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Abhishek Bose-KolanuSummer 2014

Turning out the Light on Thomson’s Lamp

In recent years the topic of supertasks, or infinite operations, has become popular once more. Mathematicians, philosophers, and physicists have been drawn into a debate over their logical possibility, physical plausibility, and computational power. Yet to date the logical possibility of a supertask’s completion has not been adequately defined. In this paper we present the first known solution to Thomson’s Lamp, providing a logically sound explanation for the supertask’s completion. We also ground the reader in the math involved, and discuss the philosophical consequences of the different types of infinities at play. We conclude with a Deleuzian reading of Frege’s Begriffsschrift to ground math as an exercise in radical plasticity with far-reaching consequences.

Introduction

J.F. Thomson’s celebrated “Tasks and Super-Tasks” is the source and inspiration for the current debate. Thomson proposed a number of supertasks and each time asked the same question: Can we determine what the state of the world is after the completion of a supertask? For roughly one decade the paradoxes therein led philosophers to conclude the answer was no, and that supertasks were self-contradictory. Then Paul Benacerraf challenged the prevailing notion by demonstrating a fault in Thomson’s construction, thereby vitiating the paradox and permitting discussion of supertasks once more.

While Benacerraf accomplished his aim of disabusing us of the notion that Thomson’s work proves the impossibility of supertasks, he stops short of answering Thomson’s thematic question: Do we have a way of explaining what it means for a supertask to be finished?

For Thomson the only way to begin answering this question is to define the state of a system after the completion of some such supertask, a point he reiterates in his reply to Benacerraf. If this baseline comparison between the system before and after the completion of a supertask cannot be established, then further discourse on supertasks is unwarranted.

We present an original answer to Thomson’s most celebrated paradox, the Thomson Lamp. In addition we discuss two other supertask paradoxes Thomson supplies as way of preparation and elaboration of the mathematical and philosophical concepts at stake. We conclude with an application of Deleuze’s concept of repetition to Frege’s Begriffschrift.

Thomson’s Paradoxes

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Thomson’s main contribution is his insistence on the following question. “Is it conceivable that someone should have completed an infinite number of tasks? Do we know what this would be like?” (2).1

Thomson takes as his inspiration the paradoxes given by Zeno of Elea, and offers several of his own to circumscribe the question of supertasks. Through their consideration we will exhume certain mathematical notions of interest, which later will form the basis for a philosophical exploration of the consequences of supertasks.

We consider three main paradoxes Thomson offers: infinitely divisible chocolate, a man running from [0, 1), and the Thomson Lamp. From the outset Thomson wants to make his inquiry’s goals clear. Immediately following the quotation above, Thomson writes “It is necessary here to avoid a common confusion…to say that some operation can be performed infinitely often is not to say that a super-operation can be performed” (2).

Thomson distinguishes between the capacity to perform a single operation an infinite amount of times and the capacity to perform a single, infinite operation. To do so he offers the first thought experiment we consider. “Suppose (A) that every lump of chocolate can be cut in two, and (B) that the result of cutting a lump of chocolate in two is always that you get two lumps of chocolate. It follows that every lump of chocolate is infinitely divisible” (2).

From this chocolate splitting he derives a means to differentiate between a set of infinitely many possibilities and the possibility of infinity itself (3). “But to say that a lump is infinitely divisible is just to say that it can be cut into any number of parts. Since there is an infinite number of numbers, we could say: there is an infinite number of numbers of parts into which the lump can be divided. And this is not to say that it can be divided into an infinite number of parts” (2). It is crucial to grasp the difference here. Though there are an infinite number of numbers of parts possible, Thomson does not accept that this means there is some number called ‘infinity’ using which we can divide the chocolate into ‘infinity’ parts. “If something is infinitely divisible, and you are to say into how many parts it shall be divided, you have ℵ0 alternatives from which to choose. This is not to say that ℵ0 is one of them” (2).

Thomson has pulled out an essential mathematical concept for us, the relationship between ℕ and ω. ℕ is the set of all natural, or counting, numbers. For us this is the set {0, 1, 2…}. ω, the first transfinite ordinal2, is the first number that comes after all the members of the set ℕ. We can say that ω is the first number after the infinity of numbers in ℕ. The ℵ0 that Thomson mentions is the cardinality, or size as number of elements, of the set ℕ. It is also the cardinality of the set ω, for as we shall see, in Zermelo-Fraenkel set theory with choice (ZFC) the standard construction of natural (and transfinite ordinal) numbers is as sets. Thus Thomson wants us to bear in mind that having a set whose cardinality is ℵ0 does not mean that ℵ0 itself is a member of that set. Here ℵ0 is used

1 Throughout, simple parenthetical cites refer to Thomson’s “Tasks and Super-Tasks” essay.2 We follow von Neumann’s construction of the transfinite ordinals, as is standard.

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somewhat imprecisely, since what Thomson is really talking about is ω, which is a point he will clarify later on. However, the question of whether or not one can slip between the cardinal and the ordinal will form a crucial point of application for our later philosophical explication of the concepts at stake in this intervention.

First, however, we must finish with our chocolate. Thomson blames the imprecision of everyday language for why some think supertasks possible. “If I say ‘It is possible to swim the Channel’ I cannot go on to deny that it is conceivable that someone should have swum the Channel. But this analogy is only apparent” for the reasons given above (3). For us what is important to note is the tense of this example: Thomson is concerned with whether “someone should have swum the Channel,” or, whether someone should have completed a supertask (3). What is being pointed at is the first moment upon completion of a supertask. Throughout, this is point in time is the nexus for Thomson.

To illustrate mathematically the difference between “an infinity of possibilities” and “the possibility of infinity” Thomson continues dividing his chocolate (3). However, he does so in such a way as to render his results mathematically invalid. Recall that Thomson says “there is an infinite number of numbers of parts into which the lump can be divided” (2). Further, “each of an infinite number of things can be done, e.g. bisecting, trisecting, etc.” (3). Finally, “the operation of halving it or halving some part of it can be performed infinitely often” (2, emphasis mine).

What does it mean for an infinite number of numbers of parts to be possible? There is some slippage here as well. Thomson began by claiming “to say that a lump is infinitely divisible is just to say that it can be cut into any number of parts” (2). Strictly speaking this is not true. We could re-specify the thought experiment to require that chocolate may be divided into any number of lumps, provided the total number of lumps does not include the number three. In this manner we could start with one lump, halve it into two, halve those into four, etc. We would still have infinitely many numbers of parts (since the series can progress without upper bound), but we would have excluded a possible number: three. This is because ω - 1 = ω. While at first this statement may seem dubious, a close reading of the formal definition of ω later on will banish any doubts. (As a first blush of the argument, it is simple to see that if ω - 1 ≠ ω, then there is some number one less than ω that nonetheless is also greater than all other members of ℕ. However, this contradicts the definition of ω as the first such number.)

Given the three quotations above, Thomson imagines a chocolate-cutter capable of dividing any lump of chocolate with any number of cuts at any step in the process. Thus our series of chocolate, expressed in total lumps of chocolate, could be: 1, 2, 3, 4, 5, 6…. Advancing by 1 is given by cutting only one of the previous two sub-lumps. Or it could progress 1, 2, 4… by halving each sub-lump at each stage. Or a particularly obstinate chocolate-cutter might refuse to cut at all, yielding 1, 1, 1, 1…. Or, we might choose to cut at some unspecified time X, suddenly transforming from 1 into 2 in a manner analogous to grue: 1, 1, 1… ?, 2.

