## Wednesday, August 17, 2011

### Zeno’s Paradox of Achilles and the Tortoise

"There is no motion because that which is moved must arrive at the middle of its course before it arrives at the end." The quotation, from Aristotle’s Physics, expresses the principle underlying Zeno’s parable of the race between Achilles and the tortoise. A tortoise challenges Achilles to a race, asking only that he be given a head start. Achilles scoffs at the suggestion but the tortoise explains that the race need not even take place because it would be impossible for Achilles to catch up with him once he had started running from his vantage point some way down the track.

"When you reach the point from which I started, you must admit that I will have run further still, even though it may be a small distance. When you have run that distance, I shall have run further still, and so on

At this point, Achilles should have insisted on running the race and proved empirically that the tortoise’s argument was false. This provides a clue to the counterintuitive idea that a fast runner cannot overtake a slow one, which contradicts our quotidian experience. If we stick to the tortoise’s a priori argument we are caught in a logical trap, but if we insist on an empirical refutation the status of logic is downgraded, and we must doubt the applicability of logical argument to physical events.

The tortoise’s argument is one way of defining the race but there are other ways that exclude the paradoxical trap. For example, Achilles could run the agreed distance L (which might have been 1 stadion = 185.4 metres) separately and the tortoise could then have run the shorter distance L/10 say. Timing both events would show that Achilles runs the full distance faster than the tortoise runs the shorter one, the conclusion being that Achilles must have won, without having to overtake the tortoise. The objection to this in those ancient times would be that there were no clocks capable of accurately measuring such short time spans. Zeno might have objected that this was no race at all, because the events took place at different times. If Achilles runs first, he obviously crosses the finishing line first, but if the tortoise goes first he wins if only because he crossed the finishing line before Achilles even starts to run.

So, we are forced to consider the mathematics of Zeno’s argument. We can suppose that the tortoise has a starting advantage of
K < L which is sufficiently small to allow Achilles to catch up before the race is over. Careful consideration will show that K can always be set so the tortoise can win but that this is not the case here. The decreasing distances defined by Zeno’s argument can be precisely defined in terms of the relative velocities of Achilles and the tortoise. Let these be Va and Vt respectively. Achilles runs the distance K0 in T1 = K0/ Va. The distance run by the tortoise in time T1 will be K1 = T1. Vt = K0Vt / Va. A series of geometrically decreasing distances can now be defined as (K0 + K1 + K2 + …+Kn) = K0Σn (Vt / Va)n. Since Vt < Va the sum will converge to an infinite limit, which is the point where Achilles catches up with the tortoise being D = K0Va/(Va-Vt).

We can now calculate the time taken by Achilles to cover the distance D which is TD = D/Va. At any time between D/Va and Achilles’ finishing time L/Va, we can claim that the tortoise has been overtaken. This provides an a priori rather than an empirical refutation of Zeno’s argument. However, the objection remains that an infinite summation of the decreasing differences is impossible in the physical world. A way out of this difficulty is to ignore the race track altogether

and consider the relative velocity of Achilles and the tortoise, which is Va – Vt. The time taken by Achilles to close the initial gap K is therefore K0/( Va – Vt) so the distance D = K0Va/( Va – Vt) which is the same formula derived using the infinite summation.

The cleverness or inadequacy of Zeno’s argument is that it is an incomplete model which precludes consideration of the race as a whole, focussing only on the vanishingly small distances between the participants. The argument is a petitio principii in that it assumes in advance that motion is impossible and creates a model that favours this assumption. The existence of other theoretical models which do conform to everyday experience and the intuitions based on it resolve the apparent paradox by rejecting Zeno’s model.

Water clocks existed in Zeno’s time, so the timing of a race by this means should not be ruled out. However, modern circular clocks do provide continual, empirical refutations of Zeno’s argument. The minute hand of a clock does overtake the hour hand several times a day, but the question is: at what time do the hour and minute hands of a clock coincide?

The method of calculation is similar to the one used to find the crossover point in Zeno’s race. The minute hand of the clock completes one revolution in the time it takes the hour hand to move the five minute division between one hour and the next. The relative velocity is therefore
1/12. The first coincidence after 12 O’clock is 12/11 hours = 1h 5m 27.27 sec. Peculiarly there will only be 11 unique coincidences, with no coincidence between 10h 54m 32.72 sec and 12h 0m 0sec. Since the coincidences are independent of the clock face divisions, the latter could be dispensed with. The result would be a clock which divides the diurnal cycle into 22 equal periods.

If a typical clock is observed closely, the minute hand appears to move in one second jerks rather than smoothly. The implication is that it cannot exactly coincide with the hour hand at any time other than 12 O’clock, since there are no other coincidences exactly measured in whole seconds. In this sense, Zeno would be correct in asserting that the minute hand cannot overtake the hour hand at such a coincidental point but would be wrong in concluding that it could not do so after that point. This is because we assume that ‘time’ is a continuous process, rather than more correctly realising it is the nature of the physical process that defines how time is to be interpreted. If physical process involving movement are small but discontinuous, they can be regarded as almost continuous because the smallness of the Planck constant.

The movements of Achilles and the tortoise can be regarded as individual steps, or strides. When Achilles has run the overtaking distance D, he may be in mid stride, between and n and n +1 strides, in which case he will overtake the tortoise on completion of the nth stride. His stride cannot exactly coincide with the point D unless Va is exactly divisible by Va – Vt . If it is, then he must complete n +1 strides before we can claim that he has overtaken the tortoise. In either case, he is bound to win the race unless Paris shoots him in the heel with an arrow first, but Zeno’s argument would also rule out this possibility too. We can conclude from this that Zeno did not believe Homer’s tale or indeed any myths where mythological figures get pierced by arrows. This must have been comforting since on his theory he could never be hit by a missile of any kind.

The idea of a limit that cannot be transcended appears in Einstein’s Theory of Relativity. He argued that, as the velocity of a mass approaches the speed of light, the energy required to accelerate the mass approaches infinity. Since the speed of light is a constant, he concluded that the mass must

increase without limit, which is impossible. Like Zeno’s argument, this mathematical proof seems counterintuitive, but empirically true according to certain scientific observations. Using the earlier Newtonian model, there appears to be no reason to doubt that a mass could reach the speed of light and possibly exceed it. The point is that different theories may lead to different truths. Putting it the other way round, two theories cannot be different and non-contradictory.

Einstein’s formula below, which relates the mass of a moving body to its rest mass, also uses the idea of a limit that cannot be exceeded because of an infinite process. The interpretation is that a moving mass increases without limit as its velocity approaches the finite limit of the speed of light. If we switch the rest mass
m0 in the formula with the moving mass m, the result would be a different theory where the moving mass decreases to zero as its velocity increases towards the speed of light.

Zeno’s interpretation was based on his philosophical beliefs. His teacher Parmenides believed that time was an illusion because the universe was a permanent whole that never changed. From this point of view we experience change because our consciousness is incapable of observing the absolute nature of the universe. This approach is similar to Einstein’s four dimensional space time continuum, where events are frozen by regarding time as another spatial dimension. Even the simple Cartesian diagram of parabolic motion illustrates the principle of representing time as a spatial dimension, so Einstein’s formulation was just an extension of this type of description to a fourth dimension, transcending our usual mode of experience.

