Sharing Paradoxes: Impossible Spaces, Impossible Times and Impossible Facts. The Function of Self-Contradictory Structures in Arts, Sciences and Philosophy. (Part 4)

A first step in dealing with paradoxes is then to accept their double existence as being true and false. Outside academic studies, it is a natural process implied in the appreciation of any narrative art. As it has been described about cinema, there is a point where we accept the false to be true, as if trying to find a proof reducto ad absurdum. This process is called the suspension of disbelief (Walton, 7). Proposed first by the poet Samuel Taylor Coleridge, it has extended in the study of literature, cinema and videogames to name a few. This refers to any action where the false is taken as possibly true in order to appreciate a narration and facilitate the immersion process. Youssef Ishaghpour describes the duality implied as the reality of the image and the image of reality (Ishaghpour, 8-11). The suspension of disbelief is therefore a way to conciliate this duality in order to appreciate the fiction.

The difference when working in a fictional environment rather than in a scientific one is that every time a contradiction or unearthly events appears, it is usually taken as an extension of the perceived diegetic world: when the staircase optical illusion appears in Inception, it is accepted as part of the fictional world. Instead of defying physical rules, it is simply accepted as a new information contained within the fiction. Again, as in Kierkegaard or Bohr’s vision, it is an extension of the paradigm. The same happens for multiple worlds’ diegetic construction such as previously mentioned in movies like ExistenZ or Avalon. The multiple ontological world, very similar in their nature to the Russell’s paradox are accepted as such. The suspension of disbelief catalyses the conceptual acceptation of such constructions and even changes them into interesting and pleasing artistic visions.

Cat'.s Paradox

Figure 7: The cat’s Paradox. Source: Wikipedia

The use of paradoxical constructions taken from science does not end here. Another case comes from quantic physic. Erwin Schrödinger described the nature of some quantic events by the metaphor of a cat in a box. Let say there is a cat and some poison inside a box. There is as well a 50% chances for the poison to be relieved and therefore for the cat to die. The way quantic physics works is that as long as the information about the cat has not been extract from the situation, the cat is in both states: alive and dead. Both states excludes each other and therefore it leads to a paradox that of ‘’ the living and the dead cat mixed or smeared out in equal parts.’’ (Schrödinger) What modern physics proposes as a solution the acceptation of both state for a certain period of time. This process is known as quantic bifurcation. Even if this is very difficult to accept as being true for neophytes, when transferred to fiction it leads to acceptable and interesting narrative constructions. An example of a movie using this type of multi-linear time frame is Source Code by Duncan Jones. In this movie, a soldier is sent multiple times in the past to prevent a terrorist attack. After failing multiple times, he achieves his goal and life continue normally in this new independent timeline. The use of quantic bifurcation appears in multiple science-fiction movies and communities of fans are sketching schemas to understand the structure behind the film. Movies like Primer by Shane Carruth and Looper by Rian Johnson have generate numbers of complicated charts using quantic bifurcation in order to explain these narrations. (Figure 8)


Figure 8: Looper movie chart by Rick Slusher. Source:

Another paradoxical time construction that has caused many problems is the time loop. Circular construction of time was accepted by many cultures around the world: Egyptians had circular time named Neheh (Assman, 137), the tzolkin, the Mayan calendar based on cycles of 260 days (Falcón, 19-21) and Hinduism has constructions of multiple intricate circles (Eliade, 134-136). This vision does not conciliate with the European linear construction of time, but it still easily apply to fiction.

Time being both in the future and the past appears in various fictional cases. First of all, whenever there is a time loop a cyclic time has to be accepted. Movies like Terminator (Cameron, 1984), Before the Rain (Manchevsky, 1994) or Chin Chin el Teporocho (Retes, 1976) all present this cyclic time construction. Indeed, time loops can be multiple and quantic bifurcation might again apply.

In the quantic bifurcation case as in the circular time construction, the paradoxical construction induces multivalued time states, discrete moment can be different but at a same time distance from a specific moment, the bifurcation point or those previous, or they can stand both in the future and in the past of a referential moment. It can also be seen as a specific case of multiple ontological states, as previously described, but with the specificity that the ontological state is defined by a time value.