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This malleability of when and how to cut weakens the thought experiment mathematically, for such a series is no longer a mathematical series. A series is a sequence, which is a function on a countable (has cardinality ℵ0) and totally ordered (every element is either greater, smaller, or equal to another) set. A function maps elements from its domain to its co-domain. Crucially, every element in its domain must map to one and only one element in the co-domain. However, in our example above one lump of chocolate might map to two, one, or some unspecified probability of either one or two. Two might map to three or four, etc. Hence, this example is not a function and therefore cannot be a series. The notion of a function elaborating a specific relation between domain and co-domain will aid us in our exploration of two types of infinite limits later on.

Thomson’s Lamp Introduced

The canonical example on the logical impossibility of supertasks is Thomson’s eponymous lamp. He describes to the reader “certain reading-lamps” that have a button in the base (5). Pressing the button switches the state of the lamp. If it was off, it is now on, and vice versa. Applying the infinite geometric series xn = 1 / 2n-1

(which sums to two) where n is the term of the sequence, Thomson asks what the final result is if we make one jab for each time interval (1, ½, ¼…). “Suppose now the lamp is off, and I succeed in pressing the button an infinite number of times, perhaps making one jab in one minute, another jab in the next half-minute, and so on…After I have completed the whole infinite sequence of jabs, i.e. at the end of the two minutes, is the lamp on or off?” (5). Thomson declares the question impossible to solve, since, “I did not ever turn it on without at once turning it off…[and] I never turned it off without at once turning it on. But the lamp must be either on or off. This is a contradiction” (5). Hence the terminal state of the lamp remains a mystery.

Benacerraf successfully challenges the construction of this proof by contradiction with the following critique. Demarcating the beginning3 and final instants of time for the lamp task as to and t1 Benacerraf explains, “From this it follows only that there is no time between t0 and t1 at which the lamp was on and which was not followed by a time also before t1 at which it was off. Nothing whatever has been said about the lamp at t1 or later” (Benacerraf, 768). He concludes that Thomson has been asking for the impossible by applying conditions about the state of the lamp that are prior to t1 to the lamp at t1. Thus there is no final, ωth act. Thomson acknowledges as much in his rebuttal to Benacerraf, writing “If the successive transitions of S [the system] are caused by successive acts of an agent, then for the purposes of this Gedankenexperiment we want the agent not to perform an ωth

act; it would only get in the way. But then there is still a last uncaused transition, and it is this that we want to inquire about” (ZP, 134).

From this we take that Benacerraf is right to vitiate Thomson’s proof on the grounds that the condition of the lamp’s state immediately switching state only applies to moments in time before t1. However, Thomson is also correct in his rebuttal to note that the central question remains unanswered by Benacerraf’s challenge: what state is the lamp in at the

3 Throughout, ZP refers to Thomson’s essay rebutting Benacerraf in Zeno’s Paradoxes.

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conclusion of this supertask? If no such description of the lamp’s final state can be given, then we can follow Thomson and rightly conclude that it may not be sensible to speak of supertasks, or at least of this supertask.

Running [0, 1)

Thomson gives another example of a supertask, a runner along a racecourse from [0, 1). Reconstructed, the thought experiment is as follows. Consider a runner who must pass through all the mid-points between the interval Z that is [0, 1). He must touch ½, ¾, etc. Since the interval is closed on the left and open on the right, 1 is not part of Z, but 0, his starting point, is (10). “Further: suppose someone could have occupied every Z-point without having occupied any point external to Z. Where would he be? Not at any Z-point, for then there would be an unoccupied Z-point to the right. Not, for the same reason, between Z-points. And, ex hypothesi, not at any point external to Z. But these possibilities are exhaustive” (10).

Benacerraf’s response to this paradox is exactly the same as his response to the lamp. Since some time after the completion of the supertask must come (else, the supertask’s time to complete is unbounded and therefore it does not complete), conditions about the position of the runner given by the supertask can apply only to positions before time t1. In his response to Benacerraf Thomson later acknowledged he thought of the [0, 1) example “as a kind of joke,” since “[t]he resulting situation is simply that [the runner] occupies 1” (ZP, 130). However, he acknowledges the force of Benacerraf’s counter-argument, both with respect to his construction of the lamp and his attempt to analogize the [0, 1) paradox to the lamp (ZP, 130-31). Nonetheless, he maintains that there exists “some conceptual difficulty about the idea of a lamp having been turned on and off infinitely often, because, roughly speaking, of the question about the state of the lamp immediately afterwards,” and that this concern is independent of the particular way in which he sought to find a contradiction via his lamp example (ZP, 130, 131-32).

There is still something to be gained from Thomson’s thought experiment on the interval [0, 1), and that has to do with the status of ω and ω + 1 as ordinals. Thomson notes that the runner who successfully runs an infinite sequence of midpoints from 0 to 1 (and arrives at 1) “is not a sequence of type ω but a sequence of type ω+1 (last task, no penultimate task), the sequence of the points 0, ½, … , 1 in Z’s closure” (12). Our sympathies lie with Thomson here, for it is precisely the closure of the interval [0, 1) through the addition of the point 1 that is brought about taking the series from ω to ω + 1. This is the same reason why in his rebuttal Thomson points to the fact that the ωth transition remains exactly on the bound, with ω + 1 providing the first moment in time after the completion of the supertask. To clarify what is meant by ω and ω + 1, we need now to pause and consider numbers.

Numbers, Transfinite and Ordinal

As previously stated, ω is the first transfinite ordinal, the first number larger than all members of ℕ. ω + 1 is simply the next number. But how shall we define numbers, and

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how can we define ω? Following the standard construction of numbers in ZFC, we hold that the ordinal numbers are sets that include all the numbers below them starting from and inclusive of zero.

For example, the number 5 is {0, 1, 2, 3, 4}. Likewise, the number ω must include all the members of ℕ. Note that a number is not present in its set (5 does not appear in 5). Note also that the cardinality of the set is equivalent to the name we give it (the cardinality of the set 5 is in fact 5), a point we will return to in our solution to Thomson’s Lamp.

Given these constructions, how does one move from one number to another? Movement up the chain of natural numbers is given by the successor function: S(a) = a {a}∪ , with 0 defined as the empty set, ∅. Thus 1 is ∅ {∪ ∅}, which yields {∅},the set containing the empty set. Alternatively we may write 1 as {0}. To get 2 we reapply the successor operation, yielding S(1) = 1 {1}, which gives {0, 1}, etc. However, ω presents a ∪different case.

ω is a special ordinal because it is a limit ordinal. A “limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if and only if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists an ordinal γ such that β < γ < λ. Every ordinal number is either zero, or a successor ordinal, or a limit ordinal” (Wikipedia, “Limit ordinal”). We will comment on this extraordinary definition shortly, but first we must ask, how can we generate a limit ordinal?

As we can see, a limit ordinal cannot be given by the successor operation, since every ordinal given in such a manner can produce another ordinal greater than it by repeated application of the successor operation. The operation required to produce a limit ordinal is the supremum. A limit ordinal is the supremum of all the ordinals below it, which is given by taking their union. In the case of a normal, finite number we can see that the supremum exists conventionally speaking, as the maximum. For example, the set 5 being {0, 1, 2, 3, 4} gives a supremum of 4, which is the ordinal immediately prior to 5 (and does in turn contain all the ordinals below itself). However, for the limit ordinal ω there is no ordinal immediately prior to it, by definition. It is against this infinite succession of ordinals that one must take the union, and in so doing produce a supremum where no maximum exists, thereby generating ω. Once generated, ω is free to appear as an element in the set of a larger ordinal via the successor operation, thereby paving the way for ω+1, ω+2, etc.

This generation of a supremum without a maximum may seem scandalous. Mathematically speaking, it stands on solid ground. Given ZFC, it follows that each of the ZFC axioms must be accepted axiomatically. Present in the ZFC set is not only an Axiom of Infinity (an infinite set exists), but also an Axiom of Union (defining the union operation). ω is what happens when infinite successor operations are possible and the supremum of that infinite sequence is taken. It is a supremum of a set with no maximum, made possible by an Axiom of Union that is every bit as powerful as the Axiom of Infinity.

