The Tin Can Telephone

By Bobby Neal Winters

When I was a kid, there was no such thing as trash service in rural areas.  You burned your trash to minimize its total volume.  Then, when your burn barrel was full of things that would no longer burn, you hauled it off an dumped it in a isolated area where no one was looking.  I’m not proud of it, but that’s the way it was.

Sometimes we dug tin cans out of the trash and made phones out of them.  The idea is simple and I am sure many of you have done it--or something similar--yourselves.  You take two cans, put holes in the center of the bottom, and attach the cans with a light string.  You then holed the cans so the string is taut and talk into one can while someone listens in the other.

The model for communication theory is only a little more sophisticated. You have the equivalent of the two cans: call one the transmitter and the other the receiver.  And you have the string: call in the channel.

Instead of talking on one end and hearing on the other, you are sending symbols on one end and receiving them on the other.  When we say symbol, you can think what you want; the model is abstract enough to admit just about anything.  In practice, the folks who do this sort of thing will think of a symbol as being a string of ones and zeros.

The channel--the string, as it were--brings in an little more complication because it is a device through which we can add noise to the signal.  Those of us who have used the tin can telephone know that sometime the wind would whistle through the string.  This model will allow for that, but it will also allow for electromagnetic disturbances disrupting those ones and zeros being transmitted.

As an exercise, think about the following situation. Agents have captured an enemy operative.  She is a beautiful blond bombshell, a perfect exemplar of the “Bond Girl.”  You send a message, “Kill the prisoner.”  As you do, lightning strikes and your agents receive, “Ki** the prisoner.”  

There is ambiguity in the message.

While it can be reconstructed correctly, it can also be reconstructed as, “Kiss the prisoner.”  Depending upon the proclivities of your agents, they might find this message more attractive.

One value in creating a system to communicate effectively is to minimize the chance of this sort of ambiguity.  One way around it is to create a code wherein only certain things can be said.  This book, possibly, wouldn’t include the possibility of kissing an agent.  In practice, the symbols of ones and zeros are constructed so that only a few strings of ones and zeros are acceptable and corrupted ones are no longer in the alphabet, as it were.

The military does this with they so-called phonetic alphabet.  Interpreting strings of letters over a telephone line can be difficult.  The letters ess and eff  can sound the same, for example.  Instead of saying “Ess eff,” which could be heard either as “ess ess” or “eff eff,”  using the military phonetic alphabet you would say “Sierra Foxtrot.”  A set of symbols has been created so that, even when transmitted over a noisy channel, there is a reasonable chance of recovering the original symbols.  

So you could say “Kilo India Lima Lima” the prisoner and that wouldn’t be heard as “Kilo India Sierra Sierra” the prisoner.

What we’ve done here is to start talking about using a code.  The word code is often used to mean hiding the meaning of a message as when we say that people are talking in code to one another.  This is what mathematicians refer to as encryption, which is a different sort of thing. Encryption is about hiding meaning, but codes are about trying to transmit messages accurately.  I won’t chide you about blurring the distinction in casual speech, but in this context I will keep the distinction.

One practical issue that does occur in communication is whether the transmitter and receiver have the same code book.

I was watching a television show the other night where a young woman invited her date for the evening in for “a cup of coffee.”  His code book interpreted that phrase to mean an invitation for a hot, caffeine containing drink.  In her code book, it was intended to convey the possibility of insuring wakefulness by other means.

This is by no means an artificial example, nor is it unique. When adults are talking to children, the children have a different code book, as there vocabulary is smaller.  Communication is possible between parent and child though there is sometimes frustration in both direction.  There is also much comedy, as in the preceding paragraph, based on the characters having different code books.

It seems to me that an important element in basic communication is for the transmitter to know as well as possible what code book the receiver has and to craft the message accordingly.

The folks in marketing are masters at this.  They will tailor their messages to a particular demographic, folks with a particular code book and get their message through to that market.

In talking so much about the transmitter and receiver, let’s not forget about the channel.  There is only a certain amount of information that can be sent across a channel.  What is not sent can be as important as what is sent.

There have been time when I’ve met people in church.  They’ve got nice clothing on.  They are driving a late model car.  The overall impression, their image, is one of prosperity.  The truth is that they live in a modest home and that the car isn’t paid for and the clothing are saved for special occasions.  They can make themselves look rich by hiding their bank accounts and their homes.  It’s not only what is seen; it’s what’s not seen.

Celebrities make use of this as well.  They have their image, their public persona, but they have their private selves as well. For them, the image is a commodity that they sell just like a farmer sells produce.  They must master what is seen and what is unseen.

It is a mistake, though, to believe that only celebrities have images.  Each of us has an image as well.  We use different language for it; often we call it a reputation.  It doesn’t take long to get one, and once you’ve got a bad one it can take a while to improve it.

We build our images, our reputations, by the signals we send.  Some are masters of image creation.  It is relatively easy to convey an image of being prosperous; you just have to be sure to spend your money where people can see it.  It is relatively easy to appear to be intelligent; much of time it consists of keeping your mouth shut.

Beyond that you have to know your market and what code book they have.  It also helps to be rich, smart, or whatever you are trying to portray yourself to be, but because of the narrowness of communication channels, it’s not always necessary.

