"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?)"

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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?