A Proper Argument

Very recently it was vigorously brought to my attention just how far out of the mainstream I am.  I have views on how to properly conduct an argument that are at variance with a lot of people.  That is perhaps not surprising.  But I find that I hold views that are at variance with pretty much everybody.  Someone whose thinking I thought was not wildly different from mine turned out to in fact be wildly different from mine.  That was both disappointing and deeply distressing.

I have put more than fifty years of thought and effort into my thinking on the subject of how to determine what's true and what isn't.  I have tried very hard to figure out what works and what doesn't in the context of this pursuit.  I think everybody should value the truth and am disappointed when I come across people who don't.  But it turns out my focus was too narrow.

A lot of my blog posts over the years have been attempts to correct the record.  If there is something floating out there that I think is wrong and others are doing a good job of getting the correct information out there, I leave it to them.  I try to stick with situations where there is either insufficient effort being made to correct the record or where everybody seems to be missing something important.

And I am actually humble when it comes to a monopoly on the truth.  I screw up all the time.  But I figure that if I have gotten something wrong then the only way someone can set me straight is for me to say what I think.  That way people know where I am off base and, therefore, need correction.  I take it as a plus when somebody sets me straight about something I have gotten wrong.

This seems to me to be a reasonable way to approach life.  And I know all about the white lie designed to soothe someone's feelings.  I know that a well placed white lie can smooth out many a social situation.  I am just bad at it.  I know this often holds me back in social situations.  I would dearly love to be much better at it.  I have just never been able to find a way that I can consistently pull off.

But there are social situations and there are social situations.  I try to not be abrasive in purely social situations.  But what about a discourse on the issues of the day?  Is disagreement permitted in these situations?  I would have said the answer to this question would be "yes".  Apparently, I am wrong.

There is a lot of discussion of "echo chambers" and "people talking past one another".  This is universally decried as being a bad thing.  I agree.  But what's the remedy?  Before going into that, at the risk of coming off pedantic, let me restate the problem.  The problem is that disagreement is not allowed.  Beyond that, no one directly engages with the other side's arguments.

The "fix" now becomes obvious.  People should stop engaging in the problematic behavior.  People should be allowed to disagree not only with what the other side says but with what their side says.  Further, both sides should understand and engage with the other side's argument.  And all disagreements should be with the argument, not the person making the argument..

I don't think there is much disagreement with anything I said in the previous paragraph.  (I will go into why there is not across the board agreement below.)  I have now outlined exactly how I proceed.  And I am in trouble for doing so.  Before continuing I am going to make a digression.

Lots and lots of people have outlined roughly the same "fix" as the one I outlined above.  But far too often they add something.  And this is most common when politics is being discussed.  They say some variation on "both sides do it".  This is misleading.

It is true that to some extent both sides do this.  But one side does it a lot more than the other side.  This "both sides do it" argument provides cover for the side that is doing it the most.  They can say "we are only doing it because they are doing it".  I don't think that's true, but as a tactic for getting off the hook, it works great.

Now I could be wrong.  When engaging with this "both sides do it" claim I say "here's why I believe they do it a lot more than we do it".  All you need to do to destroy my argument is to provide evidence that my claim is false.  But nobody ever does that.  Instead they get mad at me.  In other words, they treat me as being on the other side then they fail to engage with my argument.  I have a blind spot.  I am always surprised when this happens.

I think having the argument is critically important.  So there need to be "rules of engagement" for how to conduct a proper argument.  The rules I try to follow are:

• Understand your argument and the evidence that goes along with it.
• Understand the other person's argument and the evidence that goes along with it.
• Engage with the evidence, the data and analysis.  Do it to both sides' argument.
• Do not confuse the argument with the person who is making it.

Stated this way, I think most people would agree.  But I find that often people don't behave that way.  I find the last item critically important.  I never confuse the argument with the person making it.  But this concept is honored in the breach far more often than I thought it was.  I wasn't expecting that.

I very carefully separate the arguments from the person making it.  Just because I disagree with an argument someone had made I don't think they are a bad person or stupid or ignorant.  I just think that in this specific case they have gotten it wrong.

On the other hand, maybe I have gotten it wrong.  If you point out the error of my ways then I am better off for it and that's how I see it.  I am well aware that not everybody operates the way I do.  But I still think it's the best way to operate and I am disappointed when someone who I thought operated that way doesn't.

And I know a big source of my divergence from the norm.  I spent a lot of time interacting with computers.  To state the obvious, computers are not people.  I find that I get along much better with computers than I do with people.  Computers play by rules I am comfortable with.  People often don't.

Computers are good at giving you instant and unambiguous feedback.  I will write and run a computer program.  It will either behave the way I want it to or it won't.  Here's the thing.  Computers don't hold grudges.  If I run a program and it messes up badly the computer, in effect, says "here's the story".  I look at it and try to figure out what I did wrong.  Then I fix it and try again.  The computer doesn't remember what happened last time.  It just notes what happens this time.

I have gone through this "try - fix" cycle so many times I long since lost count.  In each case I soldiered along until the program did what I wanted it to.  And the computer is a neutral arbiter.  It just follows the instructions my program contains and lets me know what happens.  It does not denigrate my looks or ancestry.  It just does what I tell it to.  If I told it right then the right thing happens.  If I told it wrong then the wrong thing happens.  But the computer doesn't even venture an opinion with respect to the right or wrong of what it was told to do.

I flourished in that atmosphere.  I never took it personally when the computer told me I got it wrong.  I just dug in and tried to do better next time.  I was also okay with not receiving praise from the computer when the program worked.  In short, there was an implicit "nothing personal" about how the computer behaved.

So, what's all this have to do with a proper argument?  Just this.  Computers taught me to get comfortable with criticism of my argument/program and to not take it personally.  In the real world, there is the argument and the person that is making the argument.  They are two different things.  Even though most people don't have the computer background I have I thought thoughtful people knew that.  Apparently I got that wrong.  Silly me.

I have no problem separating the person from the argument they are making.  Maybe it's my computer experience.  Maybe I am just wired that way.  But it just seems so obvious to me that I don't continuously say anything about it.  I think objecting to an argument is NOT objecting to the person making the argument.  But apparently way more people than I thought always see objecting to an argument they are making as some kind of personal attack on them.

It would be nice if this didn't matter but it does.  The Greeks made a distinction between "logic" an effort to determine what is right and true, and "rhetoric", the best tactics to use if you want to win an argument.  Their study of rhetoric focused on what was effective.  But along the way they identified both fair and foul ways to be effective.

If "winning is the only thing" then, by all means, use whatever works.  (These are the people who would not go along with the list of principles I outlined above.)  But we should all be able to identify when someone is using one of those foul means to advance their position.

One of the most common foul means is called the "ad hominem" argument.  "X" and "Y" have a difference of opinion.  "X" says "I'm right because of blah, blah, blah".  "Y" says "X is a bad person so you don't have to pay any attention to what he said".  If a person quickly resorts to ad hominem arguments I assume they are in the wrong unless I see substantial evidence to the contrary.

But, since nothing is ever as simple as I would prefer, sometimes an ad hominem argument is justified.  If a person says "I'm right because I'm and an expert and I have studied the situation carefully" but an opponent presents evidence that the person is not an expert and has not studied the situation carefully, then it is appropriate to take the characteristics of the person making the argument into account.  This all assumes, of course, that the opposition provided credible evidence to back their claim up.

Ad hominem attacks are deployed in order to avoid engaging with the meat of a person's argument.  Unsubstantiated or easily disproven ad hominem attacks are the worst.  They should be routinely denounced.  But this almost never happens.  Instead, we are subjected to ad hominem attacks all over the place.  I try my best to make things better, not worse.

The problem is that in the present environment, bad behavior works.  The most generic version of this sort of thing is called "going negative".  When someone runs for public office they should advocate for their positions and qualifications.  If they instead say "my opponent is a bad person", that's going negative.  And this kind of attack is often extended to "my opponent and all of his supporters are bad people".

A couple of generations ago "going negative" was widely derided.  But it worked and it kept working.  It turns out that voters are happy to support a candidate who go negative.  When it became apparent that going negative was effective everybody started doing it.  I never liked going negative but that is an argument I lost a long time ago.  How long ago?

An early proponent of going negative was Richard Nixon.  He used it successfully to get himself elected to the US House of Representatives.  He later used it to a lesser extent to win a Senate seat and then a slot as Vice President on a winning ticket.  In 1960 he decided to run for President.

He also decided to run a positive campaign.  He was obviously more qualified than his opponent, a relatively inexperienced Senator named Kennedy.  So why not win fair and square?  He lost.  If you look at the debates they engaged in, you will find that their positions were little different.  And conventional wisdom had it that Nixon won the debates if you talked to people who heard them on the radio.  But TV viewers gave the nod to Kennedy.  He looked handsome and confident.  Nixon looked swarthy and untrustworthy.

