12 06 2013


Maçoro, July 2008

Singularities & Multiplicities

Singularity that means ‘being single’ is not the only meaning that the word refers to; the meaning of the notion in question is also related to the situation of being ‘singular’ as in being non-ordinary, remarkable, and unique. Singularity is the most basic and hardcore element of creative process in nature referring to a ‘dynamic’ that holds the potential of shaping and organizing the space around its extension. Progressive Differentiation and Individuation which are the sub elements of the Morphogenetic Process- which can be examined through the approaches of topology such as ‘Topological Transformation and Folding’- are the extensions of the conceptual notions of singularity.

Singularity in mathematics can be explained with trajectories that are attracted and curved through the point of attraction. A variable that is adapted to the equation changes the closure of the curvature to the point that this variable refers to (De Landa, 2002). The variable works like a differential, representing a point or points in three dimensional world, encloses the curve which is formed by the equation to itself, re-forming the curve: The potential of singularities to shape and organize the space around can be understood through these examples of mathematics.

Claude-Paul Bruter from  ‘Département de Mathématiques Université Paris’ gives examples by making an analogy through geographic world that can be guideline to understand mathematical notions in question, additionally helping the intension of connection of the conceptual notions with the life of existence:  “Mathematical examples of basic singularities arise from hydrodynamic and geographic analogies : rivers are analogous to trajectories, and singular trajectories which are reduced to points, are called sources, sinks, and saddle points. The meaning of the terms source and sink is clear : from a source, rivers or trajectories goes out, while they end their lives at a sink. A saddle point has both the properties of a source and the properties of a sink : the river or trajectory may be first directed to that point which acts as a sink, but then, approaching the saddle-point which acts now as a source, it is repelled towards another direction” (Bruter, C-P). The system works like a manifold, it is not simple as those singularities getting together to form multiplicities, but, within a space of multiple points of attractions with multiple trajectories multiplicity is formed. These points of attractions often called as topological singularities: Where singularities are called ‘two dimensional manifolds’, the n dimensional curved surfaces that are formed by multiple trajectories and multiple points of attractions, are called manifolds or multiplicities. Manuel De Landa supplies that, “the structure of this n dimensional space of possibilities is the universal singularities, and all the  multiplicities and singularities form the virtual, plane of immanent, this is a conception of spiritual, a spiritual world that is impersonal”(De Landa, 2007)

“The formations in space, a galaxy or a black hole are the examples of singularities of a great scale that the center of them as “cores” have  the role of a point of attraction.

Singularity can also mean a singular state of high stability. Its character can be sensed through the extraordinary values of its parameters that insure fixity (Bruter, C-P). As it was scrutinized before, singularity has the dynamic that holds the potential of shaping and organizing the space around its extension- is through this intrinsic self accord situation of being stable and keeping. For example a leader of a tribe is selected, or has become a leader ‘by force’ – through the dynamics of being tougher, wiser or being descendent of a royal blood, are his intrinsic potentials of becoming that works like a point of attraction where his destiny would be a curve that is attracted through this point which would be a path to leadership, giving him a higher, stronger and stable (stability is a dependant to time situation that is valid only at a certain time- that through time that may be lost) position and the power to rule as this position ‘is a situation of a singularity’ for him, both for being singular as in meaning single, both for being unique, non-ordinary and remarkable and being in a state of higher (relatively) stability. This example can be applied to any ‘becoming’ in nature, animals living together has their leaders and are selected through similar processes that are examples to the intrinsic potentials and dynamics of creative process in nature.

The leadership example is only a one of the nearly infinite creative processes in nature, every formation in nature goes through these similar processes. All the transformations, forms, mutations and evolution are depended on the dynamics that are related to the processes in question. Manuel De Landa supplies with an example to a naturally formed shape of a soap bubble that may help to comprehend the notions better with a different perspective of singularity as an element of creative process in nature: Like many physical structures, soap bubble forms its component to meet certain energy requirements. An example to meeting certain energy requirements is withholding minimal energy to extend the life cycle to maximum. (De Landa 2002).  For the soap bubble it is indispensable that it has to keep its surface tension to minimum to keep its existence- taking the best form as a ‘sphere’ to keep the tension equal through its surface, and adjusting the size of the diameter to keep its surface tension minimum a singularity is achieved The point of attraction is its center where the tension is zero, the trajectory is its liquid density, the main affecter to take its form at certain size under certain circumstances. It’s the singularity that defines multiplicity through the genesis of a physical form.

Even a soap bubble has its own potential of becoming, for singularity is its main engine of self creation, though it may be a smallest example to the formations within the great assemblage, infinite singularities co-exist and ‘affect’ each other to form multiplicities that may be a metaphor of being the essence of all surroundings.


