# Thomson's lamp: a useless paradox?

Thomson's lamp was mentioned at How to understand numbers that become really large? (as well as a number of posts elsewhere on SE). I have mentioned elsewhere that in addition to Cantorian infinite numbers (ordinals and cardinals), there are infinite numbers of NSA such as nonstandard integers, which could be called ringinals as they are elements of a ring. From this point of view, Thomson's lamp paradox (impossibility of specifying the status of the lamp at t=2) appears to be due merely to a confusion of cardinals and ringinals.

What creates the paradox is that an infinite cardinal is used where an infinite ringinal should be used, instead. If a ringinal is used, the paradox disappears: the lamp is on or off in accordance with the parity of the ringinal.

More precisely, one fixes a nonstandard integer H, and clicks Thomson's switch at geometrically decreasing intervals, but stops at rank H (thus after "infinitely many" clicks). The lamp will then remain on or off in accordance with the parity of the last integer H, which in particular specifies its status at time t=2. The fact that nonstandard analysis is more expressive than traditional non-infinitesimal analysis is well known; a classical example is continuity expressed as "infinitesimal changes of input always produce infinitesimal function increments"; for further examples see this answer: https://math.stackexchange.com/a/1836900/72694

Is there any depth to this famous paradox beyond such a conflation of kinds of infinite number?

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## 8 Answers

The resolution to the paradox is that it violates the laws of physics. It would take an infinite amount of energy to move the lamp switch so fast. "Would the lamp be on?" is a question about physics, which in principle must be answered by following the differential equations of physics. If you introduce singularities (such as infinite kinetic energy in the moving switch), it's not surprising that the equations can't give a meaningful result.

Perhaps you object that this is an imaginary lamp, not a physical one. In that case, calling it a "lamp" does nothing but invoke inappropriate intuitions about how a lamp works - intuitions derived from physical lamps following the laws of physics. The intuition that lamps are on or off depending on how many times the switch was flipped is inappropriate here, for example, since the switch is not flipped an integer number of times. We should call it something else, perhaps a "blonker," and then the paradox can only be answered by following the axioms of "blonkers," which you'll have to define yourself, and which have no relation to reality as we know it.

Thomson introduced this paradox as a way of arguing for the impossibility of "supertasks," tasks where an infinite number of "steps" are completed in a finite amount of time. His argument was that since we have no way to say whether the lamp is on or off at the end, the task must be impossible. Of course, that's not the only reason the task is impossible!

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==
The paradox arises from confusion between an open set and the closure of that set. It also makes the common error of confusing infinity with a large integer.

The total time that the switch is being moved is the sum of the elements of an open set. 2 minutes is the closure of that set but it is not an element of the set. As the time interval between on and off gets shorter, the time only approaches 2 minutes. It never reaches it.

It's exactly like telling somebody to write down every positive integer twice as fast as the previous integer, and after 2 minutes, when he's reached infinity, ask him if the last number he wrote was was even or odd.

There IS no "last number he wrote." It's an illusion. It's stage magic.

It's not a paradox. There are no paradoxes, really.

FREE BONUS FACT!

This is also a common mistake in understanding black hole event horizons. The edge of spacetime there is an open set. The event horizon is the mathematical closure, but it doesn't actually exist.

And because relativity makes you get shorter and slower as you get closer, you can never reach it.

Or consider: Mass at the event horizon would move at c and stop moving through time. It's not only impossible for mass to move at c, but (per Feynman) it's also impossible for mass to stop moving through time.

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It's similar to the Achilles-and-tortoise paradox: The situation is described as an infinite series of points in time, which converge on a given time t-limit; since the description does not define what happens beyond t-limit, asking what happens after that time given just the description is meaningless.

For Achilles and the tortoise, the situation can be easily extrapolated beyond t-limit: Achilles will overtake the tortoise, the description was artificially phrased in a way that ends at t-limit but the situation is completely linear.

For Thomson's lamp, there is no such extrapolation: The Grandi series 1-1+1-1+1... diverges, i.e. fluctuates between 0 and 1 without ever stabilizing.
One could state that at an arbitrary interval between 0 and t-limit, the lamp is on and off 50% of the time, respectively, so it is effectively half-on and half-off, but I think even that does not converge but alternates between 0.25 and 0.75.
So, no, there isn't a state assignable to t-limit for Thomson's lamp.

So, the answer is: it is not a paradox but a divergent series, and any attempt at assigning a limit value to it is useless.
It would be useful for teaching, and more or less useless depending on what kinds of axioms you use to define the situation.

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You ask:

Is there any depth to this famous paradox beyond such a conflation of kinds of infinite number?