People didn't decide based on the quality of the candidate.  They decided based on likability and personality.  Nixon also didn't go negative when he ran against a far less well qualified candidate to become Governor of California in '62.  He lost again.  In '68 he went back to his "tricky dick" tactics and won the Presidency.  He won big in his reelection campaign in '72 by using even less savory tactics.

It is hard to fault Nixon for reverting to type.  Playing fair was not a successful strategy for him.  It was the voters who decided what worked and what didn't.  He just decided to go with what worked.  For a while the thought was that Nixon was an outlier.  But then more and more candidates went negative and won as a result.  Voters decided that going negative was okay.  If they had decided otherwise we would now be in a far different place.

I have known for a long time that going negative works when it comes to elections.  But that hasn't meant that I liked it.  And it has not worked when it comes to my vote.  But I do confess to being typical in that I make my decisions based on many factors.  I don't just go with the candidate that is the most honest or the most competent.  I do, however, accord those factors a lot of weight.  But elections are not the only thing we argue about.

This is not my first run at this subject. Back in 2014 I wrote a blog post called "Faith Based Conflict Resolution".  Of all my posts, it is the one I am most proud of.  Here's a link to it:  http://sigma5.blogspot.com/2014/12/faith-based-conflict-resolution.html.  Looking back at it I find that I was too optimistic.  I just assumed the whole business about separating out that argument from the person making the argument was commonly accepted and just focused on the mechanics of the argument.  Before moving on, here's the meat of the argument:

Ultimately the only tactic that is effective in this environment [a "faith based" environment] is the power tactic.  And do we really want to decide all conflicts by a test of power?

A little later I partially answered that question.  I pointed out that my preferred approach, the scientific one, frequently leads to embarrassment.  Then I said:

Well, there's the whole "inconvenient" thing.  In the world of science it is frequently true that everybody is wrong.  An outcome where everybody is wrong is the only one that is worse on our egos than an outcome where we are wrong.

 I knew that this approach would not appeal to everyone.  After all, some people are more interested in being on the winning side than they are on getting the facts or the tactics right.  But I truly believed that there were lots of people who shared my "facts first" attitude.

But the whole "how should conflicts be resolved" issue presupposes that that it is possible to go about the business of disagreeing without it instantly and inevitably turning personal.  Lots of people are comfortable engaging in ad hominem attacks.  Turning all disagreements into something personal is something they are comfortable with.  Apparently more people are comfortable with ad hominem attacks than I thought.  That's bad.  I still think it is important to be able to disagree without it getting disagreeable.

So is all lost?  Actually, no.  I take hope from the most unlikely of sources, sports.

People take their sports and their favorite teams very seriously.  And you don't have to look far to find examples of fans getting totally out of control.  But mostly the opposite is true.  Sports bars are everywhere.  And they serve alcohol, which usually makes things worse rather than better.  But things getting out of hand is actually the exception rather than the rule.

On any day in any city you can find lots of sports bars full of rowdy fans.  And many of these bars are populated by heterogeneous groups.  One group consists of fans for one team or athlete.  Another group consists of fans of another team or athlete.  And they are often very vocal when it comes to their opinion.  And large quantities of alcoholic beverages are consumed.  But at the end of the day almost all of these rowdy fans go home peacefully and quietly.

This actually happened to me.  Many years ago my then girlfriend and I visited the "Cheers" bar in Boston.  Locals take the Sox very seriously and there was a game on between the Sox and the Seattle team when we arrived.  When patrons found out that we hailed from the land of the enemy they derided our team and exalted theirs.  But then the Sox lost quite unexpectedly.  Things could have gone south at that point but they didn't.  Instead, all sides were good sports about it.

So what's going on?  I'm not much of a sports fan.  But I do routinely skim the sports section of the paper.  You know what it's full of?  Facts and data.  Sure, there are opinion pieces.  But page after page is full of box scores, statistical breakdowns, and all kinds of detail about teams and players.  And ask the typical fan in the typical sports bar.  They can reel off statistics and figures until your eyes cross.

Sports fans are deeply knowledgeable about their passion.  Couple that with an unambiguous result.  This team or player won or lost.  The score was whatever.  Modern day sports coverage is deeply analytical.  And that means that sports fans are intolerant of BS.  Even the opinion columns have to back up their opinions with facts and data.  Fans get into arguments with other fans all the time.  But "'cause I say so" just doesn't cut it.

And, while a lot of trash talk goes back and forth, no one gets upset by it.  At the end of the day it's mostly "no hard feelings" and "see you at the next game" rather than "I now hate you from the bottom of my heart".  Sports fans, even drunk ones, have mastered the art of separating the argument from the person making the argument.  That makes them role models of a kind we badly need.

And I think the fact that sports and sports coverage is now so data driven that is a major contributing factor.  Michael Lewis wrote a book called "Moneyball" way back in 2003.  The book discussed something called "sabermetrics", an effort to replace emotion with data when it came to evaluating baseball players.

Baseball fans will no longer tolerate a team that doesn't adopt a sabermetric approach.  And many other sports have since adopted similar approaches.  Fans now demand no less.  A team that now tries to take a "seat of the pants" approach can count on such a decision being greeted with scorn and derision from their fans and from the press.  So sports and sports fans have adopted a scientific approach to their fandom.

Sports is definitely the better for it.  And sports betting is about to become ubiquitous.  It will soon be easy for a fan to lose a lot of money by betting from the heart rather than from the head.  And this will provide additional inducement for fans to behave responsibly.

Sports is supposed to be less important than politics.  But more people invest more time and effort in sports than they do on politics.  Unfortunately, it shows.  Politics would be better off if it adopted the kind of data driven approach that is now common in sports.  Where's the call for a "sabermetrics of politics"?

And people who are not that into sports need to behave more like sports fans do.  Remember!  You heard it here first.

60 Years of Science - Part 16

This post is the next in a series that dates back several years.  In fact, it's been going on so long that I finally decided to bite the bullet and update the title from "50 Years of Science" to "60 Years of Science".  Same series, just an updated title.  And, ignoring the title change, this is the 16th entry in the series.  You can go to http://sigma5.blogspot.com/2017/04/50-years-of-science-links.html for a post that contains links to all the posts in the series.  I will update that post to include a link to this entry as soon as I have posted it.

I take Isaac Asimov's book "The Intelligent Man's Guide to the Physical Sciences" as my baseline for the state of science when he wrote the book (1959 - 60).  With the new year it is now fully sixty years since the book came out.  In these posts I am reviewing what he reported then noting what has changed since.  For this post I am moving on to a chapter he called "The Waves".  I will be reviewing two sections:  "Light", and "Relativity".

Light is fundamental.  As Asimov notes, the first words in the bible are "let there be light".  But for a long time the nature of light was a complete mystery.  Two early ideas were that it was emitted by objects and that it was emitted by the eye.

CGI, Computer Generated Images, now a staple of the movie making business, was not a thing in Asimov's time.  A single CGI shop now has more computer power than existed in the entire world until some time in the '80s.  But one of the techniques employed by CGI is called "ray tracing".  And one way to do ray tracing is to start with the eye of the viewer and trace light paths back to the virtual objects in the CGI image.  So the latter idea is not as nutty as it now sounds.

Little was known about light.  It traveled in straight lines, hence ray tracing.  When it was reflected, say off of a mirror, the angle of the reflected light was equal but opposite to the angle of the incident light.  The transition between materials, say from air to water, caused light to bend or "refract".  That was pretty much it until Newton came along.

Newton published the results of his experiments on light in a book called Optics.  Unlike Principia, Optics is easily understood by regular people.  The experiments he performed and analyzed are clearly described and elegantly analyzed.  This is the complete opposite of the situation that I found when I dived into Principia.  In Optics, it is easy to follow along with him and nothing he has to say is hard to understand.

Newton investigated light's characteristics by completely covering the windows in a room.  Then he poked a small hole in the covering, thus letting a narrow beam of sunlight enter his now darkened room.  The then placed objects, primarily lenses and prisms, into the beam to see what happened.  Using this simple and easy to understand (and reproduce) approach, he was able to determine many of the properties of light.

Both the lenses and the prisms bent light.  And, in the prism's case, it broke light up into a spectrum of colors.  Water droplets in the air do the same thing.  The result is a rainbow.  Lenses curve light so that it either converges to a point or diverges to a band much wider than the original sunbeam.

Newton proved that sunlight is actually composed of a mix of a whole lot of different colors.  He was even able to break light apart into its component colors and then put the colors back together again.  He did this by first guiding the sunbeam into a prism, which broke the light into colors.  He then guided the output of the first prism into a second prism that had been turned the other way.  This reassembled the colors back into white light.  He also observed that the degree to which a lens bent light depended on the color of the light.