Expressivity is an act of nature to express its states. The expressions of nature are the signals that are to be receipted by its members for manipulating their moves, keeping the self accord situation of the nature, which is a key to logic of affordance that triggers evolution and creative process.

Art, Architecture, Literature are tools for expressivity of humans. It is important to understand not only the humans are capable of expressing themselves, but all of the elements of the nature, animals, plants and even the “non-living” have their way of expressivity; prior to that, the morphogenetic process, evolution, the formation in nature- as assemblage, is due to a complex and dynamic relations and processes in universe where singularities and multiplicities are examples to that.

The non-human expressivity functions as a signalization system, forming the base of the affordance -the main guide of morphogenesis. The water being “hot” is the expressivity of the molecules forming it are filled with energy vibrating, so may be harmful for a living organism to swallow, understanding that it should not touch it, wait until the heat drops, the energy of the  molecules lessens… This is a simple form of expressivity of the non-living, as it may get more complex in various situations.

An old mountain is smooth, where the rocky, bald and the tougher one would cry out that “I am young”, a volcanic mountain would signalize that it is dangerous by the look of it

Manuel De Landa points out that “Geology is the beginning of non-human expressivity” referring the conception of Gilles Deleuze “Geology of Morals”: An old mountain is smooth, where the rocky, bald and the tougher one would cry out that “I am young”, a volcanic mountain would signalize that it is dangerous by the look of it, guiding the animal not to build a nest near the opening. The form of the clouds is three dimensional expressivity or the colors of the materials and components are the signals of the characters of materials as different forms of expressivity, telling about the hardness of it, the lightness of it (De Landa 2007)

A simplest form of expressivity is territoriality. For example, a dog marks his territory with urine, a signature like words: “This is my territory”(De Landa, 2007). But the expressivity of the non-human living organisms can be formed of signatures and actions in more complex way. Bacteria for example are capable of sensing and responding knowing that it should avoid from a toxic area. It is important to remind that the signal of toxins- as a non living expressivity is sensed by the bacteria, as this response of avoiding is another expressivity,-expressivity of a living organism, showing an intimate relationship of folding expressivities.

As this may get more complex, due to genetic variations, some animals are capable of expressions in changing colors for example a changing color of the skin of an animal signaling that it its ready to mate or if it is poisonous or not… etc. But more interestingly there is another type of expressivity in animals that it is “chosen expressivity”: Bower birds for example, are able to decorate their nests to seduce the female. There are three types of bower birds: One of them has a very bright blue colored skin, easily attracts the female, builds simple nest. The skin of the bird, sometimes changing color is its internal expressivity that is due to genes and hormones. The Second type of Bower bird have only a few of its feathers blue, that it builds a much more complicated nest prior to the first one, begins to express itself externally, due to the lack of internal expressivity. The last type of bower bird that have lost all its blue feathers builds a magnificent nest, even seeks some blue ribbons, blue bottle caps around to bring them to decorate its nest to impress the female, this action is not only due to instincts that genetically determined, bird becomes an artist- a simple one, but an artist. The chosen expressivity is not only the ability of the humans, but also the ability of the simpler life forms. It can be said that, when the first human beings were painting the caves they were already surrounded by this environment of conscious expressivity. (De Landa, 2007)

Bower Birds are creatures being one of the best examples to chosen expressivity

Affordance & Assemblage

Montage is to add numerous things together straightly; assemblage however is slightly different that it is things getting together in the line of adaptation and formation. For example if a wooden shelf is connected to the wall with a screwdriver, that is simply montage. If we think the relations in the same way, but more complex, to wear hat is also a montage, but the hair is that comes out of your head through your skin is an adaptation as a part of a morphogenetic process that is a new formation, therefore is an assemblage. Nature (as in universal extend) is the greatest assemblage that its members are assembled with each other in a line of adaptation related to the notions of affordance.

Prior to expressivity, affordance is the act of morphological adaptations in nature that works like a tuner of the nature, keeping the assemblage accorded and triggering evolution.