It can be presented in a more mystic way as in the movie Voyage in Time by Tarkovsky and Tonino Guerra. In this case, the movie shows the directors talking about the film they will make about a trip they once had. The anecdotes supposedly in the past appears as well in front of the camera and therefore the time of the movie is triple, it stands for the past, the present and the future as in Three Sundays in a week, but without the logical explanation.

Perhaps the most well-known results about paradoxes is Gödel’s incompleteness theorem. After Cantor and Russell discoveries, logicians have tried to build a perfect and complete system for logic. The project happened to be more problematic than expected and new set of axioms surfaced. The outstanding result obtained by the mathematician Kurt Gödel changed radically the conception of logic and left the community in crisis. The incompleteness theorem states that no matter how many axioms we add to a logic system, there will always appear some statements that will be undecidable, meaning it will be impossible to prove them right or wrong (Nagel, 19-20). This is a perfect example of Paz’s perspective of grasping dual objects as such instead of considering them as problematic undefined concepts to reach a better understanding of it. In this case, the conclusion obtained by paradox is that paradoxes are inherent part of complex logical systems.

Paz’s consideration encompasses a big range of logical instance and, as seen previously, they apply to a wide variety of paradoxical objects: from optical illusions to narrative charts passing through quantum physics. It still does not hold for a type of undecidable statements. Some facts are not necessarily true or false; they stand somewhere in between as a result of incomplete definitions. They work as ambigrams but instead of offering mainly a finite amount of elements, they offer a continuous range of possible information. Such problems are common in everyday life since more situations are not clearly defined. For instance, we can pretend the sky is blue but it can’t be proven without adding precisions to the statement offered; at night the statement does not hold for instance. A relatively new branch of mathematics dedicates itself to such logical system. The idea behind this fuzzy logic, as it is coined, is to attribute truth values that varies continuously between the usual zero and one (Kandell). Therefore allowing any probability of truth ranging from zero to 100%. Such logical system coincides with perspective of quantum physics allowing diverse states with various probabilities. It is the case for instance for electrons in the atomic model were they navigate through a probabilistic area instead of following a precise trajectory.

Finally, paradoxes can appear within humoristic or philosophical functions. The twist are often used in usually called intellectual humour such as Woody Allen’s work. In Allen’s quote from Annie Hall ‘’ The food here is terrible and the portions are too small’’, the double statement stands in the contradiction that, in fact, if the food is terrible there is no reason to ask for more, but complaining about small portions implies asking for more food. This kind of construction can be found as well in Annie Hall: ‘’ Life is full of mystery, loneliness, and suffering –and it’s all over much too soon’’. The role of the paradox is then, in this case, to release a tension constructed around the paradoxical statement. In this situation, the contradiction, or double truth value, stands as a sign that the joke has reached its climax. The contradictory aspect of the logic involves is to be read as a sign to character does not make sense anymore, therefore the humoristic relief. The humour can follow as a comment on a paradox: way before Russell, Lewis Carroll underlines that no one can contain himself because of excitement because nothing can contain itself (Benayoun, 84). These considerations follow the seriousness of Ambrose Bierce’s definition of logic as the art of thinking within human capacities (Benayoun, 113), the presence of this limit is in itself both humoristic and a relief.

Paradox Humour

Figure 9: Paradox humour

In a broader perspective, the same applies to koans, small stories or statement present in the zen tradition. The sentences serve to increase doubt and questioning. The simple logic behind the koan ‘’What is the sound of one hand clapping’’ is similar; it states the possibility and impossibility of the referred sound. Possible since it is stated there is a clap sound and the impossibility by the uniqueness of the hand producing it. In this case, the paradoxical information serves again to release tension. The same holds for the koan: If you have a stick, I’ll give one to you, if you don’t I’ll steal it from you. The tension is released with acceptation to work outside a strict logical frame, to accept our humanity as proposed by Bierce.

This work outside logic may serve as well, paradoxically, for theological arguments. To understand we have to go back to the unliftable rock paradox. God, being almighty, should be able to create an unliftable rock, but then if he can lift the rock he is not almighty. An easy solution to this problem is to state that God’s work beyond human based logic.