Zeno’s paradox of Achilles and the tortoise relied upon generating an infinite process which frustrated finite resolution, at least until mathematicians devised a satisfactory theory of limits. When it came to Cantor’s thoroughgoing examination of infinity, a great many paradoxes arose which had to be dealt with. Behind his approach to the theory of infinity lay the kind of absolutism espoused by Parmenides, which was a belief in the existence of absolute infinity rather than Aristotle’s comparatively timid potential infinity. Cantor’s achievement was to show that a hierarchy of infinities had to be defined as a consequence of his set theoretic approach. One of his best known achievements was to prove that the set of irrational numbers could not be counted by the natural numbers. He did this by means of his diagonal argument.

Zeno’s argument was Aristotelian in character, relying on the principle that it was impossible to add up all the increasingly tiny distances that Achilles had to run to catch up with the tortoise. Cantor’s starting point was to assume that the set of natural numbers, like any set, must have a totality, which he called Aleph null. Like Zeno’s argument, Cantor covertly employs the trick of introducing a limit that cannot logically be transcended, to wit the diagonal of digits in a finite square adopted for the purpose of demonstration. This involves the belief that the square can be expanded as far as Omega (the ordinal version of Aleph null) thus preserving the property of excluding certain combinations of digits.

Once the trap is accepted, that the square cannot be greater than the arbitrary number Omega, the result follows that the power set of the natural numbers is non-denumerable. Since the list of irrational numbers has been defined as a permutational power set, the argument degenerates into a

petitio principii. The point being that there is no diagonal associated with a non-square list, so the whole argument is a tautology signifying nothing. While cantor’s diagonal argument is logically sound, like Zeno’s, the model he constructed was erroneous. The consequence is a flawed set theory riddled with paradoxes.

The general issue is the extent to which we should accept counter intuitive arguments on the basis of special definitions which purport to render them logically sound. In the case of Zeno’s imaginary race, this should not be done without careful examination of alternative models. The corollary is that sound logical argument is not a sufficient criterion for truth, since the devil lies in the details hidden in the definitions and covert assumptions lying beneath the surface.

Tony Thomas

August 2011

## Friday, April 8, 2011

### The Idea of Truth

“Beauty is Truth, Truth Beauty, -That is all
Ye know on earth, and all ye need to know.”

John Keats, Ode on a Grecian Urn

The last two lines of Keats’s poem have been the subject of much erudite analysis, and it is presumptuous for an undistinguished writer to put in his twopenn’orth, but I will anyway. The intention, however, is not to add to an already overburdened debate but to use this example as a cave entrance to the even more labyrinthine consideration of the nature of truth in general.

In relation to the statement, “Beauty is Truth, Truth Beauty,” The critic IA Richards had warned against trying to take poetic statements too seriously. TS Eliot responded: “on re-reading the whole Ode, this line strikes me as a serious blemish on a beautiful poem, and the reason must be either that I fail to understand it, or that it is a statement which is untrue.” It is unlikely that Eliot did not understand the several possible meanings of the line, so we must conclude he did believe it was false. I agree with his judgment, albeit from a strictly logical interpretation of the last two lines of the poem.

As one would expect from a great poet, the multiple assertions in the concluding lines are bound together in a complex construction which does not immediately yield up an unequivocal meaning, either to the casual reader or even to prolonged analysis. Some may consider this to be a poetic virtue, akin to any artistic or mystical mode of thought. However, the resulting ambiguity confounds the kind of clear and unique interpretation demanded by the logical mind.

The two lines can be broken down as follows, at the risk of destroying any beauty or truth they express in their original form:

S1: Truth is Beauty.
S2: Beauty is Truth.
S3:S1 and S2 is all ye know.
S4:S1and S2 is all ye need to know.

Before attempting to interpret each of the four propositions it is worth noting the following points. ‘Truth’ and ‘beauty’ are both abstract nouns that are used as adjectives (attributes) in S1 and S2 as well as nouns. This raises the problem of whether attributes (qualities) can properly have other attributes asserted of them and whether nouns can properly be used as attributes. S1 and S2 appear to be universal statements although they are not explicitly quantified ie “All truth is beautiful” and “All beauty is truthful”. If this were the poet’s intention, then we could logically conclude that beauty is in some sense equivalent to truth and vice versa, a proposition that can be tested by substituting one attribute for another in any and all possible propositions containing these attributes.

Consider the proposition, “Helen of Troy was (a) beauty”, which would become “Helen of Troy was (a) truth”. This provides an immediate contradiction of the equivalence because the second statement is hardly meaningful. We could elaborate and say, “Helen of Troy told the truth” but I am not sure if this was the case. In general, I am inclined to reject the assertion that either abstract or concrete entities are necessarily true because they are beautiful. One reason for this is that beauty is a subjective judgment which does not command universal agreement. What some find beautiful others find uninteresting or even ugly. Truth, however, has a better claim to objectivity whether it is established empirically or as a valid deduction from first principles. Truth, therefore, is not properly comparable with beauty at all.

Mathematicians and scientists might object to this negative judgment by declaring that beauty is a good if not infallible guide to truth. The justification for the scientist is that nature determines what we find beautiful, both internally through the mind and externally through the senses, and that this natural beauty is an expression of the perfect harmony that exists in nature. Clearly, this was the sentiment that Keats expressed in his ode, although the urn was a man-made artifact rather than a natural one. The mathematician has an even stronger claim for using aesthetic criteria as a guide to truth. The declaration that Euclid’s proof of the infinity of primes is beautiful is certainly meaningful, and the generalisation that mathematical intuition depends on the detection of such beauty soon follows from this aesthetic point of view. However, beauty is being used here as a guide to enquiry but not as a final criterion of truth. The subjectivist objection might be that truth and beauty are generated in the human mental system and so are not proper attributes to the real world as a noumenal domain. In this case, there might very well be a strong connection between aesthetic and veridical judgments.

Putting aside the inappropriate comparisons of truth and beauty in S1 and S2, the assertion that knowledge is limited to these two statements is clearly false. Even if beauty cannot be separated from truth, this would not justify ignoring all else that we take for true knowledge of our world. Indeed, if truth and beauty are equivalent then we can drop one of them as redundant. To reject all other knowledge is surely epistemological hubris. Most of the humble facts of daily experience, as well as much of scientific knowledge would be excluded from Keats’s idealistic garden of perfect delight. Of course, he may only have intended to assert what he felt was a special relationship between beauty and truth, rather than to follow up the inherent contradictions of this profession of aesthetic idealism.

The fourth proposition reinforces the third by insisting that no further knowledge is required beyond the equivalence of beauty and truth. This could only make sense in a metaphysical system of which S1, S2 and their equivalence were the axioms, from which all else could be deduced in the perfect world of the poets imagination. From this exalted point of view, the poet seems to have been striving after the kind of ultimate truth that philosophers had long sought after and latterly rejected.

In Keats’s time the vogue for Greek art was still highly influential. That supreme example, the Parthenon, had incorporated the highest geometrical knowledge of classical Greece, and so expressed a formal beauty derived from it. Given that mathematical principles are embodied in nature and strongly influence our ideas of beauty, Keats’s perception of the relationship between beauty and truth assumes a clearer meaning. What we mean by beauty is the expression of mathematical form intuitively observed in nature through the senses. One might have expected the nature loving poet to have observed the perfidious function of natural beauty. All manner of deceits are dressed up in nature’s colourful finery. Butterflies open their wings to display imitation eyes and beautiful sexual displays are widely used generally to lure a mate. Such strategies are clearly inconsistent with displays of truth combined with beauty in nature. Deception therefore has long preceded the emergence of mankind and its ability to develop such abstract categories as truth or beauty. But even without such deceptions, there are the illusions that derive from imperfect perception. The fly with its compound eyes is presented with multiple views of its world, and even humans learn the clever trick of seeing a world transformed from the inverted image that falls on the retina.

