Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
ASCS09: Proceedings of the 9th Conference of the Australasian Society for Cognitive Science Visual paradox and cognition Chris Mortensen ([email protected]) Department of Philosophy The University of Adelaide, SA 5005, Australia Abstract This paper offers a systematic approach to describing the socalled visual paradoxes. The particular example treated is what is known somewhat incorrectly as the Penrose triangle. The analysis is in three parts: an inconsistent logical theory which demonstrates the existence of a paradox, a cognitive explanation of how the logical theory is internalised, and a mathematical model which validates the logical theory. Keywords: visual; paradox; inconsistent; Penrose triangle; cognition. Visual Paradoxes There has been much activity on the subject of so-called “impossible images”. Almost all of it has been in the twentieth century, mostly by artists (notably Oscar Reutersvaard and M.C.Escher), some by psychologists, mathematicians and philosophers. Impossible images, better termed visual paradoxes, are impossible in the sense that the visual system projection to 3-D is of an apparently contradictory object. Such a thing cannot exist. This is the key question: what is the 3-D content in virtue of which we judge it impossible? Clearly the answer is that the visual system constructs an inconsistent theory. Only then can it be proved that it is a paradox, ie. a contradictory theory. This takes the tools of symbolic logic to prove, as we see below. The second task is to explain why it looks to us in this way; that is, what cognitive structures as exemplified in us in virtue of which we identify it as impossible. This goes along with the classification problem, namely these images come in natural kinds. See Thro (1983). When are they essentially the same principle, and when are they different? The internalized cognitive tendencies must have something to do with this. Finally, mathematics can make a contribution to the classification problem by providing models which validate the sentences of the internalized inconsistent theories, and not other sentences. This part takes some mathematical development, and is not strictly relevant to the cognitive story, so it is omitted here. The Triangle is a Paradox Consider the triangle, discovered by Reutersvaard in 1934, re-discovered by the Penroses in 1958. See Fig 1, (with letters added). Article DOI: 10.5096/ASCS200937 Figure 1: The triangle (with letters) Axes x and y are the usual axes in the plane of the paper, the z axis is normal to the plane of the paper. Now one key intuition about Fig 1 is that it constantly receded from the observer as the eye traverses the image anticlockwise. For a closed loop to do this is impossible. To show this rigorously, we adapt some psychological findings by Cowan and Pringle (1978). The corner A, considered in 3-D, tilts into the page when it is traversed anticlockwise. This means that points further along the face AB are further away in the z-direction than points not so far along. Thus we may deduce that: (1) A is closer in the z-direction than B Traversing the corner B anticlockwise, the face BC similarly tilts into the page. Thus: (2) B is closer in the z-direction than C By similar reasoning: (3) C is closer in the z-direction than A We now make the natural assumption, that the relation closer is transitive. (4) Trans (closer). It follows that: (5) A is closer in the z-direction than A But clearly, by observation (or by definition of closer): (6) A is not closer (in the z-direction) than A. The statements (5) and (6) contradict one another. This is the proposed proof of the paradoxicality of the triangle. 245 ASCS09: Proceedings of the 9th Conference of the Australasian Society for Cognitive Science The Triangle is an Occlusion Paradox Now we proceed with a first step on the classification problem, by proving that it is an occlusion paradox. This will take several sections. An occlusion paradox is defined as a paradox which can be rendered non-paradoxical by reversing one or more occlusions. Not all paradoxes are occlusion paradoxes. The proof that the triangle is an occlusion paradox requires two reversals of occlusions. The first reversal is on the bottom left hand corner. Fig 2 shows the reversal and those edges whose occlusions are reversed not the same as the previous proof of inconsistency, the previous one won’t work. This is confirmation that these are 2 different types of triangular form. Keeping x and y axes as usual, the z-axis is chosen as a normal to the AB face. This allows us to say: (1) A is the same distance in the z-direction as B Now the B corner still turns in, when traversed anticlockwise. So: (2) B is closer in the z-direction than C1 Now, by inspection, C1 is part of the same plane as the AB face, therefore with the same normal. Hence: (3) C1 is the same distance in the z-direction as A Now we invoke a weaker form of transitivity, a form of functionality, namely that if A is the same distance as B and B is closer then C1, then A is closer than C1. So: (4) A is closer in the z-direction than C1. But this implies the negation of (3). This is a contradiction. It follows that the second triangle is likewise paradoxical. Both Triangles are Occlusion Paradoxes Figure 2: First reversal