Numerous Numbers

By Bobby Neal Winters

What is man, that thou art mindful of him?
And the son of man, that thou visitest him?
--The Psalms

The Psalmist asked the timeless question “What is man?” thousands of years ago. The answers have come back in many forms.  Darwin said man is an animal; Freud said man is a sick animal. Others would say that man is an animal sick enough to care about math.

At least some of us.

Some of us care about mathematics.  Some of us care about numbers. The modern, mathematically-minded psalmist might ask: What is number that man art mindful of it?

For most people, that is a truly strange question.  Numbers are those things that are written on your bills.  You write them in your check register; you add them up at the end of the month; none of them has more than two decimals.  

Other people had encountered numbers in a somewhat more sophisticated way.  They’ve been in science classes and have encountered Avogadro’s Number, Pi, and the speed of light.  Still these numbers are, in most minds, yoked, nay, identified, with their decimal expansions.  Our teachers do tell us--and the sicker ones of us do care--that Pi can’t be completely captured by its finite decimal expansion, but for most the distinction is not made between the decimal expansion for the number and the number itself.

In a certain way of looking at the world, the failure of making that distinction is not a bad thing.  If you putty over the difference between the two, you can build the pyramids, create the hydrogen bomb, and work on cold fusion.  If make the distinction, you might not be worthy any activity besides mathematics.

Mathematicians are careful about making such distinctions and precise about language because they need to prove their assertions.  Mathematicians prove their assertions not only so that people will believe them but so their students will understand.

One means of laying the ground work for proof is setting up a system of axioms.  Those of you who’ve been through a course in geometry have experienced a system of axioms.  Axioms are statements about the objects in your system that allow you to do proofs.  What can be done for geometry can be done for the real numbers as well.  

This is called a synthetic description of the real numbers.  My aim is to stay as un-technical as possible so I won’t go too deeply into detail, but the axioms for the real numbers state the properties of the four arithmetic operations and how they deal with each other and with the order properties of the real numbers.  These axioms can be packed into the phrase that the real numbers are a complete, ordered field.

Dealing with mathematical objects synthetically, i.e. by listing properties in the form of axioms is clean.  It can be tricky because sometimes one must be rather clever.  It is much like trying to tie your shoes when you are too fat to see your feet: you have to be patient and have a good imagination.

There is also the danger that the object you are describing with your axioms might not actually exist.

There is a joke about a woman who went into a store to find a husband.  She came to two doors.  The one on the left said choose this door for men who are kind and the one on the right said choose this door for men who are kind and make a good salary.  She chose the one on the right.  

She then came into a small hall that again had two doors. The one one the left said choose this door for men who are kind and make a good salary and the one on the right said choose this door for men who are kind, make a good salary, and are handsome.  She again choose the one on the right and again she was in a room with two doors.  

This time the one on the left was one like she had just chosen but the one on the right said choose this door for men who are kind, make a good salary, are handsome, and are fantastic lovers.  Very excitedly, she chose the door on the right and found herself back out on the street.

Whatever point the one who made this joke had, mine is that sometimes you can put so many conditions upon an object, making them rarer and rarer, until they disappear entirely.

Mathematicians like to have at least one non-trivial example of whatever class of objects they are talking about.  These examples have to be described in terms of other well-understood mathematical objects and the language of set theory.  This is referred to as making a model.

One means of creating a model of a the real numbers is to begin with the rational numbers.  As I said earlier, the real numbers are a complete ordered field.  The rational numbers are simply an ordered field; this is to say they lack the property of completeness.

Completeness, in the way of mathematical words, has a very precise, very technical definition. One can discern from the meaning of the ordinary English word completeness that a complete ordered field, such as the real numbers, has something that an ordered field that is not complete, such as the rational numbers, lacks.  What is this?  

A quick and--to the cognescenti--smart-alecky answer to this is the irrational numbers such as the square root of two and Pi.  This is smart-alecky because it ignores a the very real need that the rational numbers have for those irrational numbers.  The incompleteness of the rational numbers--again in the English sense of the word--signifies a lack, a deficiency.  This lack can be described in two different ways.

The least technical of these two ways involves the existence of least upper bounds.  The set of positive rational numbers whose square is no more than two does not have a least upper bound that is a rational number.  This fact--in different language--was discovered by the Pythagoreans in ancient Greece some time in the Sixth Century B.C.   

The more technical of these two ways involves certain sequences of numbers.  You may have heard of infinite sequences of numbers such as ½, ¼, ⅛, and so forth.  This sequence of numbers converges to zero.  There is a certain type of sequences that are referred to as being Cauchy.  All sequences that converge are Cauchy, but not all Cauchy sequences of rational numbers converge to rational numbers.  Again, one can easily find Cauchy sequences of rational numbers than converge to Pi and to the square root of two.

What mathematicians do in these two cases is to construct models based on the rational numbers.  In the first case, special sets of rational are created and the arithmetic functions are extended to those sets.  The objects in this model are no longer rational numbers but sets of rational numbers. In the second case, the objects in the model are sets of sequences of rational numbers.

These two different models of the real numbers are clearly different from each other in terms of what they are, but both of them satisfy the axioms.  Each as a complete, ordered field.  We call that field the real numbers, and there is a very precise mathematical sense in which that definite article is justified.

But I am becoming a mystic.  There are more numbers than we can know.  Our need for numbers springs from the world around us in numerous ways and I wonder if by drilling down to one idea of the real numbers if we are missing other things.

But the timeis late, and I want to go home.