Our quotidian experience involves continual judgments, both conscious and unconscious, about the present. Standing upright or walking requires such adjustments which may only enter into consciousness when we stumble, and experience error. In going about our business we make innumerable judgments according to habitual criteria, consciously learned or otherwise, which may turn out to be wrong. Some of these criteria are of a sufficiently high order to qualify as beliefs, although many of them may have been acquired as received knowledge rather than consciously examined and granted the status of truth. So, we live among a welter of beliefs which guide our actions and only occasionally get examined for their veracity.

For higher animals, truth is intimately bound up with memory, and may be marred by its imperfections. Memories of past events can be compared with present situations and accumulated knowledge acquired through memory and concept formation used to make judgments in the present. This process gives rise to the idea of repeated similarities between prior and current events. One overarching truth we observe is that events, though similar, are unique. The clouds we see today may resemble those of yesterday but cannot possibly be identical because of their random structure. Other more permanent forms, such as rocks, trees, and common animals exhibit a stronger degree of resemblance and form the basis for the belief that their identity persists over time, even while they gradually change. This very complex state of affairs is the basis on which we experience and formulate ideas of truth.

The fairly recent discovery of non-linear mathematics has demonstrated that nature mimics infinite forms, as in the case of each unique snowflake, river channels, human bronchi and mountain ranges. The corollary is that limitations are placed on human ability to understand and to adapt to a world of potentially infinite complexity.

One approach to the truth enigma is to ask, what kind of things the attribute ‘true’ can meaningfully apply to. One answer is that it applies to beliefs, but these must at least be encoded in some way, usually in a language, before such judgments can be made. Furthermore, the resulting statements need not be believed at all, or any beliefs in them suspended for purposes of logical or semantic analysis. The essential point is that beliefs can be divided into true and false beliefs, so that the fact of believing something is distinct from the fact that the proposition expressing the belief is true or false. In other words, belief has no effect on whether a state of affairs is true or not, except in those cases where human physical or mental performance is influenced by belief, eg to win a race or pass an examination.

In the Keats example, there was some uneasiness about whether statements like “truth is beauty” are even meaningful, as in Chomsky’s example: “colourless green ideas sleep furiously”, which is grammatically correct but meaningless and contradictory. To judge whether a state of affairs is true or not, it is necessary to encode the situation clearly, unambiguously, grammatically and meaningfully. The expression of ideas in a language exacerbates the problem by transferring the focus of truth judgments to propositions and away from unformulated beliefs or situations that the propositions represent. For example, if we assert “The King of France is bald” several linguistic problems arise: there is no present King of France, we do not know which past king is being referred to or to what degree of hair paucity constitutes baldness. The power of language is very great, and innumerable propositions about a subject may have to be composed before any factual truth criteria can be considered. The simple assertion, “the earth goes round the sun” is readily understood by our educated minds, but some thought would be needed by the non-specialist to demonstrate this scientific ‘fact’ to a disbeliever. Indeed, it was not until 1838 that the astronomer F.W. Bessel was able to measure the parallax of a nearby star to show that the earth was at a different place from the time of the first measurements compared with observations six months later. Einstein, of course, threw doubt on what the term ‘different place’ might mean.

In this astronomical example, the truth was established by making not only accurate observations but also by relying on the validity of trigonometry, whose truths are of quite a different kind from facts established by observation of what we call the real world. Fortunately, the delicate observations did not depend on Einstein’s as yet to be discovered facts about the behaviour of light under strong gravitational fields but would have been affected by the refraction of light through the earth’s atmosphere under different atmospheric conditions. The point here is that establishing a fact in one area of science may depend on believing a host of other facts. If any of these should prove to be untrue, there may be significant ramifications for those experiments or theories which assumed they were true.

From this point of view, our most certain fields of knowledge are a contingent house of cards that must be continually be maintained. Similarly, when we make our casual judgments about our ever day lives, we rely on innumerable beliefs, most of which are received knowledge and many of which we would be incapable of adequately demonstrating as true.

It is often the case that we form opinions, particularly about people, based on their appearance, manner and a few instances of social behaviour. Hypotheses are formed and tested by observations until beliefs about them become firmly established. Such opinions may be influenced by received ideas about how people of their class, colour, creed, occupation, or physiognomy usually behave. In this arena of unavoidable social interaction and judgment, beliefs are founded intuitively rather than through any systematic assemblage of consistent propositions, as would be required in a science. Human behaviour is very complex, a fact which renders difficult the social sciences from psychology to economics. The veracity of statements in these fields of knowledge is commensurately less certain and often characterised by probability rather than certainty.

It should be clear even to the most exalted mind that significant truths are hard to come by and that, consequently, we live in a fog of beliefs that fall short of the highest standards of veracity. To make matters worse, psychology makes clear that the human mind can be unreliable in making even simple judgments about recent or even current events. The conjuror makes use of this deficiency by exploiting the tendency of the mind to fill in the gaps between what actually occurs, i.e. as recorded on a video camera, and what they think happened. It seems that the control of the senses by the mind produces a mixture of fact and predictive fiction as it attempts to assess what is likely to happen on the basis of quickly varying events.
Truth, then, depends on the formulation of memories and beliefs and the comparison of these with some kind of independent criteria. The most obvious comparator in the case of everyday events is the memories and beliefs of independent witnesses, as well data from any recording devices. This raw material of quotidian events produces quite a different class of truths from the important generalisations about the human condition and the physical environment on which humans depend. Classification, abstraction and generalisation form the basis of useful knowledge, which is different from the truths of individual observations and events. The truths of science require the additional assumption that what is true in several cases can be generalised to apply to all similar cases, the basis of induction. The failure of a hypothesis might be due to imperfect observations about a few situations of the required type or it might be due to unexpected or unknown factors. A grander scientific assumption is that identical circumstances (ceteris paribus) must produce the same results, or be due to different or improperly controlled factors not included in the theory. Establishing the truth of such fundamental ideas poses a special problem for epistemology.

The empirical procedures and inductive analyses which form the basis of scientific enquiry, together with informed hypotheses have led to general truths of incalculable benefit to humanity. It is hardly surprising, therefore, that the truths of science have replaced the once ubiquitous religious and philosophical beliefs that sustained earlier societies. The question, what is scientific truth, could be answered by saying that it is the body of knowledge, albeit provisional, that has accumulated as a result of applying legitimate scientific methods to the highest possible standards, and confirming the results by the process of peer review.

This is a very high standard indeed when compared with the way we form common beliefs in our business and private lives. In general then, truth is the body of the best knowledge that humans are capable of producing in the present state of cultural development. Even in the span of a century or so, great changes have occurred in both philosophy of knowledge and the processes of science and technology which have led to what may be classified as truths.

Several philosophical theories of truth have been formulated. The correspondence theory assumes that the truth predicate applies to beliefs, and further supposes that every true belief corresponds to a fact. This assumes that there are such things as true facts and that their veracity can be established. The Pragmatist William James objected that this approach was just a lexical trick that did not discuss the nature of truth at all. The salient point is that the idea of truth enters into both beliefs and what are regarded as facts and it is not clear what is meant in either case by saying that a belief or a fact is true, apart from applying the attribute to a fact and a corresponding belief simultaneously.