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What are we to make of Thomson’s argument when he writes that the runner at position 1 “is not a sequence of type ω but a sequence of type ω+1 (last task, no penultimate task), the sequence of the points 0, ½, … , 1 in Z’s closure” (12)? While earlier we agreed with him, perhaps this question is worth a second look. Why does the runner ending up at 1 require a task of order-type ω + 1, rather than a task of order-type ω? Were the task to remain an order-type of ω, it would still involve members of ℕ, which means that ω itself would never be reached. This much follows readily from our construction of ordinals as containing the whole numbers below them, starting from and inclusive of zero. But what does it mean to generate ω, that is, to reach ω only through a task of order-type ω + 1? It means that the supremum alone is insufficient, but instead a supremum plus an additional successor operation, S(ω), is required. Seen in this light it becomes clear that for all supertasks, the act of their performance is none other than the Axiom of Unity threading the supremum of infinite successors brought about by the Axiom of Infinity. And further, that all questions about the completion of supertasks must necessarily be about order-types ω + 1, the infinite supremum plus an additional successor.

We can see that the completion of a supertask requires an evaluation of some sort, but what exactly is being evaluated? We know the evaluation must take place at point ω + 1, but is it an evaluation of the function representing the supertask, or an evaluation of the state of the world? If it is the former, the function is undefined at ω + 1, which is exactly the reason why Benacerraf’s critique is trenchant. If it is the world, then we must ask what is meant precisely by ω + 1 in the context of the world. Is it not instead world + 2 minutes (assuming the ω tasks are accomplished, along with the final transition of the system at moment ω, within two minutes)? It must obviously be world + 2 minutes, which somehow means the world is capable of permitting the inherence of supertasks within it within finite time bounds. The following quote from Thomson’s rebuttal to Benacerraf is worth reproducing in full.

“In general, the idea of an ω-task arises from consideration of a bounded ω-sequence of points on the real line. If the sequence is not bounded there is not even the semblance of a likelihood that the task can be completed; if it is bounded, the sequence will have a least upper bound and there will be some question to ask about the state of the world or about what happens at some time corresponding to that bound” (Thomson, ZP, 134).

Therefore, if we conclude that supertasks are not only logically but also physically possible, we must be forced to acknowledge the system we call world can only be represented with cardinality greater than or equal to 2ℵ

0. That the cardinality is greaterthan or equal to 2ℵ

0 follows from the definition of the power set applied to the natural number line (or equivalently to the set ω).4

4 In general the cardinality of the power set, or set of all subsets of a given set, is given by 2 raised to the cardinality of the set. So if our set has three elements, the cardinality of the power set is 23 = 8. A simple proof can be given by replacing each element of the set with either 1 or 0 if it is in or not in the subset currently being considered. The empty set and full set are therefore represented by a subset with all 0’s or all 1’s.

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Solving Thomson’s Lamp

Can we establish that there exists a supertask that is logically possible? We maintain the following original solution constitutes the first known solution to Thomson’s Lamp ‘on the merits.’ Whereas Benacerraf’s solution involved pointing out a contradiction in the particular proof Thomson attempted to construct after assuming the existence of his lamp, our solution takes as given the logically defensible portion of the original thought experiment and proceeds to answer the nexus question: What state is the lamp in after two minutes of infinite button presses have elapsed?

As we have shown, there are compelling reasons to continue to ask this original question Thomson posed, and Thomson himself corroborates the possibility of a proof’s appearance, writing “I am now inclined to think that there are no simple knock-down arguments to show that the notion of a completed ω-task is self-contradictory” (ZP, 131). Given the original experimental setup, and ignoring Thomson’s original and invalid claim that no solution is possible due to contradiction, what state is the lamp in after two minutes of infinite button switches?

As Thomson notes, an even number of switches will maintain the lamp's state while an odd number will change it (5). The only question that requires answering is, “Is infinity even?” It is, when we use the set-theoretic definition of even to mean capable of subdivision into two disjoint subsets of equal cardinality, or size. As we will show, this set-theoretic definition of even is equivalent to the numerical, “divisible by 2” definition to which we are accustomed.

It is trivial to show that a countably infinite set can be divided into two disjoint subsets of equal cardinality. Consider the set ℕ of natural numbers. From the definitions for even and odd we know that if a given integer j is even, j+1 is odd. From the definition of ℕ we know that for all j, j+1 exists. Accordingly we can subdivide ℕ into two disjoint subsets of evens and odds respectively, both of which have the same cardinality as each other.5

One might object that the set-theoretic definition of even may not apply, since Thomson was asking a question about the number of tasks performed, and of what number (1 or 0) would represent the lamp’s final set. However, the two definitions are equivalent, once the construction ‘number’ is understood set-theoretically. We traditionally define even and odd as follows.

Eqn 1.1 An even integer n = 2k where k is some other integerEqn 1.2 An odd integer m = 2k + 1 where k is some other integer

We can rewrite these definitions in terms of the set-theoretic definition given above. Let us define a natural number r as a set s that contains all natural numbers that precede r,

5 They happen to also have the same cardinality as the original set ℕ, since all three sets are countably infinite. This quality is at the root of Galileo’s famous perfect squares paradox.

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starting from and inclusive of 0. So again we have the integer 5 defined as the set {0, 1, 2, 3, 4}. Note that the cardinality of s is equivalent to r. Let us use the set-theoretic definition of odd and even to define the oddness or evenness of set s. If set s can be divided into two disjoint sets (whose union contains all its elements) that are equinumerous (whose cardinalities are equivalent), then we say set s is even. If not, set s is odd.

Let us denote the sets u and v as the two disjoint subsets into which we divide s. Let us denote the cardinality of u and v as w and x respectively. If and only if w = x, can we say that the set s has successfully been divided into two disjoint subsets of equinumerality. However, if w = x, then we can express r, the cardinality of set s, as 2 * w. If not, we express r as 2 * w + 1, since at most the count can be off by one (since we have been choosing from among two subsets, and thus if we had two extra items left in our parent set, then each subset would have received one, etc.). Thus we have arrived at our so-called ‘numerical’ definition of even from the set-theoretic.

We can see that the two disjoint subsets into which we divide the button presses of the Thomson Lamp can be none other than the sets of ‘switch to on’ and ‘switch to off,’ just as they appeared in Thomson’s use of the Grandi sequence6 (6). What is novel is the knowledge that the termination of the supertask depends, in this case, on the initial configuration. If the lamp was on, it remains on at the conclusion. Else, it remains off.

One might object that even though the cardinalities of the two subsets are equivalent, both are ℵ0, which is the same cardinality as our original set ω. Hence, it might be impossible to determine which of our ℵ0 subsets terminates ‘after’ the other, voiding our application of evenness to the proof. However, such worries are easily put to rest once we view the evenness proof as a restatement of 2ω = ω. That 2ω = ω is clearly understood once order-types are applied rigorously. Since ω is an ordinal we know it is specifying a positioning, or sequencing of elements. It is the order-type ω. Hence, to say that 2ω = ω is simply to say that ω pairs in sequence possess the same order-type as ω singletons in sequence.

At first this explanation does not seem to extricate us from our quandary, since the pairs are obviously (switch, switch) pairs of button presses (Thomson’s claim of never once turning the lamp on without turning it off and vice versa). However, we can use our initial knowledge about the lamp’s state at the beginning of the experiment to fix the state of the lamp upon completion. Since the ω pairs of (switch, switch) will take place regardless of what state the lamp is in at the beginning, and since we know that an even number of state transformations preserves state, we can see that the initial state of the

6 The sequence is 1 – 1 + 1 – 1…. Thomson assigns the states ‘0’ and ‘1’ to ‘lamp off’ and ‘lamp on,’ and asks what the sum of Grandi’s sequence can tell us about the lamp’s final state. In our opinion the question is misguided, as the sum of this divergent series is given as ½, which is not a reason for supertasks being internally self-contradictory, as Thomson supposes, but instead is due to the particular definition of a Cesàro sum, whereby a second converging series is constructed by the partial means in order to find a consistent value to assign as the sum for certain divergent sequences.