As we have shown, the contradictory dialectic raising from paradoxes only cause problem within its own paradigm of binary logical values as being true or false. When grasped as specific concepts gathering both values, or, even infinitely many values ranging from absolutely true to absolutely false, many applications can be found. Accepting such condition standing in between these poles is what allows us to enjoy narratives in different ways; first to enhance the emotive effect of a diegetic world by accepting the ontological quality of fiction as being an image of reality that is itself included in and presented as a simulacra of reality, secondly as to define science fiction or fantastic narratives as legitimate by extending the accepted diegetic world. Logical statements sharing both truth and false value are integral parts of human scientific and cultural knowledge[1]. It is hoped that more research to consider paradoxes in their social appearances will be provided as to understand better their functions as a fundamental part of human thinking rather than solely as odd mythological thoughts gravitating in the abstract spheres of philosophy and logic.

Felix Lambert

First version September 2015

[1] As it is the case in Dialetheism. For a good review of this as a philosophe, the reader is invited to read the entry in the Stanford Encyclopedia of Philosophy by Francesco Berto.



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Sharing Paradoxes: Impossible Spaces, Impossible Times and Impossible Facts. The Function of Self-Contradictory Structures in Arts, Sciences and Philosophy. (Part 3)

We can classify visual paradoxes into two categories: the simply self-contradictory ones and the ambigrams. In looking at most of Maurelius C. Escher’s optical illusions, the spatial design seems to be both true and false. As pointed out by Biederman, since the shapes preserve adequate angle constructions, the objects seems to be credible. That is a direct result of the importance of line intersection in object recognition. Nevertheless, the overall conceptualisation of the object seems not acceptable as being true (Biedermann, 135-140). In this case, these visual paradoxes are simply self-contradictory because the macroscopic veracity statement contradicts the microscopic ones (as in the two sentence: The next sentence is true. The last sentence is false (Hofstadter, 19)) Many artists have found multiple interesting ways to produce these types of paradoxes. Escher indeed is well known for such constructions, but we can name as well graffiti artist Damien Gilley, Dutch artist Ramon Bruin and Istvan Orosz (Figure 3) Having met quite a wide popularity, it is normal that these simply self-contradictory visual objects have appeared in various situations. For instance, Penrose’s triangles has appear on post stamps, tattoos and many everyday objects.

Inception stair scene

Figure 4: Inception.s stairs scene by Christopher Nolan

These self-contradictory visual objects can contribute solving other artistic problems. In Christopher Nolan’s movie Inception the infinite stairs illusion is used and as real instance within the diegetic world to trap the enemy. In this case, it is a use of a visual paradox to solve of narrative problem. (Figure 4) On his hand, Cameron Browne have found interesting ways to merge the optical illusion construction with another old problem; the paving of the plane (Browne, 2007). Browne has constructed infinite patterns of self-contradictory visual objects that can be used to fulfill the entire plane. He worked as well with contradictory fractal structures (Figure 5)

Camron Browne.jpg

Figure 5: Impossible Fractal by Cameron Browne. Source:

Ambigrams are figures that show two incompatible information at the same time, inasmuch a paradox, they work on a scale more nuanced than the dichotomic paradoxes[1]. The figure of a young-old lady is a popular example of ambigram due to H.H. Hill (Delahaye, 91). In this image we can actually perceive two different portraits. One pictures a young lady and one offers the profile view of an old lady. It is a paradox since it contradicts itself, not in the previous case straightforward manner, but by ricochet. If it is a young lady, then it is impossible to be the old lady at the same time and vice versa. In the other hand, we can interpret the image as containing two informations, two different images. Ambigrams, working on a larger scale; they can contain more information. For instance, in figure 5 one can find six apparitions of the word palindrome, all put upside-down and to be read in both directions.

The advantage of using a word like ambigram is that it underlines an interesting property of paradoxes, that fact of being containing multiple statement that would usually not appear in general in a coherent manner. To go further in this sense, we have to go back to an analysis made by Mexican sociologist and writer Octavio Paz. In an analysis of complementary and dual concepts existing in various societies, Paz stresses the importance of considering such pairs as a whole by focusing on the relation between them. For instance, body and no-body are not to be considered as specific meanings except to express contraries (Paz, 55). This perception of duals as a whole can lead to interesting results when applied to paradoxes.


Figure 6: Ambigram

[1] We mean by this that the information is not straight opposite, like true and false, up and down, etc.