In the coherence theory of truth, the objection is that individual statements are incapable of capturing truths about ‘reality’, since states of affairs may be described in different ways from different perspectives and motives. Furthermore, descriptions of events are infected by the meanings inherent in the language used, which imports ready-made concepts into the description of what is supposed to be the factual criteria. Only a wide theoretical context will suffice to judge whether a statement is true or not. This mirrors what we actually do in making casual truth judgments, where comparisons are made with our existing knowledge base to see if any inconsistencies arise from accepting a new idea or supposed fact. William James’s pragmatism supposed that true beliefs were those that we must act upon in order to survive or advance the welfare of humanity, which is a rather partisan notion that fits in well with the American predilection for social Darwinism, as opposed to an impartial and unselfish search for truth for its own sake.

Tarski’s semantic conception of truth applies truth and falsity to sentences, and consequently focuses on their meaning. He pointed out that a statement such as: (It is true that (Socrates was wise)) are meta-lingual statements, where an assertion is made about another statement in what he called an object language. He provided the gnomic example: “snow is white” if and only if ‘snow is white’. One could interpret the first statement to be a belief and the second to be a statement of an empirical fact, which doesn’t seem to advance matters much beyond the rejected theory of correspondence. The objection to this approach is that the so called real world is lost in the process of linguistic and logical formulations. It is worth observing here that pure logic is not at all concerned with empirical meaning and so can provide no guidance whatever on the vexed relationship between an empirical fact and its description as a thought or a belief.

The distinction between truths of the mind, a priori truths, and observed or a posteriori truths is an important one. An early system of a priori truth was Euclid’s geometry, which was based on five axioms and five elements of construction. This proved capable of generating innumerable mutually consistent theorems, including the well-known truth that the angles of any plane triangle add up to two right angles. The truth of many of these theorems is not immediately obvious to the uninformed observer of geometrical figures and provided essential knowledge for ancient architects and surveyors. How wonderful that every triangle inscribed in a semi-circle is a right angled triangle, and that every triangle inscribed inside an arc defined by an arbitrary chord generates triangles with a constant angle. It is this power to generate truths about infinite cases which distinguishes mathematics from empirical science.

The idea that a few basic ideas and rules of logic could generate new knowledge was a powerful one, which dominated philosophy until quite recent times. The false idea was that knowledge of a few fundamental principles in philosophy or science would be sufficient to deduce all possible knowledge. A corollary of this idea is that the resulting totality, realised or not, represented a perfect and consistent body of truths, just like the totality of all possible theorems derived from Euclidean Geometry. An obvious drawback to this epistemological programme is that it could only apply to a priori systems, which would confine the resultant knowledge to logic, mathematics and related disciplines. The unfortunate truth was that there is no obvious connection between empirical truths and the logical means of elaborating facts about the physical world.

This idealistic view persisted until recent times, until empiricism and the rise of science gradually confined it to speculative philosophy. It still remains a philosophical question as to how the observed behaviour of the physical world conforms to Euclidean geometry or to the more advanced systems of mathematics that are so essential to modern science. However, some philosophers of science think this need have no bearing on how science is conducted in pursuit of knowledge. However, what does seem to restrict the progress of scientific theories is the lack of sufficiently powerful mathematical theories used to formulate and describe them. The calculus is an obvious example in relation to mechanics and other areas of physics, as is matrix theory and probability theory in their many applications to science. This indicates the strong dependence of empirical theories on specialised languages.

A peculiarity of axiomatic systems, both logical and mathematical is that different axiom sets can lead to different and sometimes inconsistent theorems. A simple example is that the angles of a triangle drawn on a sphere add up to more than two right angles, and so does not even include the corresponding Euclidean theorem as a special case. The distinction between Newton’s and Einstein’s theories of space and time is another example of differing but true systems. This difficulty extends to formulating logical languages for demonstrating the consistency of mathematical truths. The avoidance of paradoxes had doomed Russell’s and Whitehead’s attempts to provide a reliable logical language as a basis for all mathematics. Kurt Gödel upset the apple cart by proving that it was not possible to construct such a language that was both consistent and complete, so that there would always be, potentially, true but unprovable theorems describable in the system. This put an end to the dream of creating a final system of a priori knowledge, and further undermined the kind of certainty associated with grand systems of universal truth, whether empirical or not.

From one point of view, the pursuit of knowledge requires both the general principle of truth and the philosophical, logical, mathematical and empirical means of establishing the ever changing facts which are generally agreed to be true. Because the body of such knowledge is now so vast, it has become inaccessible to the ordinary citizen, if only because of the cost and deficiency of educational systems and the limits to individual knowledge. One unfortunate result of this divide is that beliefs in the general population fall far short of the knowledge that is available to those with the inclination, means and opportunity to obtain access to them. In other words, general ignorance is a source of discontent and social upheaval, often exploited by unscrupulous politicians for personal power and by corporations who wish to preserve an ignorant and subservient population.

Access to these huge knowledge bases has been greatly increased through the medium of the Internet, but this has also provided access to many bodies of pseudo-knowledge and speculative thought that lies outside the strict borders of academic and professional knowledge. The term truth has been stretched accordingly to accommodate this burgeoning diversity. One particular example of this is the reactionary movement of creationism which seeks to re-establish forms of authoritarian knowledge prevalent in earlier societies. The tension is between restrictions on free thinking beneath the shadow of the now enormous tree of accepted knowledge and allowing a tangle of speculative and redundant thought to thrive in the jungle outside its shadow. Such a wilderness has often nurtured the kind of mavericks who have contributed greatly to scientific and cultural knowledge, so tolerance of a robust unorthodoxy is preferable to an epistemological monoculture. The counter argument that academia already encourages such diversity of thought is less convincing now that corporations finance and control so much of what constitutes advanced education. The corollary is that the truth, whatever it might be, ought not to be controlled by orthodoxy or by whatever dominant philosophy may declare it to be.

We continue to search for truth without knowing exactly what it is or how we ought to go about it. The combined efforts of countless generations have shown that such a search is both valuable and necessary for human welfare, and latterly perhaps, continuance as a species. It seems obvious, if not proven, that adherence to truth in its many forms is our best hope for a satisfactory social life and that weeding out false beliefs, albeit in a kindly way, must be an ongoing project. Notwithstanding that great monuments to truth have been erected, their continual replacement, however costly, ought to continue if civilisation is to remain accessible to a majority of humans. The balance of false beliefs seems to be gaining ground and might easily derail the beneficial presence of true if not final beliefs that assist humanity on its journey into the unknown.

Tony Thomas
April 2011

## Friday, October 16, 2009

### What is Knowledge?

Like many fundamental questions, 'what is knowledge' is surprisingly difficult to answer in a satisfactory way. This is because it is an important philosophical question that forms the basis of the division of philosophy called epistemology. A dictionary definition yields the following:

Knowledge (n)
1 information and skills acquired through experience or education - the sum of what is known. Philosophy true, justified belief, as opposed to opinion.
2 awareness or familiarity gained by experience.
- PHRASES come to one's knowledge become known to one. to (the best of) my knowledge 1 so far as I know. 2 as I know for certain.
- ORIGIN ME (orig. as v. in the sense 'acknowledge, recognize')

The most basic meaning is of being aware that some thing or situation resembles another one that has been previously experienced and exists as a memory. For example, recognizing a face is a skill that does not require conscious, volitional learning but happens automatically: hence familiarity gained by experience, which is not peculiar to humans but is present in all animals who modify their behaviour in response to pain or pleasure. The usual example given is the pain a child feels before learning to avoid contact with fire. This becomes encoded in language as the proposition, "fire burns" or the more general form, "fire is hot", signifying that fire belongs to a more general category of phenomena which can be harmful or useful to mankind. This introduces the fundamental role of knowledge as not only helpful to the conduct of life but essential for survival. In this sense, both humans and animals are knowing beings.

