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
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.
"The intuition of the bare two-oneness" is also a fundamental concept in L.E.J. Brouwer's philosophy of mathematics. He believed in the mind's primal intuitive awareness of time, originating from the "falling apart of moments of life into qualitatively different parts, to be reunited only while remaining separated by time". (2015, Brouwer)
Brouwer states that the "Two" (Separated) represents the discrete experience of a moment passing away and another becoming, the distinction between the current instant and the memory of the previous one. This, he argues is the origin of the discrete units in mathematics.
He then goes on to describe the "Oneness" (Reunited), as representing the continuous, seamless flow of time that connects these separate moments in a unified experience. This is the origin of the continuous in mathematics (the linear continuum). The "bare two-oneness" is therefore the abstraction of this experience, devoid of its emotional content, forming the foundational phenomenon of mathematical thinking. Something that I find interesting, as this is exactly at the point where I would insert or highlight the emotional moment that this recognition is associated with. As we draw ourselves out of the sea of time that all other animals swim in, we become aware of time's passing and this itself creates an emotional engagement as it arised alongside concepts associated with life and death.
Brouwer considered intuition to be the sole basis for all of mathematics, enabling the mental construction of mathematical objects. From this intuition, the natural numbers (one and two) arise, and indefinite repetition generates all finite ordinal numbers and the smallest infinite ordinal number.
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:
Joseph Farrell (2010) Financial Vipers of Venice Port Townsend: Feral House
See also:
Mathematics, rightness and underlying beauty
See also:
Mathematics, rightness and underlying beauty