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lamp can be removed without affecting the even number of state transformations that take place. And in fact, it is removed, since the initial state does not count as a button press. Hence, the initial state remains the final state as previously described.7

A particularly vicious reader might inquire if 3ω = ω, thereby demonstrating an arbitrary decision to subdivide into two disjoint subsets instead of three. Perhaps this argument points to slippage between the set-theoretic and numerical definitions of parity, which earlier were claimed to be equivalent. No such slippage is evident. That a number (set) may also be divided into thirds does not mean it cannot also be even.

That many possibilities for subdivision exist should not dissuade us from maintaining the possibility for subdivision into two disjoint subsets of equal cardinality. All that remains is to show that the possibility for a quantity to be subdivided in more than one way does not void the property of subdivision into two in which we are interested.

Division into singletons requires division into ℵ0 many subsets. Division into pairs also requires ℵ0 many subsets. So does division into thirds, or fourths, or fifths…. Division into one subset of size ℵ0 is the case where the subset is equivalent to the parent set, and no division needs take place. Strictly speaking, division into equinumerous subsets of any size < ℵ0 requires ℵ0 many subsets.

What this means is that our first inclination, to hold on to the evenness of ω despite the fact that it is also divisible into subsets of many other sizes (indeed, infinitely many), is correct. If this poses a problem for the reader it is due only to the limitations of language and imagination, rather than to some logical inconsistency. In the first case, with the challenge of dividing into a sequence of triples instead of pairs, we can easily imagine a number that could be divided either way, but which nonetheless possesses the property of evenness in which we are interested, e.g. six. Were six button presses to take place, we could order them as {(switch, switch, switch), (switch, switch, switch)}, or as {(switch, switch), (switch, switch), (switch, switch)}. In either case, the fact that six is capable of subdivision into two disjoint subsets with equivalent cardinality is sufficient to guarantee that the final state of the lamp remains the same as its initial. We are also protected from an instance of a subdivision into triples that is not also possible of subdivision into pairs, since we know that there are infinitely many button presses, and thus for any button press i that happens before the two minutes is up, i+1 exists (because ω - 1 = ω). Or put another way, for any set of j triples, we know that j+1 triples exists, and thus there is no division into subsets of an odd size that cannot also be divided into pairs. It is for this reason that ω, like all limit ordinals, is even.

7 For another way of thinking through this proof, we might consider what was at stake when Cesàro sums became the accepted way of fixing the sum of infinite divergent series. The need for a way to fix the sum was precisely because alternative methods, all of which seemed reasonable, gave inconsistent results. If one took Grandi’s sum as 1 + (-1 + 1) + (-1 + 1)… then the answer was 1. If one grouped the terms as (1 - 1 ) + (1 - 1)… then the answer was 0. We can view the initial state of the lamp as an instruction on how to group the terms.

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Can Thomson’s Lamp Generalize?

We can now put to rest Thomson’s anxieties about having a “method for deciding what is done when a super-task is done” (6). However, the generalizability of our results is an open question. “One difficulty, then, about the question of whether ω-tasks can be completed is that there are different kinds of them, and there is no reason to think that in regard to completability they stand or fall together” (ZP, 136). Nonetheless, it is a significant result to be able to declare that there exists a supertask for which a logical rendition of the state of the world following its completion may be given. That this supertask happens to be the ur-supertask is a happy coincidence.

As initial guidelines for considering a categorization of supertasks, we offer the following schematization of the Thomson Lamp. There are a finite number of states (two), transition between the states is deterministic (it always inverts the lamp’s value), there is only one transition mechanism (pressing the button), the starting state of the lamp is given, an infinity of transitions are completed in a bounded time interval, and the question concerns the state of the lamp upon the supertask’s completion. In passing we note the task may be modeled as a deterministic finite state machine.

On the question of the relative difficulty of supertasks, we note one example from Thomson’s original paper that to us seems manifestly more problematic than the lamp. Thomson analogizes the lamp experiment to a machine capable of writing all the numbers of π in two minutes. To this machine he adds another, “parity-machine” that outputs 0 or 1 if the digit it is fed is even or odd (5). He asks, “what appears on the dial after the first machine has run through all the integers in the expansion of π?” (5).

On our view the analogy is inadmissible, as the initial state of π’s decimal expansion gives us no information about the end state (final integer). As we have shown, in the lamp experiment the starting state of the lamp fully determines its ending state. There is no additional information necessary in order to arrive at a determination of the end-state other than the setup and the initial state. Despite possessing the setup and the initial state in the case of the π and parity machines, no such final statement is definitively possible.

Consider the following example of an original supertask whose difficulty may also exceed the Thomson Lamp, for different reasons. Consider again the interval Z, from [0, 1), and the runner who must proceed by mid-points. Now let us add an additional constraint: the runner is to avoid every point in the sequence of third-points (1/3, 5/9, etc.). If our linguistic sensibilities are strained, we can imagine the runner jumping over such points. This supertask requires not only the completion of a supertask (proceeding by midpoints), but also the completion of a negative supertask (avoidance of third-points). How does the inclusion of an infinite exclusion contribute to or lessen the difficulty of a supertask?

There is no convincing a priori reason for assuming that all supertasks are alike, nor that all will exhibit tractable solutions for determining “what is done” when they are done. Further study on this question will no doubt make use of the rich resources available in

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the study of computational complexity, in particular with respect to such constructions as Turing machines with oracles (black boxes capable of completing operations from specific complexity classes), finite and infinite state machines, quantum models of computation, etc. With a logical proof of Thomson’s Lamp in place, the way is now clear for an earnest discussion of hypercomputation freed from the trappings of pure fancy.

Order (Position) vs. Size (Distance)

Now that we have a handle on how to generate and use ω, how can we talk about it as an ordinal? Recall that the point of an ordinal number generally speaking is to denote position or place, as in first, second, third etc. ω, then, is the first position available to us upon completion of an infinite set of operations. More generally speaking, the transfinite ordinals do exactly what their name suggests: provide us a way to talk about position that transverses infinity.

To see what is meant by ‘transverses’ more clearly we shall consider the formal definition of limit ordinal given earlier.

“an ordinal λ is a limit ordinal if and only if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists an ordinal γ such that β < γ < λ” (Wikipedia, “Limit ordinal”).

ω is a limit ordinal precisely because there is an ordinal smaller than it, and for all ordinals smaller than it there is an ordinal larger than that smaller one that is still smaller than ω. One would be justified to say that the number of such ordinals is infinite. ‘But wait!’ an impassioned reader might cry, ‘what about ω being the first ordinal after ℕ?’ Herein lies the beauty of ω. There is no largest member of ℕ and yet ω exists. In fact, we define ω in precisely this fashion.

Thus ω is a moving target. If we rely on the intuitive notion of tasks as discrete operations with a beginning and end, then ω will never come since by definition there is always one more task to complete. Hence the word operation is ill-suited to speaking of ω or ω+1, since ω is constantly advancing into the distance, as it were. Benacerraf is right to point out that the confusion surrounding Thomson’s apparent paradox may be due in no small part to the insufficiency of language colliding from two different domains, here the notion of a ‘task’ with the notion of the ‘infinite’ (Benacerraf, 782). Nonetheless, by bracketing this infinite movement of ω within a definition, we may then talk about and even beyond ω, which is the beauty of the mathematical form. It is a logical and powerful consequence of the Axioms of Union and Infinity taken together.

