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In Aemilia’s Papafilippou “Liquid Sky”, a red grid (of Cartesian coordinates?) coexists with elusive wavy shapes that bring to mind, reflections of sun light on spilled engine oil. The grid attempts to measure and define a slippery and contradictory world, while it is half sunk in it.  

It could be an invitation to ask about the ways, art and mathematics each try to come to terms with our experience of reality.

Mathematics produces structures that are useful in carving up and re-composing the world. Art mirrors the world in a unified manner, it is immediate and impacts firstly and directly on the senses. Mathematics seems distant and hard to understand. Could it be though, that things are not exactly like that? Could it be that the senses are opaque and mathematics just “gives it all away”?

 

 

Immediate Art-Transparent Mathematics: On the Edges of the Spectrum of Concepts

Yiannis Vlassopoulos

 

Whoever has seen a red apple can easily bring its image in their mind. What is though the relation between this image and the apple? Physics for example, tells us that the apple we consider solid is actually mostly void since it is made up of atoms and every atom has a nucleus and electrons much smaller than the space it occupies. 

The way we perceive an optical image is as indirect as trying to capture the shape of a statue in total darkness, by throwing pebbles on it and detecting the way they bounce off. In the case of the apple, photons bounce off it, they penetrate the iris and end up on the retina where they stimulate light-sensitive cells that then send electrical signals to the brain. Millions such signals are used by the brain in order to produce the optical model that we call, red apple.

The brain incessantly uses information that it receives via the senses, in order to produce models of external reality. This information is the result of experiences, namely interactions with the environment. 

Air oscillations caused by movements are transformed into auditory models, chemical reactions between specialized cells and various molecules are transformed into olfactory or gustatory models and pressure between skin molecules and other surfaces leads to haptic models. Moreover, we may say that models reflecting our internal states, appear as emotions. All these are created automatically, without effort, they are part of the hardware of the average person. 

The models produced in this fashion do not of course remain isolated, since the brain has the ability to relate them and form clusters/networks. They are thus organized in structures that reflect the relations of things, as they follow from experience. Recognizing these clusters/networks of simpler models, as new entities and naming them, appears as another “sense”, namely abstraction, which acts on this internal scene. The new entities arising through clustering and abstraction, are then used as nodes in more complicated networks. For example, the model “red” is abstracted from what is common in apples, roses and the sunset, while the model “human” is a network whose nodes are other complex models like “face”, “voice”, “blood”, “thought” etc.

Moving on to the society level, civilizations are structures that are made up of models and that function partly as “repositories” but mostly as environments in which we live. If we need a certain model, we may often pick one “ready to wear”, from this cultural environment. Sometimes it’s close by, other times it’s far away and occasionally, it just doesn’t really exist yet.

According to this more or less Aristotelian outlook (of dividing up knowledge in more and more refined categories and inversely, clustering categories to make more complicated ones) the total conscious model of reality, produced by the brain, appears as a network with nodes and connections among the nodes. To each node corresponds a certain model which, in its turn is another network with nodes corresponding to other models and so on… Can we tell where and indeed whether, this process ends?

A prima facie negative answer to this question follows from the fact that, no matter how deeply we go into our analysis, we will never get a complete description, precisely because we have no idea how models are created by our senses. In fact the brain carefully hides this from us, as we see for example in the workings of vision: there is a blind spot in the eye, where the optical nerve is located but the brain “fills in” the image at this point by extrapolating from nearby points, so that the sense of continuity is not lost. 

Therefore, it seems that, the information (i.e. structure), contained in the maximal possible conscious network of  models does not lie only, on which node is connected to which, but lies also in the interior of each node, where it remains hidden and unconscious.        

In other words, it looks as if, after the most refined possible analysis, information is still contained, not only in conscious relations but also in the things that are related. Of these things, we do have experience via the senses but they ultimately remain unknown to us.