Sharing Paradoxes: Impossible Spaces, Impossible Times and Impossible Facts. The Function of Self-Contradictory Structures in Arts, Sciences and Philosophy. (Part 2)

Russell’s paradox is an example that shows how thinking about paradoxes can be fruitful. Bertrand Russell studied set theory at a time when it was still in a naïve simple form at a time the vary basics of logic were studied in philosophy and mathematics. He realised the idea of set was really powerful but was still problematic since it involved the following paradox equivalent to an extension of Grelling’s paradox: separate all sets into two categories, the sets including themselves and the sets not including themselves (Reinhart, 23). As in in Grelling’s case, the paradox appears when we try to classify the set of all sets not including themselves. The paradox works in the exact same fashion as Grelling’s one. Finally, Russell added some axioms to the naïve set theory and included levels of inclusions for a theory called type theory (Vidal-Rosset, 17). A set defined at a certain level of inclusion cannot be taken as a set of a lower level. Therefore, the question of including the set of all sets not including themselves within itself or its dual is not a legitimate question since it becomes a prohibited operation.

Although this paradox has been solved in a logical point of view, it still holds in many fictional construction[1]. In constructions that Brian McHale, based on Gerard Genette’s theorical work on narratology, defines as being on different ontological levels. McHale provides with many examples of novels and short stories based on this idea. Many science-fiction movies are also based on this principle of multi-stage inclusions: it is the case for EXistenZ by David Cronenberg or Avalon by Mamoru Oshii. In both cases, as underlined by McHale for one of Robbe-Grillet’s short stories Project for a Revolution in New-York, the reader cannot identify the ontological level on which the action is happening (MacHale, 117). This narratological strategy helps staging a Daedalus that extends to many ontological levels and reinforces the impression of being lost. This state of delocalisation of the protagonist has a double impact: the loss of referent implies by the fact that they can’t identify on which ontological level they are, and by secondly by erasing the value of the initial referent as the real initial and authentic state. This construction of infinite ladder of diegesis is constructed from confounding a set with a higher (or lower) level set.

The section of mathematics dealing with set, their properties and their axiomatic use as a starting point for mathematics is called set theory. Set theory also had to deal with other conceptual problems at the time and this had to do with bijective functions: correspondence between the elements of two sets where an object is related to one and only one object on the second set (Grimaldi, 279). As an example, we can set such a function between the set of positive integers and the set of negative integers by linking every number to its negative equivalent. No matter how many numbers there are, an infinity as a matter of fact, we will always be able to construct the function. The problem arises when we put in relation sets that seem to clearly have a different number of elements. This time, we can create a bijective function between every positive integers and every positive even integers. In this way, one is linked to two, two is linked to four, and three is linked to six and so forth. Since we have an infinite number of integers we will always have enough to construct the relation. This is counterintuitive since all the elements of the set of even numbers are present in the set of integers but the converse is false. We can set other kinds of strange relations between finite length segments and infinite length segments. The two-dimensional stereographic projection provides such an example. We proceed as follow: we set a circle on an infinite line and from the top of the circle we trace rays that cross the circle at a point and then continues until it hits the line. By proceeding as such and scanning the rays on 180 degrees, every point on the circle will be linked to a point on the line and this time the reverse holds. Even more surprising, it has been shown that the line segment can be put in relation with the square which at the time seemed very curious since both geometrical objects did not even share the same number of dimensions: the line is one dimensional whereas the square is two dimensional (Sagan, 115).

These paradoxical constructions led Georg Cantor to the creation of the transfinite numbers theory. Cantor defined different types of infinity: the countable and uncountable (Cantor, 1976) Countable simply means they can be put in a bijective relation with the natural numbers. As mentioned previously, the set of even numbers is countable for that very reason. The two sets, natural numbers and even numbers are then of the same cardinality because they are both infinite countable. The same logic holds for the circle and the infinite line: both of them are infinite uncountable and therefore are of the same size. The expansion of the theoretical frame for infinite numbers explained as well the natural relation between the line and the square. In 1635, Bonaventura Cavalieri already proposed the idea that plane figure were made of infinitely many line segments (Alexander, 70). With the work of Peano (1890), the idea of filling the square with a single curve spread widely and many mathematicians proposed such curves. These curves where not bijections, they were in fact surjections; points of the square were actually covered many times by the same curve.

