Vi Hart on Pythagoras

Here is another great bit by Vi Hart.

You Can Count on It

By Bobby Neal Winters

I’ve been revisiting an old friend this semester, or should I say and old opponent; when you get older sometimes the two are the same.  This isn’t a man or a woman.  It’s a book: Topology by J. R. Munkres.  I could quote Heraclites here and it would be half true.  Even though it is a second edition, the part of the book I covered in class hadn’t changed.  I have.

I only have a tithe of the energy I had in 1983 when I took the course from this book, and my mind is not as quick as it was in days of old.  I do, however,have almost 30 more years of experience now so my energy is focused better.  It’s like the joke told about the old bull and the young bull.  

The young bull said, “Let’s run down the hill, jump over the fence, and breed a couple of those cows.”

The old bull countered this, “Let’s walk down the hill, go through the gate, and breed all of those cows.”

It’s about planning and priority.

I suppose my lower energy level makes my mind go more slowly. I pause over ideas I would’ve pushed on by before.  I try to get the point of what I understood only on a technical level before.

Recently my mind has been focusing on the concept of the uncountably infinite.

Infinity is one of those concepts we really don’t get even when we “get it.” There is a gestalt of a sort when we play games with children asking them to name the biggest number they can.

“A thousand million billion trillion zillion,” they say.

“I can name a bigger one,” you reply.

“Uh, uh,” they reply.

“Oh, yeah,” you return. “What about a thousand million billion trillion zillion plus one?”

You better be careful doing this, because if they don’t have the gestalt the game can go on a long, long time.

Uncountability goes beyond this game of always being able to add one to get a bigger number.  It is a stranger critter.  

The first kind of infinity we encounter is the infinity of the so-called natural numbers, the numbers we use for counting: 1, 2, 3, and so forth.  Sets that can be listed out, each element having a unique natural number for a label, are said to be countably infinite. We can count the numbers and will eventually get to every one of them even if there will never be a time when we counted all of them.

Look back at the last sentence.  If you’ve read it carefully and are not a mathematician, you may feel a little green, but it’s written as I meant to write it.

A set is uncountable if you can’t list them in such a way that you will eventually get to each particular one of them.  This can all be put in precise, technical language, I assure you.

Don’t make me do it.

To show that an uncountable set exists, all one has to do is construct a set all of whose elements can’t be listed.  My favorite such set is the set of all sequences of zeros and ones.

A sequence is a list itself.  A sequence of zeros and ones would be like the following: 1,0,1,0,0,1,0, 0, 0, and so forth.  Note that here I’ve attempted to make something with a pattern.  First a 1 and then one zero; then another 1 followed by two zeros; then 1 followed by three zeros.  Continue in this way.  These sequences needn’t have a pattern.  I could have a sequence: 1,1, 0, 1,0, 1,1,1,1,0 and so for with no pattern.  These sequences would be different.

Now I claim that it is impossible to list out all sequences like this even if the list goes on forever.  The way I show that making this list is impossible is through the method of proof by contraction.  I assume that I can but the show there is always at least one left over.  

Assume there is such a list.  Then make another sequence of zeros and ones whose nth item disagrees with the nth item on the nth list.  Such an element is clearly not on the list and since the list was assumed to be exhaustive there is a contradiction.

You say you could just put it on top of the list.  I say, “Bah! By your own lying words you claimed if was already there! Die you varlet.”

Sometimes doing this stuff makes me sound like I am talking to a pirate.  Let’s just push on.

What I am going to say now will sound strange.  What else is new?  Anyway, I like this example because it is so concrete.  Seriously.  It is just lists of zeros and ones.  School children can make lists of zeros and ones.  I appreciate this now, better than when I was a kid those three decades ago, because of a theorem from Munkres I am going to teach my students tomorrow if they don’t derail class by bringing donuts or something, which is a constant risk with these youngsters.

The theorem states that if a topological space is compact and Hausdorff and contains no isolated points then it is uncountable.

You are no doubt saying to yourself, “Make sense to me.”