All these and many others (I am just skimming the surface of what he so clearly lays out in Optics) make light sound like it is made up of waves.  Nevertheless, Newton concluded that it was actually composed of tiny particles he called "corpuscles" that traveled at very high speed.  (He decided that refraction was caused by a speed change in light corpuscles as they transitioned from one medium, say air, to another, say water.)  This "corpuscular" idea set off a battle over whether light was made up of particles or waves,  That battle took hundreds of years to resolve.  Moving on, . . .

Huygens was an early proponent of the "wave theory".  Waves have a "wavelength", the distance from one peak to the next.  If various colors of light have different wavelengths then many of the attributes of light can be explained.  Refraction, the bending of light, and the color dependence of refraction (light of different wavelengths is bent more or less, depending on its wavelength) could be explained this way.  But particles don't have a wavelength, or so everybody thought.

But the wave theory of light had problems, which I am not going to go into.  The wave people could knock holes in the particle people's analysis.  And the particle people could knock holes in the wave people's analysis.  Both sides believed that the holes in their theory could somehow be patched up but the wholes in the other side's theory were fatal.  So the battle continued until new ideas were introduced.

One experiment that tilted thinking toward the wave theory was the "double slit" experiment pioneered by Young.  Light is passed through two narrow slits.  After that it strikes a screen forming a pattern.  It is easy to do an experiment with waves in a water tank or guns and a target.  One shows the pattern expected if light is waves.  The other shows the pattern expected if light is particles.

The "two slit" pattern shouted "waves".  The experiment was easy to do.  So lots of people tried various adjustments.  The variations allowed the computation of the wavelength of various colors of light.  The numbers turned out to be extremely small.

Fresnel was the first to show that if an object was about the same size as the wavelength of light (bacteria turn out to be too big) then a "diffraction" pattern results.  (His ideas also resulted in the creation of "Fresnel lenses".)  So the particle theory of light is dead, right?  Not so fast.  But first a digression (by scientists, not me).

Now that we know the wavelength of light it should be possible to determine the speed.  Galileo was the first to try.  Flashing lamps from the tops of hills, even hills that were miles apart, didn't work.  What did work was carefully studying when various moons of Jupiter got eclipsed.

Newton had provided a way to very precisely calculate orbits so the expected eclipse times could be very accurately calculated.  Careful observation by Roemer looking for moons eclipsing earlier or later than Newton said they should provided a number, 192,000 miles per second, that is not far off the true number.

Now that they knew what they were up against others were able to bring things down to earth.  If you shine a light between the teeth of a disk that is spinning very fast you can detect extremely small time differences.  Fizeau did just that in 1849.

Foucault introduced some clever modifications that allowed him to come up with a speed of 187,000 miles per second.  His technique was precise enough that he was able to get different results if light traveled through different materials (water versus air, for instance).

Michaelson added more improvements and measured the speed of light in a vacuum as 186,282 miles per second.  In Asimov's time "atomic clocks" and "masers" (the predecessor to lasers) were available.  This degree of accuracy permitted light to be used to measure distances.  In Asimov's time this trick could only be used to measure astronomical distances, millions and billions of miles.

Today we can use it to measure "down to earth" distances.  The speed of light is roughly one foot per nanosecond (billionth of a second).  It is now easy to count nanoseconds and, depending on how much money you have and how much effort you want to put into it, much shorter time durations.  So measuring distances of a few feet using light delay is now easy.  That's how GPS works.  And smartphones can easily do GPS.

If light is a wave the question becomes what's waving?  Sound waves cause air to move.  What's moving in the case of light?  How about something called the "luminiferous aether"?  (This was often short-handed to "ether".)  Let's say some kind of "ether" permeates everything and light works by vibrating it?  This sounded reasonable so scientists went looking for it.  The stuff turned out to be quite elusive.

But its fundamental property was that light propagated through it.  So it should be possible to detect it by carefully measuring the speed of light in multiple directions.  (You can calculate what direction and speed air is moving in by very accurately measuring the speed of sound in multiple directions.)

The thinking of the time was that the ether was fixed in space and the earth moved through it.  The speed of the earth was tiny when compared to the speed of light.  But Michaelson had refined his procedures to the point where it should be detectable.  And everybody knew that the earth moved.

He teamed up with Morley and started making measurements using something called an "interferometer".  The problem is that no matter how hard they looked the speed of light turned out to be the same no matter what direction you measured it in.  If the earth was moving though the ether this was impossible.  Oops.

Newton had developed the idea of a "preferred frame of reference" in Principia.  The idea was that in some sense the universe did not move.  He showed how to translate measurements in one frame of reference to another frame of reference in simple situations.  But he always assumed that there was such a thing as a fixed frame of reference that wasn't moving.  It was very hard to square the Michaelson/Morley results with the ides of a fixed, preferred frame of reference.

The ether was supposed to provide the proof that such a frame of reference existed.  But the experiment that was supposed to once and for all demonstrate the existence of the luminiferous aether failed completely.  In Asimov's time the same experiment cold be performed with a much higher degree of accuracy.  The results were the same.  We can now do it far more accurately than was possible in Asimov's time.  It still fails.  And that failure led to the subject of Asimov's next section, "Relativity".

The first step in moving from what we now call "Classical" or "Newtonian" mechanics was taken in 1893 by Fitzgerald.  He posited that space "contracted" in the direction of motion.  This process became known as "Fitzgerald contraction".  Mathematically, the idea was a great success.  It used a simple mechanism to exactly match experimental results.  Since the fundamental stuff of the universe was affected it meant there was no experiment that would detect it.  That was unsettling.

A side effect of Fitzgerald's work was that, if what he was saying was true, then the speed of light in a vacuum was a universal speed limit.  Nothing could go faster.  That was perhaps even more unsettling.  And Lorentz extended Fitzgerald's work by saying the mass a a particle traveling at neat the speed of light would increase.  In fact, it would go to infinity if it actually reached the speed of light.

This provided a mechanism for enforcing the speed limit.  F=MA, Newton's old formula, was how you "accelerated" particles.  If the Mass went to infinity then the amount of Force necessary to provide that last scintilla of Acceleration would also go to infinity.  Since infinite Force was not available, acceleration all the way to the speed of light was impossible.

All this seemed totally nuts at the time.  (It still does.)  But results like these made it harder and harder to argue that were now called the "Lorentz-Fitzgerald equations" were not only nuts but wrong.  And then there was the annoying fact that all the sensible ideas has been conclusively proved wrong by this point.  It was a good thing that experimental results came to the rescue just when they were needed most.

As noted above, there was no experiment that could detect the Fitzgerald contraction.  However, there were experiments that could be done to detect the absence or presence of the Lorentz effect.  Electrons could be accelerated to very high speed.  And the mass of a fast moving Electron could be measured.  Kauffman did the experiment in 1900.  The Lorentz effect was real.

The "real world" that Newton had explored looked sensible.  It looked "natural".  This world that scientists were now uncovering looked truly weird and very unnatural.  If what was "natural" was that which conformed to the experience and intuition of people going about their every day lives, then scientists' understanding of how the "natural world" worked was diverging more and more as the twentieth century unfolded.

If the results of the Michaelson/Morley experiment had been all that scientists were coping with, that would have been one thing.  Unfortunately for fans of the old understanding of "natural", there was more, much more.  Another problem cropped up almost immediately in what seemed to be an entirely unrelated place.

We are all familiar with the fact that when you heat something up it often glows.  And you can roughly estimate its temperature if you know what the material is and what color the glow is.  For good but obscure reasons to be explained below, scientists call this the "black body problem".

The color/temperature problem can be divided into two parts:  the type of material and the color. We can assign a magic number to the type of material.  If we back this number out of the calculation then the rest of the problem always looks exactly the same.

So scientists picked a mythical "black body" as their name for the "always the same" part.  They then developed tables of magic numbers for specific materials.  They could then back this number out and consult their "black body" calculations for the rest.  That greatly simplified the search for a theory to explain the behavior of their mythical black body.

Black body theory came together quickly after that.  If heat was vibration then they had a formula for translating that vibration into color.  Temperature X should produce color Y.   And it worked, mostly.  But the actual situation was more complex.  Materials did not all vibrate at the same frequency.  Instead there was a frequency distribution.  That resulted in a color distribution.  But there was a reference temperature and a reference color so everything could be tied together.

And the main part worked.  Experiment tied a reference temperature to a reference color just like it was supposed to.  The problem was with the distribution.  It wasn't right.  The detail are complex so I am going to skip them.  Instead I am going to cut to the chase.  A man named Max Plank came up with an idea that fixed the distribution problem.  It's just that his solution was one of those "worse than the problem" solutions.

He decided that the energy involved was "quantized".  It was natural to think that things were vibrating at every frequency, more at this frequency and less at that frequency.  But there would be some vibration at every frequency, even if it was not much.  Plank said "no".  Only certain frequencies were permitted.  If you did the calculations based on this idea then everything came out exactly right.