These curious objects also brought light on other concepts that were taken for granted like the idea of dimension. As a result, many definitions for dimensions have been proposed and objects called fractals have been found having non integer dimensions. For instance, the Koch curve, a well-known fractal, has dimension 1.2619 (Mandelbrot, 36). The Koch curve is famous as well for being paradoxical to the notion of continuity. When Cauchy developed the concept, he believed that continuity implied derivability, i.e. the existence of a tangent line (Wallace, p. 187). Bolzano and Weirstrass constructed such curves, but von Koch, unsatisfied by the too analytical model of these curves decided to construct geometrically his now famous curves with the property of being continuous but nowhere differentiable. (von Koch 1904-1905) Various other examples from that time shared similar paradoxical value over the canonical comprehension of continuity and dimensions. They participated in the birth of the notion of fractal geometry by French mathematician Benoît Mandelbrot in 1975. Again, redefining paradigm from arising paradoxes led to improvement of various theory in mathematics.

Another common way to use paradoxes appears in mathematical proofs. In a proof technique called reducto ad absurdum a statement that seems to be false is taken to be truth. The proof holds if a contradiction with the hypothesis is to be found. A common example of such proofs is Aristotle proof that √2 is not a rational numbers. It concedes the number a rational form and by dividing by all possible cases of appearance of even or odd occurrences for the numerator and denominator, contradictions appears in all cases. The hypothesis is therefore impossible (Boll, 31-32). Euclid showed in a similar way that there is an infinite amount of prime numbers. He started by setting the highest prime number on the theoretically finite list and then show he could construct in bigger number not divisible by any of the finite prime number list (Grimaldi, 222). The list here could be very long but the result would be the same: paradoxes can be use actively in search of knowledge. It maps this knowledge by defining areas of impossibility, therefore implying areas of certainty.

This strategy holds for other scientific area. In neuroscience, paradoxes often helps understanding the way the brain gather and compute information. A typical case is the Ames room, named after the American ophthalmologist Adelbert Ames Jr. In this particular construction, a room is deformed in such a way that, viewed from a specific perspective, it seems to be a normal room. In that sense, it is a three-dimensional anamorphosis (Baltrušaitis, 7). When looking at people in the room from that specific point of view, it seems that they are changing sizes like balloons. The Ames rooms creates a falsifical paradox: we clearly know that these people are not changing size, but our brain is unable to actively interpret the visual information as such. This experiment shows how our brain treats information and take some shortcuts for granted instead of computing them all the time. In this case, the granted fact is that rooms are rectangular prisms regulated by right angles. (Figure 2)

Elm's room 1

Ames Room

The importance of the intersection of lines in our perception has been proved again to be of prime importance by Biederman in 1987. In his study, he shows how the brain is faster in recognizing drawings from which no intersection segments have been erased compared to drawings where totally random segments have been withdrawn. By this mean, Biederman also explains how work optical illusions, objects that we can define as visual paradoxes.

Istvan Orosz

Figure 2: Istvan Orosz Source: pour la science 2005, nu 330.

[1] We do not imply here that these narratives are a direct result of Russell’s work, we simply state that overall shape of its structure is similar to these narratives. More detailed historical researches could lighten this question of direct implication.

Sharing Paradoxes: Impossible Spaces, Impossible Times and Impossible Facts. The Function of Self-Contradictory Structures in Arts, Sciences and Philosophy. (Part 1)

By Félix Lambert


Relativity by M.C. Escher

Logical twists and games have always seemed to intrigued thinkers from all times and civilizations. As being mostly curiosities, they appeared in an unorganised fashion in many disciplines, arts and games. One such logical twists is the paradox. From Antiquity’s philosophy to modern mathematics, paradoxes have brought various questions and, in some cases, answers about human knowledge. The main goal of this paper is to demonstrate how paradoxes have proven useful in various cases in sciences and narrative arts, therefore justifying them as proper object of study instead of being considered simply as odd singularities. Different definitions of paradoxes will first be discussed and a series of paradoxes will be presented. We will then use Octavio Paz’s discussion over dual concepts to approach paradoxes. Finally, we will come back to some examples where a dualistic study of some paradoxical structures has been useful. This will show how paradoxes now constitute a significant part of contemporary knowledge, art and to a certain extent, mythology.

The first common point found in various definitions of paradoxes is the self-contradictory aspect of its claim. One of the most common paradoxes that clearly exemplifies this fact is the liar’s paradox. It seems to first have been proposed by Epimenide when saying that all Cretans lie. The problem appears when we realise Epimenide is himself native from Crete and therefore two options are possible: first, if he lies then his statement is right and therefore we cannot trust his saying, secondly if he tells the truth we because he is Cretan then he tells a lie. A more condensed version of this paradox was expressed by Eubulibe of Millet (Vidal-Rosset, 31): ‘’I’m lying’’ of which the literary equivalent is ‘’This sentence is false’’. The common point of all these statements is that they all appear as contradicting themselves, leaving us incapable of deciding the rightness of their claim.

