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Algorithmic “unconscious” – Part 1

 

This two-part essay offers an investigation of the “depths” of the algorithmic workings of digital visualisation systems. The focal point of this study is digital “memory”, as explored in relation to the Algebras of “Logic”, the notions of process paradigm, pre-representational pixel-grids and emergence.

In this way the algorithmic “unconscious” of digital visualisation technology – i.e. the inherent paradoxes and unknowns, the uncontrollable data flaws, the hidden inconsistencies, the exchanges and the potentialities of the system – is unveiled. New possibilities and ways of creating more than mere visual ‘phenomena’ emerge.

Histories & Theories: An Algebra of Logic?

An algorithm is an automated language of sequential instructions, through which, a computer carries out calculations. Computer algorithms are fully automated and thus devoid of meaning. The calculations are carried out through the use of multi-adjustable variables, various generators and the binary substratum. The latter constitutes the ultimate digital reduction of information. One of the most important issues debated in contemporary cyber-discourse, concerns the problems that are related to the addition of higher abstraction levels, such as, linguistics, to digital technology, as they ‘hide’ the substratum and increase complexity. Nevertheless, there have been strong disagreements on whether digital technology is purely calculus-based, whether it is structured in a more ‘architectonic’ than mathematical or semiotic mode, and on the qualities of its pre-representative “environment”.

An investigation of the histories and theories that concern the foundations of digital technology, is necessary for elucidating its roots, workings, flaws and potential. Essentially, the roots of digital technology can be found in George Boole‘s reduction of Aristotelian syllogism into formalistic analogies drawn between algebra and logic, resulting in meaningless propositions. Boolean ‘logic’ was proved to be a-syllogistic, as Boole mistakenly believed that an algebraically formulated equation suffices as a proof of a statement.

As Sriram Nambiar explains, Boole applied an axiomatic treatment to logic.[1] The mathematician aimed to show that the processes of reasoning could become fully explicit in the form of algebraic equations. In this way, he attempted to achieve the mathematical formulation of logic, for creating an “algebra of logic”. [2] In fact, Boole deciphered a merely ‘formal agreement’ between language and algebra, without investigating their relationship in depth. [3] According to him, the laws of reasoning and algebra have identical symbols laws, processes, and axioms, but their subjects and interpretation are different. [4]

Boole used primarily ‘and’ = ‘+’, ‘not’ = ‘-’, ‘or’ = ‘x’ as binary operations, each of them describing the relationship between a particular set of premises.[5] Boole however, failed to provide a consistent definition of how each sign of an operation was to be interpreted. In their separate reviews John Corcoran and Michael Dummett note that no explicit interpretation is given for the operations ‘+’ and ‘–’.[6] Consequently, as Corcoran explains, Boolean ‘logic’ cannot be verified, as it lacks the principles of logical deduction: “Boole… was more interested in solving equations arising from his algebraic representation of Aristotle’s logic than he was in the details of the deductive processes presupposed in algebra” [7]

As both Nambiar and Corcoran conclude, Boole’s arguments are unverifiable. [8] Corcoran clarifies that Boole’s main error is to overlook that “a solution to an equation is not necessarily logically deducible from, or implied by, the equation”. [9] Consequently, both Theodore Hailperin and James W. Van Evra note that Boolean operations are not truly based on deduction, but yield meaningless conclusions. [10] Contrary to Boole’s intention therefore, the process of reaching a conclusion could not be proved, as the relations between propositions could not be demonstrated.

Another important shortcoming is that Boolean classes were not explicit and thus a verifiable conclusion could not be reached. Aristotelian syllogism derived from the known concepts and logical distinctions of real things. Unlike Aristotle, Boole rejected ”the imperfection of senses” and thus a geometrical and any other type of observation and proof, as he believed that they would jeopardise the verification of “universal truths”.[11] Consequently, Boole decided to established ‘0’ and ‘1’ as the premises of Boolean ‘logic’. Those premises represented entirely arbitrary classes. In particular, as Corcoran clarifies, ‘1’ was an arbitrary class that included all elements of a hypothetical argument, while ‘0’ was the empty set that resulted from a false proposition.[12] Nevertheless, as Corcoran and Nambiar note, Boole did not provide a clear definition of those premises.[13] For these reasons, Boolean premises involve unknowns and are arbitrary.[14] Unlike syllogism, Boolean premises are abstract and refer to hypothetical classes.

