Thursday 7 January 2016

Abstraction, mathematics, metaphor and creation

I have recently been looking at ways to write about the moment of a drawing's becoming, trying to find a way of articulating an image’s inception. In my post of January 1st I wrote this sentence as an attempt to get at what I meant, ‘The grounding of the image in the materials of its making, coming together with the force from which the image arises’. See 
However on reflection the writing didn’t really get to grips with what could be called the ‘ontology’ of being faced with the empty vacuum of a clean sheet of paper. Ontology is a metaphysical concept concerned with the philosophical study of the nature of being, becoming, existence, or reality. Metaphysics is a branch of philosophy concerned with explaining the fundamental nature of being, therefore ontology could be seen as the area of study where you get right down to the core of what’s going on.
A mathematical structure is in its essence a set of abstract entities with relations between them. I would argue that we live in a relational reality, our awareness and understanding of the world around us comes not from the properties of its basic components (quarks/electrons/gravity/electromagnetic force etc), but from the relationships that exist between these things.  Therefore my idea that an image arrives from a relationship between materials and some sort of energy or force still to some extent holds water but as an idea it is not as yet stripped down to its conceptual root. In order to do this I am going to attempt to use some mathematical theory, an area I tend to avoid because as an artist I am you could say “mathematically challenged” but as an artist I am also able to poke my fingers into any discipline that helps me understand something or help me with the development of an idea, so bear with me.

The following proposition is a response to George Spencer-Brown’s ‘Laws of Form’, written in 1969.

Let the blank page ☐ denote True or False, because it is a space for possibilities to happen and let a ≠ symbol be read as Not or not as things were. I.e. if the blank page was a sort of truth, then once ≠ was added to it, things will have changed, this difference would mean that what was a truth is now something else, it is not true. Conversely if the blank page is in some way false, by adding ≠ to it, it is now something different and if it is not false it can now be true. 
Then the primary arithmetic would have the following sentential reading:
If ☐ = false, then ☐ ≠ = not false = true
If ☐ = true, then ☐ ≠ = not true = false

This interested me because I hadn’t thought about mathematicians being interested in the blank page. Read in this way the moment before a mark is put on a blank piece of paper is either true or false, however once a mark is put down some sort of truth or falsehood is made and then interestingly when a second mark is put down, that first truth or falsehood is questioned, so we now have an untruth or a new truth. 

The ≠ sign symbolises the essence of how we think about ideas, it indicates the capability of differentiating a "this" from "everything else but this." It represents the drawing of a "distinction", and can be thought of as signifying the act of drawing a boundary around something, thus separating it from everything else; it also represents that which becomes distinct from everything by drawing the distinction or boundary as well as the crossing from one side of the boundary to the other.

So in effect the first mark we make changes things, we have made a distinction between one thing and another, we have created a difference, and in creating this difference a new possibility is born.

This is reminiscent of another area of theoretical conjecture as to how to visualise the moment of creation itself, that moment of the Big Bang; which as Stephen Hawking has stated, was ‘a quantum fluctuation out of nothing’. In the 1960s John Wheeler and Bryce DeWitt combined quantum mechanics and general relativity into a mathematical framework now known as the Wheeler-DeWitt equation. By integrating Heisenberg’s uncertainty principle into the Wheeler-DeWitt equation He, Gao and Cai have argued that a small empty space can come into existence probabilistically due to fluctuations in what physicists call the metastable false vacuum.
‘When this happens, there are two possibilities. If this bubble of space does not expand rapidly, it disappears again almost instantly, but if the bubble can expand to a large enough size, then a universe is created in a way that is irreversible’. (He, Gao and Cai, 2014)
Perhaps there are very close parallels between the moment of a drawing’s inception and how we think about the coming into being of the universe.

Spencer-Brown takes us into some interesting areas. This is a thought experiment taken from the beginning of the first chapter of ‘Laws of Form’.