The first meaning distinguishes between information and skills. The latter may be motor skills, like sawing a piece of wood or playing the violin which involve a different process of repetitive practice from learning, say the multiplication tables, the latter requiring some minimum understanding of numbers and their properties. When Bertrand Russell was a young child he was reduced to tears by his inability to learn the multiplication tables and he records, in his autobiography, that what delighted him most was learning that blue mixed with yellow produced green. Here is what he had to say about learning some basic algebra.

The beginnings of Algebra I found far more difficult, perhaps
as a result of bad teaching. I was made to learn by heart:
"The square of the sum of two numbers is equal to the sum of
their squares increased by twice their product." I had not the
vaguest idea what this meant, and when I could not remember
the words, my tutor threw the book at my head, which did not
stimulate my intellect in any way.

This example of the failure of rote learning by a child of exceptional intelligence shows that the acquisition knowledge is affected by the form in which it is presented to the student. The translation of the symbolic form of the mathematical expression into words, instead of explaining the relationship between the parts of the algebraic expression (a + b)2 = a2 + b2 + 2ab, was the cause of the problem, since Russell had no problem learning his Euclid. For Russell, the linguistic proposition was not knowledge because it was meaningless relative to his understanding. It follows that, for the student, knowledge must be understood and meaningful in relation to existing knowledge. Knowledge, therefore, exists in clusters, not in isolation.

It is clear from the rest of the definition that, for the philosopher, empirical knowledge is always provisional and subject to revision. This has become a basic principal of science but is often ignored in daily life where certainty about common knowledge is usually assumed. Even apriori knowledge may prove inadequate, as non-Euclidean geometry shows. The lesson here is that absolutely true knowledge is always subject to prior assumptions as expressed by the term ceteris paribus. The experimental scientist must restrict his variables by excluding possible but highly improbable factors from the variables. The discoveries of empirically established 'facts' become knowledge, but always subject to the conditions of the theory and experiments within which they are constructed.

Outside the laboratory and the University a great deal of information is taken on trust as factual. The inadequacy of human records and memory becomes clear when they are disputed in the adversarial system of the courts, where arriving at the truth is a tedious process of forensic examination. Such analysis is usually impractical in business or government, where highly educated individuals are trusted to use their skills and experience in making informed judgments about what are true facts and what are doubtful propositions. The idea of facts being beyond reasonable doubt or of high probability has become an essential tool of administrators as well as scientists. The necessity for scepticism about unusual proposals is well understood and undermines the simplistic assumptions that we always deal with true facts rather than those that common sense and experience suggest are probably true. For practical purpose, therefore, one is always dealing with facts that may turn out to be false, albeit with a small probability.

The corollary of all this is that the bulk of what the non-specialist knows is a mish-mash of well established common knowledge, which is probably true, and a great deal of assumptions deduced from this knowledge which might be flawed due to inadequate reasoning. While most intelligent people reason correctly about everyday affairs, when faced with complex circumstances involving a huge number of facts and assumptions, this reasoning proves inadequate, and intuitive judgments are applied instead. While humans are capable of reasoning, it would be wrong to assume that they are predominantly reasonable in their judgments as opposed to intuitive, reactive and emotive.

The distinction between knowledge and assumptions is latent in human affairs, and the incorporation of statements in printed or electronic media raises the status of such information from provisional to established facts.
Editorial or peer review, or the category opinion rather than fact, goes a long way towards ensuring the quality of recorded knowledge, but the vast quantity of such widely available information reduces the overall quality of such knowledge.
Disputes about whether a certain body of knowledge is true are commonplace. Prior to the 18th Century is was unthinkable or at least unwise to challenge religious orthodoxy, whose vast repositories of doctrine went largely unchallenged, except within the upper reaches of the various churches. Theological debate, like today's science, was the preserve of specialists. In our world, specialists abound and represent a process of creating bodies of knowledge that can be widely applied. Fields of doubtful knowledge abound, including psychoanalysis, literary theory, new age regurgitations of Eastern philosophy and many more fields where pseudo-knowledge stakes its claims. This only matters when they affect important government or business decisions, or individual welfare. Familiar examples are the definitions of human personality that underlie the vexed debate over contraception and abortion. This is a case of a conflict between new knowledge or understanding versus old knowledge based on redundant religious beliefs. A more important example is the more complex question of global warming, which turns upon a vast number of facts and theories developed by many different scientific specialists relevant to understanding climate change.

Big questions like 'has peak oil been reached' or 'will average temperatures rise by more than 3% in the next 20 years' do not admit of definite answers, only informed opinion arrived at by scientific or technical experts. The problem of deliberately distorted information about these important questions is evident in the biased reports and lobbying that is currently taking place. It is clear that politics, at all levels, has an impact on what can be accepted as true, just as religion is often the enemy of truth in both past and present. Indoctrination by the mass media on these and many other sensitive issues shows that many facts commonly held to be true are artefacts of propaganda. In other words, orthodox knowledge is manufactured by ruling elites to serve their interests. An examination of what constitutes knowledge must take account of this fact of life.

The import of this is that finding reliable sources of information depends not only on doing enough research but making judgments about these sources based on ones own flawed judgment. When faced with hundreds of thousands of seemingly intelligent people who question, say, Darwin's theory of evolution, it becomes evident that untutored opinions are the norm rather than the exception. Like the esoteric theological doctrines of the past, advanced knowledge of, say, biotechnology lies beyond the competence of the average person. Furthermore, advanced specialist in one discipline lack competence in most others fields of advanced knowledge.

What is knowledge and what is contestable theory is an ongoing problem when viewing human knowledge as a whole, no more so than in the field of economics, where the ever increasing complexity of society casts doubt on theories widely adopted by governments. Knowledge in the social sciences is generally agreed to be less reliable than in the physical sciences, where controlled experiments are possible. Unfortunately, it is in the human sciences that solutions to social problems exist and the need for reliable knowledge can be critical to control by both government and management. The very idea of such controls calls up the philosophy of ethics, which is yet another field of knowledge which can only be established by making arbitrary assumptions or appealing to 'facts' derived from social sciences such as psychology and anthropology.

In today's complex societies, stability increasingly depends on narrow specialist, but also on generalists, usually managers or administrators, who specialise in generality, or comprehending and judging the work of specialists and coordinating it towards the achievement of certain goals. Such generalists have always existed as eminence grises in the corridors of power. The training of administrators in Britain and France, as well as other nations has long been focused on the task of making future mandarins. What kind of special knowledge should they be taught to equip them to deal with the great generalisations appropriate to the government of millions? Beginning with the works of Plato and Aristotle seem somehow inadequate but choosing alternatives is a difficult educational decision in itself. Who will train the Guardians and then protect us from them is an insoluble problem.

From a philosophical point of view, knowledge is the totality of true propositions. Unfortunately this set is potentially infinite and its contents mostly irrelevant to the imperatives of individual or collective purposes. From this point of view, it is the engine of human purposes, rational or not, that create the demand for existing knowledge or for the creation of new knowledge. Knowledge, then, is a mental artefact, discovered or created for human use. The body of pertinent knowledge is what is actually in current use or is being created by science or other disciplines. As pointed out above, a great proportion of this knowledge will be doubtful or false, if only because humans have a preference for the useful or interesting rather than the true.

