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 ad infinitum. "

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

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