I did not want to go classical to quantum without due cause, as it is an
intellectual arms race, but once you start referring to particles as
having individual wavelengths, you are talking quantum, and I must
respond like Pakistan. My apologies.
The easiest way to point out the problems with finding an exact
position is wavelength is the notorious Heisenberg uncertainty
principle. I almost hesitate to bring it up as it is the only element of
quantum physics that has made it into pop physics, (along with e=mc^2
and that bit about Galileo dropping things off the Tower of Pisa)and
thus is the most misunderstood theorum of quantum physics, as all the
others are merely unknown. This is not because it is complex, it is
because people tell lies about it (or perhaps .2 Truths).
Unfortunately I have no mathematical symbols at my disposal, so words
must do.
The uncertainty principle states that it is impossible to precisely
measure a particle's momentum and position simultaneously. If one is
measured precisely, the other becomes unbounded. This has practical
significance towards the question of whether a photon is or is not blue
according to whether it lies within the range of 540-560 nm. As the
wavelength of the photon may lie arbitrarily close to the cutoff point
of 540 nm, the wavelength must be specified to infinite precision. The
wavelength of a photon is determined by the relationship x=h/E, where x
is the wavelength, h is Planck's constant, and E is energy. I will
further translate E into mc^2(which describes all aspects of momentum
but the direction). x=h/mc^2. Planck's constant is constant, as is
the speed of light. The wavelength is therefore determined by the mass
variable. The observational task at hand is to pin down the mass to
infinite precision so that it can be determined whether or not the
stated photon is or is not blue. However, once you have accomplished
this the position of the photon becomes unbounded! Since the photon is
just as likely to be in the Andromeda galaxy as on the surface of the
Earth, I do not think it possible to draw any conclusions about the
color of the sky from your precise measurement of wavelength.
In practice both precision and momentum can be constrained to a very
small range, although there is always a remote possibility (assuming
non-infinite precision of momentum) that a photon emitted from your
flashlight will end up in the Andromeda Galaxy. In our blueness
defintion this suggests that there is actually some ambiguity at every
wavelength, although in most cases infitesimally small. We can perceive
blue because the cones in our eyes are tolerant to small fluctuation of
wavelength,and the sheer number of photons that enter our eyes make
precise positions unimportant due to statistical mass.
>> The change
>> over time destroys any attempt at certainty.
>Hardly. We can see this from a macroscopic example: if I lead you in a
>simple waltz on the Mexican border facing north, my left foot will be
in
>America during the first measure and in Mexico on 2 and 3 of the second
>measure. (Ignore the fence.) It will truly be wherever it is at any
>given time, however.
The question is not the truth value at points well within boundries. It
is what truth value to prescribe exactly at the border. I will not try
to extend the analogy to your macroscopic example,as feet are not point
particles and will overlap the border. Photons, when well behaving, are
point particles and do not cause this sort of confusion.
>
>
>> In fact, it seems that allowing for a discrete range of
blueness
>> has made the ambiguity twice as bad. If blue was defined as light
with
>> a wavelength of 550nm, at almost every point the Tarskian truth value
of
>> the statement 'le laser est bleu' would be false, except at 550nm
which
>> would be ambigous. By creating a range, there is an area of clear
>> truth, clear falsehood, and TWO ambigous points. It does not seem to
me
>> that Tarskian logic is the path to finding absolute truth if there
are
>> necesarily ambiguities wherever one tries to draw boundries. Indeed,
the
>> only way to create a genuinely universal Tarskian logic statement
would
>> be by defining blue as being the set of all wavelengths, i.e., an
>> infinite statement.
>As noted, in light of the above definition, this discussion of
ambiguity
>is fishy. Since particles have a discrete location at a given time,
and
>since their instantaneous velocities ARE given by calculus, there's no
>ambiguity at all on the Newtonian scale.
Unfortunately Newtonian physics had to be abandoned because it does
not adequately explain the behavior of light (and other things, but
light was the initial problem). Your explanation works slightly better
because it is not Newtonian. Photons cannot have frequencies
independent of other photons within Newtonian physics.
The gist of my argument is that when attempting to map all
statements (in a metalanguage) onto the world, wherever boundries are
drawn there will be fundamentally unavoidable ambiguities. At some
point (in this case at the points of 540 and 560 nm), it will be
impossible to give a truth value of either true or false. I suppose if
you insist on maintaining the Tarskian diagram, you could say "le laser
est bleu' is Truly True or False, but I don't think this is what Tarsky
had in mind! If you find multivalence truly abhorrent, I find no
objection to making a Tarskian diagram with three Truth values, True,
False, and Ambiguous (or Not Applicable). However, as I pointed out
above, the uncertainty principle would make Ambiguous the correct Truth
value at every point (each photon may or may not be blue). While
accurate, it doesn't do much good to anybody.
You could get around the problem of Ambiguity by redefining blue in
terms of complex numbers, but your new defintion would not bear any
relation to what anybody else thought of as blue.
-Christian Whitaker
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