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 in...
I don't think the "paradox" has anything to do with physics. As a physical problem it total BS, of course. But it could be formulated as a mathematical problem and then at x=2, the step function doesn't have a well-defined value. I don't quite see what would be interesting about this (unless perhaps you do invoke Conway's surreal numbers?).
@mudskipper Non-convergent series were well known before Thomson; Thomson's idea was that at the end the lamp must be either on or off so that we can't just say it's undefined. That's an idea that derives from our (in this case faulty) intuition about physical lamps, not from the math of non-convergent series.
@Miss_Understands There is no mathematical paradox. The paradox arises only when you apply the faulty intuition that because it's a lamp, it has to be on or off at the end as a result of the sequence of switch flips. That faulty intuition is based on physics, not pure mathematics.
@mudskipper He used his physical intuition about how a lamp is always either on or off and its final state depends on how many times the switch was pressed, while ignoring the physical intuition that infinitely fast switches are impossible.
But if you define instead a function f(x) whose support follows the time intervals of the problem and which alternates between the values zero and one, then the argument would be that its value f(2) is not defined. So you’d still have a paradox independent of physics. If I’m not mistaken the paradox is another way of talking about divergent series, i.e. of the f(x) not having a limit at x=2.
@bob Divergent series were very familiar to mathematicians at the time of Thomson's writing as well (1954). Calculus was fully modern long before then. The only novelty of Thomson's lamp is that it's a lamp, which brings in the (unwarranted) physical intuition that it should be on or off at the end. In fact, Thomson himself discouraged the use of mathematical convergence in analyzing his scenarios: "The impossibility of a super-task does not depend at all on whether some vaguely-felt-to-be-associated arithmetical sequence is convergent or divergent."
@causative Funny how everything in math about infinities is about series, sets and limits, and then someone comes along and says "forget about convergence". It simply makes no sense. But he's not the only one who gets that wrong. I guess there's an infinite supply of people using faulty intuition to assign false results to undefined limits...
Even if the lamp doesn't work, it could be interesting to analyze why not - where the intuition breaks. If it's the infinitely fast switch - why does it break down there? (It probably ends up being because a switch that fast would need infinite energy or something like that?!)
@kutschkem The second sentence of this answer already does address that, with the exact example you give. It may even be more intuitive to think in terms of power. Assuming the switch takes approximately the same energy to flip each time, you could easily see the required power per flip explode at a faster rate than just the net energy of all the flips.
This answer might be improved (especially given the content of the comments!) by noting explicitly that paradoxes in physics are reductions to absurdity of an unknown premise. And that all thought experiments which involve measuring the real state of a real system are thought experiments in physics, even when the person is not a physicist.
Put another way: a contradiction resulting from the application of a model indicates that: the model is one or more of: false, or inapplicable, or wrongly applied, or wrongly interpreted, or applied to premises held by the physicist - usually inexplicitly - which are themselves contradictory.
One thought I had is that you should be able to define a probability of the function (or lamp) being in a given state at a given time and then take the limit of that probability as you approach time 2 (or x=2). My hunch is that that probability limit will be defined and that you could construct a limiting probability distribution at x=2 and thus say that at x=2 the function f is undefined but is instead in a superposition of all states governed by this distribution. Thus the superposition providing intuition as to why the function is undefined at that point: because in some loose sense it’s…
…in all states at once. You don’t need a physical lamp here to reach an unintuitive state nor to (if my superposition interpretation is valid) to resolve it. A function like this being undefined at a point as a consequence of the properties of the function isn’t weird in the sense that it’s just an accepted part of math but it is deeply unintuitive because it’s so different from our daily experience in life.
@bob, For any given time in the half-open interval [0, 2) other than the exact time of a switch, you can compute whether the switch is on or off with probability 1. Perhaps you were thinking of something like computing the probability of the lamp being on at a time chosen randomly from that interval or a subinterval. My intuition is that if you did that over smaller and smaller tail intervals then the probability would converge to 50% as the interval size went to 0.
I wasn’t speaking precisely enough. You’re right. I meant that at x=2 you have a probability distribution and was using my convergence of probability distribution argument as a hand-wavy way to show that there is a probability distribution at x=2 that describes the behavior there, not that the distribution exists for other values of x. Thanks for showing me the need to clarify.
@JohnBollinger just as soon as you invent an infinitely precise clock to specify the given, and an infinitely fast camera shutter to measure the whether.
Discussion on answer by causative: Th…
Discussion on answer by causative: Th…