Yes. It plays a role in metaphilosophical discourse in speaking to the power and limits of intuition (SEP) in the philosophical method. The point for a philosopher of mathematics, physics, or computation, for instance, might be to attempt to resolve the paradox. For a philosopher interested in exploring the philosophical method itself, it raises questions about the reliability of philosophy itself. In An Introduction to Metaphilosophy, for instance, Overgaard et al. explore the use of intuition in Chapter 4: The Data of Philosophical Arguments.

On the face of it, it's a question that examines reasoning about computation in physical systems (SEP). If one considers computers in virtue of their physical implementation, then one must tackle certain philosophical challenges regarding logic truths versus nomological truths. Supertasks are a class of thought experiments that push the limits of intuition to explore how what is logically possible might violate what is nomologically possible. In the case of Thompson's lamp, for instance, although one can accelerate the activity of switching mathematically indefinitely, the same cannot be said of building physical machines to do so.

Consider that the flow of electricity cannot occur infinitely fast physically because of the constraints of the speed of light, and yet the switching of a lamp without taking this nomological constraint in effect knows no limit. Thus, Thompson's paradox and other supertasks are philosophically not mathematical questions, but questions of metaphysical grounding (SEP) since they impinge on questions about how explanations among various classes of theories and claims work.

A selection of the various answers others provide demonstrate the value of challenging intuition. Intuition skepticism in metaphilosophical literature is a position in the epistemological function of intuition (SEP) that rejects the use of intuition in a philosophical method. One might argue Thompson's paradox supports intuition skepticism because it shows that our intuitions can lead astray without careful consideration of the mathematics of nomological constraints. Of course, a philosopher who defends the use of intuition can point to Einstein's thought experiments as a support for the use of intuition. But either way, the attempt to dissolve the paradox and the questions that the paradox raise on method are important to understanding the nature of philosophy.

I am with Causative on this.

This is a hilarious example of creating faux profundification by mixing immiscible categories:

• empirical (physics) with rational/Platonic (math)
• more importantly, pure math with math modelling

In about 34 steps we are in picosecond flips — 'nuf said?

If one insists, it's a couple more steps to Planck time.

34 < ∞ methinks!

At the 2 minute mark the lamp is being turned on and off simultaneously and so it's in a superposition of being both on and off. At the first instant after the 2 minute mark the superposition collapses and the state is observed.

You'd probably need a few robust lamps and a willing superbeing to say more about the probability distribution of the final state and it seems reasonable that the state would depend on the specific lamp. But supposing a perfect platonic lamp then it seems hard and pointless to argue that it would be anything other than 50/50.

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Edit-This answer was provided before Mikhail Katz finally agreed to edit his 'question' to explain how he modifies Thomson's lamp scenario with the help of nonstandard integers to get a 'last (nonstandard) switching' of the Thomson--Katz lamp. It appears that Katz allows for 'someone' to decide to stop switching the lamp after a nonstandard integer H of switchings. I'll let everyone decides for oneself whether this condition (namely, to decide to stop switching the lamp sometime before the mark t=2, be it infinitesimally before) is logically an extra assumption or even a self-evident assumption with regard to Thomson's plausible intent. In any case, the current answer was provided without this assumption, the conclusion being that if one does not decide to stop the lamp at an infinitesimal time before t=2, then the transfer principle implies that the lamp should continue to switch, and the paradox remains.

This is an interesting thread! Whether or not Thomson successfully argued to the effect that "super-tasks are not possible of performance" [Thomson's 'Tasks and Super-Tasks', p.5], the variety of answers in this thread shows that Thomson's paradox acts as a useful (meta)philosophical tool.

I shall share here some thoughts about the mathematical aspects raised in the original question to this thread, especially the relation of Thomson's paradox (and other super-tasks paradoxes) to non-standard mathematics. Most of what I'll say has already been said (at least implicitly) by other commentators on this thread, but I'd like to round up a few different points in this answer.

The physical aspect of Thomson's paradox. There is a sense by which Thomson's paradox is not a mere mathematical question, and that it has a (meta)physical aspect to it. Indeed, I doubt that anyone would argue against the possibility of "mathematically performing", i.e. of defining, the function f(t) defined over 0 ≤ t < 2 as 'f(t)=1 if 2 - 2^{-2n} ≤ t < 2 - 2^{-(2n+1)} for some nonnegative integer n, and f(t)=0 otherwise'.

Granted that Thomson's problem involves some implicit physical aspects, I don't think that the physical irrealizability of Thomson's lamp scenario offers a satisfactory solution to the paradox, for Thomson's argument aims at disproving the possibility of (some) super-tasks at a deeper, more general level.

