Re: The Pythagorean View and the Lamp
[Warning: maths ahead :-P and I've just stumbled into this thread, so I
might be waaay off; my apologies if that's the case]
I don't think there's necessarily an answer to this question. The
history of the lamp is a series of points in the set {ON, OFF}X R_[0,2],
but you have only defined it for {ON, OFF} X R_[0,2), yet the question
asks wheter the history includes (ON, 2) or (OFF, 2). As you defined the
history as a function F:R_[0,2) -> {ON, OFF}, one is tempted (and I
guess that's what you were thinking about) to exted the function in a
continuous way. With a suitable topology for {ON, OFF}, the domain of F
is dense, so if F were continous such an extension would exist and be
unique. Sadly, F isn't continuous unless you give {ON, OFF} the
indiscrete topology, but then you lose the power to distinguish between
ON and OFF in any case [the lamp's history converges in the limit to
both (yep, in non-Hausdorff topologies [ie, "too coarse], limits are not
necessarily unique)].
So, ignoring the physics of the issue [of which I understand nothing,
I'm afraid :-(], mathematically my guess off-the-cuff is that the
problem has no solution [as the setting doesn't give explicit
information about t=2, and either you can't extend in a natural way the
lamp's function, or if you can't it's at the price of saying anything at
all with that extension].
open to comments --- I'm fairly new at this in any case,
Marcelo
PS: A fascinating problem, really.
> Welcome,
> I've been looking for an idiot savant to answer this question: 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?)
> Norman
--
Marcelo Rinesi | [EMAIL PROTECTED]
People were stupid, sometimes. They thought the Library was
a dangerous place because of all the magical books, which
was true enough, but what made it really one of the most
dangerous places there could ever be was the simple fact that
it was a library.
-- Terry Pratchett, "Guards! Guards!"
Re: The Pythagorean View and the Lamp
Dear everythingers
Reading Bruno Marchal's last email, I realized that it may contain the
answer that Norman seeks.
The answer I think is simply that the set of states for the lamp is
incomplete under the operation of turning the lamp on and off an
infinite number of times as described by Norman.
Just like the paradox of the square root of minus one was resolved by
adding a new type of quantity, i, we may resolve the Thompson problem
by adding a new state, ONF, which is neither ON or OFF but the result
of the infinite process. We now have a lamp capable of being in three
states: ON, OFF and ONF. No more paradox. We have also upgraded our
lamp. With this new lamp capable of being ONF we can do all kinds of
things. For example if reading a newspaper requires the lamp to be ON,
what could you do to with the newspaper with the lamp ONF? And if
having sex requires the lamp to be OFF what would you do with the lamp
ONF? This is something we should really worry about instead of
worrying about the lamp!
George Levy
Norman Samich wrote
Welcome,
I've been looking for an idiot savant
to answer this question: 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?)
Norman
Bruno Marchal wrote:
At 06:30
14/07/03 -0400, PaintedDevil wrote:
However - what mainly interests me is what
reason one could have for not
taking the Pythagorean view, which does, after all, explain why the
universe
exists (or appears to exist).
Perhaps because most people believe that the Pythagorean view
has been refuted.
Just consider the "little Pythagorean view" according to which
-Every length can be measured by integers or ratio of integers
This has been refuted by the Pythagorean themselves when they
discovered that the square root of 2 *is* not given by any ratio of
integers.
It is the discovery of the irrational numbers, a long time ago.
Now Pythagore could have "corrected" his doctrine with:
-Every length can be measured by integers, ratio of integers or
radical of integers.
But this would have been refuted by Abel's discovery in 1824 that
polynomial
with degree greater than 4 can have solutions which cannot be described
in
term of ratio and radicals.
Now Pythagore could have corrected his doctrine again with
-Every length can be measured by polynomial's zeros.
But then Pythagore would have been refuted by the discovery of the non
algebraic numbers: the transcendant numbers like euler e, and PI.
Perhaps at this stage Pythagore would begin to think his Pythagorean
view
could may be not work.
And then he would have been destroyed by Cantor's discovery, who showed
with his famous diagonalization, that the set of reals (the lengths) is
not enumerable.
But then Pythagore would perhaps have postulated the comp hypothesis,
thinking that *algorithmic* real , which should be obviously
enumerable, exist and are easily defined.
