Friday, March 21, 2014

Re-post

I notice two posts on this blog, "Women in science," and "All men are rapists," have been getting a lot of traffic.  Certainly they make nice "click-bait," but this blog was not meant to be political.  Of course it is the arrogance of scientists and philosophers to claim that they deal in "objective" or "eternal" or "universal" (or whichever other superlative you care to use) truths, rather than pieces of the moment.

As other posts make clear, I am interested in moral philosophy and the morality of equality is one of the most important developments in Western thought whose ramifications have yet to be fully worked out.  The idea, however, is taking considerable abuse from modern feminists.

In any case, here is an older post, that I think much more clearly reflects the "spirit" of this blog.

On the negation of modal verbs


I recently learned (it goes to show how diligently I've been practising my German) that Germans use the phrase "must not" in the opposite sense of English speakers to mean "need not." Thinking about this, I realized that there is an implicit "or" in any statement involving modal verbs and the sense of the negative depends upon which part of the logical proposition is negated.

For example, I would translate the phrase:

"You must do A"

into a logical proposition as follows:

^A -> P

where P is some form of punishment. Or:

A or P

We could negate the phrase either by negating the whole thing:

^A and ^P

Or we could negate only one part:

A or ^P = P -> A
(e.g., "You (must not) play in the street.")

^A or P = A -> P
(e.g. "You must (not play in the street.)")

The first example would seem to be how the Germans use the phrase since whether we do A or not, we will not get punished for it, while the third form is more in line with how we use the phrase, that is, A implies punishment. The second example seems rather more ambiguous and in fact inverts the construct: now, getting punished implies that we have done A.

It has rather deep implications, since all of ethics, law and morality is related to the use of modal verbs. Can we use this idea to justify breaking the Ten Commandments?

I translate,

"I shall go to the store,"

as:

F (uture) -> S (going to the store) = ^F or S

The sixth commandment becomes:

You shall (not kill.) = F -> ^ K = ^F or ^K = ^(F and K)

You (shall not) kill. = ^F -> K = F or K

or perhaps,

^(F -> K) = F and ^K

The second says that if there's a future, there may or may not be killing while the first and third say what we want them to say: if F is true, K must be false.

Thursday, March 13, 2014

My dream

Recently I asked a friend where she and her family go to walk in the woods, not realizing that not everyone considers it as necessary as breathing (the jury is not out--it is).  During Bible study one night, we discussed the question, "How did you used to find meaning in life?"  "How do you find it now?" with the implication that before finding religion, it would be normal to find meaning in drugs, sex, partying, etc.  Though I said nothing (the meeting was in Germany and my German is rather weak), I thought to myself that I was always closest to myself, that I found the greatest joy and meaning in life while experiencing the outdoors, whether walking, cycling, skiing, snow-shoeing, and I still do to this day.

The green-spaces in my vicinity are drying up.  Luckily there is one small patch of wood nearby and the owner has even generously cut trails through it.  I keep meaning to go meet him and thank him as I suspect this tiny wood may have saved my life.

Now I don't mean to be ungrateful, but I also believe the configuration of this space, which is right beside my house, is somewhat inauspicious.  It's not an easy concept to explain, but lately I've become interested in Feng Shui, not as some mystical art, but as basic common sense.

For instance: where do you camp?  If you park your tent along a sometime game-trail in a corridor between the trees on a windy night, as my sister and I once did, you will not sleep well.  In Feng Shui, we think of "energy" or "chi" moving out from the camp-site along the trail, but I think a better word for it would be "spirit."  Energy already has a precise definition in physics.  Spirit, by its very nature is ethereal--difficult to measure and without physical form.  Surely the low level of stress of the wind threatening to blow away the tent and the possibility of feral animals wandering into the camp-site would serve to reduce our spirit?

Quite the opposite of a camp-site, parks ought to "breathe," the spirit flowing uninterrupted through trails and other connections, but with sheltered pockets where it can "pool."

Living in Washington D.C., there were at least two routes out of the city that weren't open to motor vehicles.  There was the W & OD trail, built on an old rail line, that ran West along the Orange Line through Falls Church out to Leesburg, ending up just past the first hump of the Blue Ridge mountains and just shy of the Appalachian Trail (AT)  Then there was the C & O canal towpath that ran along the Potomac, joining the AT for a short distance.

I used to take the commuter train or cycle along the W & OD to escape the tension of the city and hike along the AT.  These long distance trails were linked to a network of smaller trails: through Rock Creek park, along the rivers, yes, even through my own, ghettoized neighbourhood in Temple Hill near Anacostia.

