# 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)

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.

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.

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