Many authors have proposed classifications for paradoxes. One interesting proposition has been Willard Van Orman Quine’s tripartite classification. Quines distinguishes falsifical paradoxes, veridical paradoxes and antinomy (Vidal-Rosset, 105). The first category includes paradoxes that are finally proved to be false. Such an example is given during the Renaissance by Guido Ubaldus finding that 0=1, which was interpreted at the time by implying that matters can be creating out of nothing. The claim seems to contradict itself by giving two different values to an integer, one being equal to zero and of course to itself. The proof uses infinitely many addition of ones and zeros by reorganising them in such a way to obtain the result (Stewart, 578). The paradox is false because it does not use the allowed operations between infinite series (Labelle, 262-264). Another common example is 1=2 obtained by a division by zero, which is of course prohibited. The second type of paradoxes, the veridical ones, contains paradoxes that seem false but end up being true. The most common example is the Monty Hall paradox. A player is offered three choices of doors behind which one of them a price is hidden. The player picks a door. After the choice, one of the two remaining doors is opened and shows no price. The player is then asked to choose again. Although it is commonly believed that the chances on the last pick are even, it is in fact false: it seems that there is a 50% chance of winning when there is actually a higher chance to win if the player switches their choice. We can compute all possible options and the results shows the player stands better chances, in fact 2 out of 3, if he doesn’t change his mind. This paradox is a veridical paradox because we can prove it to be true. (Figure 1)

Monty Hall Paradox

Figure 1: The Monty Hall paradox

Another veridical paradox has been used by Edgar Allan Poe in one of his short story “Three Sundays in a Week’’. In this story, two young lovers want to get married but the uncle in charge decides it will only happen when a week will have three Sundays. A year later on a Sunday afternoon, the couple meet the uncle with two captains that traveled the globe in opposite directions at such speed that they respectively lost and gained a day, therefore thinking that Sunday was the day before or the day after and fulfilling the uncle’s condition (Poe, 225-232). Yet again, the statement seems contradictory but an analysis on the matter shows in fact in it is a veridical paradox; Poe constructs the story around the fact that the referential for the day was not specified.

Finally, there are paradoxes that can be both, true or false. Grelling’s paradox falls into this category: let us divide all adjectives in two sets, the autological and heterological ones. The first ones are those that describes themselves, for instance short is a short word. Heterological are those that does not describes themselves: long is a short word. The paradox arises when we try to classify the adjective heterological: if its autological then it describes itself, it is then heterological which contradicts the statement. On the other hand, if it is heterological, then it cannot be in its own category, it must then be autological, but we already showed that it can’t be (Vidal-Rosset, 26). We will see later that another paradox by Bertrand Russell is similar to this case.

Trying to classify paradoxes into one of these categories is the first way to turn paradoxes into useful abstract objects enlarging the scope of knowledge. The categorisation implies gathering enough knowledge and understanding of the paradigm in which the paradox is stated to be able to classify it into the proper box, but this procedure might redefined the paradigm or lead to new theories. Although, it is quite natural to take some assumption as true -as axiomatic- in part of building knowledge, but the interrogation about these keystones of knowledge really comes unavoidable when paradoxes are found. This thought is well express by many great thinkers in sciences and philosophy. As Niels Bohr said when working on theoretical model for the atom ‘’How wonderful that we have met with a paradox. Now we have some hope of making progress.’’ (Moore, 196). The idea of meditating on the meaning of paradoxes have even brought to an almost mystical perspective in Kierkegaard’s’ writings: ‘’ The paradox’s really the pathos of intellectual life and just as only great souls are exposed to passions it is only the great thinker who is exposed to what I call paradoxes which are nothing else than grandiose thoughts in embryo’’ (Slaatté, 64). In these cases, paradoxes technically mean self-contradiction, but it is seen as well as a possible extension of knowledge, and wider possible range for the concerned paradigm. An interesting and humoristic quote sometimes attributed to Mark Twain expresses this state of mind about knowledge: ‘’All generalisations are false, even this one’’.