I couldn't help but think of these trails as a lymphatic system in the body of the Capitol region.  Transporting the bad spirit from inner city crime, drug gangs (not to mention the dealings of the Whitehouse), and racial tensions out into the surrounding countryside, where it could be purified.

Back to the wood behind my house: it is bordered on three sides by houses.  On the fourth side you used to be able to ski to a golf course, but now the owner of the next property over (not the immediately adjacent one) has erected a fence which blocks you.

The last time I skied in Gatineau park I took lunch in one of the "chalets": wood-heated cedar cabins with picnic benches inside.  I'm certain that these were copied from the Scandinavian countries where skiing is a way-of-life.  I'm also certain that they lost something in the translation.  I imagined these being, not lunch stops for spandex-clad racers bombing along meticulously-groomed snow-highways that form useless circuits, but rather way-points for travellers skiing a single-track trail through the middle of the woods.  A trail that usefully connects two points, that actually goes from one place to another.

I'm trying to imagine a world where I can put on a pair of skis starting at my house and ski for the rest of the day without crossing or retreading the same stretch of trail.  Where I can sling a backpack and hike through the woods until I actually get somewhere, somewhere that I need to go, rather than back again to the noisy, inefficient motor-vehicle that transported me to the trail-head, ten times the distance I ended up hiking that day.

Wednesday, March 5, 2014

Rewriting Maxwell's equations

When I was in my fourth year studying physics, I felt a bit overwhelmed by the E&M courses.  Every thing we learned in these courses was a corollary, ultimately, of Maxwell's equations.  Yet if I had gone into the final having only memorized them or written them on my cheat-sheet, I was sure to fail.

Fast forward several (many) years and I'm working on atmospheric science stuff.  To do atmospheric science successfully requires an intersection of knowledge from many different branches of physics: classical and fluid dynamics, thermodynamics, electricity and magnetism.  We even use some quantum mechanics.  Here again, it's mostly fairly narrow theorems and rarely delving into fundamentals.  In the area of quantum mechanics, for instance, I've never touched a wave-function, just used spectral lines in radiative transfer calculations.

Some of the most interesting work I've done (although I didn't think so at the time) has been simulating surface emissivities, specifically sea water and sea ice.  It's a very fundamental problem: how electromagnetic radiation interacts with matter, but you wouldn't think so from the way it's actually done in the field.  Here again I was using corollaries of more fundamental theories, specifically the Fresnel equations which can be derived from Maxwell's equations.

Here's where it gets interesting: I started off with a radiative transfer equation that was derived fairly directly from Maxwell's equations.  But this equation had at least two errors so I re-derived it much more simply using the Fresnel equations as a basis.  This and a couple of other things got me thinking about Maxwell's equations.  The model requires a series of dielectric constants for the media being simulated.  These constants are quite difficult to calculate, even for homogeneous media such as salt water, and most of the models for them are at least semi-empirical.  For sea ice, it's almost impossible and most of the models handed down to me have only a passing connection to reality.

So it really got me thinking about how to simplify the laws to make them more tractable and also wondering, are they really complete (or perhaps I should say complete enough, since I'm not sure any physics law can ever be complete, efforts to find a "final theory" not withstanding) if they need these finicky and difficult to calculate constants?  Physics laws like Maxwell's equations have deep "roots" and people would sooner question their own measurements or application of the laws rather than the laws themselves.  I've experienced this myself: you keep beating away at the solution until you get the "right" answer, that is the one that agrees with your measurements.

The first thing I came up with was the plane-wave solution: why do we always throw away the imaginary part?  Why not keep it: this is the other field.  If your solution is for the E-field, then the imaginary part is the B-field and vice-versa.  The vacuum equations become:

div(R)     =  p

curl(R)   =    i    d R
                   -----  ------
                  c^2  d t

where R=E+Bi and p is charge.  In light of the equivalence of the electric and magnetic forces--an electric field turns into a magnetic field under a Lorentz transformation--merging the two fields would seem to make sense.  It works for the plane wave solution and as I understand it, every other solution can be decomposed into a series of plane wave solutions.

This is about as far as I got with it.  As you can tell from my ham-fisted attempts to analyse my "Vernier" clock, I'm not that good at theory, although I'd like to do more of it.  To be fair my earlier mistake was more because I couldn't be bothered to take the time, sit down and work things out properly.  I'd like to write more about that later.

There are still problems with this: you can't really merge the electromagnetic permittivity and magnetic permeability into one constant because there will be cross terms.  So obviously I haven't made any progress towards understanding the electromagnetic properties of sea ice--unless that's the answer: there really are cross terms and we haven't accounted for them!


