6 Jan 2009

Deleuze and Posthumanism Paper, Part 2 "The Criterion of Our Critique," of "Do Posthumanists Dream of Pixelated Sheep?"

by Corry Shores
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[May I thank Lorenz Demey for his help with clarifying Tarski's isomorphism. Lorenz is an incredibly talented analytical philosopher. He was awarded the highest possible honors at Katholieke Universiteit Leuven, which almost unheard-of.]

"Do Posthumanists Dream of Pixelated Sheep?

Mental Uploading under Deleuzean Critique"

Part 2: "The Criterion of Our Critique"

What is the successful result of mental uploading? We address this question first, so to orient our investigation. Uploading purportedly produces a computer-rendition of our minds so that we may exist independently of our bodies. The abstract formal elements constituting our minds – their tendencies, personality, abilities, etc. – need to be perfectly copied to the computer; that is to say, the mind and its replication must be isomorphic (equal, iso; form, morph). Although defining this term requires we trudge through some technicalities, it is essential that we clarify the basis of our critiques.

For posthumanists, a perfect replication of the mind serves to successfully duplicate one’s selfhood; yet, from a Deleuzean perspective, it is an indication of failure. Either way, we ask if the copy is functionally isomorphic, as Hilary Putnam terms it; which is to ask, ‘regardless of how they are constituted, do the original and its copy function alike?’

[To better explain isomorphism, let us brave the technicalities of the more basic (and perhaps seminal) formal definition Alfred Tarski provides in his (1938) “Der Aussagenkalkül und die Topologie” (“Sentential Calculus and Topology”).[i] His purpose in this particular essay is to correlate logic with topology (a branch of geometry studying space). As part of that effort, he proposes his Matrizenmethode (‘matrix method’). A matrix is a system of networked elements (for example, numbers connected by their mathematical relations). Nonetheless, his method itself is not our concern; we merely examine his definition for isomorphic matrices:[ii]

In the English translation:[iii]


symbolizes any given matrix. The

means that the matrix is a system composed of certain elements and logical operations.[iv] The

stands for the set (collection) of all the elements (objects) in the system. For example, a mathematical matrix contains such elements as ‘2 + 2 = 4’ and 4 = 3 + 1’ (each equation is taken together as one formula). We may refer to such formulae more abstractly with generic names (a capital letter) such as A, X or Y, which are called variables, because they can stand for any such unique formula. Tarski ends his definition with



signifies inclusion, so these variables are included within the (first) set. ­We see that next to the W set-symbols are



By specifying two such variables, we assign them a privileged role (and soon we see why). The rest of the symbols between brackets are the logical operators. (These particular markings correspond to their usual forms: , , , ~). Thus the

indicates the logical operation of implication.

is disjunction.

means conjunction;


stands for negation.

These logical operators may conjoin our numerical formulae:

2 + 2 = 4’ 4 = 3 + 1’ 2 + 2 = 3 + 1’

This new formulation is akin to a syllogism, because it means: if both ‘two plus two equals four’ and ‘four equals three plus one,’ then ‘two plus two equals three plus one.’ A sequence such as this instantiates one of Peano’s arithmetical equality axioms, formalized as:[v]

X = Y Y = Z X = Z

What interrelates the two matrices


is a function (F), which assigns an object or operation in one system to its correspondent in another, by means of a process called ‘mapping.’ We might consider how a road-map’s markings index their respective physical locales. Thus, when Tarski writes that there must be a function that ‘maps’

in a ‘one-one’ fashion, he means that, for every element in the first set, there is exactly one in the other set to which it corresponds. Yet, this condition does not by itself specify what objects match which other ones. Let us consider, for example, two different matrices: the Roman and Arabic numerals (more specifically, the system of mathematics performed using Roman or Arabic numerals). We might know that there are Arabic counterparts for each Roman numeral, without knowing yet their specific correlates.

The next condition,


requires that the linking-function map our two specified objects of both sets. This relation serves as the ‘anchor’ against which all other relations will be compared. For example, we may orient ourselves by finding our respective location on a map, and by means of the chart’s proportional distance-relations, we can determine the physical positions of actual locales surrounding us. Likewise, so to establish the correlates between the Roman and Arabic numerals, we should designate a specific point of origin, perhaps: F(4) = IV; that is to say, ‘4’ correlates with ‘IV.’ This way, if we have both sets arranged in their proper series, then we know that the successor of ‘4’ corresponds with the successor of ‘IV,’ without any other indication that ‘5 and ‘V are equivalent.