James Gibson, one of the most important psychologists in the field of visual perception, has very important studies on animal behavior and how animals perceive their environment. Gibson has invented the term affordance As long as environment (all the members of it, nonliving / living are included) expresses its capacities to affect and to be affected with certain signals that animals/us to perceive, it guides its members; all the opportunities and the risks of the environment (biological or geological) affords an animal with, or supplies an animal with, or provides an animal with… Manuel De Landa have told in his book Intensive Science and Virtual Philosophy “Ecosystem is a complex ‘assemblage’ of a large number of heterogeneous components” Expending the term ‘affordance’ introduced by James Gibson within the context of a theory of ecological interactions. For example water, for an insect may be a surface to walk or leave its larva, but for an elephant it’s impossible to walk: lake is insect’s affordance. Or the ground may only be a walking surface for an animal, where for another it can provide a place to hide: the ground has afforded an animal with providing a hideout. (De Landa 2002, De Landa 2006) This guidance or providence as you may call it, is the trigger of the evolution, tuner of the nature as an assemblage, and provides this assemblage to be self-accorded, is the machine of the creative process of the self organized nature – as in Kant’s beautiful nature, by supplying a ready to transform base structure for the construction of the universe of unlimited possibilities.

The lake has afforded water strider a walking surface were an emptied log provides a hiding place for snake

The nature consists of singular and irregular construction, and in much deeper than a reflection of a self accorded formation that is capable of stranger mechanism of morphology. The point of attractions, drags the trajectories in the space that creates singularities, singularities come together and forms multiplicities, filled with intensive processes, progressive differentiations cause new formations in nature, chain reaction process of neoteny as key to individualization, conceptual notions of creative process in the nature. Looking through these concepts, it may be understood that the process is unplanned, everything is unfinished-ready to evolve. Through this, a dynamic, non-stop, creative process is in question.

Evolution and Darwinian Approach

The nature is morphologically pregnant to all possibilities of formations. The dynamics of its formation can be explained through complex mathematical and physical notions, basically as “Singularities are the most basic and hardcore elements of creative process in nature referring to a ‘dynamic’ that holds the potential of shaping and organizing the space around its extension” , consisting the notions of  “Progressive Differentiation and Individuation which are the sub elements of the Morphogenetic Process- which can be examined through the approaches of topology such as Topological Transformation and Folding”. Keeping all these notions in mind, the formation of life due to a protein molecule (DNA) can be comprehended where the question will be “How and through which creative processes such complex molecules like proteins are formed nearly out of nowhere?” which is an important one for understanding the logic of evolution.

The genetic code that withholds all the definition that defines a living organism is due to chromosomes consists of DNA spiral which is a very complex and large protein. Like every protein, DNA is a complex molecule. To be reminded that molecules are the main structures of every formation, are made up of atoms: The simplest structures which are elements.

The whole of the living organisms of today have formed of molecules that has the ability to replicate themselves that were formed through chemical processes. The chances of the formation of a working protein ‘intrinsically’ is so small that its possibility is same as a plane would formed by itself out of a  whirlwind winding within a scrap field (Charlesworth, 2006). Through experimental processes, scientists have obtained some of the basic chemical structures of life like sugars, fats, some amino acids and foundational elements of DNA and RNA, by giving electrical waves to simple molecule solutions or exposing them to radioactive waves. There were small steps taken to obtain molecules that are more complex similar to DNA’s or RNA’s, but scientists were not very successful to make those molecules replicate themselves (Charlesworth, 2006).

Looking to the age, scale and dynamics of the universe, the possibilities, of elements coming together to form a complex molecule like protein, which is associative with its surroundings, and the chances of formation of a complex organism which consists of various complex molecules that are functioning as the elements of a larger assemblage would not be small or unimaginable. The morphology of life beginning from the simplest life forms to the most complex ones is the most sophisticated example to the natural creative process that can be comprehended with evolutional theories.

As it was emphasized in previous chapters that the trigger of the evolution is affordance that keeps the assemblage tuned in a line of adaptation: The idea of evolution, proposes various natural processes which describes the variety of species and the characteristics of their adaptation to their natural habitat through no celestial explanations. The evolutional processes depend on the emergence of new and evolved forms of living organisms called ‘mutations’. Mutation is so rare and a by coincidence situation that it occurs nearly in one percent of a hundred thousand members of a generation of a population. For evolutional change to occur, various mutational formations have to be repeated through generations (Charlesworth, 2006). In relation to these processes, the most important dynamic of evolution is ‘natural selection’. For survival of a species the adaptation is absolutely necessary. If, mutations occurred by coincidence, improves the survival situation of a species in a specific habitat, for it (mutations) will be transferred through generations by genetic code (DNA), the survivors that have the specific individuals of the species will extend their life easier and their percentage in numbers will dominate the population.

Specific mutations that is essential for survival of an individual in its natural environment is a form of adaptation that through natural processes only the members of species that has the specific mutation may survive in various situations: is natural selection; that, evolution may seem rather a process of an action of purposeful intention, however it is a very important milestone natural creative process which proposes a ‘half-by coincidence’ model and an important explanation to the formation of the natural environment.





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