ultrafinitist approach to utopian theories (and even more so to utopian practices) much earlier than to problems of mathematical logic.
5.5 The vanishing point
Father Ted:Now concentrate this time, Dougal. These (he points to some plastic cows on the table) are very small; those (pointing at some cows out of the window) are far away. . . Small. Far away.
Father Ted[438] In the discussion of the psychology of mathematics, the problem of infinity is unavoidable; our visual perception of infinity is partic- ularly puzzling; I have mentioned it in Section 5.1; here, I wish to return to the subject.
I have stated earlier in this chapter that our first encounter with infinity does not happen when we see railway tracks merging at the horizon; the reason for this is that our brain resolutely refuses to recognize that the parallel rails converge at the vanishing point. Instead, your brain reminds you, like Father Ted to Father Dougal in the famous episode of the cult TV series, that, although the cow in the field projects onto a tiny spot on the retina in your eye, while a toy cow in your hands projects onto a much larger retina image, it is the cow in the field that is actually bigger. Similarly, your brain continues to remind you that the tracks are actually parallel all the way to the horizon. The brain actively fights geometry!
The ability to use and read linear perspective in simple schematic drawings is the result of cultural conditioning. (It is different with photographs where the perspective, with its vanishing points, is present due to the physical laws of geometric optics.) In ancient Egypt, for example, the graphic culture was quite different:
In seeking to represent three-dimensional objects on a plane surface, whether a drawing board or an area of the wall, the Egyptian avoided the perspectival solution of the prob- lem which alone of the nations of antiquity, the Greeks ul- timately reached by the fifth century B.C. Their vision of the world, seen from a certain standpoint at a certain mo- ment of time, would have seemed to the ancient Egyptian as presumptuous, and concerned only with illusion, a mere distortion of reality. The Egyptian was concerned not with presenting an evanescent personal impression, caught in an instant, but with what he regarded as eternal verities. [. . . ] His non-perspectival vision placed the Egyptian artist in harmony with the world that he knew to exist. His percep- tion of the forms of nature was derived from a fusion of sev- eral aspects recollected in the tranquility of his mind and
not captured as an instant revelation to the seeing eye. [418, p. 13–15]
In the Eastern Orthodox tradition of icon painting (which orig- inated in Byzantium), for example, an over-elaborate and techni- cally sophisticated system of reverse perspective is employed: the parallel lines intersect in the foreground of the painting, creating a very soothing, comforting feeling of a finite world which embraces the viewer, Figure 5.6.
This paradox is even more remarkable because the image on your retina, in accordance with the simple laws of geometric optics, is similar to photographic images and is something like the one in Figure 5.7, with tracks meeting at the vanishing point on the horizon. But, in the environment which shaped the evolution of our ancestors, the vanishing point was of no importance; for survival, it was much more important to recognize that a tiny speck of grey on your retina was the Big Bad Wolf.
Fig. 5.6.Trinityby Andrei Rublev, c. 1410, arguably Russia’s most fa- mous icon. The reverse perspective closes the composition around the viewer and creates a powerful emotion of unity. For Rublev’s contempo- raries, the message was not only spiritual and dogmatic, but very much political: in the historical context, it was a call for the political unity of Russian principalities. Source: Wikipedia Commons. Public domain.
5.5 The vanishing point 103
Fig. 5.7.The railway tracks converging at the vanishing point. c°2003 by Tomasz Sienicki, licensed under Creative Commons Attribution 2.5 License. Source:Wikipedia Commons.
One way to make your brain accept the existence of the vanish- ing point is to dramatically simplify its task of visual processing. The best feeling of geometric perspective is achieved when you look into a long dark tunnel with a tiny spot of daylight at the opposite end, or (as I did when I was a child) stare into a deep, deep well, with a tiny reflection of blue sky in the water. The brain is no longer forced to recalculate the actual sizes of objects, because there are no any in the field of view; the vanishing point, which was of no im- portance, becomes the only source of light and dominates the field of view.
Inna Korchagina aged 8 The same happens in the “tunnel of light” illusion (or hallucina-
tion), caused by the shutting down of light receptors in the retina, as a result of oxygen deprivation or the effects of drugs. I experi- enced it as a boy when I broke my arm in a skiing accident and was given rather barbaric ether narcosis; the last thing I remem- bered about the real world was the surgeon saying to the nurse: “Add a bit more”—and ether dripping from the gauze mask on my face. Then the flight started. In a rational reconstruction, the light sensors shut down one by one from the periphery of the retina to its center; this is perceived as if only the receptors at the center of the retina receive light, and is deciphered by the brain as the light at the end of the tunnel. Adding to that illusion, the vestibular apparatus, knocked out by ether, reports to your brain the sensa- tion of free fall, while the feeling of time is also suppressed—and you have the astonishing, overwhelming out-of-body experience of a flight through the endless tunnel towards the Light—and into Nothingness.
Fig. 5.8.Paradisio(plate 35) by Gustave Dor´e.—Illustration to Paradiso by Dante Alighieri, Canto 31, Verse 1–3. Source:Wikipedia Commons. Public domain.
Paradoxically, in order to recognize infinity in the form of the vanishing point of perspective, your brain has to be so seriously impaired that the infinity is no longer mathematical.
Not surprisingly, the “tunnel of light” could be a powerful reli- gious and spiritual experience and as such, is well documented— seeParadisio, Gustave Dor´e’s illustration to Dante’sDivine Com- edy, Figure 5.8. But religious and spiritual experiences are strictly individual and not reproducible—and this draws the line which separates infinity, and how it is understood and perceived both for-