Understanding what qualifies as knowledge is an example of what Wittgenstein called a language game, where meanings are not separated by sharp boundaries. A final, complete and perfect definition of knowledge for all purpose, therefore, is not to be expected. Far more useful is to address the prior question, why do I want to know what knowledge means and what use will such a definition be to my present purposes. This question is relative to the individual and would produce a commensurately different set of propositions about what knowledge means to them.

Tony Thomas
October 2009

## Wednesday, October 14, 2009

### Nietzsche's eternal recurrence

Nietzsche presents the idea of the eternal recurrence in three chapters of Thus Spake Zarathustra (TSZ). These are: On the Vision and the Riddle, The Convalescent and The Seven Seals, the second chapter containing the most detailed explanation; the third being a note on eternity.

Before examining the 'doctrine' of eternal recurrence it is worth asking whether anything in TSZ can be taken at face value or as genuine philosophy. The aim of the work seems to be to establish Nietzsche as a great and unique human being through the poetic fiction of his proxy, the prophet Zarathustra.  If the whole work is intentionally ironic, in explaining the difficulties that a great philosopher has in getting his ideas across, then it is entirely acceptable. However, if Nietzsche actually believed in the myth presented in TSZ, it would appear to be the work of a megalomaniac, albeit a significant one of poetic fantasy.

It seems probable that Nietzsche did attach importance to the ideas expressed in TSZ and was convinced of his own greatness to an unhealthy degree. This is consistent with his notion of creativity, where the positive affirmation of the individual and of life in general is a sign of greatness. This positive attitude towards achieving greatness permeates TSZ, and runs counter to the mainstream of philosophical discourse since Descartes, which is characterised by scepticism and doubt. Zarathustra affirms life, with all its suffering, and finally exults in the certainty that the superman will lead humanity to a better life. Contrasted with this is the pessimism and disgust with life that precedes his epiphany in The Convalescent. The role of the eternal recurrence is to render the coming of the overman certain in the way that Christians use their doctrinal framework to ensure the second coming of their saviour. Marx's belief that socialism was inevitable relies on a similar pattern of thinking.

On the Vision and the Riddle

Part 3 of TSZ was written in 1884, a few years after Nietzsche's decline in health had forced his retirement from academic life. However there is no suggestion that he was not mentally competent when it was written. However, the chapter entitled The Convalescent may well have been related to the several severe complaints he suffered from, which included migraines and stomach illnesses. It is interesting that Zarathustra suffers a physical and mental collapse in this chapter but it is in the earlier On the Vision and the Riddle that a psychologist is given an insight into Nietzsche's state of mind several years before his actual collapse.

The chapter opens with Zarathustra on a sea voyage, but quickly turns to an address to, "you bold venturers, adventurers", who are complimented: "because you do not want to probe along a thread with cowardly hands; and because where you can guess, there you hate to deduce". This remark suggests that the story Zarathustra is about to tell is not for the ears of pedantic philosophers but is aimed at more adventurous intellectual spirits.

Zarathustra is then depicted as climbing with great courage and difficulty up a steep mountain path (an exercise that the backpacker Nietzsche was familiar with) impeded by having to carry a "half dwarf, half mole, lame, paralysing, dripping lead in my ear, lead-drop thoughts into my brain." This dwarf aspect of Zarathustra then criticises his efforts rather cleverly, by pointing out "You hurled yourself so high - but every hurled stone - must fall!" This can be read as Nietzsche's internal dialogue convincing himself that he can achieve greatness if he can rid himself of the dwarf, characterised by gravity itself, that is holding him back. The dwarf, then, is his self-doubt. Nietzsche confirms that the dwarf is an aspect of his psyche by having Zarathustra say, "and being at two in such a way truly makes one lonelier than being at one."

Zarathustra continues to bewail his struggles and concludes by saying, "Dwarf - you or I!" He asks, "Was that life? Well then! One More Time!" This appears to be the subtle announcement of the doctrine of eternal recurrence. Although Zarathustra's life has been hard, oppressive and agonizing, he decides that he wants to have more of this life.

Part 2 of the chapter begins with Zarathustra confronting his dwarf with the words, "Stop, Dwarf! "I - or you! But I am the stronger of us two - you do not know my abysmal thought! That - you could not bear!" The dwarf hops down from his shoulder and Zarathustra proceeds to explain to him his abysmal thought. The text continues as follows:

"See this gateway, dwarf!" I continued. "It has two faces. Two paths
come together here; no one has yet walked them to the end.
This long lane back: it lasts an eternity. And that long lane outward -
that is another eternity.
They contradict each other, these paths; they blatantly offend each
other - and here at this gateway is where they come together. The name
of the gateway is inscribed at the top: 'Moment.'
But whoever were to walk one of them further - and ever further and
ever on: do you believe, dwarf, that these paths contradict each other
eternally?" -
"All that is straight lies," murmured the dwarf contemptuously. "All
truth is crooked, time itself is a circle."

The dwarf rightly notices that the metaphor is linear, ceteris paribus, but this is only Zarathustra arguing with himself. The statement that "no one has walked them to the end" seems pointless, since the paths are eternal and humans are mortal. Also, the idea of walking back in time seems impossible whether the path is eternal or not. It is interesting that Nietzsche confounds time and space in this model, as if he were anticipating Einstein's concept of space-time. Similarly, the arbitrary 'moment' where Zarathustra and the dwarf are conversing anticipates the relativistic origin of a local space-time system of coordinates. The two paths contradict because of the 'arrow of time' constraint, but it is unclear whether Nietzsche intends this to be the answer to his question. It seems that the dwarf does not like the linear concept of an infinite path, from indefinite past to indefinite future, but prefers a crooked or circular one. Elsewhere in TSZ, the dwarf is referred to as the Devil, so the confrontation described above can be seen as a variant of the dialogue between Christ and Satan in the desert.

"You spirit of gravity!" I said, angrily. "Do not make it too easy on
yourself! Or I shall leave you crouching here where you crouch, lamefoot -
and I bore you this high!

It is not clear from this remark whether Zarathustra prefers a linear eternity or a circular one, but later it becomes clear he does prefer circularity by adopting a wheel of time model.

See this moment!" I continued. "From this gateway Moment a long
eternal lane stretches backward: behind us lies an eternity.
Must not whatever can already have passed this way before? Must
not whatever can happen, already have happened, been done, passed by
before?

Here the doctrine is stated, that what can happen, must have already happened. The italicized 'can' is critical, and seems to suggest that anything is possible as opposed to the repetition of some restricted set of possibilities.
The implication is that the universal event space comprises everything that is not impossible. For the mathematician, this idea can be expressed as a four-dimensional possibility space, but for the physicist this has to be constrained by those mysterious regularities called laws of nature.

And if everything has already been here before, what do you think of
this moment, dwarf? Must this gateway too not already - have been here?
And are not all things firmly knotted together in such a way that this
moment draws after it all things to come? Therefore - itself as well?
For, whatever can run, even in this long lane outward - must run it once
more! -

It is not clear whether Nietzsche had really thought the consequences of his metaphor through. The statement, "everything has been here before" is too general. The small event that is the gateway and its environs when the conversation takes place is not really a 'here and now' but an ongoing (literary) event that ceases when it passes away into new events. Nietzsche's use of 'everything' seems to suggest that the particular space-time juncture is associated with every conceivable event, which is clearly erroneous. The best that can be said is that the construction is just a fanciful figure of speech that does not bear close analysis. The alternative is that Nietzsche is too clever to be understood by all but those who espouse his obscure idea.