Draw a distinction.
Call it the first distinction.
Call the space in which it is drawn the space severed or cloven by the distinction.
Call the parts of the space shaped by the severance or cleft the sides of the distinction or, alternatively, the spaces, states, or contents distinguished by the distinction.
Let any mark, token, or sign be taken in any way with or with regard to the distinction as a signal.
Call the use of any signal its intent. (Spencer-Brown in Farrell, 2010, p. 47) 

Going back to Hawking’s ‘quantum fluctuation out of nothing’, we would have to think about ‘nothing’ as existing infinitely and extended everywhere in every direction. However just to be able to think about this is impossible unless you use a metaphor and in our case it is the empty sheet of paper.

This rectangle represents a blank sheet of paper that itself represents an empty box with no sides that exists infinitely and extends everywhere in every direction. The dashed line used to make the rectangle, as explained in the previous post, representing an idea that is not solid, or to some extent invisible. This infinitely extended ‘no-thing’ can be mathematically expressed as an empty hyper-set like so: Ø. Within this space we can draw a simple distinction: (Based on Farrell, 2010)

We now have two spaces, one inside and one outside the circle. The circle in effect is a circumscription of a space. It is though important to remember that what is now inside the circle is actually also the same as but now defined as different from what was circumscribed. Because the original space was infinite the selection would also be infinite.
So a crude mathematical expression of this would be:

If Ø1 represents the original empty hyper-set, then Ø2 could be used to represent the simple distinction or selection and Ø1,2 the boundary that is in effect the common surface between the two spaces that have now been distinguished. We now have three distinguished nothings. A primordial trinity that always includes Ø within it because everything is always composed of what was the original infinitely extended no-thing. This abstract topological construction can now be used as a metaphor for a moment of becoming, of making something from nothing.

This is where mathematics and religion begin to fuse together. The Vedic Padama Purana states:

‘In the beginning of creation the Great Vishnu, desirous of creating the whole world, became threefold: Creator, Preserver, Destroyer. In order to create the world, the supreme spirit produced from the right side of his body himself as Brahma then, in order to preserve the world, he produced from his left side Vishnu; and in order to destroy the world he produced from the middle of his body the eternal Shiva.  (Wilkins, 1991, p. 116) In Christianity we have God the Father, God the Son and God the Holy Spirit’.

In my post on Mathematics, Rightness and Underlying Beauty I mentioned that mathematical order, was a 'supreme value', one that is basic to the whole universe. Perspective as a tool is uniquely able to synthesise the practices of observational drawing and geometry, its basic triangular format allowed it in Renaissance times to be used symbolically to represent the Christian ‘Trinity’. As Panofsky states, ‘perspective transforms psychophysiological space into mathematical space’, (p.31), and in its turn mathematical space can be itself subsumed within a theological space, the vanishing point of perspective, effectively becoming ‘the eye of God’, a point from which all others can be derived. See 

What I’m skirting around is that old conundrum, the one that Duchamp was getting at when he decided to reject ‘retinal’ art, I’m looking for structures that can hold meaningful concepts, trying to explore the possibilities for complex thinking within existing frameworks and looking for pointers towards alternative ways of communicating ideas. By moving between disciplines it forces us to use areas of our brains that can become lazy, but perhaps more importantly we can be open to new ideas because we are able to see what we are doing from a different position. In particular I’m interested in how value is made, exchanged and transformed across disciplines. From Marx’s idea of the ‘fetish’, to the concept of money, from art as an honorific term suggesting that it gives a particular value to things, to religious concepts of good and bad. All of which seem at one point or another to slip between meanings and to become at times substitutes for each other. Number becomes a way of measuring the world; money is used to transfer value, itself becoming an abstract measure for the worth of ‘real-life’ transactions. Original sin becomes a debt we all have to pay for, but religious art can command some of the highest prices for commodities that exist in our world. What makes a Van Gogh so expensive? Where does its value lie, and how is an art concept evaluated, weighed and measured against a human life? Well mathematics and order do appear to give underpinning resonance to concepts and Pythagoreans did stress how all life could be underpinned by this mathematical order, including their support of monetary systems which operated using the fact that life's actions such as work could be actually valued/measured as currency. Something that didn't have value before now has one. Monetary value was created out of something that was about life and action, a material, such as silver, is given its validity by the stamp put upon it as it is minted. Paper is stamped with assurances that it is now worth something and we make drawings on paper that may well be worthless, but may also be meaningful. So I'm touching on that something, but not yet putting my finger hard on it, which is quite a good spot to be in, as whenever I try and press too hard down on something it can shy off like a tiddlywink. 