The problem was that scientists could think of no reason why only certain frequencies were permitted while others were forbidden.  This was another step away from natural and toward weird.  Trust me, if scientists could have thought up something that worked and was natural, Plank's ideas would have immediately been discarded.  But they couldn't.

Plank's idea was extremely simple.  He said there was a fundamental unit of energy he called a "quanta".  Everything had to be done in quanta or exact multiples of a quanta.  It turns out that Plank's quanta is extremely small.  So color or temperature can take on a lot of values.  As a result things look like a smooth or continuous variation is present.

It's only if you look hard that you see that things are actually not smooth.  And the fact that the effect of the quantization of black body radiation is only apparent when you look very closely is why it was not initially apparent.

But that didn't make quantization any less necessary in order for the math to work.  And if it had only been this one small corner of physics that got the quantum treatment then we wouldn't be talking about it.

But this quantum business turned out to be ubiquitous in the world of the small, the world of atoms and subatomic particles.  (That's why the field is now called "quantum mechanics".)  It's now almost impossible to get away from it.  And that means that every part of the world of the small is weird -- really, massively, seriously, weird.

Plank's quantum theory was announced in 1900.  At first it didn't make waves.  Nobody liked it.  Everybody wanted it to go away.  But after Einstein published several papers in 1905 it was too late.  Einstein attacked a couple of seemingly different problems.

One is called the "photoelectric effect".  If you shine a light on the surface of a metal you expect it to kick things like electron and photons of light loose.  That happened but it didn't happen the way people thought it should.  It was another distribution problem.

The details aren't that hard to understand but it would take too long.  So, I am again going to cut to the chase.  Einstein, in one of his 1905 papers, applied "quantum theory" to the problem and out popped a solution that exactly matched the experimental results.  (He later got a Nobel prize for this paper and not Relativity.)  All of a sudden, this "quantum" business was a lot harder to ignore.

Speaking of Relativity, in another paper published in 1905, he introduced what we now call "Einstein's theory of Special Relativity".  In it he introduced the concept of the "Photon".  A Photon sounds like a particle and under some circumstances Photons act like particles.

But a Photon also has a wavelength so Photons act like waves in other circumstances.  In reality, a Photon is neither pure particle nor pure wave.  It has some attributes of either and some attributes of neither.  It's just its own thing.  And, by the way, photons are quantized.

This reformulation of how light worked into this entirely new thing, the photon, allowed Einstein to provide a single coherent explanation for all that was then known about light.  Since everybody -- well, all the scientists working in the area -- had been tearing their hair out because everywhere they looked, they saw problems, it was hard to ignore what Einstein had come up with.

As part of Special Relativity Einstein turned something inside out in an unheard of way.  As I noted above, if you do the Laurence-Fitzgerald thing you come up with a reason why things can't go faster than the speed of light in a vacuum.  But this seemed true "purely as a practical matter".

Einstein turned this inside out.  He said it was a fundamental characteristic of the universe that nothing could move faster than the speed of light in a vacuum.  From that principle he showed how you could derive the Laurence-Fitzgerald equations.  They followed from the absolute speed of light limit.  It was not the other way around.

This inversion might have seemed unimportant.  But Einstein used his view of how things worked to show how a bunch of other things followed from it.  One of these things was the effect on time.  Until Einstein everybody assumed that there was something called "absolute time".

Time worked the same everywhere, right?  It might be hard to synchronize clocks in two places but that was just a practical matter.  If you got it right you would see that all clocks in all places could be used to calculate the time and the time would be the same everywhere.

Einstein said the fact that time didn't always flow at the same speed meant that properly functioning clock didn't always run at the same speed.  He expanded the Lorentz-Fitzgerald equations to include time as well as space and mass.  He then showed how to translate from one frame of reference to another.  There was no such thing as an "absolute frame of reference".

Inherent in the idea of an absolute frame of reference was the idea of absolute time.  But if time could be sped up or slowed down then there was no such thing as absolute time.  And that meant that there was no such thing as an absolute frame of reference.  All frames of reference were always relative.

Einstein's Special Relativity equations showed how to translate between any two frames of reference as long as neither of them was accelerating. In Asimov's time there was still carping in the scientific community about this whole business of time speeding up and slowing down.  It just seemed so unnatural.  There was some evidence that the speed up - slow down was true at that time, but only some.

Now we can measure time much more accurately.  This pertains to both long and short periods of time.  As a result we can easily measure time with enough accuracy to confirm that it behaves exactly as Einstein predicted.  The most obvious example is GPS.

GPS satellites include code to adjust for the fact that they move around the earth at a relatively high speed.  Ignoring this "relativistic effect" would quickly cause the GPS system to get the time wrong.  And that would produce easily detectable location errors.

Moving from the practical to the esoteric, scientists now have access to clocks that are so accurate that raising one of them a single additional foot above the ground is enough to make a measurable change in how fast time flows.  Proof of the veracity of Special Relativity is now unavoidable.

Ten years later in 1915 Einstein came up with General Relativity.  All you need to do Special Relativity is High School Algebra.  That is well within the capabilities of many millions of Americans.  The mathematics of General Relativity are beyond he abilities of all be the most capable mathematicians.  I freely admit it is beyond me.  So we are not going to go there.  But some of the key ideas of General Relativity are easily understood.  They are just super weird.

Remember when Einstein did that inversion and said the constancy of the speed of light was not the effect but the cause.  Well, he did the same thing with Gravity.  Newton said that absent some kind of kick (rocket motor) or drag (friction) things went on at a constant speed in a straight line.  Einstein said that was completely true.  So why do planets like Earth circle the Sun rather than going in a straight line?  Because space is curved in such a way that a "straight line" causes the Earth to orbit the Sun.

This is again one of those things where looking at things this way gets you to the right answer.  But it sounds like a trick or shortcut, rather than who the world really works.  But over time evidence has built up that this actually is the way the world really works.

Special Relativity showed how to translate from one frame of reference to another as long as acceleration was not at play.  General Relativity shows how to translate from one frame of reference to another when acceleration is at play.  Not surprisingly, the math gets Hella complicated.

And scientists would have run from General Relativity except that Einstein was able to make predictions.  (He had this fantastic track record but still, the theory was beyond weird and the math was obscenely difficult.)  I am only going to cover two of those proofs.

Newton had shown how to calculate the orbits of planets.  But predictions based on Newton's equations yielded the wrong answer when it came to Mercury.  The difference between prediction and reality was small.  But astronomers had been tracking it for decades as it got larger and larger.  Einstein was able to apply his equations to get the answer that exactly matched observation.

The problem with Mercury's orbit was a well known one.  Maybe he cooked the books knowing the answer that needed to pop out at the end.  But what if he made a prediction about something that no one had imagined was even possible?  He predicted that in a certain situation something would be a certain amount.  If asked, anyone else would have predicted that nothing would happen.  The answer would, in effect, be zero.

Einstein predicted that if a photon from a star passed very close to the Sun on its path to Earth the path would bend by a certain specific amount.  This would cause the star to appear out be of place for a short period of time.  A star was found and a handy eclipse allowed the confirming observation to be made.

Asimov doesn't even mention Black Holes.  In his time they were considered a quite speculative possible consequence of General Relativity.  But at that time there was no solid evidence that they actually existed.  Gravity Waves were another possible consequence of General Relativity.  But there was no solid evidence for their existence back then either.

A few years later a celestial body called Cygnus X-1 was investigated.  Many astronomers concluded that it contained a black hole at its center.  But for a long time this conclusion was controversial.  But we keep getting better and better and hunting for and finding Black Holes.

We now believe that many, perhaps all, large galaxies contain a supermassive Black Hole at their center.  Our Milky Way contains one that is several million times the mass of our Sun.  Andromeda, a neighboring galaxy, is thought to have one a several billion times the mass or our Sun.

And we have recently been able to detect gravity waves.  The first detection involved the merger of two large Black Holes into one.  Since then dozens of Gravity Wave events have been detected.  But there is an even more interesting post-Asimov development in the General Relativity area.

Einstein applied General Relativity to the fate of the universe.  In his time the universe was assumed by most cosmologists (the people who studied this question) to be in a "steady state".  But evidence piled up that it was evolving from a Big Bang (now estimated to have been about 13.5 billion years ago) through several stages to its present state.

Einstein couldn't get a steady state to come out of his equations until he added a "Cosmological Constant".  He later thought this was an idiotic idea.  It soon became clear that the universe was expanding.  (That was one reason Einstein thought that the Cosmological Constant was a bad idea.)

But, if the Cosmological constant is set to "just right", the expansion of the universe will stop, but only after an infinite amount of time.  Another value causes the universe to expand indefinitely.  Still another, causes it to expand for a while, then collapse back to a "Big Crunch".

For a long time it looked like the universe was expanding at that "just right" speed that would cause it to expand forever.  Now it looks like it expanded slowly for a while but is now expanding faster and faster.  All this can be modeled by fiddling with Einstein's much maligned Cosmological "Constant", which may not even be constant.