However, we must attend to our dear reader’s passionate objection. Such a reader would be right to note something of massive significance for the philosophical consequence of Thomson’s Lamp, and transfinite ordinals more particularly. Namely, they inaugurate a concept of position or order decoupled from size or distance. Intuitively speaking, we conflate order with size regularly. The first place has the smallest distance from zero. The second the second smallest, and so on.

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What must be stressed is the intimate relation between distance from zero and size. Recall our nested definition of numbers, whereby a number is the set of all numbers below it starting from and including zero. Recall as well that the cardinality of such a set is equivalent to the number that set is. Distance from zero is enumerated via the successor function, which successively adds one element to the set. This enlargement by one corresponds to a positional increase by one as well, moving us from an ordinal to its successor ordinal. The correspondence between increase in size and increase in position is necessary, logical, and irreversible.

Only in the limit ordinal does the relationship between the cardinality of the set in question and its place as an ordinal become unmoored. By definition the number of ordinals less than ω is infinite. Specifically, it is countably infinite. To say a set is countably infinite is to say that it can exist in bijection with ℕ. Bijection identifies a one-to-one correspondence between terms. Following Cantor, whose work birthed the study of the transfinites, we say all these sets have cardinality ℵ0. Even the sets ω+1, or ω2 (the second limit ordinal) have cardinality ℵ0 (the first transfinite with cardinality greater than ℵ0 is ω1).

Strictly speaking, then, we cannot say that the number of numbers in the set ω is greater or less than the number of numbers in the set ω+1. Yet, we can say that ω+1 occupies a different position from ω. This is the first time in our discussion of numbers’ positions that two numbers with equal cardinality have been identified as occupying different positions. It is likewise the first time that size and order are no longer co-extensive, hence our earlier criticism of Thomson’s slippage between ℵ0 and ω in our discussion of chocolate.

The Metrical and The Ordinal

What are we to make of this decoupling of position and size? In reality, the alliance between size and position masks a much more profound alignment absent in Thomson and Benacerraf. Both pass over a remarkable relation between two very different kinds of limits employed: the metrical, and the ordinal.

A metric, or distance function, provides a distance between elements of a set. A metric space is a set with a metric. The real number line (Euclidean space of dimension one) is the metric space at work for the sum of infinite geometrical series as discussed in Thomson, e.g. the runner running the distance [0, 1). What must be emphasized here is the crucial difference between the limit at work in the provision of a sum for an infinite series and the limit at work in the first limit ordinal, ω.

Consider the definition of the metrical limit.

“We call x the limit of the sequence (xn) if the following condition holds:

For each real number ε > 0, there exists a natural number N such

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that, for every natural number n > N, we have |xn - x| < ε.

In other words, for every measure of closeness ε, the sequence's terms are eventually that close to the limit. The sequence (xn) is said to converge to or tend to the limit x, written xn x or limn∞xn = x.

If a sequence converges to some limit, then it is convergent; otherwise it is divergent” (Wikipedia, “Limit of a Sequence: Formal Definition”).

What is actually at work here is two infinites, moving in opposite directions. We have first the infinity of the increase in input size, the n in xn. For this infinity only a cardinality of ℵ0 is necessary, so the set ℕ could suffice. We then have as second infinity the closeness measure ε, which may be made arbitrarily small provided it remains greater than zero. What our definition says is that no matter how small we make ε, there exists some natural number beyond which the distance between outputs of the function and our limit is less than ε. Thus, no matter how close to the limit we wish to force our output, we are capable of doing so.

However, this second infinity, that of the closeness measure ε, requires an infinity of cardinality 2ℵ

0, that of the set of reals ℝ. In each of Thomson’s supertask examples that we have examined (chocolate, [0, 1), the lamp) there has been a tacit acknowledgement that there is a domain of ℕ for the function representing the supertask. This domain has functioned indexically, providing the current position in the execution of the supertask. The co-domain for the output of this function has been different in each case: ℕ (total lumps of chocolate), ℝ (position on the interval), or {0, 1} (lamp off or on) respectively. However, in the latter two supertasks there has also been a second function, that of the timekeeper. This function maps the index of the supertask’s operation to the geometric sequence xn = 1 / 2n-1, which is taken to provide successive moments of time in our two minute sequence to infinity.8

As we have seen, any supertask requires a bounded period of time for its execution, within which infinite tasks must be executed. Thus, there will always be an infinite series on the reals to provide a decreasing time per execution. It is because the set ℝ of our metric space, to which our distance function applies, is infinite that we can produce arbitrarily small distances, permitting an infinite timekeeper series to accelerate our supertask to completion. And in fact, this set must have cardinality > ℵ0, since acardinality equal to ℵ0 would not provide for arbitrarily infinitesimal intervals to be cut.

Hence the limit in this sense is a measure that only makes sense with distance (the metric); whereas, the limit of ω was precisely one that forced us to decouple position (order) from size (distance). So when Thomson asks us to consider the sum of an infinite series as a limit, and uses that to speak of a set of ω tasks, he appears to be equivocating 8 In the [0, 1) example this function also gives the runner’s position, so the supertask function and timekeeper function are mathematically identical. It should also be noted that strictly speaking, the chocolate division was not a supertask as its operation was not finitely bounded by time.

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between infinity treated in two very different manners. In point of fact, this equivocation is likely the cause for Benacerraf’s critique, which spends most of its time concerned with the approach to an open upper bound on a segment of the reals (e.g. the vanishing genie). Can we find a way to unify the two infinities?

We maintain that the definition of ω as a limit ordinal is directly related to the definition of the limit of a function, once the difference between number and variable is properly understood. It is precisely the inexhaustibility of ℕ (necessary for ω) that allows us to state definitively that there is some sufficiently large n ∈ ℕ that gives an xn argument for the function whose output satisfies all possible closeness measures greater than zero (necessary for our metrical limit). What is seen at work here is the Axiom of Infinity, whose under-appreciation accounts for Thomson’s difficulties with imagining an infinite series’ summation.

He objects to the possibility of summing an infinite series only because he requires the sum to inhere as a discrete, accomplished task, rather than an ongoing operation. He states as much when he writes, “The belief that one could add together all the terms of an infinite sequence is itself due presumably to a desire to assimilate sums of infinite sequences to sums of finite ones,” here chastising another philosopher (9). What the definition of a function’s limit for metric spaces shows us is that the ‘sum’ here is understood in the only way possible, in relation to a function’s output as its input increases without bound. What is up for discussion is the behavior or trend of a function as its input is manipulated. This is the behavior of a function and its variable, not a finite sum and its numbers.

This shift in perspective is foregrounded as necessary the moment one pauses to consider what it means to take the sum of an infinite series. The number of terms of the series is permitted to grow without bound, or ‘diverge to infinity.’ But what is the series? The series itself is a sequence, which is a function. Recalling our discussion of chocolate, functions have certain properties. What we are to understand here is that from the outset, once a function is invoked, we are dealing no longer only with numbers (nor still only with finite numbers), but instead with arguments. And once a function’s domain is identified as infinite, by extension its argument is permitted to diverge to infinity. With this established, we must be aware that what we are dealing with is not a function defined over a discrete, finite set of numbers but instead a function whose argument is properly variable.

What the metrical limit notion of infinity does for us when calculating an infinite sum is to take advantage of the distance function for an infinite space to permit us to increase the input size without bound while infinitely decreasing the distance-to-the-limit represented by the closeness measure, provided it remains greater than zero. That the limit is capable of satisfying any arbitrary measure of closeness is what guarantees it is ‘never reached’ by the sum of our infinite series.