This of course usually causes no problem in communication, since we all have some basic common models from our senses and more complicated ones from the civilization we share. If for example, I want to speak to someone about an “apple”, I don’t have to explain much because most likely, they also have more or less the same optical, gustatory and haptic model for the apple, as I do.

If though they have never seen an apple, then I will have to describe it as a fruit which is red, hard and juicy. If they live in a strange place and they don’t even know what a fruit is, then I’ll have to find other common communication codes. This is where the creative process begins, for me in trying to explain, as well as for them in trying to create in their brain, a model of what I am talking about. In the end, I will have externalized and crystallized my model and they will have theirs in their mind. Our models will perhaps be similar but perhaps not so much. We can eventually figure this out, if someone finally brings us a red apple! 

One obvious problem then, is: what happens when we want to think or communicate about things for which we have no common model from our senses or from civilization?

This problem certainly appears when we want to think about or communicate something new, original. What we then do is, try to use well known models attaching to them new meaning, either by “deforming them” to a certain extend or by relating them in a different than usual manner. There is a whole spectrum of such situations, from every day communication to philosophy, social sciences, psychology etc.

At the edges of this spectrum, as far as their end products are concerned, are, I believe, the arts and mathematics.

Indeed, with the arts we try to say new things and create new models, “deforming” or relating in new ways already known ones (e.g. colors, symbols, images) but at the same time, staying as close as possible to the senses. 

The power of art lies, to a great extend, on the immediacy of the experience. Such immediacy as only the senses can provide. 

Even though the models made by art have the immediacy of the senses, they still manage to give new meaning to already known models. Obvious and extreme examples of that, are, Warhol’s “Cambells’ soup” or Duchamp’s bicycle weal. As for “deformations” of old models, the literally deformed clocks of Dali come to mind.

On the other hand, just as experiences coming from the senses can’t be further analyzed (since the brain hides the way the senses function) so do works of art and especially visual art, resist to being further decomposed into simpler models and to storing part of the information they contain in the assembly of the parts. They are in a certain sense, indivisible.

This way, even though works of art have references to abstract models taken from the repertoire of civilization, they still give the impression that they refer directly to the senses and therefore have a strong and immediate influence.

On the other hand with mathematics we focus on the ability of the brain to relate. Models are therefore constructed in such a way so that, eventually, all the information contained in them, lies in the relations and not in the things that are related! The things that are related are signs-characters, to which no internal structure is assigned. Thus all the information of the model is contained in the relations among these characters. In geometry the initial ingredients are geometrical points, which we assume to have no internal structure and in algebra they are some letters which we consider to contain no information.

Initially, geometry is more directly accessible because of visual images but ever since Descartes, we have more and more confirmation that algebra and geometry are equivalent fields.   

For example the visual image of a half-line (a straight line that has a beginning but extends to infinity) is clear, but algebraically we can also construct the set of its points as follows: consider ten characters denoted by: ,, ,, ,, , ,,, with no a priori meaning. Then it turns out that the points of the half-line correspond, one to one, with the pairs of sequences of these ten symbols such that, the first sequence does not have , at both of the first two positions and the second sequence does not have , at both of the last two positions. One such sequence is for example: (,,,, ). The only structures we need are: the correlation (clustering) of some of the characters (i.e. that they belong to the same group) and the order of the characters in the group.

There is nothing mysterious about this correspondence, it’s enough to imagine a piece of the half line joining two cities A and B on a map such that city A is at the beginning of the half-line. Traveling along the half line from A to B, every point has a distance from city A. If the distance is 153.270 (say Km) we associate to this point the pair of sequences ((,,), (,,)). Indeed the order matters: it’s different to be 153.279 Km away, than to be 513.702 Km away. 

To complete our description of the correspondence, we also agree that if the decimal part of the distance is nothing but 0 from some digit onwards, we only keep one of them. For instance distance 12.00000…. corresponds to the pair ((,), ()). Finally, in order to cover all of the half-line, it’s enough to keep taking B further and further away from A. [1]

In our explanation, we have picked up the thread from the middle, as is usually the case with long stories: in order to complete our network of models, we must look inside the node called “distance” and see what model is hidden there, but this too is possible.