Okay, let me gloss that a bit.  A topological space is a set with structure: its open sets.  Compactness is a technical condition that gives us a certain type of control.  Hausdorff, other than being the German work describing a dorff belonging to a particular haus, is a condition that gives us another sort of control.  Again, I could make this even less readable by rolling out the technical definitions, but my point is these are simply abstract conditions, as is the condition of having no isolated points.

It is at the opposite end of the abstractness spectrum of our example of the set of sequences of ones and zeros, but, and here is the kicker, the spirit of the proof is the same.  You assume the points of the space are in a list and then construct a point that is not in the list.

Shazam!

More interesting yet is the fact that--if I had a taser and another semester with these kids--we could use the same technique to prove the Baire Category Theorem which, to mind young, energetic mind of three decades ago was not related to either of these results.

Arithmetic, Calculators, and Work

By Bobby Neal Winters

Let me start out by saying that I am not now and never have been any good at arithmetic. Something goes wrong in my brain when I trying to multiply number two numbers together. I do have a good memory and I can remember things like 5 times 5 is 25 and 25 times 5 is 125 and 5 times 125 is 625.  And I can use this to remember that one-half is 0.5, one-fourth is 0.25, one-eighth is 0.125, and one-sixteenth is 0.0625.

But this is something that I’ve learned from years of teaching mathematics. For most things, I resort to a calculator and have since they were invented.  Yes, young people, if there are any of you out there, I am that old.

I am so helpless with arithmetic, that I am the reason my high school accounting teacher, Mr. Billy R. Scott,  began the practice of allowing his students to use calculators.  Yes, I am that bad.

Needless to say, I’ve allowed students to use them as long as I’ve been in front of the classroom.  True, many of the classes I teach are so theoretical that calculators are of dubious value, but I allow their use anyway.  

And to tell you the truth, I’ve not really worried about it very much for the last 29 years. Yes, I have been teaching math for that long because, yes, I am that old.

But recently there’s been a problem.  Student’s are having problems reading their calculators.  You see, here’s the thing about calculators: you’ve got to learn how to use them.

When I teach now, I tell my students to develop a loving, mutually supportive relationship with their calculators.  You can’t just ignore it for a month and expect it to perform for you on the morning of the test. You only get out of it what you put into it.

Seriously.

Even though I was a bust with arithmetic, I could make a calculator sing.  I could do things with it that the folks who made it didn’t know about.  Back in those days, and maybe yet, teachers put questions on tests that required arithmetic that the calculator couldn’t do like exact answers with no decimals.

I could do those on the calculator because I was that good.  I was that good because I spent time on it. Somehow, the chicks didn’t dig it, though.

You wouldn’t dream of waiting to the day of recital to play your piece on the piano for the first time; you wouldn’t think about climbing behind the wheel of a car for the first time on the day of the test for your license; why would you even think about not turning on your calculator until the day of the test?

I look back at that and see the bit about the recital.  There are a lot of student’s who’ve not had that experience.  I never did. I only think of it because my kids--my middle-class kids--have. They’ve had recitals and calculators and computers.  They’ve had parents harping at them to practice as well, and I do know that you can hear parental harping at a distance, even from beyond the grave.

Many of the students I teach haven’t had any of those things.  This isn’t an excuse.  They’ve got an opportunity to go to college now and it’s my job to be a navigator.  

I chose that metaphor pretty carefully.  Students have to do the work themselves.  They have to care themselves. I can’t work for them.  I can’t care for them.

I can navigate them along a path that I’ve followed successfully myself. I can mark the finish line. They have to run.

Notions of Sameness

Notions of Sameness

By Bobby Neal Winters

Mathematicians use sets and structures on sets to model realities that they perceive only in their minds.  I dealt with this in the matter of topology in one of my recent essays.  These structures induce special properties that we want to preserve.  I am still interested in this mainly in the case of topological spaces, but I will work my way up to it via a series of examples from other areas of mathematics.

The first example is that of sets with no structure in which we can only discuss the concept of cardinality.  This is to say, how many elements does a set contain?

With finite sets we can say that a set is composed of five elements or seventeen elements or two trillion five hundred fifty four billion two hundred twenty seven million three hundred twelve thousand five hundred and two elements.  There is a number we can associate with them.  