Needless to say, scientists now take Relativity, both Special and General, as givens and try to expand on them in various ways.

To Infinity and Beyond - Bonus Content

I recently posted on the subject of Infinity.  Let's have some fun with what we learned there.  And, by "there", I mean:  http://sigma5.blogspot.com/2020/01/to-infinity-and-beyond.html.  Everybody has heard of the following conundrum:

What happens when an irresistible force meets an immovable object?

What do we mean by "irresistible" and "immovable"?  The obvious answer is "infinitely irresistible" and "infinitely unmovable".  Now that we are experts on infinity, let's see if we can shed some light here.

Mental pictures are helpful.  So let's picture our irresistible force as the Mongol Hordes and our immovable object as the Great Wall of China.  Neither, of course, is actually of infinite extent.  But we can take our mental picture of a large group of Mongol Hordes milling around and mentally replace it with a Horde with an infinite number of members.  Similarly, the Great Wall of China is long, very long.  But it isn't infinitely long.  But in our imagination we can extend our picture of it so that it is infinitely long.

Okay.  That's progress.  So what do we mean by "meets".  Well, in our mental picture we can now see this as "meets in combat".  Our infinitely large Mongol Horde attacks our infinitely long Great Wall of China.  That works, so we have made more progress.

So what happens when they meet, as in "fight it out".  Who knows?  But let's now ask the question in the context of our discussion of infinity.  How about this?  When an element of the Mongol Horde attacks an element of the Great Wall of China those two specific elements mutually annihilate each other.  In this context we can now define "winning".  Elements of each group meet and annihilate each other.  If one side has some remaining elements after this process is complete then that side wins.

And, more specifically, we can characterize combat as a process of bringing the two sets of combatants into "one to one correspondence".  If all the elements of one set can be brought into "one to one correspondence" with elements of the other set and if, after we are done, there are elements in the other set that are not matched, for which there is no "one to one correspondence", then that set is larger and that side wins.

Imagine our Mongol Hordes lined up in front of our Great Wall of China.  Say each Horde member occupies a file one yard wide.  And, to keep things fair, we divide our Great Wall of China into segments that are one yard wide too.  So in a particular one yard wide file we look to see if their is a Horde member and a Wall segment.

If both are present it's Horde to Wall and both are annihilated.  If there is a Horde member present but no Wall segment then the Horde member wins.  If there is a Wall segment present with no Horde member in front of it then the Wall wins.  If this was somehow real then the rest of the Horde could pour through any breach in the Wall.  But we aren't going to allow that.  Everyone has to stay in their assigned file.

And we assume that the Horde spreads itself out so as to cover as many Wall segments as possible.  If we run out of Horde members before we run out of Wall segments then that means there are more Wall segments and the Wall wins.  Similarly, if there are more than enough Horde members to cover every segment of the Wall that means there are more Horde members and the Horde wins.

We now have a complete mental picture of what's going on.  So what does go on?  The process of placing Horde members in front of Wall segments is just putting elements of the Horde set into "one to one correspondence" with elements of the Wall set.  So what happens depends on the rule we use to create our "one to one correspondence".

We went into this in some detail in our "Infinity" post.  It is easier to deal with Aleph Naught sets so let's do that.  It also seems right.  The number of members of our imaginary Mongol Horde is a natural number.  We start counting, one two, three, . . .  With a real Horde we would eventually get to the last Horde member.  Thus the highest "counting number" we would reach would be some finite, specific, number.  But we extended out Horde to infinity so we would end up with Aleph Naught Horde members.

In a similar manner, we could measure the length of the Great Wall in yards.  With the real Wall we would eventually reach the end.  So we would stop at a large, but finite and specific, counting number.  But, again, we extended the length of our imaginary Great Wall to infinity.  So we would decide that our great Wall was Aleph Naught yards long.

We could now cut to the chase.  But that's no fun.  So let's imagine that each side has a commanding General.  And that General's job is to come up with a "one to one correspondence" rule.  (We already know that trying to do it by hand, assigning a specific Horde member to a specific Wall segment, is impossible.  So, we'll have to use the "rule" method.)

In any case, let's say the Horde General says "I am going to count my Horde using Integers but the length of the Wall is obviously a natural number.  Then I will match each positive integer on my side with a natural number on their side.  So when we are done I will only have used up the positive integers.  That leaves zero and all the negative integers on my side unmatched.  So I win".

Sounds like a plan, right?  But if the Wall General is smart enough he can overcome this strategy.  He can win even if he lets the Horde General have Integers and he keeps Natural numbers.  What?  Well, he could propose the following rule:  Match the natural number "2" with the integer zero.  Then match the natural number "4" with the integer "+1".  Now match the natural number "6" with the integer "-1".  Keep going.

Using this method we can create a "one to one correspondence" rule that matches only the even numbers in the set of natural numbers with all of the numbers in the "integers" set.  After we have done this each and every integer will be matched up with an even natural number.  But all the odd natural numbers are unmatched and, thus, left over.  So the Wall General wins.

Our Generals would spend forever arguing about which "one to one correspondence" rule to apply.  And that's why mathematicians decided that the cardinality of two sets was identical if even one "one to one correspondence" rule could be found that matched all elements of one set with all elements in the other set.

So the answer to our conundrum is that both sides are evenly matched so both sides would be totally annihilated.  Unless, of course, one General could show that the cardinality of his set was Aleph One while the cardinality of the other General's set was only Aleph Naught.

See, wasn't that fun?  And here you thought that the mathematics of infinity had no real world applications.

The Iowa Primary

Some TV talking heads were droning away the other day when a particular exemplar made reference to "The Iowa Primary", not once but twice.  Taking heads are employed and given air time because of their expertise, right?  Except there is no such thing as an Iowa Primary.  What Iowa does is an entirely different type of political cat.  It's a Caucus.  I like to do posts on subjects where I can clarify and illuminate.  This boneheaded bungle alerted me to just such an opportunity.

And I'm not going to dive into the issues or spend time explaining what I like or dislike about any of the candidates.  I am going to stick with the mechanics.  What exactly is going on?  Why does this and the New Hampshire Primary even exist?  What are we to make of them?

The Iowa Caucus is the bastard stepchild of the New Hampshire Primary.  So what's the story behind that event?  It has been going on for a long time.  And for a long time I am convinced that the only reason it got any coverage at all was so reporters could write off their vacations.

New Hampshire is not far from New York City, still the headquarters of the US press.  North of NYC is Vermont, home of various ski resorts that New Yorkers like to frequent in the winter.  Next to Vermont is New Hampshire.  I am convinced that writers would head to Vermont for some skiing.

While they were there they would make a very quick detour over to New Hampshire.  They would later whip together a "New Hampshire Primary" story which they would subsequently file after they returned to work.  This ploy allowed them to charge all, or at least a big chunk of, their ski vacation off as a "business" expense.

So, for a long time the New Hampshire Primary survived by being kept alive by the cynical desire of multiple NYC based reporters to be able to write off their ski holidays as a "legitimate business expense".  Then 1968 came along.  Lynden Johnson seemed poised to easily cruise to re-election.  Then something happened that the press decided was newsworthy.

As expected, he won the New Hampshire Primary.  So that in itself was not especially newsworthy.  But it was a solid win, not the landslide that everybody expected.  A nobody named Eugene McCarthy finished a strong second.  This marked the beginning of the end.  Eventually, Johnson decided to not even run for re-election and that's how Nixon became President.

After that, the New Hampshire Primary became, and has remained to this day, a big fucking deal, at least in the eyes of the press.  And, seeing how well it had worked for New Hampshire, in about 1972 some political types in Iowa asked themselves "is there any way we can horn in on this?"  And thus the Iowa Caucus was born.

New Hampshire had deliberately positioned itself to be the first step along the road to the White House.  That was an integral part of the scheme, a scheme whose real goal was to raise the visibility of an otherwise insignificant state.  Iowa decided to try to jump in front of New Hampshire.

If they could pull it off it would definitely be good for Iowa and, by implication, bad for New Hampshire.  To cut a long story short, after years of squabbling, the two states eventually worked out a deal.  Iowa would be the first Caucus and New Hampshire would be the first Primary.  And both of them would work together to thwart plans by any other state to horn in on the whole "First" thing.

And, from a historical perspective, Iowa turned out to be a much condensed version of New Hampshire.  No one paid much attention in '72.  But in '76 an unknown peanut farmer from Georgia won the Iowa Caucus.  That peanut farmer went on to become President Jimmy Carter.  And with that, Iowa also became a big fucking deal.

But, as I said, a Caucus is a different kind of cat from a Primary.  A Primary is easy to understand.  It's just an election.  Someone wins.  Someone loses.  End of story.  Caucuses are much more complicated.  More accurately, what we are actually talking about something called a "Precinct Caucus".