The rest of the conditions require that the logical operations be preserved too. These operators are necessary for constructing the axioms defining the systems’ shared relations. So for example, we have this pairing of conditions which require that implication and conjunction relations in the first set be found among correlate objects in the second:

Hence we compare:

2 + 2 = 4 4 = 3 + 1 2 + 2 = 3 + 1

II + II = IV IV = III + I II + II = III + I

Here we see that both system’s objects and logical operations correspond; and thus, the same principle of equality may be expressed in both matrices. (The definition’s two remaining conditions require the same for negation and disjunction).] In sum, two isomorphic systems share the same essential constitution, even if its internal parts appear different.

[click images for enlargements]

For Hilary Putnam, isomorphism is a criterion for artificial intelligence. Yet, we must first expand our notion of function. Putnam still means it as an assignment relation between isomorphic systems, but the ‘systems’ in this case are not static like sets of numerals, but instead are sequences of successive states, as though one necessarily brings-forth the next. To illustrate, consider when we enter ‘2’ ‘+’ ‘2’ into an electric digital calculator and then press ‘=.’ After doing so, ‘4’ displays on its screen. Likewise, when we move two abacus-beads, then two more, the result is four beads grouped together. Here we might say that there are two sorts of functions: trans-systemic functions and intra-systemic functions. Trans-systemic functions are the ones that assign states in one system with states in another (4 displayed on the calculator screen; four abacus beads). Intra-systemic functions are ones that assign within the same system an ‘output’ state to an ‘input’ state, if you will (we assign the resulting four beads to the movement of two sets of two).

So, Putnam defines functional isomorphism in general as “a correspondence between the states of one and the states of the other that preserves functional relations.” (This correspondence between states is a trans-systemic function; the preserved functional relations are intra-systemic functions). Thus, if “state A is always followed by state B” in one system, there will be corresponding states in its isomorphic counterpart: the digitally displayed

correlates to the abacus’


because both follow unique states that themselves correspond with one another.[vi] We might also apply the more common usage for the term function, and note that these devices function equivalently – but not identically – insofar as both are capable of the same calculations.

Putnam notes that these devices are physically realized in utterly different ways; for, the calculator’s integrated circuit bears little resemblance to a rack of beads. Hence his provocative conclusion: “so a computer made of electrical components can be isomorphic to one made of cogs and wheels or to human clerks using paper and pencil;” for, humans too can begin in an equivalent state A (receiving the mathematical problem ‘2 + 2’) and result in an equivalent state B (writing the number ‘4’). Hence, if a machine, software-program, alien life-form, or any other such alternately physically-realized operation-system were functionally isomorphic to the human brain, then we may conclude, says Putnam, that it shares a mind like ours.[vii]

Moravec and artificial intelligence theorists also regard a functionally-isomorphic relation with the brain as validation that something has a mind. From the Deleuzean perspective, however, functional isomorphism would not indicate a true human self, and later we discuss why. Yet, we begin first with the subject of our critique.

[i] One may safely skip to the right-bracket at the bottom of page 7 to avoid these technical matters. We merely conclude that, for Tarski, isomorphic systems share the same essential constitution but have internal parts with dissimilar appearances.

[ii] Alfred Tarski, Der Aussagenkalkül und die Topologie, (Fundamenta Mathematicae, XXXI, 1938), p. 107.

[iii] Alfred Tarski, “Sentential Calculus and Topology,” in Logic, Semantics, Metamathematics: Papers from 1923 to 1938, Transl. J.H. Woodger, (Oxford: Oxford University Press, 1956), p.425.

[iv] Tarski defines a concept by first giving a formalized list of the terms which will define it: “The question how a certain concept is to be defined is correctly formulated only if a list is given of the terms by means of which the required definition is to be constructed,” Tarski, “The Concept of Truth in Formalized Languages,” in Logic, Semantics, Metamathematics, p.152.

[v] Martin Goldstern & Haim Judah, The Incompleteness Phenomenon, (Massachusetts: A.K. Peters, 1995), p.80.

[vi] We know he must be speaking of both trans-systemic and intra-systemic assignment functions, because he writes, “To start with computing machine examples, if the functional relations are just sequence relations, e.g. state A is always followed by state B, then, for F to be a functional isomorphism, it must be the case that state A is followed by state B in system 1 if and only if state F(A) is followed by state F(B) in system 2. If the functional relations are, say, data or print-out relations, e.g. when print p is printed on the tape, system 1 goes into state A, these must be preserved. When print p is printed on the tape, system 2 goes into state F(A), if F is a functional isomorphism between system 1 and system 2.” Here we see that the functional relations are the “sequence relations” within one system, and the functional isomorphism F is the correlation-relation between both machines. Hilary Putnam, “Philosophy and Our Mental Life,” Mind, Language, and Reality, (Cambridge: Cambridge University Press, 1975), p.292.

[vii] Putnam, p.293.

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