For infinite sets, the above does not work; we have to do a flanking maneuver. We avoid the word number and use the term cardinality.  We say that two sets have the save cardinality if they can be put into one-to-one correspondence with each other.  

We use the language of functions to make such notions precise.  Functions are thought of as having a sending set (the domain) and a receiving set (the range).  The range is like a target.  When two sets have the same cardinality, there is a function from one set to the other that is one-to-one and onto.  A function is onto if every point in the receiving set is matched with a point in the sending set.  A function is one-to-one if no set in the receiving set is matched twice.

A function that has both of these properties at once is christened with the high-falutin’ label of bijection.  It’s technical and pretentious but remember it anyway because it’s important.

If two sets have a bijection between them, they have the same cardinality.  If one (and therefore the other) of those sets is finite, they will also have the same number of points.  By this mathematical slight of hand of using a particular type of function to create a new notion of “sameness” we’ve expanded the notion of number/size into the notion cardinality/size.

This is done in the realm of algebra as well.  And here I must be careful because I am using the word “algebra” differently than many use it.  If you had pre-algebra and algebra in middle and/or high school, or if you have had intermediate algebra or college algebra in college, then rest assured that I am talking about something that is related to that in the way that a Tomahawk Missile is related to, well, a tomahawk.

Algebraists, and I am over-simplifying, study sets with binary operations on them.  For example, they might study the real numbers with the operation of addition or the positive real numbers with the operation of multiplication.  (Notice I put the word “might” in there because they aren’t really interested in those two particular sets, but I will use them because I don’t want to talk about a lot of algebra now.)

The real numbers numbers with addition and the positive real numbers with multiplication are examples of groups.  Groups are sets with binary operations that obey certain rules.  Don’t worry about what those rules are; if you can’t help it, look it up on Wikipedia.

Those of you who have had a course in college algebra might remember the logarithm.  I put emphasis on the word might there because I’ve no doubt that you have seen the logarithm.  You might remember it; you might wake up in cold sweat screaming it at the top of your lungs; your lover might have attempted to comfort you afterwards asking, “What is a logarithm, Sweet heart?” only to have you deny ever having heard the word.

In any case, the logarithm is a function from the positive real numbers to the real numbers that is bijective, i.e. it is one of those important one-to-one correspondences.  The important thing about it is that it changes the operation of times to that of plus: log(a times b) is equal to log(a) plus log (b). In mathematical ages B.C.--before calculators--the logarithm was used as a means of doing multiplication by turning it into addition.  You may not have notice, but addition is a lot easier than multiplication.  

Such a function that is bijective and preserves the operations on the groups--in this case turning multiplication into addition--is called an isomorphism.  The groups are said to be isomorphic to each other and a mathematician in the capacity of algebraist can’t tell the difference between them.

I could go on listing the various areas of mathematics and the functions they use to preserve there important concepts, but instead of doing that let me focus on topology.

A topological space is a set along with certain special subsets that we call “open” sets.  Topological concepts are those concepts that can be defined in terms of open sets.  As a consequence of this, we are interested in functions that preserve open sets.  A bijective function between two topological spaces that takes an open set in the first to an open set in the second is called a homeomorphism.  A pair of spaces between which there is a homeomorphism are said to be homeomorphic.  Spaces that are homeomorphic are the same as far as a topologist is concerned.  

Most frequently, I see homeomorphisms defined differently.  We usually define a homeomorphism to be a continuous function which is invertible and whose inverse is continuous.  This is mathematically equivalent to the other definition I’ve just given, but it’s shorter.  

Continuous functions are interesting in their own right.  Indeed, one might argue that topology was created in order that we might better study continuous functions.  There is also quite a well-developed theory of continuous functions that allows us to check whether a particular function is continuous.  For example, it is quite easy to use calculus to check that the logarithm and its inverse the exponential function are both continuous; it might be less easy to show they preserve open sets.

But in our desire to be efficient with our definitions, we should take care not to obscure for the beginner the very thing that homeomorphisms do: they preserve open sets.  As they preserve, nay set up a one-to-one correspondence between the open sets of each space, any properties defined in terms of open sets will be preserved.  That is the point.