Most states are broken up into precincts.  (The rest have something that amounts to the same thing.)  Everybody in a specific precinct votes in the same place.  And, on Caucus night, voters in Iowa go to their individual precincts and "caucus".  Iowa has 1678 precincts so they have 1678 caucuses.  Each precinct has about 370 Democratic voters in it.  (There are 1678 separate Republican Caucuses happening at the same time but, like everybody else, I am going to focus on the Democrats.)

That means that there will be 1678 separate events in Iowa on Caucus night.  And to participate you have to show up in person.  And you have to make it to wherever your assigned precinct is meeting.  There is no such thing as an absentee ballot in a Caucus.  If you don't show up at the right place at the right time you don't get a say.

So what happens at a caucus?  Well, given that it's a political event, a lot of hot air is expended.  But for our purposes what happens is that at some point in the evening attendees are segregated into groups, one group for each candidate that has at least one supporter present in that particular precinct.

Or voters can join the "uncommitted" group.  In a caucus you can support "none of the above" or, more accurately, "a candidate to be decided upon later".  It is also important to know that this is all done in public.  There is no secret ballot.  You literally have to stand for your candidate, which might be "uncommitted".  But you have to publicly pick a side.

Then the 15% rule kicks in.  If your group does not represent 15% or more of the attendees at your particular precinct caucus then your group must dissolve and its members have to join one of the "15% or more" groups.  Or, of course, you can just go home, leaving behind your ability to influence the process.  So when that's done, we are all set, right?  Not by a long shot.

What each caucus actually does is select delegates to the "district" convention.  Each delegate is committed to vote for whichever candidate (or, in the case of the uncommitted contingent, a candidate to be named later) they stood up for at the caucus.  The Precinct Caucuses are just the first step.  They are followed by "District Caucuses" and finally "State Conventions".

The process at the district caucus is pretty much the same except that even more hot air is expended.  But the delegates that actually show up again group by candidate.  The 15% rule is then applied.  The result, after a whole lot of bickering and the expenditure of vast amounts of hot air, is the election of a slate of delegates to go to the state convention.

The composition of the delegation mirrors the relative strength of support among district caucus attendees.  There is typically some falloff in attendance between the precinct caucuses and the district caucuses.  There is a whole "alternate" business.  But in practice it doesn't work.  So some delegates from some precincts don't show up.  And if you don't attend you can't vote.

This is followed by the State Convention.  Delegates selected at the district level tend to be more dedicated so there is usually very little falloff between the district caucuses and the state convention.  There, the same "group by candidate" business again happens.  The 15% rule is again applied.  Finally, delegates to the National Convention are selected.

A Presidential candidate is actually selected by the delegates attending the National convention.  There are rules governing how all the preceding steps operate.  But their entire job is to eventually feed delegates to the National Convention.  Depending on the year, the Iowa delegation will or will not closely track the numbers from the precinct caucuses.

If all this sounds exhausting, that's because it is.  Until relatively recently this process was designed to find and promote committed political types who would provide the blood, sweat, and endless amounts of time necessary to run a political party.  And, as such, it worked pretty well.  But then this whole "picking a Presidential Candidate" business got grafted on top of it.  This is supposed to be a better process than the old "smoke filled room" method previously employed.

So, at this point (focusing on Iowa for the moment) we are getting all kinds of polling and "informed speculation" about what will happen on Caucus night in Iowa.  All this can be indicative but that's it.  Iowa may come out the way the smart money thinks it will.  Or it may not.

Beyond that, the press will assume (because that's what they do every single time) that whatever numbers emerge from Caucus night in Iowa will at least tell the complete tale of which candidates will get how many delegates "from the Great State of Iowa" at this year's Democratic National Convention..

In reality, we won't know what the correct numbers are until the State Convention happens many months from now.  The better observes will monitor the district conventions to see what happens there because changes may creep in.  The process is pretty predictable from there on so extrapolation from that point is justified.

But let me circle back to Caucus night.  As I said, there are about 370 registered Democrats in each precinct in Iowa.  Should we expect 370 people to show up?  Of course, not.  How about, say 180, about half?  (That's close to the percentage of registered voters that voted in 2016.)  Probably not.  The actual answer is "nobody knows".

Washington State has, for the most part, been a Caucus state.  That's why I know so much about the process.  In off years attendance at the Precinct Caucuses I have attended has been in the single digits.  In hot years it has run as high as 40-60.  So 40-60 is a good guess for Iowa for this time around.  But it might go higher.  People are extremely revved up.

And, for context, if 20 people attend then 15% is 3 people.  Also, for context, it is impossible for more than 6 candidates to simultaneously hit the 15% threshold.  There are 12 Democratic candidates still in the race, at least according to the New York Times, so that means that half of them are going to get aced out.  And if there are 3 undecideds then that's another candidate who won't get anything.  And if a candidate gets 30% that means another candidate gets the boot.  So in most precincts three or perhaps four candidates will make the cut.

But a candidate can make up for a loss in one precinct if the same candidate does very well in another precinct.  Candidates tend to do relatively better in some places and worse in others.  So there should be some averaging out.  But the 15% rule and other rules are there to funnel most support toward a few candidates, maybe even only one.

By now it should be obvious that in Iowa it's all about the ground game.  A supporter who stays home might as well not be a supporter at all, as far as the results of the Caucus are concerned.  A campaign expecting to do well in Iowa must be good at getting their supporters to attend a caucus.  And absolute numbers don't matter.  A thinly attended caucus in one precinct sends the same number of delegates as a heavily attended caucus in another precinct.  And all that matters is delegates.

Carter put a lot of effort into Iowa while other candidates didn't.  He was able to identify people who would support him and who would stick with him all the way through the process.  He made sure they showed up at caucuses.  Other candidates didn't invest as heavily in their Iowa "ground game" and they ended up losing out.  But everybody has long since figured this out.

This "strong ground game in Iowa" strategy works best in years when the energy level is low.  If other candidates are coasting along then a lesser known candidate can sneak in and run away with Iowa even if he doesn't actually have that much support in the state.  If the supporters of other candidates stay home then it takes only a small group of dedicated followers to make a big splash.

But, given how much coverage is already focused on Iowa this year, all the candidates know that having a strong ground game in Iowa is more critical than ever.  A whole lot of second and third tier candidates are hoping that their Iowa ground game will catapult them up in the rankings.  If that happens we will know within twenty-four hours of the end of Caucus night.

On the other hand, if you are a top tier candidate and you don't do well in Iowa, expect to see a lot of "candidate in trouble" ink immediately spilled.  The good news for these candidates is that Iowa is not the whole story.  If you "bounce back" in New Hampshire then a poor showing in Iowa will soon be forgotten.  If you are a top tier candidate and you do poorly in both Iowa sand New Hampshire, then you are toast.  Or so the conventional wisdom has it.

Both Iowa and New Hampshire are atypical states.  They are small and rural and white.  They have hung on to their prominence because of their track records.  Historically, if you do bad in both state events, you are toast.  Given that they are so atypical, lots of people have advocated for diminishing their power.  There is a simple way to do this.  Tarnish their track record.  And this is the year that might happen.

Biden has been "the one to beat" for more than a year now.  He could do fairly poorly in both Iowa and New Hampshire (not a prediction, just a thought experiment).  He is currently polling extremely well in South Carolina, the fourth of the "big four" Presidential contests that come before Super Tuesday.  (I'll get to what Super Tuesday is in a moment.)  Besides the two states I have already discussed, Iowa and New Hampshire, the third "big four" state is Nevada.  If he actually does well in South Carolina and then follows that up with good success on Super Tuesday then he could end up with the nomination.

An even more interesting possibility is Bloomberg.  He has made no secret of his strategy.  He is, in effect, skipping all of the "big four" early contests entirely.  Not surprisingly, the "smart money" expects him to do badly in all of them.  Instead, he is betting heavily on doing well on Super Tuesday.

Super Tuesday happens exactly a month after the Iowa Caucus.  The Iowa Caucus is on February third.  Super Tuesday is on March third.  On Super Tuesday twelve states hold primaries.  Included in this list are Texas, California, and several other large states.  So it differs from the "big four" events in pretty much every way possible.  They are "one state at a time" contests.  Super Tuesday is a "many states at a time" event.  They are small, homogeneous states.  (At least Iowa and New Hampshire are.)   Super Tuesday includes large states and states with diverse populations.

Combined, the number of delegates at stake on Super Tuesday is enormous.  Iowa and New Hampshire, in particular, will each send small delegations of delegates to the Democratic National Convention.  They have such an outsized impact because of the press coverage they get and not because of either's actual direct effect on the outcome.  (BTW, the Washington State Primary is on March 10, a week after Super Tuesday.)

If Bloomberg cleans up on Super Tuesday he will be in good shape to snare the nomination.  He has been spending heavily for several weeks already.  And he has enough money to spend everywhere, not just in the "big four" early states.  If he then goes on to win the nomination (or if Biden follows the path I have outlined above) then he will have shown that there is a path to the nomination that does NOT go through Iowa or New Hampshire.  And that will substantially diminish their clout.