And this slow spider that creeps in the moonlight, and this moonlight
itself, and I and you in the gateway whispering together, whispering of
eternal things - must not all of us have been here before?
- And return and run in that other lane, outward, before us, in this
long, eerie lane - must we not return eternally? -"

The concept is clarified here, by affirming that the precise 'state of affairs' described will be exactly reproduced, presumably at some other time. This raises the major difficulty of understanding how the space-time of a particular event could be holographically reproduced and distributed throughout Zarathustra's eternal cosmos.

Thus I spoke, softer and softer, for I was afraid of my own thought and
secret thoughts. Then, suddenly, I heard a dog howl nearby.
Had I ever heard a dog howl like this? My thoughts raced back. Yes!
When I was a child, in my most distant childhood:
- then I heard a dog howl like this. And I saw it too, bristling, its head
up, trembling in the stillest midnight when even dogs believe in ghosts:
- so that I felt pity. For the full moon had passed over the house, silent
as death, and it had just stopped, a round smolder - stopped on the flat
roof just as if on a stranger's property -
that is the why the dog was so horror-stricken, because dogs believe in
thieves and ghosts. And when I heard it howl like this again, I felt pity
once more.

This passage seems to be a feeble attempt to justify the concept anecdotally, but the relevance of the childhood experience and the dog is doubtful.

Where now was the dwarf? And the gateway? And the spider? And all
the whispering? Was I dreaming? Was I waking? I stood all of a sudden
among wild cliffs, alone, desolate, in the most desolate moonlight.
But there lay a human being! And there! The dog jumping, bristling,
whining - now it saw me coming - then it howled again, it screamed: had
I ever heard a dog scream like this for help?

Now Zarathustra reveals that it was all a dream, but links the dream to his present by reintroducing the dog motif.

And truly, I saw something the like of which I had never seen before.
A young shepherd I saw; writhing, choking, twitching, his face distorted,
with a thick black snake hanging from his mouth.
Had I ever seen so much nausea and pale dread in one face? Surely he
must have fallen asleep? Then the snake crawled into his throat - where
it bit down firmly.
My hand tore at the snake and tore - in vain! It could not tear the snake
from his throat. Then it cried out of me: "Bite down! Bite down!
Bite off the head! Bite down!" - Thus it cried out of me, my dread, my
hatred, my nausea, my pity, all my good and bad cried out of me with one
shout. -

Zarathustra now launches into a new story, but is not clear whether this is just another fancy. The passage is redolent with symbolic significance but it is not clear how Nietzsche intends the symbols to be interpreted, if at all. The Shepherd may stand for Christ, Apollo, Dionysus or some other mythological figure. If it were based on an actual dream, the obvious interpretation is a childhood experience of fellatio, whether performed on the writer himself, or some relative or merely imagined. The snake is traditionally a symbol of wisdom, so the meaning may be that the shepherd, as some kind of leader, is choking on his own wisdom. The biting off of the head and spitting out of the snake may therefore mean the rejection of old knowledge.

In Psychology and Alchemy (p.137) Jung says of a dream about a snake, ""This is brought about by the ceremonial use of a reptile, presumably a snake. The idea of transformation and renewal by means of a serpent is a well-substantiated archetype. It is reported of the mysteries of Sabazius...(A golden snake is let down into the bosom of the initiated and taken away again from the lower parts). Among the Ophites, Christ was the serpent...The shepherd's experience with the snake in Nietzsche's Zarathustra would accordingly be a fatal omen (and not the only one of its kind - cf. the prophecy at the death of the rope dancer)." It is notable that Sabazius was confounded with Zagreus, the Roman Dionysus. The Shepherd however is to be found in the Hermetica in the person of Peomandres, the Shepherd of Men. Whatever Nietzsche's sources may have been, the important result of the incident is the trauma and nausea that Zarathustra feels as a result of the 'incident' and the subsequent transformation of the shepherd into the superman.

You bold ones around me! You searchers, researchers and whoever
among you ever shipped out with cunning sails onto unexplored seas!
You riddle-happy ones!
Now guess me this riddle that I saw back then, now interpret me this
vision of the loneliest one!
For it was a vision and a foreseeing: what did I see then as a parable?
And who is it that must some day come?
Who is the shepherd into whose throat the snake crawled this way?Who
is the human being into whose throat everything that is heaviest, blackest
will crawl?
- Meanwhile the shepherd bit down as my shout advised him; he bit
with a good bite! Far away he spat the head of the snake - and he leaped
to his feet. -
No longer shepherd, no longer human - a transformed, illuminated,
laughing being!

Zarathustra has moved on from his perfunctory explanation of the eternal recurrence to the birth of the superman. The shepherd has been transformed by his experience into a "transformed and laughing being". This rebirth is reminiscent of the birth of Dionysus from Zeus's thigh or the birth of Athena from his head.

Never yet on earth had I heard a human being laugh as he laughed!
Oh my brothers, I heard a laughter that was no human laughter - and
now a thirst gnaws at me, a longing that will never be still.
My longing for this laughter gnaws at me; oh how can I bear to go on
living! And how could I bear to die now! -
Thus spoke Zarathustra.

Having witnessed the birth of this superhuman being, albeit in a vision, Zarathustra realizes his own irrelevance but cannot bear to accept his mortality - he must go on living, to glory in the superman.

The Convalescent

At the opening of this chapter, Zarathustra suffers an epiphany, behaving like a madman he "screamed with a terrifying voice and behaved as though someone else were lying on his bed, who did not want to get up". He then refers, presumably, to the eternal recurrence by saying, "Up, abysmal thought, out of my depths!... listen! Because I want to hear you!" This latter remark reinforces the separation between the two personalities within, the 'you' being his "most abysmal thought".
The first section finishes with "Hail to me! Here now! Give me your hand - ha! Let go! Haha! - Nausea, nausea, nausea - oh no!" The uniting of Zarathustra with his great thought leads first to nausea and then to the complete collapse described in Part 2 of the chapter. From a psychological point of view, Zarathustra seems to have suffered a personality split, with his 'abysmal thought' becoming the dominant identity.

After a convalescent period of seven days, Zarathustra is persuaded by his
animals to "step out of your cave: the world awaits you like a garden. The wind is playing with heady fragrances that make their way to you; and all brooks want to run after you." He returns to the idea of the eternal recurrence by saying, "aren't words and sounds rainbows and illusory bridges between things eternally separated? To each soul belongs another world; for each soul every soul is a hinterworld." He continues with this subjectivist position by saying, "For me - how would there be something outside me? There is no outside!"

The animals then take over the discourse and tell Zarathustra about the wheel of being, "Everything dies, everything comes back: the wheel of being rolls eternally. Everything blossoms again, the year of being runs eternally." All this is just conventional medieval philosophy about seasonal change, and its illusion of certainty. But then, "Everything breaks, everything is joined anew; the same house of being builds itself eternally. Everything parts, everything greets itself again; the ring of being remains loyal to itself eternally. In every Instant being begins; around every Here rolls the ball There. The middle is everywhere. Crooked is the path to eternity."