The truth is that the organization of mathematical structures is not exactly that simple. Gödel taught us that no matter what initial relations (called axioms) we start with[2], there will be true propositions (relations), which cannot be proven to be true within the system. Nevertheless, if we forget such non provable propositions, the network that remains containing propositions that can be proven, is useful and interesting and has the property that every node consists of a character containing no information, while the whole information of the network is in its assembly structure. 

 Complicated structures can of course be constructed in mathematics and in their turn they are abstracted and considered as new entities which are then used with their name, without keeping always in mind the definition in full detail. Mathematicians try to create metaphors and similes, so that the senses can be used. We say for example that a space is curved, that an algebraic construction is smooth, that another one is like a ladder or that in a certain space we have a flow. Many times new things are found via appropriate parallelisms, using intuition. In the end though, once a result is completely formulated and proven, one can unwind all the structures until they are expressed only by relations among characters that have no further internal structure and in that sense, mathematics is completely transparent!

Mathematics therefore, provides a sort of positive answer to the question we posed earlier, namely if the subdivision of knowledge into finer and finer categories, has a limit. The answer though is by a kind of “leap into the void”, where we first renounce all connection to reality, by considering characters empty of structure (i.e. information) and then build step by step, fully conscious constructions that at the end, may (or may not)  be good models for situations that we have experience of. Mathematics is, as a consequence, maximally divisible and stands in contrast to the strong tendency of art to be indivisible. At the same time, the cost for the total transparency of mathematics is, its initial remoteness from the senses, until at least a minimum of experience can be acquired.

With mathematics we build a tower of structures, since new constructions contain previous ones as special cases. There is a danger that it becomes the tower of Babel. In order to avoid this (but also for reasons of aesthetics and pleasure) we look for ways to see things so that, what was previously complicated, becomes obvious. Usually this happens by inserting it in an appropriate bigger structure. It’s as if, one was previously looking through a tiny keyhole and seeing something round with blue-green colors, levitating and then finds the key, opens the door and realizes that it is the eye of a person, sitting right behind.  

In our quest to know ourselves, our ability to relate symbols or just signs-characters, is perhaps one of the simplest and most accessible. As we incessantly try to construct models of ourselves, we have created a machine that emulates our ability to handle models, where all the information is contained in the relations among characters. This machine is of course the digital computer. The brain is not expected to function as a common digital computer because, for example, a digital computer usually crashes when the assembly of its hardware is changed or the software is modified, while the brain is much more flexible. There are nevertheless, several directions in trying to understand the operating principles of the brain (see 1,2,3,4), some of which are based on statistical and combinatorial methods. 

If we finally understand how the brain constructs models of the external and internal realities, using the electrical signals of the neurons, in a way so that the whole information is contained in the connection patterns of nodes, that have no other internal structure, then the edges of the spectrum of communication and creation, namely art and mathematics, will be united through artificial intelligence.

Until then, the connection exists at the level of mutual inspiration, through the opaque for the moment, natural intelligence.

 

 

 

References

 

1.      Gerald M. Edelman: “Second Nature: Brain Science and Human Knowledge”, Ed.: Yale University Press, 2006.

 

2.      Gerald M. Edelman: “Wider than the Sky: the Phenomenal Gift of Consciousness”, Ed.: Yale University Press, 2004.

 

3.      Mikhail Gromov: “Structures, Learning and Ergosystems: Chapters1-4, 6” http://www.ihes.fr/~gromov/PDF/ergobrain.pdf

 

4.      John Von Neumann: “The Computer and the Brain”, Ed.: Yale University Press, 1958.

 



[1] To be precise, some points correspond to two different pairs of sequences, for example the distances 0,9999... and 1, correspond to the same points on the half line but we can add one more simple rule, in order to chose one of the two expressions.

 

[2] As long as the integer numbers can be described with these axioms.