Finally, there is a little noted impact of caucuses.  Besides being responsible for each state's role in selecting Presidential Candidates, this whole Precinct - District - State business is also responsible for putting into place all of the official that run the local, regional, and state party apparatus.

In 1988 Pat Robertson took advantage of this in Washington State.  On the Republican side it was a low energy year so few people showed up on Caucus night.  That allowed his small but committed group of supporters to take control of all of the Republican party apparatus in this state..

They sent a slate of delegates consisting almost entirely of Robertson supporters to the Republican National Convention that year.  But they also put their people in place to run the Republican party in this state.  Their control lasted for many years.

Pat Robertson was a major figure in the religious right.  His bid to snag the Republican Presidential nomination went nowhere.  But the Republican party in this state fielded candidates that were closely associated with the religious right long after Robertson and his people faded from the national scene.

They were able to do very well in various regions in the state but pretty much struck out in state-wide elections.  Our last Republican Governor was John Spellman.  He left office in 1985.  It's not that the Democrats keep putting strong candidates up.  It's that, due to the control exerted by the religious right, for a long time Republicans put up very weak ones.

Their lock on the levers of power within the state Republican party has finally weakened.  But then the state has drifted decidedly blue in the past few years too.  This has allowed Democrats to continue to have great success in statewide races.  But if Republicans put up strong candidates, who knows what will happen?

Anyhow, you should now have a deeper understanding of what's what with the Iowa Caucus.  Is that going to affect the outcome?  Nope.  But it should better equip you to deal with the BS that is and will continue to be spewed by people who are supposed to know what they are talking about.  That's all I can hope for.

To Infinity and Beyond

This post is about mathematics but it is aimed more at the "I'm bad at math" crowd than it is at those having substantial expertise in the subject.  That doesn't mean the mathematically inclined won't enjoy the post.  They should.  At least, that is my hope.  On to business.

All of us are exposed to Arithmetic from an early age.  That is, the basic business of addition, subtraction, multiplication, and division.  Most of us got some exposure to Algebra somewhere along the line in school.  For many it represented a fork in the road.  My reaction to Algebra was "cool".  That put me irrevocably on the path toward nerdiness.  Others found it a slog, or worse.  They became part of the "I'm bad at math" crowd.

Beyond Algebra is Trigonometry.  And beyond that is Calculus.  The big four:  Arithmetic. Algebra, Trigonometry, and Calculus, provide a "degree of difficulty scale ranging from Arithmetic (easy) to Calculus (hard) that we are all familiar with.  But there are other kinds of math.

I wasn't exposed to set theory until I got to High School.  But it has now migrated down to elementary school, or so I'm told.  Geometry seems non-mathematical but generally gets lumped in with math.  Logic is also some kind of weird step-child of math.  So these oddballs, set theory, geometry, and logic, are kinds of mathematics that many people have at least some exposure to but which they also view as something other than "pure" math.

It turns out that once you get past the basic four there are lots of these oddball branches and offshoots of mathematics beyond the three I listed above.  And these other oddballs and offshoots are generally lumped together as "higher mathematics".  The general perception is that all of them are really hard.  After all, they are called "higher" mathematics because, with few exceptions, no one studies them until after they have mastered Calculus and Calculus is really hard.

And, for many kinds of higher mathematics, this is a totally accurate characterization.  Matrix Mechanics makes Calculus look like a walk in the park.  Yet it is the mathematical foundation of everything Particle Physicists do.  And that's why 99.99% of us are not Particle Physicists.  But there are also parts of higher mathematics that are not all that complicated.  They are just ideas that regular people never get exposed to.  One of them is infinity.  And that's what I want to expose you to.

We all have some idea what infinity is.  It's the number beyond all numbers, it's so big.  Or, to put it into more concrete terms, you start counting.  You go one, two, three, and so on.  You keep going and going and going.  And at some distance beyond the last number you count up to is the last number, infinity.  It's bigger than any number you can actually count to.  That's people's idea of infinity.

People have an intuitive idea that "infinity" is a weird number.  But they tend to underestimate its weirdness.  For one thing, it's not a specific number.  Say it was a specific number.  Call it "I".  Now add one to "I".  This "I+1" number is also a specific number and it's bigger than infinity.  But infinity is the biggest number there is.  So that's impossible.  So forget about the idea of infinity as "a number like any other number".  It just isn't.  It's these sorts of problems with infinity that make it hard to wrap your arms around it.

But mathematicians wrestled with all these kinds of issues and found a way to work with infinity anyhow.  They resorted to set theory.  A "set" is just a group of objects.  One of the fundamental attributes of sets, the mathematical construct, is that you need to have a way to determine if a particular object, called an "element", is in a particular set or not.  There are several ways to do this but I am only going to very briefly cover the two most common.

First, you can just list all of the elements of your set.  You can say "my set consists of 'A', 'B', 'C', . . ., and 'Z'".  The second way is to provide a rule:  "My set consists of all of the letters in the alphabet".  For many small sets the "list all the elements" method works just fine.  But we are going to be dealing with large sets so we will use the "rule" method.  And the elements of the sets we are going to be talking about will consist solely of numbers.   So no "the set of all Presidents" or "the set of all cars I have owned".

Another attribute of a set is the number of elements in the set.  It may be zero.  The set may be empty.  That is perfectly legal.  But all of our sets will have elements in them, lots of elements.  In fact, they will have an infinite number of elements in them.  And the technical term for the number of elements in a set is its "cardinality".

And we can compare sets.  Two sets may be equal (each consists of exactly the same elements) or unequal (there is an element that is in one set but not the other).  We can also have subsets (all the elements in one set are found in the other set) or supersets (some of the elements in one set are not in the other set but all of the elements in the other set are in the one set).

There is lots more that could be said about sets but that's all we need.  Oh!  There is a bit more we need to know about cardinality.  If two sets are equal (contain exactly the same elements) then the cardinality of the two sets is identical.  But what about sets that don't contain exactly the same elements?

Let's say that we have two unequal sets but we want to know if the cardinality of both sets is identical.  If we can make up a rule that associates exactly one element of the first set with exactly one element of the second set then we can say that those elements have been put into a "one to one correspondence".

And let's say that we make up a rule that creates a "one to one correspondence" for every single element of the first set with an element in the second set.  And let's say the same rule can be used to do the reverse, to create a "one to one correspondence" for every element of the second set with an element of the first set.  And to be clear, there is only one "one to one correspondence" for each element of each set.  We don't ever use the same element twice.

Anyhow, if we succeed in doing this then the cardinality of the two sets will be identical.  And you are probably saying to yourself at this point "this all seems needlessly baroque".  But it is necessary to deal with the fact that infinity is not a single specific number.  With that out of the way, let's go to work.

The "counting" numbers:  one, two, three, etc., are called "natural" numbers by mathematicians.  So if we create a set containing all of the natural numbers its cardinality will be infinity.  That seems like a lot of work to get somewhere that seemed obvious from the get go, but trust me, it's all necessary.  Because now the fun starts.

The next step up in the hierarchy of "numbers" from the natural numbers is to "integers".  Take each of the natural numbers, and add "plus" to its name and add it to a new list.  Then add the new number "zero" to the front of the list.  Finally, take each natural number in turn again.  But this time add "negative" to it's name before adding it to the list.  So, add "negative one", "negative two", "negative three", etc.  And let the negative numbers increase as we move left and also put them on the front of the list.

Now, instead of having a list of "natural" numbers starting at "one" and marching off to the right as they get bigger and bigger, we now have a list with a central number, "zero", and two lists of numbers, one marching off to the left and the other marching off to the right.  It's just that the list of numbers marching to the right now have a "plus" added to each name and the list of numbers marching to the left now have a "minus" added to each name.  That all seems straight forward.  But now let me throw a monkey wrench into the works.

Let's make a second set consisting of all of the integers.  The question is:  Is the cardinality of both sets (natural numbers and integers) identical or not?  The obvious answer would be "no".  The natural numbers are a subset of the integers.  We can form a "one to one correspondence" between each of the numbers in the set of all natural numbers and an element in the set of all integers.  Just match "one" from the set of all natural numbers up with "plus one" in the set of all integers.

Proceed step by step with "two" and "plus two", "three" and "plus three", and so on.  (This would take an infinitely long amount of time.  But that doesn't matter.  Obviously the process works.  And that's all we need.)  We end up with everything lining up nicely with respect to the "natural numbers".  But at this point we also have "zero" and all the negative numbers left over in the "integers" set, and they are not matched up with anything.  So obviously the cardinality of the set of all integers is greater than the cardinality of the set of all natural numbers.