So far so good, but have we really answered the most basic objection to the summation of an infinite series, that it could never take place? Should we fail here, must we not also

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abandon our proof of solution to the lamp? Consider what happens if we set the closeness measure to zero. The only argument capable of satisfying that closeness measure would be to evaluate the function at xω, the first position after the set of natural numbers. We see that the only number capable of reducing distance between series and sum (metrical limit) to zero is precisely the first number that decouples distance from position (limit ordinal). Paradoxically (but only to the non-mathematician), the number that could guarantee distance travelled, as in the Thomson experiment of walking the midpoints from [0,1), is the number that requires us to let go of distance.

Towards Deleuze

One might object that the apparent paradox we have unearthed concerning the decoupling of size from distance has to do with mere intuition and has no place in serious discussions of math. Such skeptics might be among the few to believe themselves when claiming nonchalance at the knowledge that a larger distance is not, in fact, larger. We contend that the matter raised concerns the very heart of an epistemology of mathematics that reveals itself only in play, as we have just done with Thomson’s Lamp. For play is the essential genetic field of math.9 To understand this epistemology of play, we turn to Deleuze’s concept of repetition.

Difference and Repetition is a notoriously difficult book, and one that Thomson and Benacerraf likely did not have access to when they were writing (Paul Patton completed the English translation in 1994). We will take the briefest of glances at the titular concern, showing first a naïve treatment of the terms followed by a more essential interpretation. In so doing we will expose what we consider the essential genetic element of math: its radical plasticity. This radical plasticity we find in the most unlikely of places, none other than Frege’s Begriffsschrift.

ω is Repetition

From the beginning Deleuze is focused on finding “a concept of repetition in which physical, mechanical or bare repetitions (repetition of the Same) would find their raison d’être in the more profound structures of a hidden repetition in which a ‘differential’ is disguised and displaced” (DR, xx).10 Continuing, Deleuze writes, “generality expresses a point of view according to which one term may be exchanged or substituted for another. The exchange or substitution of particulars defines our conduct in relation to generality” (DR, 1). On the other hand, authentic “repetition is a necessary and justified conduct only in relation to that which cannot be replaced. Repetition as a conduct and as a point of view concerns non-exchangeable and non-substitutable singularities…it is no more possible to exchange one’s soul than it is to substitute real twins for each other” (DR, 1).

We can see that two poles are being established: bare repetition (generality, Identity or the Same, representation) and authentic repetition (irreplaceability, the differential). In

9 A more extended reading would benefit from Huizinga’s Homo Ludens, in particular on the relation between play and the Law. 10 Throughout, parenthetical citations marked ‘DR’ refer to Difference and Repetition.

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the first we have repetition and difference subordinated to Identity. “For difference implies the negative, and allows itself to lead to contradiction, only to the extent that its subordination to the identical is maintained” (DR, xix). The subordination of difference to Identity requires the operation of a Law that maintains its power through a regime of generality, founded on differing degrees of resemblance and empty variation (DR, 1-3). On the other hand, “[i]f repetition is possible, it is due to miracle rather than to law. It is against the law: against the similar form and the equivalent content of law. If repetition can be found, even in nature, it is in the name of a power which affirms itself against the law…underneath laws, perhaps superior to laws” (DR, 2). This event of authentic repetition subordinates Identity to Difference, such that “a concept of difference without negation” becomes possible (DR, xx).

The construction of the members of ℕ follows a simple law, the repetition of the successor function. In this manner starting from 0 all the natural numbers can be given. As previously discussed, the emergence of ω necessitates a new operation. As a limit ordinal infinite successor operations are permissible (indeed, required) below ω, however the existence of ω can only be guaranteed with the application of the supremum: a union over the infinitely many successively generated ordinals. After this supremum the successor operation resumes, producing ω + 1, ω + 2, etc.

ω appears to cleave quite closely with Deleuze’s feelings on authentic repetition as autochthonous and originary. “This is the apparent paradox of festivals: they repeat an ‘unrepeatable.’ They do not add a second and a third time to the first, but carry the first time to the ‘nth’ power. With respect to this power, repetition interiorizes and thereby reverses itself: as Peguy says, it is not Federation day which commemorates…the fall of the Bastille, but the fall of the Bastille which celebrates and repeats in advance all the Federation days; or Monet’s first water lily which repeats all the others” (DR, 1).

Repetition concerns that which cannot be replaced, such that an initial event repeats in advance those that would come after it. One cannot establish an unbroken chain of equivalences between a historic event, its commemoration in the first celebration, in the second, in the third, etc. The fourth annual celebration is not the same as the third, nor still the first. Though there is a degree of resemblance it is not “the qualitative order of resemblances and the quantitative order of equivalences” under generality (DR, 1). That it is not the quantitative order of equivalences has been shown. The case of qualitative resemblance is more tempting, though Deleuze cautions us that “[r]epetition can always be ‘represented’ as extreme resemblance or perfect equivalence, but the fact that one can pass by degrees from one thing to another does not prevent their being different in kind” (DR, 2). A better candidate for qualitative resemblance might be a mass produced commodity.

Seen this way, ω is the limit ordinal that repeats in advance all the limit ordinals to follow. It is the first closure of an infinite set, ℕ, permitting the repetition of ℕ in ω2 (second limit ordinal), ω3 (third limit ordinal), etc. ω2 and on are reached in exactly the same way ω was, through repeated successor operations forming ω + 1, ω + 2…. At first it appears like ω is a facile repetition, belonging to the order of equivalences under

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generality. Several factors stand against that reading. First, ω requires the first application of a supremum as opposed to one more successor operation. Second, ω inaugurates a rupture between the coextensivity of position and size. With ω the position of an ordinal is no longer indexible with reference to its cardinality. What has been disrupted is the applicability of a distance metric, which at most could cover ℕ.

But what is the application of a distance metric? It is the regular application of a general function for giving distance between all points in a given set. As we saw, the applicability of a distance function to a set is what makes possible a metric space. In this mathematicization of space, it is precisely the reproduction of the Same through an empty variation that creates distance. This space is produced in the subordination of difference to Identity. Hence, the appearance of ω appears as a compelling candidate for an authentic Deleuzian repetition.

ω is Not Repetition

Is the difference evinced in ω strong enough to constitute an autochthonous, genetic repetition of the miraculous kind Deleuze espouses? We know for the repetition to be authentic it must be “due to miracle rather than law,” and “against the law…underneath laws, perhaps superior to laws” (DR, 2). Yet there exists a way in which ω is simply another application of law. This reading begins where our former, naïve interpretation leaves off: with the supremum. As previously discussed, the supremum is possible precisely because the Axiom of Union is every bit as sovereign as the Axiom of Infinity. One might think this is a situation of one law replacing another. In fact there has been only one Law and it remains in perfect operation: ZFC.

Further, ω itself repeats in ω2, ω3 up to ωω (ω2), and on to ωω and even higher. What appears at first as the end of a regime comes to be seen as variation on a relentlessly applicable set of axioms, endlessly extending the Law’s reach in the very numbers we previously received as miracles. Is there any way to rehabilitate what has transpired in the service of an authentic encounter, a miraculous event that might constitute something other than the bare repetition against which Deleuze writes?

In proper Deleuzian fashion the answer is to be found in an enculage of Frege, the master logician. The overall project of Begriffsschrift is developing a “pure logic…disregarding the particular characteristics of objects, [that] depends solely on those laws upon which all knowledge rests” (B, 5).11 Frege takes specific aim at establishing logical foundations for arithmetic, though his work remains focused on “laws of thought that transcend all particulars” (B, 5). That these laws should be generalizable to all fields of thought Frege finds evident. Explaining the subtitle of Begriffsschrift, Frege writes, “Since I confined myself for the time being to expressing relations that are independent of the particular characteristics of objects, I was also able to use the expression ‘formula language for pure thought’” (B, 6). The ideography Frege establishes is to support his pure logic, and he sees it continuing Leibniz’s search for “a universal characteristic…a calculus philosophicus or ratiocinator” (B, 6). He even goes so far as to explain how, from the

11 Throughout, citations marked ‘B’ refer to Begriffsschrift.

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ideography of the Begriffsschrift, we may proceed to establish thought formulas for geometry, and by extension physics and the laws of motion, as well as for philosophy generally (B, 7).