I can conceive of other physical scenarios that share the same 'functional aspects' as those of Thomson's lamp, and whose performance does not require the use of an infinite amount of energy. Those scenarios are physically irrealizable for other reasons--because of differents aspects of the currently known physics of our world, and because of the limits of my own ingenuity--but they would make sense in a perfectly 'classical physics' world (and perhaps also in a 'future physics' world). I feel that a more satisfactory and robust solution to the paradox would cover these counterfactual worlds.

The mathematical formulation of Thomson's problem and the assumption of determinism. Notwithstanding the importance of some physical aspect to Thomson's problem, much as in the case of Zeno's paradox, I think it is the mathematical abstract formulation of Thomson's scenario (more than the physical irrealizability of Thomson's scenario) that helps best to pinpoint a difficulty with Thomson's paradox: for on what grounds should we suppose that the 'behavior' of the function f(t) over 0 ≤ t < 2 determines an extension of f over 2 ≤ t?

From a mathematical perspective, it is clear that several functions extend the given function f; in other words, the extension is not (uniquely) determined. Note that the same can be said about Zeno's paradox (it is only defined over a bounded open set, and not at the endpoints), but in this case a continuity requirement can be evoked to select a unique 'preferred extension'. In comparison, the above function f in Thomson's scenario admits no continuous extension.

Thomson (implicitly, as far as I can tell) assumes that (A) its lamp have a well-defined state at each instant (either 'on' or 'off') and, most importantly, that (B) the lamp's state at any time t is (uniquely) determined by its past. These are the 'physical aspects' that Thomson considers. Combined with assumption (C) this switching super-task can be performed, one reaches a paradox, as one is unable to tell the state of the lamp for 2 ≤ t.

The importance that Thomson attributes to assumption (A) and (B) transpires in the following quote [p.6]:

What is the sum of the infinite divergent sequence +1, -1, +1, ...? Now mathematicians do say that this sequence has a sum; they say that its sum is 1/2. And this answer does not help us, since we attach no sense here to saying that the lamp is half-on. I take this to mean that there is no established method for deciding what is done when a super-task is done. And this at least shows that the concept of a super-task has not been explained. We cannot be expected to pick up this idea, just because we have the idea of a task or tasks having been performed and because we are acquainted with transfinite numbers.

Thomson only attaches sense to a lamp being in state +1 or -1 (assumption (A)). And without assumption (B), one could certainly not decide (or rather predict) what is done when a super-task is done.

However, assumption (B) is a nontrivial one: even in the realm of Newtonian physics, there are 'simple-tasks' that are not uniquely determined. (For instance, read this Wikipedia section (in French) around the Cauchy--Peano--Arzelà theorem.) Such 'simple-tasks' might also be difficult to realize physically, but they do not strike the mind as utterly impossible or artificial.

Hence an important ingredient for Thomson's paradox appears to be assumption (B). (This point seems to have been raised by Paul Benacerraf.)

Relation to non-standard mathematics. I am no expert in non-standard mathematics (NSM), but I fail to see how it could solves Thomson's paradox.

(1) My first point is merely a heuristic one. NSM is a 'conservative extension' of standard mathematics (SM): roughly put, this means that NSM proves exactly the same results 'about' SM than what SM (+ the axiom of choice) can prove 'about' SM. To me, this suggests that NSM behaves in many respects like SM, and thus that one should spontaneously be at least somewhat skeptical when first encountering claims that NSM can solve things that SM cannot solve.

Of course, a (vague) proposition such as 'f admits no preferred extension to t=2' is not a purely SM statement, because of the word 'preferred': the greater richness of NSM over SM might provide some further criteria that allow to single out a preferred type of extension for f. However, considering that within SM there is no distinguished extension of the aforementioned function f to the interval 2 ≤ t, I still spontaneously feel that NSM cannot do much differently.

(2) What I can imagine NSM to allow, via the transfer principle (which is a mathematical implementation of Leibniz's suggestively named 'law of continuity'), is to have 'non-standard' or 'ghost' switchings of the lamp: for instants t = 2 - s where s is any (positive) infinitesimal, the lamp could continue to switch.

Let's delve a little more into the details of this. Although I won't use exactly the same vocabulary, I'll phrase things essentially in the enriched language described in this reference of Mikhail Katz (where ringinals are defined as the 'unlimited nonstandard integers').