There is just this one tiny little problem.  Take "zero" from the "integers" set and match it up with "one" in the "natural numbers" set.  Take "plus one" in the "integers" set and match it up with "two" in the "natural numbers" set.  Now take "minus one" in the "integers" set and match it up with "three" in the "natural numbers" set.  Keep going matching "four" in "natural numbers" to "plus two" in "integers", "five" in "natural numbers" to "minus two" in "integers", and so on.

It turns out that we can create a "one to one correspondence" between the "integers" set and the "natural numbers" set.  So what should we do?  If we do the "one to one correspondence" one way we decide that the cardinality of one set is greater than the cardinality of the other set.  If we do it another way we decide the cardinalities are identical.  And, if we are clever, we can even come up with a rule for creating a "one to one correspondence" that makes it look like cardinality of the "natural numbers" set is greater than the cardinality of the "integers".

What's really going on all goes back to the fact that infinity is not a single specific number.  And what mathematicians decided to do about that was to decree that as long as at least one rule existed that put every element of one set into exactly one "one to one correspondence" with every element of the other set and that as long as the "one to one correspondence" went both ways then the cardinality of the two sets was identical.  To say that this is not intuitive is a vast understatement.  But there it is.  But wait.  There's more.

"To infinity and beyond" was the signature line of Buzz Lightyear, a character in the movie "Toy Story".  Anyone who has seen the movie (and everyone should) knows that Buzz was not the sharpest tool in the box.  Still, Buzz brings up an interesting question.  Is it possible for anything to be "beyond" infinity?  And, spoiler alert, the answer turns out to be "yes".  What's beyond infinity turns out to be infinity.  But this second kind of infinity is distinct from the first kind.

Mathematicians call this first kind of infinity, the one based on integers and natural numbers, "Aleph Naught".  "Aleph" is the first letter of the Hebrew alphabet.  "Naught" is just British for "Zero".  So, Aleph Naught can be roughly translated as "A Zero".  And you will also sometimes see "Aleph Null" or "Aleph Zero" used.  They are just variants for the same thing and "Aleph Naught" is the form most commonly used by mathematicians, so it's the one I am going to use.

And mathematicians asked themselves:  "is there a bigger infinity than Aleph Naught infinity?"  Their early tries at finding one were a failure.  Like us, they started with natural numbers and moved on to integers.  Their next step was to try "rational" numbers.

Any number that can be represented using the combination of an integer and a fraction consisting of a natural number in the numerator and a natural number in the denominator is a rational number.  Minus twenty-seven and five thirty-sevenths is an example.  So we get all the integers plus all the numbers between two adjacent integers that can be created by making use of an additional fraction.

And there would seem to be a whole lot more rational numbers than there are integers.  There are literally an infinite number of rational numbers between each and every pair of adjacent integers.  And remember, there are an infinite number of integers.  So it would seem that the cardinality of the set of all rational numbers would be infinity multiplied by infinity.  So the cardinality of the "set of all rational numbers" surely must be greater than the cardinality of the "set of all integers".

Alas, it was not to be.  I am going to skip over the details but some clever mathematician found a way to create a rule that put the "rational numbers" set and the "integers" set into a "one to one correspondence".  So, nope.  The cardinality of the rational numbers is Aleph Naught.  But wait.  It gets worse.

What we have been talking about so far can be thought of as the "number line".  All of the rational numbers represent dots on a single straight line.  How about if we go multi-dimensional?  How about a "number plane", or a "number space", or even a "number hyperspace" with many, perhaps an infinite number of, dimensions?  Still, no joy.  It turns out that there is a rule that will put all those points in all those dimensions into a "one to one correspondence" with our original set of integers.  And that means the cardinality of the whole mess is still Aleph Naught.

So we're screwed, right?  There is only Aleph Naught.  Wrong!  We just haven't gotten creative enough yet.  Beyond the rational numbers are the "real" numbers.  It turns out that if you take a continuous line and plot all the points on it that can be described by rational numbers (integer plus a fraction consisting of a natural number in the numerator and a natural number in the denominator) not all the points on the line are marked.

The rest of the points on the line represent the location of various "irrational" numbers.  (The "real" numbers are what you get when you combine all the rational numbers with all of the irrational numbers.)  And there are lots of irrational numbers.  The most well known irrational number is pi.

And what we generally associate with pi tells the tale.  Pi is equal to 3.14159 and on and on forever.  In fact, it doesn't matter how many digits we list, the value will still not be exactly correct.  The exact value of pi, expressed as a decimal number, is a number with an infinite number of decimal places.

But when it comes to rational numbers there is a trick.  With this trick we can represent any rational numbers with complete accuracy.  And to do this we only need to list a finite number of digits.  We do this by breaking the decimal representation of a rational number into two parts.  Each of these parts contains a finite number of digits so the whole contains a finite number of digits.

To represent a rational number with complete accuracy in the usual way requires an infinite number of digits.  But the decimal representation of a rational number contains a quirk that is always present. With an irrational number like pi the digits never repeat no matter how far you go.  With a rational number the digits start repeating if only you wait long enough as you work your way along the number.  And once a particular number has gotten into the repeating part that's all it does forever after.

We can use this quirk of rational numbers to our advantage.  The process is pretty obvious.  We break the number down into a non-repeating part on the front and a repeating part on the back.  We list the non-repeating part as is. It will usually contain an integer sub-part to represent the "whole" part of the number and the part of fractional part that does not repeat.  (Either or both of these sub-parts may require listing no digits.)

This non-repeating part is supplemented by a repeating part.  The repeating part consists of a finite number of digits that are repeated an infinite number of times.  Mathematicians indicate this by drawing a bar across the top of the first set of repeating digits.  Then they just drop the rest of the number.  Note that we are getting rid of an infinite number of digits while retaining complete accuracy.  Nice trick, isn't it?

So, in example above (minus 27 and five thirty-seventh's) we would have "-27.".  That's the non-repeating part.  This would be followed by "135" with a bar over it (or so my calculator tells me).  I don't know how to get the blog formatting software to put the bar over the three digits so you'll just have to use your imagination.

As indicated above, it turns out that any rational number can always be rendered exactly in decimal form by splitting it into the non-repeat and repeat parts.  (BTW, sometimes the repeating part is just "0" repeated an infinite number of times.)  I'm not going to go into the process necessary to turn a rational number into a decimal number in this "non-repeating part plus and infinite number of repeating part" form.  Nor am I going into how to do the process in reverse.  Just trust me that it can be done.

Now we are in a position to note that an irrational number is one that has no part that repeats indefinitely when rendered as a decimal number.  Given two specific rational numbers rendered in decimal form it is easy to see how to construct lots of irrational numbers that lie between the two.  (Hint:  The rational numbers have repeat parts.)

Annoyingly, it is also easy to construct as many rational numbers as we want to, such that they lie between any two given irrational numbers.  You just note the early parts that coincide.  Then, starting with the decimal place where they diverge, you just toss in repeating sections that result in numbers that lies between the two.  This is another example of where this whole "not a single specific number" business makes life complicated.

Anyhow, a mathematician named Gregor Cantor came up with a way to crack the case by making use of the fact that real numbers can be represented accurately as decimal numbers if you are willing to include an infinite number of decimal places.  In 1874 he came up with a proof that a number bigger than Aleph Naught existed.

The easiest way to explain what he did is now referred to as the "diagonal argument".  Write down a list of all real numbers in decimal format.  Don't forget to include all of the infinite number of digits necessary to represent them with complete accuracy.  It doesn't matter what order the list is in.  It is just important that the list includes ever single real number.  (This is impossible to do in real life but this is what is called a "thought experiment".)

Now consider the first number in the list.  Change the first digit in the number to anything other than the correct digit and save it as the first digit of a new number we are constructing.  Move on to the second number.  This time change the second digit to anything other than the correct digit.  Again save this digit as the second digit of the new number we are constructing.

Move on through the list.  For each number we change the next digit along to be anything other than the correct digit and save it in the correct place in the new number we are constructing.  Again, this is impossible to do in the real world but the process is easy to understand.

When we are done (we never will be but, again, thought experiment) we will have constructed a number that does not exist anywhere in our original list.  (Remember, we constructed the new number so that it differed from every single number in the list of real numbers by at least one digit.)

But this is impossible.  Our original list was carefully constructed so that it included each and every real number.  This contradiction means that the cardinality of the set of all real numbers is greater than the cardinality of the set of all integers.  So there is an Aleph One and it is bigger than Aleph Naught.

So there are at least two infinite numbers.  There is Aleph Naught and Aleph One.  And it turns out that there is a way to construct Aleph Two and so on.  (I'm not going to go into the process because, frankly, I don't understand it.)

So there you have it.  There is a "beyond" beyond infinity.  "Infinity" presumably refers to Aleph Naught and there is an Aleph One that is "beyond" Aleph Naught.  You have now mastered some way cool mathematics that is so far beyond Calculus that you could say "it's infinitely far beyond Calculus".  So go forth and win bar bets and wow your friends and family at parties.

Pretty cool, hunh?