I seem to have rambled into some odd territory, this post was supposed to be about moments of becoming and how to think about creation from nothing. All is I suspect relational, or context dependent; I cannot but bring my own concerns to the table, but in doing so perhaps I can trigger off an interest in someone else, someone with more of a mathematical grasp of possible readings might be intrigued enough to take these ideas further, someone perhaps like Hanne Darboven. 

Hanne Darboven

As an artist Hanne Darboven was able to integrate mathematical logic with her artistic preconceptions. Her drawings often consisted of rows and rows of number sequences. She made lists of numbers from complicated additions or multiplications of personally derived numerical sequences. She developed ‘checksums’ that she used to expose the way that we were driven by the year’s calendrical progression. She developed a drawing system to represent time as both the continuous flux of life and an all-embracing order. ‘Her installation Cultural History 1880–1983, reduces the Gregorian calendrical notation to only forty-two denominations for each century, weaving together cultural, social, and historical references with autobiographical documents, postcards, pinups of film and rock stars, documentary references to the first and second world wars, geometric diagrams for textile weaving, a sampling of New York doorways, illustrated covers from news magazines, the contents of an exhibition catalogue devoted to postwar European and American art, a kitschy literary calendar, and extracts from some of Darboven's earlier works’. See 
Hanne Darboven: ‘My systems are numeric concepts that work according to the laws of progression and/or reduction in the manner of a musical theme with variations.’



Hanne Darboven

Darboven’s attempt to grasp the huge variety of life and its images within a mathematical structure reminded me of what Frances Yates called ‘memory theatres’. One of the chief protagonists in her wonderful book, ‘the Art of Memory’ is Giordano Bruno, his writings not only dealt with rhetoric and associated memory training, but also postulated an alternative to orthodox Christianity, a position that would eventually see him burnt at the stake. Both his works on memory and on cosmology refer to structures, that both impose order and give meaning at the same time. His influence has been long and continuous, James Joyce (1925) stated that ‘His philosophy is a kind of dualism – every power in nature must evolve an opposite in order to realise itself and opposition brings reunion'. He had a concept of God that subsumed in itself the multiplicity of existence, a concept not unlike the idea of a primordial unity of nothingness that infinitely extended throughout everything, which was where I started from, creation and the blank sheet. 

As I make a shape on a piece of empty paper, I am aware that this shape becomes a figure, the surface a ground, this figure and ground relationship develops and changes as I add more shapes, eventually the interrelationship between these two elements may shift, what was a series of shapes may now become a form that operates as a ground, be this abstract or figurative this is a dance out of nothing, a fluctuation of nothings that become somethings, and so it goes, on and on, each time an image is made, each time someone thinks of creation, someone will become fascinated by the process and in this fascination may open a new doorway into their own existence.


 Kant spoke about ‘an intuition of the bare two-oneness’ he was thinking about the time before mathematics came into being, a time when someone began to intuit the idea of counting. Perhaps we are in a similar position when regards to thinking about creation itself, best perhaps to simply create, and not get mixed up with trying to think too much about how to communicate what it is.

References

Dongshan He, Dongfeng Gao, Qing-yu Cai (2014) Spontaneous creation of the universe from nothing Cornell University Library arXiv:1404.1207v1
http://arxiv.org/abs/1404.1207v1 accessed on 7. 1. 15

Erwin Panofsky (1991) Perspective as symbolic form New York: Zone Books

George Spencer-Brown (1999) Laws of Form Leipzig: Bohmeier Verlag

James Joyce, Letter to Harriet Shaw Weaver, 27 January 1925, Selected Letters, p. 307

W. J. Wilkins (1991) Hindu Mythology New Delhi: Heritage

Frances Yates (2014) Giordano Bruno and the Hermetic Tradition London:Rutledge

Frances Yates (2014) The Art of Memory London: Bodley Head 

Joseph Farrell (2010) Financial Vipers of Venice Port Townsend: Feral House

See also:
Mathematics, rightness and underlying beauty









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