The statement that the above function f switches at times t_n = 2 - 2^{-n} where n is a natural integer can be expressed as follows: for all nonnegative integer n, for all real number t such that 0 < t < 2^{-(n+1)}, we have |f(2 - 2^{-n} - t)|=|f(2-2^{-n}+t)| = 1 and f(2 - 2^{-n} - t)=-f(2 - 2^{-n} + t).

By the transfer principle, f extends to a function F defined on (some) nonstandard real numbers and this extension satisfies: for all nonnegative nonstandard integer n, for all nonstandard real number t such that 0 < t < 2^{-(n+1)}, we have |F(2 - 2^{-n} - t)|=|F(2-2^{-n}+t)| = 1 and F(2 - 2^{-n} - t)= - F(2 - 2^{-n} + t). We take this statement to mean that F switches at every nonnegative nonstandard integer. Of course, for n a ringinal, 2^{-n} ± t are infinitesimals, so F extends f to nonstandard reals that are arbitrarily close to the left to t=2.

(3) At this point F(2) is still undefined, but all we need is that F admits a 'distinguished' extension to t=2. Unfortunately, I can't see any such extension.

Mikhail Katz commented to this answer: "Once the lamp is turned on at the last nonstandard integer, it stays on in particular at t=2, contrary to Thomson's claim that the state of the lamp at t=2 cannot be specified. Similarly for the case of the opposite parity." I take this to mean that (i) there exists a last nonstandard integer N, and (ii) for t≥ 2-2^{-N}, simply set F(t) = F(2-2^{-N}). I agree that granted (i), prescription (ii) would yield a preferred extension of F.

However, it seems to me that (i) is false. Indeed, we have the standard proposition: for all integers n, there exists an integer n' > n of a different parity than that of n. Via the transfer principle, one deduces that there is no greatest (even or odd) ringinal.

Besides, one can formalize the infinite number of switchings of f as: for all 0 < t < 2, there exist t < t_1, t_2 < 2 such that f(t_1)≠f(t_2). By the transfer principle, we obtain: for all 0<t<2 [infinitesimally close to 2], there exist t < t_1, t_2 < 2 such that F(t_1)≠F(t_2). So there seems to be no stabilization of F near t=2 that would allow us to pick a 'preferred' value for f(2).

(4) That said, evoking NSM in the context of super-tasks paradoxes remains an interesting idea, especially in relation to Benardete's paradox and the Grim Reapers' paradox. Indeed, in the scenarios of thoses paradoxes, one should perhaps deduce the presence of 'non-standard' or 'ghost' gods/grim reapers to explain the 'spontaneous' obstruction/death of the traveler. In other words, explicitly specifying the 'dynamics' of the system at all standard times implicitly determines the 'dynamics' at infinitesimal times too.

• Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on Philosophy Meta, or in Philosophy Chat. Comments continuing discussion may be removed. Commented Aug 8 at 9:24
• Your answer is still marred by misconceptions and errors, with regard to conservativity, "last integer", and expressive power. At least please try to read the question carefully. Commented Aug 8 at 14:55
• @MikhailKatz Thanks for finally providing details on your modified lamp in your question; for as much as I want to read carefully, I am unable to read absent words! I agree with you: if God wants to stop the lamp at the hyperreal instant 2 - 2^{-H}, then the state of the lamp will be fixed afterwards. But I guess this only shifts Thomson's point: what if God wants to continue switching at every ringinal rank H? Is this hypertask able of performance? BTW speaking of misconceptions: SE isn't a place for rhetorical questions, and expecting people to know your hidden assumptions isn't rational. Commented Aug 8 at 16:55

No cardinal or other "infinite" numbers are required in the following approach to this "paradox."

The state of Thomson's lamp without any real-world physical limitations, is specified for only those times less than 2 minutes. Its state is not specified for time t >= 2 minutes. It is in the ON state in the time intervals (in minutes): [0, 1), [3/2, 7/4), [15/8, 31/16), .... It is in the OFF state in the intervals: [1, 3/2), [7/4, 15/8), [31/16, 63/32), .... The time t = 2 minutes occurs in neither of these intervals.

EDIT

There is nothing particularly "natural" about Thomson's Lamp. It amounts to nothing more than a function f mapping the rational numbers in the interval (0, 2] to {0, 1}, and having a countably infinite number of discontinuities in that interval. If we extend the definition by defining f(2)=0, we might say that the so-called "super-task" here has been somehow "completed" at the time t = 2 minutes.

• Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on Philosophy Meta, or in Philosophy Chat. Comments continuing discussion may be removed. Commented Aug 8 at 9:26
• The question how to "extend the lamp" naturally at t=2 is precisely the issue, which you do not address. Commented Aug 8 at 14:57