I'm thinking you might be able to take it further merging the reduced set of equations and so further reduce them to only one by generalizing the curl and incorporating time into a four-space.  We did something similar in fourth year--I dug it out recently but all the details escape me.

Two other things bother me about the Maxwell equations.  You can solve the equations using a complex permittivity.  In this case, the imaginary part is the conductivity (or some part of it--I'm pretty sure the two are not precisely equivalent).  But conductivity isn't part of the Maxwell equations.  Why is it directly proportional to the electric field, instead of related through the first derivative as in Newtonian physics?  This is obviously an approximation--an empirical hack.

Of course real physicists don't bother with the classical E & M equations since we've long since moved beyond them, although I'm not sure how much use quantum electrodynamics is for real world problems such as the one above.  I still have a bunch of really dumb questions related to E & M and light, such as: What is the quantum formulation for a single photon? (Can you write out the Schroedinger equation for a photon?  How come we never did that in undergrad?) and how does it relate to the classical formulation for light?  How much information does a single photon transmit?

Tuesday, March 4, 2014

More on the Vernier clock

I've noticed on the statistics page that people are visiting the "Vernier clock" post.  All I have there is a simple animation of a clock with a Vernier scale but no explanation, so I thought I'd write a few more words about it.

Many years ago, before most of us were born in fact, watches with a second hand were expensive because they were difficult to produce.  Meanwhile, those of us who took physical science degrees before every device in sight became digital are very familiar with a type of scale called a Vernier scale.  With a Vernier scale, you really only need one hand on your clocks and watches, or at least, one dial, or, if you desire a second hand without any added machinery, you can turn the minute hand into a Vernier scale.  Or, perhaps you have second hand, but want it to read in hundredths of seconds.

The idea behind a Vernier scale, is that you make the measurement based not on which tick mark is being pointed at, but on which pair of tick marks--one on each side of the dial or sliding scale--line up.  In this way, it's not important how far apart the tick marks are, what's important is the difference in spacing between the first set of ticks and the second.

Lets go straight to the application to clocks, because it's actually simpler since the dials are cyclical.  Suppose we are dealing with a minute hand and we want it to read seconds as well.  On the outside, there are the usual number of sixty (60) stationary tick marks, one for each minute or second as the case may be.  Inside, there is a dial that turns around once every hour.  This dial has 59 tick marks.

Since the dial must move 1/(59*60)=1/3540 of an arc to go from lining up along one outside tick to lining up along an adjacent one, you'd think you're reading in 1/59 of a minute rather than seconds, which of course are 1/60 of a minute.  The seconds readout, however, is centred around the current reading for minutes, thus it moves forward every minute, producing an extra reading.  (Gaah!  this is wrong)

(Edit: I lied.  It's actually simpler for linear scales.  Take the number of divisions in the smaller units and either add or subtract 1 larger unit divided by the number of divisions to give you the spacing of your secondary scale.  E.g. if the units are millimetres and you want a reading in tenths of millimetres, your secondary scale (i.e. Vernier) will have a spacing of either 1.1 mm or 0.9 mm, depending upon which direction you want to take the reading.  It doesn't actually work for cyclical scales, that is you can't divide the dial up evenly and still have readings in minutes and seconds.  So my animation below is complete bollucks--the virtual dial is not circulating at a rate of one revolution per minute (this would be true regardless), and it's not reading in seconds either.)



It should also be noted that the ticks are lining up in a retrograde manner.  In the animation, the "virtual dial" is actually moving clockwise: thus there are actually 61 ticks on the moving dial--thus we aren't really reading in seconds, but in 1/61 of a minute.  I didn't really think that much about this aspect of it, however important.  I just made the animation as "proof-of-concept" and because it looks cool.

I find this concept of a "virtual dial" fascinating and think it might somehow be quite deep--sort of like a phase that travels faster than the group velocity in a wave because of how the different frequencies are interfering.  In other words, you can have electromagnetic waves that travel faster than the speed of light, although apparently you can't transmit any information with them.  I never seemed to be able to quite muster the mathematical chops to fully understand this stuff...

I thought about patenting the idea.  No dice: it was patented back in 1942 (U.S. patent US2293459)!  I also though about taking out a design patent on the overlapping slats idea, but even this is covered in the 1942 patent:

So is the idea actually workable?  Here's a guy who's actually built a couple of them: http://www.gizmology.net/watch.htm.

If a picture is worth 1000 words...


How about a picture with words on it?