Quite clearly Frege’s project is antithetical to the Deleuze of Difference and Repetition. Nonetheless, Frege approaches our solution to the predicament of ω in §8 of Begriffsschrift “Identity of content” (B, 20). Here in his treatment of identity itself we find the first stirrings of a miracle against the law, one that he decides not to follow to its ends but which nonetheless provides the genetic atom for the entire field his logic attempts to elaborate. It appears as though this section simply asserts a transparent meaning for the ‘=’ sign, which Frege labels the “identity of content” sign (B, 20). Namely, that the figures on the left and right of the identity sign share the same content. However, the field opened by the identity sign is more complex than it seems.

Professor Bar-Elli’s excellent reading of §8 opens the way for our analysis. There exists a tripled field between name, sign and content that the identity sign arbitrates. Frege is explicit, opening the section with an unambiguous sentence where he shifts from speaking of signs to names. “Identity of content differs from conditionality and negation in that it applies to names and not to contents” (B, 20). Bar-Elli describes the magnitude of this terminological shift, writing that “[t]hroughout this section, except for the last sentence, Frege speaks of identity consistently in terms of names, i.e. signs endowed with modes of determining their contents, whereas in the rest of the book he talks, where identities are concerned, simply of signs” (Bar-Elli, 357). A sign “just denotes its content; this exhausts its meaning. A name, in contrast, includes a mode of determination (Bestimmungsweise) of its content” (Bar-Elli, 357).

Hence we have in typical usage a sign that stands in for its content. “The correlation between a sign and its content is an arbitrary convention or stipulation” (Bar-Elli, 358). A simple matter of assignment, whereby a sign expressly links its content to the text, exhausted in this operation. “The semantics of names, in contrast, is ‘thick’: a name does not only denote its content, but includes and expresses a way its content is determined. This, Frege emphasizes, is an objective feature that pertains to the ‘essence of things’ (Wesen der Sache), to use his terms” (Bar-Elli, 358). How may one distinguish a name from a sign?

A name must carry its own history with it. This history is in fact an objective and essential feature of the name and its content. It forms a rich, ‘thick’ linkage between name and content that is “not arbitrary or conventional” (Bar-Elli, 358). Frege himself is quite upfront on this matter. “At first we have the impression that what we are dealing with pertains merely to the expression and not to the thought, that we do not need different signs at all for the same content and hence no sign whatsoever for identity of content. To show that this is an empty illusion I take the following example from geometry…” (B, 21). Frege then goes on to demonstrate how two different points A and B on a circle are actually the same point, once it is understood that the differing specifications given for each point yield the same position on the circle’s circumference. However, absent the proof that the two points are in fact identical, no such statement

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could be made. Thus it is not a simple matter of redundant signs for identical contents, but instead a matter of necessity that an identity of content sign determines a relation of equivalence between two contestable micro-genealogies, the ‘thick’ descriptions of the name-content relations embedded in each of the names on either side of the ‘=’ sign.

In a remarkable passage Frege writes,

“Whereas in other contexts signs are merely representatives of their content, so that every combination into which they enter expresses only a relation between their respective contents, they suddenly display their own selves when they are combined by means of the sign for identity of content; for it expresses the circumstance that two names have the same content. Hence the introduction of a sign for identity of content necessarily produces a bifurcation in the meaning of all signs: they stand at times for their content, at times for themselves” (B, 20-21).

The distinction between sign and name is no doubt the chief innovation of §8, as Bar-Elli notes. Far from a dead science of ossified signs, each anchored in a simple and direct line to its content, this logic of arithmetic reveals itself as alive. The taxonomist’s butterflies flap their wings once more, as signs begin to swarm, changing from one moment to the next. The sign responsible for arbitrating identity of content produces an inherent bifurcation in all signs. Signs come to “stand at times for their content, at times for themselves” (B, 21). What youthful flirts Frege makes of signs!

The Radical Plasticity of Math

Once more to repetition. How may our playful signs extricate us from the debate over ω as authentic repetition or bare repetition? Let us return to the metrical vs. the ordinal.

Dependent on the context, which is to say their practical usage by a subject, the import of a sign changes. Throughout the course of this investigation we have in fact unearthed a sentence that makes use of our charming identity of content sign. It came in our investigation of the closeness metric and Thomson’s objection to the possibility of summing an infinite series. While he held that the sum (the metrical limit) was never reached, we showed that it was in fact the series evaluated to ω (the ordinal limit) that permitted us to close the distance to zero. The identity so discovered can be stated as xω = limn∞xn. Here the identity of content sign bridges a chasm between two names, linking forever the micro-genealogy of each sign’s development with a logical equivalence. Two histories were made to converge, such that the strands of thought associated with each one’s internal development were suddenly exposed, as the signs came to stand “for themselves.”

We see the connection to Deleuze quite clearly. What appeared at first as simple application of empty variation under generality (the production of various signs, arbitrated as interchangeable through stipulation or convention), comes to be seen as an essential collision between two vectors of specific historicity. This historicity is brought about only when the identity is declared, and each sign experiences a genetic moment of

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differentiation, separating from itself so that it may stand, not for its content, but for itself. In so doing the sign becomes name, and its content transforms into a “way of determining,” a path with a history and a trace. With the identity established, the first means of specifying each sign (their initial definitions) now come to be seen as authentic repetitions, repeating in advance the equivalence statement to come at the end of the proof. Through the sign that one presumes would establish the sovereignty of Identity, Difference emerges and authentic Repetition occurs.

Even more scandalous for the devout logician, this miracle against the law reveals itself only in practice. It is through the act of doing math or doing logic that Identity is disrupted, Difference emerges, and authentic Repetition occurs. Which is nothing less than to say that it is within the action of thinking itself that an essential, genetic atom of pure difference erupts. Thought is this difference, displacement, differentiation.

We see as much the moment we stop to consider mathematics as practice. The epistemology of math is best seen as a kind of empirical idealism, the work of establishing creative playgrounds for exploration. Standard and non-standard constructions of entities, such as the natural numbers, stand side by side. Different formulations of axioms are shown to be commensurate, as with Dedekind’s proof of isomorphism between any two Peano axiomatizations in his 1888 “What are numbers and what should they be.” Or, different formulations of axioms are not commensurate, as for example when minor set theories persist as alternatives to ZFC.

This radical plasticity is math. Despite his keenness for mathematical language, Deleuze does not seem to be aware of the spectacular power hidden in the kernel of the identity sign. “Servien rightly distinguished two languages: the language of science, dominated by the symbol of equality, in which each term may be replaced by others; and lyrical language, in which every term is irreplaceable and can only be repeated” (DR 2).

Nevertheless, we must be aware that moments of authentic repetition are fleeting and soon fall prey to representation. Once an identity statement such as our own has been shown to be true, the names involved may soon degrade to proto-signs, as they become fodder for usage in future proofs. Nonetheless, we maintain that it is the potential for any sign to at any moment betray itself that conditions the field of possibility known as math. Without this essential ambiguity, this radical plasticity, no new math would be possible. All that would obtain is a dull echo, the thud of a carcass rhythmically beaten. By contrast, openness to a radical encounter is what conditions the working mathematician’s mind and what constitutes original and new progress within this discipline.