Despite their disadvantages, Boolean set operations have enabled the design and practical implementation of computer algorithms. Bertrard Russell explains that Boole has contributed to the ”recognition of asyllogistic inferences” from which “modern symbolic logic… has derived the motive to progress”.[15] It is interesting to see how David Hilbert describes the axiomatic formalisation of theory, as it sheds light to the formulation as well as to the true character of such “logic”: “[...] Theory is no longer a system of meaningful propositions, but one of sentences as sequences of words, which are in turn sequences of letters. We can tell by reference to the form alone which combinations of the words are sentences, which sentences are axioms, and which sentences follow as immediate consequences of others” [16]

 

On a practical level, Claude Shannon implemented Boolean set operations for creating the first theory and design of switching current in telegraph switching relays during 1937-8.[17] As Mitchell describes, computer processing is primarily based on its binary substratum and in particular, on “performing ‘and’, ‘or’ and ‘not’ (Boolean) operations on pairs of binary digits’’. [18] As Davis describes, such development has resulted in the ‘artificial language of logic’ that is practically implemented as a computer algorithm.[19] Logical statements are written sequentially as instructions, so that they can be executed by the computer through the Boolean set operations applied to those binary digits.

?evertheless, even if it seems that information has become fully ‘codable’, all operations have become countable and thus, the algorithmic ‘space’ of digital technology is now demystified and controllable, the algorithmic paradox still prevails. Despite all the aforementioned developments, the workings of digital technology and visualisation systems remain inherently inconsistent. Inevitably, such complex systems maintain an inherent degree of complexity, instability and emergence. Indeed, as confirmed by numerous cyber-researchers, thinkers, software engineers and media theorists, such as Friedrich Kittler, although programmers struggle to remedy “electronic diffusion… and quantum mechanical tunnelling” – which are considered to be noise, the inherent “physical side-effect” of the chip – machine reduction and constraints, an inherent degree of randomness, and the “ever-growing incompatibilities between the different generators” haunt digital technology even more [20].

There are different ways of dealing with such conditions, largely depending on how the relationship between hardware and software is perceived and investigated in relation to their roots and ‘evolution’. Certain limiting and superficial approaches have ranged from ignoring the inherent characteristics and flaws of digital technology, introducing a kind of pseudo randomness and emergence through the addition of special generators, external input etc., to simply observing software as a mere ‘natural phenomenon’. In fact, what plays a dominant role in digital technology, is the qualities of the ‘links’ (as contrasted to true relationships) and the ‘inter-dependencies’ that are formed between its various elements (including material and immaterial ones), through which, digital memory is operating. Digital memory primarily involves the often symbiotic processes of informational un-linking and re-linking. The workings of digital memory distort our familiar notion of time. A kind of algorithmic ‘unconscious’ emerges, as we encounter generative ‘intervals’, transitionality, composites and various degrees of reversibility and indexicality, due to the unanticipated intervention of excess and abstraction. These aspects, coupled with new possibilities and ways of creating more than mere visual ‘phenomena’, will be discussed extensively in the second part of this essay.

Notes:

[1] – Nambiar, Sriram, “The influence of Aristotelian logic on Boole’s philosophy of logic; the reduction of hypotheticals to categoricals”, A Boole anthology: recent and classical studies in the logic of George Boole, James Gasser (ed), Dordrecht, Boston, London : Kluwer, 2000, p.220 (see also p.228).

[2] – Boole, George, An investigation of the laws of thought, New York: Dover publications, 1854, p.67[50], Corcoran, John, “Aristotle’s prior analytics and Boole’s laws of thought”, in History & philosophy of logic, 24, 261 288, UK: Taylor & Francis, ,2003, http://www.acsu.buffalo.edu/~corcoran/HPL24(2003)261-288b.pdf, accessed 25/4/04, pp. 261, 281, 283, 271. 276. Nambiar, Sriram, “The influence of Aristotelian logic on Boole’s philosophy of logic; the reduction of hypotheticals to categoricals”, A Boole anthology: recent and classical studies in the logic of George Boole, James Gasser (ed), Dordrecht, Boston, London: Kluwer, 2000, p 238. Corcoran, John & S.Wood, ‘Boole’s Criteria for validity and invalidity’, A Boole anthology: recent and classical studies in the logic of George Boole, James Gasser (ed), Dordrecht, Boston, London : Kluwer, 2000.

[3] – Boole, George, An investigation of the laws of thought, New York: Dover publications, 1854, , p.37, 26 Corcoran, John, “Aristotle’s prior analytics and Boole’s laws of thought”, in History & philosophy of logic, UK: Taylor & Francis, 2003, accessed 25/4/04, http://www.acsu.buffalo.edu/~corcoran/HPL24 (2003) 261-288b.pdf, p.261, [283] Nambiar, Sriram, “The influence of Aristotelian logic on Boole’s philosophy of logic; the reduction of hypotheticals to categoricals”, A Boole anthology: recent and classical studies in the logic of George Boole, James Gasser (ed), Dordrecht, Boston, London: Kluwer, 2000, pp.227

[4] – Boole, George, An investigation of the laws of thought, New York: Dover publications, 1854, p.37-38 (6, 37-8, 46).