Alas, the more subtle Post-Turing-Markov-Church-Kleene-Godel
diagonalisation makes the algorithmic real not *algorithmically*
enumerable.
Surely at this stage Pythagore should abandon the Pythagorean view.
Isn't it?
NOT AT ALL. With *Church thesis* you can still say:
-Every length can be measured by a FORTRAN program.
Only you have a price to pay:
FORTRAN programs will measure *much more* than "length", and an
enumeration of the algorithmic reals, will enumerate the reals +
other objects, and no theories at all will give you an algorithmic way
to
distinguish the reals from the other objects. That is, the price is
incompleteness, randomness, unpredictability, etc.
(Click on the diagonalisation posts in my URL where I explain this,
with the notion of function (from N to N) in place of the reals).
But that is nice (for a realist platonic), and this shows that
Church thesis not only rehabilitates the little Pythagorean view (in
term
of length), but makes consistent the large Pythagorean view
according to which:
-everything emerges from the integers and their
relations.
And my PhD result shows that, with the comp hyp, the appearance of
physics *should* emerge in the average memory of the consistent
anticipating universal machine/program/number. And then I deriv
The Pythagorean View
At 06:30 14/07/03 -0400, PaintedDevil wrote: However - what mainly interests me is what reason one could have for not taking the Pythagorean view, which does, after all, explain why the universe exists (or appears to exist). Perhaps because most people believe that the Pythagorean view has been refuted. Just consider the "little Pythagorean view" according to which -Every length can be measured by integers or ratio of integers This has been refuted by the Pythagorean themselves when they discovered that the square root of 2 *is* not given by any ratio of integers. It is the discovery of the irrational numbers, a long time ago. Now Pythagore could have "corrected" his doctrine with: -Every length can be measured by integers, ratio of integers or radical of integers. But this would have been refuted by Abel's discovery in 1824 that polynomial with degree greater than 4 can have solutions which cannot be described in term of ratio and radicals. Now Pythagore could have corrected his doctrine again with -Every length can be measured by polynomial's zeros. But then Pythagore would have been refuted by the discovery of the non algebraic numbers: the transcendant numbers like euler e, and PI. Perhaps at this stage Pythagore would begin to think his Pythagorean view could may be not work. And then he would have been destroyed by Cantor's discovery, who showed with his famous diagonalization, that the set of reals (the lengths) is not enumerable. But then Pythagore would perhaps have postulated the comp hypothesis, thinking that *algorithmic* real , which should be obviously enumerable, exist and are easily defined. Alas, the more subtle Post-Turing-Markov-Church-Kleene-Godel diagonalisation makes the algorithmic real not *algorithmically* enumerable. Surely at this stage Pythagore should abandon the Pythagorean view. Isn't it? NOT AT ALL. With *Church thesis* you can still say: -Every length can be measured by a FORTRAN program. Only you have a price to pay: FORTRAN programs will measure *much more* than "length", and an enumeration of the algorithmic reals, will enumerate the reals + other objects, and no theories at all will give you an algorithmic way to distinguish the reals from the other objects. That is, the price is incompleteness, randomness, unpredictability, etc. (Click on the diagonalisation posts in my URL where I explain this, with the notion of function (from N to N) in place of the reals). But that is nice (for a realist platonic), and this shows that Church thesis not only rehabilitates the little Pythagorean view (in term of length), but makes consistent the large Pythagorean view according to which: -everything emerges from the integers and their relations. And my PhD result shows that, with the comp hyp, the appearance of physics *should* emerge in the average memory of the consistent anticipating universal machine/program/number. And then I derived a theorem prover for the logic of the physical propositions from that, but for reason of inefficacy of that theorem prover I can still not decide if it gives really a quantum logic, or which one ... Other weakness, I have neither a semantics nor an axiomatic for that quantum logic, only a theorem prover, and a naive semantics in term of maximal consistent computational histories. I have written 2/3 of a paper which summarize the proof and I intend to submit it to some international journal (once finished ...). Sure, there exists other reasons to believe the Pythagorean view is coming back, like more direct astonishing relations between number theory and theoretical physics. Look at http://www.maths.ex.ac.uk/~mwatkins/zeta/renormalisation.htm Bruno http://iridia.ulb.ac.be/~marchal/