It should now be clear that the radical plasticity of math conditions empirical idealism. By empirical idealism we mean nothing less than a fully rigorous practical science of ideas. That the entities worked with in math are ideas, by which we mean mental concepts (what other kind?), is straightforward to all but the most facilely materialist. However the relationship of these ideas to practice, and material reality, may elude the observer on first glance. Let us consider a beautiful section of Difference and Repetition where Deleuze clarifies his idea of empiricism.

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“This is the secret of empiricism. Empiricism is by no means a reaction against concepts, nor a simple appeal to lived experience. On the contrary, it undertakes the most insane creation of concepts ever seen or heard. Empiricism is a mysticism and a mathematicism of concepts, but precisely one which treats the concept as an object of encounter, as a here-and-now, or rather as an Erewhon from which emerge inexhaustibly ever new, differently distributed ‘here’s and ‘nows’…concepts are indeed things, but things in their free and wild state, beyond anthropological predicates. I make, remake and unmake my concepts along a moving horizon, from an always decentered center, from an always displaced periphery which repeats and differenciates them” (DR, xx-xxi).

From this passage we immediately can see a number of characteristics for what we have termed an empiricism of the ideal (empirical idealism). First, a massive, “insane” creation of concepts. Second, a relationship to the concept as an “object of encounter, a here-and-now.” Third, an encounter with a place that is not a place, an Erewhon, from which “inexhaustible” concepts pour forth. Fourth, an object of encounter so wild and free as to be beyond human subjects (“beyond anthropological predicates”). Fifth, a state of differenciating flux that displaces my situ along a mobile line of flight (“moving horizon”).

Can we not see precisely here, in this rich and frothy description of a worthy empiricism, the work of transfinitude? Our encounter with ω in this essay has been but the briefest of brushes with infinity. As was earlier noted, not only does ω2 exist, but even ωω. In fact, countably infinite many ω limit ordinals exist before ω1, the first uncountable limit ordinal. ω rightly qualifies as an insane creation of concepts. Second, has not our encounter with ω constituted an encounter, properly speaking, with its own “here-and-now?” Again we say yes. It is precisely the introjection of its own here-and-now that provided xω as the position reducing the difference between series and sum to zero, uniting the metrical and ordinal and guaranteeing our ‘distance’ in the first number that broke ground on a new conception of distance without size. Third, did we not find in this here-and-now, in fact an inexhaustible Erewhon? Indeed we did, for what was originally seen as authentic repetition came to be understood as the complex, deepening movement of the Law known as ZFC. We shall address the fourth and fifth claims forthwith.

Performing a Supertask

Let us say we desire to carry out a supertask. How shall we proceed? We know the execution requires an infinite acceleration (decreasing time per task), as per our geometrical timekeeper series. At the same time, we know infinite acceleration to be bounded in our universe by the speed of light. While black holes are known objects that accelerate particles to light or near-light speed, the problem of returning once the event horizon is crossed is insurmountable. However, there does exist a model of “a slowly rotating black hole of zero electric charge,” known as the Kerr spacetime (Etesi and Nemeti, 11).

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In addition to the traditional event horizon associated with stationary black holes, Kerr black holes possess an outer, ellipitical horizon associated with an ‘ergosphere.’ This ergosphere constitutes a region of spacetime where objects are induced to co-rotate with the black hole at speeds approaching the speed of light. Objects injected into the ergosphere may be retrieved, and an ambitious plan known as the Penrose process was formulated to harvest energy from rotating black holes in this fashion. As concerns completion of a supertask, imagine a computer traveling along a worldline within the ergosphere bounded by an upper event, such that it experiences infinite time in approaching the event. Imagine likewise an observer traveling along a different worldline towards the same upper event such that its proper time is finite. The computer sends a light pulse with the final results of the calculation to the spacetime of the upper bounding event. The observer’s trip to the event takes finite time, where she can retrieve the results of an infinite calculation (Etesi and Nemeti 12-14).

It is worth noting that the physical plausibility of such a scenario cries out for attention. However, it is beyond our scope both as specialists and in terms of this essay’s focus to offer a satisfactory treatment. Nonetheless, at the very least the concepts involved in imagining the physical realization of a supertask’s completion—no matter their plausibility to the reader—clearly evince a scenario ‘beyond anthropological predicates’ that constantly ‘displaces center to periphery.’ It is also worth noting that Etesi and Nemeti’s work has been extended, suggesting that at least some other scholars find it admissible. 12 In fact, this torsion between the logical exercise involved in solving Thomson’s Lamp and the physical somersaults involved in its material realization constitute a beautiful example of empirical idealism taken to the material. In this manner one can see how the genetic element of math’s radical plasticity erupts from the field of empirical idealism to repopulate and differentiate alternative fields within neighboring disciplinary nexūs, like a virus cross-contaminating different species.13

Conclusion

The chief contributions of this paper have been: 1) To supply the first solution to Thomson’s Lamp paradox, proving that a supertask is logically permissible. By considering the parity of ω, we have demonstrated that the final state of the lamp is identical to its initial. 2) To distinguish between and explore the philosophical consequences of metrical and ordinal limits, investigating the decoupling of position (order) from size (distance) that takes place with the transfinite ordinals.

We have elaborated on this decoupling in reading Deleuze’s Difference and Repetition as a method for an enculage of the master logician Frege’s Begriffsschrift. From this cross-reading we have established the radical plasticity of math as its genetic condition, in which signs have always a bifurcation between standing for their content and themselves. This essential flexibility conditions an epistemology we refer to as empirical idealism,

12 See Professor P.D. Welch’s “Turing Unbound: on the extent of computation in Malament-Hogarth spacetimes.”13 Such a move is precisely what is at stake in one of Deleuze’s most under-appreciated works, The Fold.

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which we in turn use to retroactively figure our encounter with ω and the transfinites. We examine as well the physical possibility of supertasks’ completion, noting the extreme and exotic spacetimes necessary for their execution, and conclude by considering the potential for math’s epistemology as authentic repetition to disrupt and renew other disciplines.

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Works Cited

Bar-Elli, Gilead. “Identity in Frege's Begriffsschrift: Where Both Thau-Caplan and Heck are Wrong.” Canadian Journal of Philosophy 36.3 (2006): 355-370.

Benacerraf, Paul. “Tasks, Super-Tasks, and the Modern Eleatics.” Journal of Philosophy 59.24 (1962): 765-784.

Deleuze, Gilles. Patton, Paul transl. Difference and Repetition. New York: Columbia University Press, 1994.

Etesi, Gabor and Nemeti, Istvan. “Non-Turing computations via Malament-Hogarth space-times.” International Journal of Theoretical Physics 41 (2002) 341-370.

Frege, Gottlob. Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought. (1879). From Frege to Gödel; a source book in mathematical logic, 1879-1931. ed. Heijenoort, Jean. Cambridge: Harvard University Press, 1967.

Thomson, J.F.. “Tasks and Super-Tasks.” Analysis 15.1 (1954): 1-13.

Thomson, J.F.. "Comments on Professor Benacerraf's Paper." Zeno's Paradoxes. Salmon, Wesley C., ed. Reprint. Hackett Publishing, 2001. 130-138.

Works Referenced

Deleuze, Gilles transl. Conley, Tom. The Fold: Leibniz and the Baroque. Minneapolis: University of Minnesota Press, 1993.

Huizinga, Johan. Homo ludens: a study of the play-element in culture.. Boston: Beacon Press, 1971.

Welch, P.D.. “Turing Unbound: on the extent of computation in Malament-Hogarth spacetimes.” British Journal for the Philosophy of Science 59.4 (Dec. 2008): 659-674.

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somethingworthreading.orgsomethingworthreading.org/.../2014/06/...on-Thomsons-La… · Web viewAbhishek Bose-Kolanu. Summer 2014. Turning out the Light on Thomson’s Lamp. In recent

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