[5] – Dummett, Michael, “Review of Boole”, A Boole anthology: recent and classical studies in the logic of George Boole, James Gasser (ed), Dordrecht, Boston, London : Kluwer, 2000, pp.80,25,33. Corcoran, John & S.Wood, “Boole’s Criteria for validity and invalidity”, A Boole anthology: recent and classical studies in the logic of George Boole, James Gasser (ed), Dordrecht, Boston, London : Kluwer, 2000, p.101.

[6] – ibid, pp.106,118, Dummett, Michael in ibid, p.81.

[7] – Corcoran, John, “Aristotle’s prior analytics and Boole’s laws of thought”, in History & philosophy of logic, UK: Taylor & Francis, 2003, http://www.acsu.buffalo.edu/~corcoran/HPL24 (2003) 261-288b.pdf, accessed 25/4/04, p. 283.

[8] – Ibid, p.261, [283]. Nambiar, Sriram, “The influence of Aristotelian logic on Boole’s philosophy of logic; the reduction of hypotheticals to categoricals”, A Boole anthology: recent and classical studies in the logic of George Boole, James Gasser (ed), Dordrecht, Boston, London: Kluwer, 2000, pp.227.

[9] – Corcoran, John & S.Wood, “Boole’s Criteria for validity and invalidity”, A Boole anthology: recent and classical studies in the logic of George Boole, James Gasser (ed), Dordrecht, Boston, London : Kluwer, 2000, p.101. Boole, George, An investigation of the laws of thought, New York: Dover publications, 1854, p.16, Corcoran, John, “Aristotle’s prior analytics and Boole’s laws of thought”, in History & philosophy of logic, UK: Taylor & Francis, 2003, http://www.acsu.buffalo.edu/~corcoran/HPL24 (2003) 261-288b.pdf, accessed 25/4/04, pp.280, [277].

[10] – Halperin, in A Boole anthology: recent and classical studies in the logic of George Boole, James Gasser (ed), Dordrecht, Boston, London : Kluwer, 2000, p. 61, Evra in ibid, p.87.

[11] – Boole, George, An investigation of the laws of thought, New York: Dover publications, 1854, pp.405,404 [403,407], 3,4.

[12] – Corcoran, John, “Aristotle’s prior analytics and Boole’s laws of thought”, in History & philosophy of logic, UK: Taylor & Francis, 2003, http://www.acsu.buffalo.edu/~corcoran/HPL24 (2003) 261-288b.pdf, accessed 25/4/04, pp.283, 273.

[13] – Boole, George, An investigation of the laws of thought, New York: Dover publications, 1854, p.17, Corcoran, John, “Aristotle’s prior analytics and Boole’s laws of thought”, in History & philosophy of logic, UK: Taylor & Francis, 2003, http://www.acsu.buffalo.edu/~corcoran/HPL24 (2003) 261-288b.pdf, accessed 25/4/04, pp.261, Nambiar, Sriram, “The influence of Aristotelian logic on Boole’s philosophy of logic; the reduction of hypotheticals to categoricals”, A Boole anthology: recent and classical studies in the logic of George Boole, James Gasser (ed), Dordrecht, Boston, London: Kluwer, 2000, pp.230, 232.

[14] – Corcoran, John, “Aristotle’s prior analytics and Boole’s laws of thought”, in History & philosophy of logic, UK: Taylor & Francis, 2003, http://www.acsu.buffalo.edu/~corcoran/HPL24 (2003) 261-288b.pdf, accessed 25/4/04, p.261, Corcoran, John & S.Wood, “Boole’s Criteria for validity and invalidity”, A Boole anthology: recent and classical studies in the logic of George Boole, James Gasser (ed), Dordrecht, Boston, London : Kluwer, 2000, p.114, [101,105-6]

[15] – Nambiar, Sriram, “The influence of Aristotelian logic on Boole’s philosophy of logic; the reduction of hypotheticals to categoricals”, in A Boole anthology: recent and classical studies in the logic of George Boole, James Gasser (ed), Dordrecht, Boston, London : Kluwer, 2000, p.217.

[16] – David Hilbert in Friedrich Kittler, “There is No Software,” Ctheory, October 18, 1995, http://www.ctheory.net/articles.aspx?id=74 (accessed September 7, 2010).

[17] – Davis, Martin, Engines of Logic: mathematicians and the origin of the computer, New York, London: W.W.Norton, 2000, p.178.

[18] – Mitchell W, M.McCullough, Digital Design Media, New York: Van Nostrand Reinhold, 1995, p.26.

[19] – Davis, Martin, Engines of Logic: mathematicians and the origin of the computer, New York, London: W.W.Norton, 2000, pp.119-121.

[20] – Friedrich Kittler, “There is No Software,” Ctheory, October 18, 1995, http://www.ctheory.net/articles.aspx?id=74 (accessed September 7, 2010).

 

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