of occlusions This reversal produces an intermediate triangular form. It is not the same as the first triangle, however. See below. Returning to our proof that the first triangle is an occlusion paradox, it suffices to show that the second is also. To do this, we reverse the occlusions in the top corner B. A Second Impossible Triangle Figure 4: Second Occlusion Changes The outcome is now a consistent triangle, as is evident by inspection. Figure 3: A Second Triangle. This triangle is impossible, but intuitively less so. This intuition has a natural explanation. The second triangle can now be rendered consistent with occlusion-changes in just one corner. The triangle we started takes two sets of occlusion-changes, involving two corners, so it is further away from consistency. We register, then, a classification into two different impossible kinds on the basis of these two examples. But it is still a paradox, ie. inconsistent. This then also deserves proof: that is the logician’s interest. The proof is Article DOI: 10.5096/ASCS200937 Figure 5: A Consistent Triangle 246 ASCS09: Proceedings of the 9th Conference of the Australasian Society for Cognitive Science Thus we can conclude that both triangles are occlusion paradoxes, but they are of differing kinds determined by the different number of occlusion reversals necessary to reduce to consistency. From Logic to Cognition Opposite parities in a figure cancel out, whereas contortions accumulate. The results were unmistakeable: mean scores of parity correlate significantly with ranking in degree of inconsistency, mean scores of contortions do not. It follows that parity is the significant variable here, and not contortion. This brings me to the topic of cognition. We would like independent evidence that the premisses of the argument for the inconsistency of the first triangle, are those which we internalize, and in virtue of which it looks impossible to us. Cowan and Pringle (1978) provide just the sort of experimental data relevant to the claim. They work with 4sided figures, but 3 sided figures are an obvious special case. They identify 27 different rectangles. Figure 7: Cowan and Pringle’s Results Figure 6: Cowan and Pringle, 27 rectangles Three groups of judges were used, each trained on a different 9 rectangles, and asked to rate the remainder by degree of possibility from 1 to 10. Clearly degree of possibility is related inversely to degree of impossibility. It is conceivable that there could have turned out to be little or no correlation between scores after training effects were corrected for. In fact there was a strong correlation between individuals. This indicates the presence of an objective phenomenon fairly constant across the population. We can do even better with these results. Cowan and Pringle postulate two different features of these images which could serve to explain the judgements of degree of impossibility. One explanation was that impossibility is related to the contortion of the figure. Contortion is the number of twists in the sides necessary for the corners to appear as they do, while the whole figure remains coplanar. The second explanation was that impossibility is related to the parity of the figure. They define the parity of a corner to be +1, -1, or zero, depending on whether, traversed anticlockwise, the corner turns into, out of, or coplanar with the plane of the page. The relation between contortion and parity is that the latter registers the direction of the twist. Article DOI: 10.5096/ASCS200937 Another explanation raised was whether judgements of impossibility were affected by stereopsis. Stereoscopic images of impossible objects are available, and the experience is well worth having. There are standard methods for inducing stereopsis. But when the experiment was run again in those conditions, not only was there no significant correlation with stereopsis, but the same parityinduced order was found with stereopsis. Hence, no extra effect. Conclusion But now we recall from the first section, that this was exactly the description employed in the proof of the first paradoxical triangle. The logical analysis depends on it. Parity registers into and out of, and it is intuitive calculations of parity that lead to judgements of impossibility, in the indicated fashion. In short, it is the cognition of parity that explains judgements of relative possibility and impossibility. Bibliography Cowan, T., and Pringle, R., (1978) An Investigation of the Cues Responsible for Cube Impossibility. Journal of Experimental Psychology: Human Perception and Performance, 4, 112-120. Penrose, L&S., Impossible Objects, a Special Kind of Illusion. British Journal of Psychology, 38, 49. Thro, E.B., (1983), Distinguishing Two Classes of Impossible Objects. Perception, 12, 733-751. 247 ASCS09: Proceedings of the 9th Conference of the Australasian Society for Cognitive Science Citation details for this article: Mortensen, C. (2010). Visual paradox and cognition. In W. Christensen, E. Schier, and J. Sutton (Eds.), ASCS09: Proceedings of the 9th Conference of the Australasian Society for Cognitive Science (pp. 245-248). Sydney: Macquarie Centre for Cognitive Science. DOI: 10.5096/ASCS200937 URL: http://www.maccs.mq.edu.au/news/conferences/2009/ASCS 2009/html/mortensen.html Article DOI: 10.5096/ASCS200937 248