Zarathustra then confirms that he is the shepherd of the earlier vision by saying, "How well you know what had to come true in seven days - and how that monster crawled into my throat and choked me! But I bit off its head and spat it away from me." He confirms that he has been redeemed by saying, "Now I lie here, weary still from this biting and spitting out, sick still from my own redemption."

After a litany of complaint, including the cruelty and lust of human beings, and their littleness, Zarathustra explains that it was "My great surfeit of human beings - that choked me and crawled into my throat; and what the soothsayer said: 'All is the same, nothing is worth it, knowledge chokes.' "

The animals seem to recognise that this nausea and nihilism is just part of the convalescent process, and urge Zarathustra to, "Go outside to the roses and bees and swarms of doves! Especially to song birds, so that you can learn to sing from them!"  The animals urge him to "fashion yourself a lyre first, a new lyre!" Presumably this is a reference to Hermes, who invented the lyre, and is none other than the shepherd Poemandres, but the deeper meaning is that Zarathustra must heal himself, "so that you can bear your great destiny, which was never before a human's destiny." The meaning seems to be that Zarathustra must assume the role of a tutelary god because, "you are the teacher of the eternal recurrence." Hermes Trismegistus was, of course, the tutelary god. The remainder of the chapter focuses on explaining the eternal recurrence.

That you must teach this teaching as the first - how could this great
destiny not also be your greatest danger and sickness!
Behold, we know what you teach: that all things recur eternally and
we ourselves along with them, and that we have already been here times
eternal and all things along with us.
You teach that there is a great year of becoming, a monster of a great
year; like an hourglass it must turn itself over anew, again and again, so
that it runs down and runs out anew -
- so that all these years are the same as each other, in what is greatest
and also in what is smallest - so that we ourselves in every great year are
the same, in what is greatest and also in what is smallest.
And if you wanted to die now, oh Zarathustra: behold, we know too
how you would speak to yourself then: - but your animals beg you not to
die yet!

There is 'great year' in Hindu cosmology, which is much greater than the 25,765 year precession cycle, known to western astronomy as the Great or Platonic year. This latter precession cycle was probably known to Aristarchos of Samos (280BC). The Hindu cycle is based on Kalpas, or one Brahma day, which endures for 4.32 billion years. The Brahma year is therefore very large indeed. The cosmos is destroyed and recreated in the Hindu system but there is no exact repetition as described in the passage above. Indeed, the stories about the Buddha's previous lives indicate the uniqueness of each reincarnation and the progression towards perfection. What is not clear from Zarathustra's exposition is how long his cycle is or if it is finite or infinite. The hourglass metaphor is quantitative only and does not explain why or how each grain of sand would exactly repeat its flow history in the hourglass.

You would speak and without trembling, rather taking a deep breath,
blissfully; for a great weight and oppressiveness would be taken from you,
you most patient one!
'Now I die and disappear,' you would say, 'and in an instant I will be a
nothing. Souls are as mortal as bodies.

The death of the soul with the body follows Aristotle's view rather than Plato's where the eternal soul returns to the World Soul.

But the knot of causes in which I am entangled recurs - it will create
me again! I myself belong to the causes of the eternal recurrence.
I will return, with this sun, with this earth, with this eagle, with this
snake - not to a new life or a better life or a similar life:
- I will return to this same and selfsame life, in what is greatest as well
as in what is smallest, to once again teach the eternal recurrence of all
things -

The view here is the scientific one, that identical causes will reproduce the same events, ceteris paribus. For example, the birth of suns requires similar conditions, but it is hardly conceivable that Earth's sun would arise for a second time in the same cosmos. If the present cosmos were to collapse into a singularity it is conceivable that a new big bang would reproduce an identical copy of the preceding one, but an empirical proof of this would be impossible. From this point of view, Nietzsche's grand idea is irrefutable but possibly false.

- to once again speak the word about the great earth of noon and human
beings, to once again proclaim the overman to mankind.
I spoke my word, I break under my word: thus my eternal fate wills it
- as proclaimer I perish!
The hour has now come for the one who goes under to bless himself.
Thus - ends Zarathustra's going under!'" -
When the animals had spoken these words they fell silent and waited
for Zarathustra to say something to them: but Zarathustra did not hear
that they were silent. Instead he lay still, with eyes closed, like someone
sleeping - even though he was not sleeping. Indeed, at this moment he
was conversing with his soul. The snake and the eagle, however, finding
him silent in this manner, honored the great stillness around him and
cautiously slipped away.

According to the eternal return the 'overman' would have arisen innumerable times in the past, as would multiple Zarathustra's have proclaimed him to mankind. In view of the prophet's attitude to women, one wonders if a superwoman arose too, or if the godlike hero would mate with human women like the Watchers, those angels who could not resist the temptation of the daughters of men after the fall.

The Seven Seals

Zarathustra explains in Part 1, "oh how then could I not lust for eternity and for the nuptial ring of rings - the ring of recurrence. Never have I found the woman from whom I wanted children, unless it were this woman whom I love: for I love you, oh eternity!" Nietzsche's lack of success with women is here exacerbated by his desire for the impossible rival of a female goddess of eternity. This seems to be a variant of the Church as Bride of Christ or of Plotinus's World Soul.

Each of the seven sections of this hymn to eternity repeats this stanza, preceded by the imagined delights of eternity. In section five, ""infinity roars around me, way out there space and time glitter, well then, what of it old heart!" and in section 7, "but bird-wisdom speaks like this: 'see there is no up, no down! Throw yourself around, out, back you light one! Sing! Speak no more! - are not all words made for the heavy? Do not all words lie to the light? Sing! Speak no more!" This desire for absolute freedom from gravity and from thought suggests that Zarathustra is not temperamentally suited to the life his author has been leading, one of intense study, but rather the life of a wandering mystic who longs to be free of earthly constraints.

Conclusion

If one were to strip away all the poetic literary decoration, what would remain of Nietzsche's two important ideas: the eternal recurrence and the superman? In the world that he inhabited and the much-changed world that we know today, there have developed many radical theories and fictions about the cosmos and the future of human beings transformed by science. The possibility of further advances in human evolution, in terms of enhanced human powers, is not too fanciful. A more complex model of the cosmos as a multiverse is also being explored. What is clear, though, is that these scientific advances are based on work mostly unrelated to Nietzsche's fanciful and poetic notions, although psychoanalysts and others have been influenced by his insights. Consequently, Nietzsche's opinion of his importance in human history is grossly overstated, at least regarding the two main ideas of TSZ. However, this is not to say that they are unimportant elements in the development of non-scientific thought, since TSZ has had a profound influence on philosophy, culture and the arts which may well overshadow the influence of less fanciful philosophers.

The importance attached by Nietzsche to TSZ and the intensity and complexity of the work suggest that he did view the idea of the eternal recurrence and of the coming superman as more than just poetic expressions upon which to hang his radical view of the human condition and the future of humanity. For a great scholar of antiquity he must have been aware of the hubris inherent in TSZ, despite the thin mask provided by its protagonist Zarathustra.

Given the extravagant idea that anything that is possibly must have occurred, and must occur again, the emergence of a superior human who can save the human race does not appear unreasonable. Unfortunately the idea founders on the shore of the vast accumulation of scientific and technical knowledge that has occurred since Nietzsche's time. The law of entropy alone precludes the kind of cyclic stability envisioned in Nietzsche's recurring cosmos and the eventual death of all stars promises a Malthusian outcome in the absence of humanity. This reduces Nietzsche's grand idea of eternally recurring future heroes to an empty Valhalla, and certainly one where all great beings have vanished without trace.