within its own internal mental processes, is the central issue in the bitter, millenial conflict between the human species and the oligarchical "Gods of Olympus." Witness the words of Aeschylus' Prometheus (1):

Not without cause did Aeschylus emphasize the earliest discoveries of astronomy, connected with the construction of a solar calender, as crucial events in the emergence of human reason as "the sure measure" of things.

Implicitly, the discoveries made by our pre-historic "colleague" in connection with the "invisible" motion of the stars, refute everything university students have been taught to believe about science and "liberal arts" since the mid-1960s. Astronomical cycles -- beginning with the "day" -- are neither objects of sense perception, nor are they "robust statistical correlations." Rather, the astronomical cycles emerge as {conceptions} created in the human mind, through a process of generation and creative solution of paradoxes.

From this standpoint, let us push our exploration of prehistorical discoveries a few steps further, to identify paradoxes which {necessarily} must have arisen, even though we do not now know the specific historical circumstances.

Our prehistoric observer notes: (i) The positions of the stars appear to undergo a constant process of change. (ii) But at the same time, certain arrays of stars, identified and fixed in memory through poetic (mnemonic) devices already from earliest times, remain seemingly unchanged throughout the course of a night, reappearing every night with the same distinct form. Also, apart from the appearence and disappearance of stars on the horizon, the overall configuration of the constellations in relation to each other in the sky -- "the constellation of constellations!" -- remains unchanged.

This paradox of "change" combined with "no change" evokes the notion, that the "invisible" motion of the stars, has an implicitly intelligible {form}. That paradoxical idea becomes a specific thought-object, undergoing its own process of evolution in the direction of a notion of a {universal, rotational action} subsuming both the process of change in the night sky, and the daily motion of the Sun.

Indeed, observation of the rising and setting of the Sun, and studying the Sun's overall using such means as observation of the shadows cast by poles (gnomon), demonstrates an overall {coherence} between the nightly motion of the "constellation of constellations" and the motion of the Sun during daytime. As singularities of the hypothesized universal action, we get (among other things) the differentiation of East, West, North and South as determinate directions on the Earth's surface.

In this way, we revolutionize the empirical notion of a "day" as a mere "yin-yang" alternation of light and darkness. Instead, we conceive the day as an astronomical cycle, subsuming an {increasing density of distinct events} within a single ordered totality. Just as the gnomon's shadow progressively transits the markings of a primitive sundial, including the meridean defined by the position of longest shadow; so the cycle of the "day" subsumes and orders the events of rising and setting of stars and constellations, and their transit across definable angular positions as defined by the sightings of a primitive stellar observatory. From the development of these methods, our predecessors established the regular division of the day, and an indispensible means for harmonically ordering the activities of society.

But, there is a far-reaching paradox embedded in this splendid hypothesis of the day's rotational cycle as a universal ordering principle! Looming long on the horizon of our prehistoric astronomer's mind, but now growing in urgency, is the realization, that the day itself is subject to {change}. For example, the array of constellations, which are visible in the sky just before sunrise, is strikingly different in winter than in summer. To investigate the origin of this difference, identify a star or constellation, whose setting in the West immediately preceeds the rising of the Sun in the East. Within a few days, we become aware of a slight delay in the appearance of the Sun, after the selected star or constellation sets in the West. The delay keeps growing: the Sun seems to be "slipping" backward in time relative to the stars! That apparent slippage constitutes a new, anomalous degree of change. Again, the question is posed: what is the exact {form} of this change?

Our prehistoric astonomer juxtaposes this solar anomaly with a whole cluster of paradoxes, connected with the empirical cycle of "the year." The empirical notion of a year as a mere alternation of "hot" and "cold" seasons, or periodic recurrence of monsoons, floods or other natural phenomena, bespeaks the nearly bestial state of Man before Prometheus bestowed his gifts. The mere counting of days between recurrence of some terrestial event, leads to erratic results, falling far short of the "sure sign" promised by Prometheus.

Worse, was the attempt to arbitrarily impose upon society, a non-existent correlation between changes in season and the cycles of the Moon. So, the Babylonians (and others) insisted on a calender based on the socalled synodic lunar month, as defined by the recurrence of the full moon after approximately 29.5 days. After the passage of a mere 18 "years" of 12 synodic lunar months each, winter now occurs in the months where summer used to be, and vice versa! The attempt to "fix" this monstrous failure with the addition of special days and alternation of "longer" and "shorter" months, while rejecting the primacy of the solar cycles and insisting on the cult of the Moon, (or some "rotten compromise" between the two) is more than typical of the psychosis which dooms every oligarchical empires to collapse. Although the present Western calender is entirely solar-based, and our months have no correlation to the phases of the moon, the term "month" still remains as an apparent relic of Babylonian lunacy.

In contrast, by adducing a new, "solar long cycle" from the {anomaly} posed by the slight discrepancy between the solar motion and daily stellar motion, our prehistoric astronomer was eventually able to invent a "sure measure" of the seasonal cycle, which remains true over centuries and even millenia! The result is best demonstrated by the spherical sundials of ancient Greek times, which registered not only the daily trajectory of the Sun, but also the cycle of variation of the Sun's (approximately circular) pathway in the sky, over a period of (approximately) 364 days. That cycle subsumes the cycle of change in the relative lengths of night and day, as well as the angles of inclination of the Sun's rays to the Earth's surface, providing in turn an intelligible basis for the variation of the seasons.

But, the manner in which the yearly solar cycle "modulates" the daily one, ordering the variations of the latter, implicitly poses a new array of paradoxes. For example: If the day is variable, might not the year be so also? And in fact, careful observation of the loci of rising or setting of the Sun and the stars, by means of suitable horizon markers and observation points, revealed a very slight -- but distinct -- anomaly in the solar cycle. From this, the ancient astronomers were able, thousands of years ago, to adduce an approximately 26,000 year-cycle of the so-called precession of the equinoxes! The result is a third, "long cycle" modulating the year. The latter, according to our best present knowledge, essentially determines the cycle of ice-ages, together with a fourth anomaly, namely the elliptical character of the earth-sun orbit.

Still another paradox exploded in the repeated, failed attempts to fit various among the multiply-connected solar cycles, as well as planetary and other cycles, into a single calender. This included the search for a single "great cycle" subsuming all the others, such that the end of such a "great cycle" would mark the simultaneous end of all the shorter cycles. The work of the Pythagorians, on the fallacy of "linear commensurability" put an end to such Babylonian "systems analysis," and posed yet a new level of paradox:

If there exists no grand mathematical system which can combine and account for the various cycles, then how can we conceptualize the "One" which subsumes the successive emergence of new astronomical cycles as apparent new degrees of freedom of action in our Universe? How do we master the paradoxical principle of Heraclitus, that "nothing is constant except change?"

---------------------------------------------------------

(1) This version is my first-draft translation of lines 436-471 in the "Reclam" series German translation of Aeschylus' drama, which seemed somewhat clearer than an English translation I had seen earlier. Bruce Director, who first called my attention to this selection, told me there are better English translations. But, I think the point is well enough made for the present purpose.