"Perhaps you've heard of Thompson's Lamp. This is an ideal lamp, capable of infinite switching speed and using electricity that travels at infinite speed. At time zero it is on. After one minute it is turned off. After 1/2 minute it is turned back on. After 1/4 minute it is turned off. And so on, with each interval one-half the preceding interval. Question: What is the status of the lamp at two minutes, on or off? (I know the answer can't be calculated by conventional arithmetic. Yet the clock runs, so there must be an answer. Is there any way of calculating the answer?)"
I've been greatly intrigued by your responses - thank you. Marcelo Rinesi, after analysis, thinks that the "problem has no solution". Bruno Marchal thinks that the "Church thesis . . . makes consistent the 'large Pythagorean view, according to which everything emerges from the integers and their relations.'" George Levy, after reading Marchal, thinks there may be a solution if there is a new state for the lamp besides ON and OFF, namely ONF. Stathis Papaioannou thinks the lamp is simultaneously on and off at 2 minutes. He thinks the problem is equivalent to "asking whether infinity is an odd or an even integer". He shows that there are two sequences at work, one of which culminates in the lamp being on, while the other culminates in the lamp being off. Both sequences can be rigorously shown to be valid. Now Joao Leao paraphrases Hardy to say that "'mathematical reality' is something entirely more precisely known and accessed than 'physical reality'" So I'm to understand that "mathematical reality" is paramount, and "physical reality" is subservient to it. Yet mathematics is unable to determine the on-or-off state of Thompson's Lamp after 2 minutes. What are the philosophical implications of unsolvable mathematical problems? Does this mean that mathematical reality, hence physical reality, is ultimately unknowable?