Sets with Structure

By Bobby Neal Winters

This semester I am teaching a course in topology after a hiatus of six years.  I am using a classic text by James R. Munkres with the title, fitting enough, Topology. This is the text I had my graduate level course from.  It’s been like meeting an old friend again after an extended separation.  Only someone who’s done that can appreciate all of the levels of meaning.

Topology is a word-like every word now that I think of it--that carries a bundle of meanings.  On the level that is most accessible to a popular audience, it is understood to mean that branch of mathematics in which a coffee cup is no different than a phonograph record. (That’s a CD to you, you young whipper-snappers!) For the sake of precision, we could make a distinction by saying geometric topology or even low-dimensional topology, but in practice clarifying adjectives or adjectival phrases get stripped off and we are left with topology left alone, forced to hide the other meanings it carries.

Today, I would like to venture into one of those areas where angels fear to tread to talk about the subject that mathematicians (especially geometric and low-dimensional topologists) refer to as general or point-set topology.

One can could say that low-dimensional topology is a sub-speciality of general topology, and I will justify the sense in which that is true in the sequel, but such a statement blurs over differences of mindset among the various practitioners.

Let me say I was drawn to my first topology course having seen the pictures of coffee cups being blithely changed into phonograph records, donuts, etc, only to find something entirely different.

A course in general topology begins with a topological space.  A topological space is about as abstract a concept as the math major will meet as an advance undergraduate or beginning graduate student.  It is a set which is paired with a special collection of its own subsets, and this special collection of subsets have a set of laws they must obey.  I won’t tell you now no matter how much you beg me.  We give a name to that special collection of subsets and call it a topology.  I told you the word carried a bundle of meanings.

The most common example of a topological space is the set of real numbers.  Topologists who’ve just read that sentence are now picking up pencils from their desks to write in “with the usual topology” between the “s” in the word numbers and the period that follows it.  I left it out on purpose just to annoy them because it is the usual topology. It is based on the open intervals that students learn about as early as middle school.  The open intervals are used to construct open sets and the set of all of the open sets of the real numbers is the usual topology on the real numbers.

The usual topology on the real numbers is such a natural thing to us--and my “us” I mean “geeky math types”--we don’t even notice that it’s there.  We use the real numbers with the usual topology first in calculus and later in analysis, and I have talk these courses without ever uttering the word topology.  Most of the basic results in those areas can be reached without naming the topological concepts explicitly.

Perhaps the concept of a topological space would never have been created had mathematicians not ventured beyond the real numbers, but--you know those scamps--they did.  They ventured into the plane, into 3-space, into sets of functions, and so forth, and they discovered sets of subsets in each of those areas that behaved like the open subsets of the real numbers behaved.

If I knew more of the history of the subject, this would be an opportunity to segue into a case study in abstraction.  Those three examples I listed above have quite a bit of structure on them.  They have ways of doing arithmetic, they have ways of measuring angles, and they have ways of measuring distance. They are groups; they are vector spaces.

When we push out to the level of abstraction required by the topological space, we forget about all of that other structure.  You can’t do arithmetic; you can’t measure distances.  You think about only the set and its topology.  You only define properties that can be discussed in terms of the members of the topology.  You only discuss functions which respect the members of the topology.

In some sense, learning general topology first requires that you forget everything else you know about anything. You become a slow thinker; you become a deliberate thinker; you always must be careful that your intuition--raised as it was in the fertile fields (nerdy pun fully intended) of the real numbers--does not lead you astray.

This sort of abstraction allows us to prove theorems that apply to a wide range of areas. It allows us to create language to see an underlying unity in diverse areas of knowledge.  It also provides a trap-door into what has been referred to as centipede mathematics, as in “How many legs can I pull off the centipede before it can’t walk any more?”

I called it a trap door, but I am not sure that metaphor works.  It makes what happens sound like an accident.  The truth is more complex.  Many--most--who are drawn into mathematics find this sort of abstraction attractive, not to say intoxicating.  Going deeper and deeper into abstraction leads us into what our appetite desires.  It is like the wind buoying up our wings, lifting us farther and farther from the ground.  Here the story of Icarus is attractive, but also inaccurate.  We don’t go so high that the sun melts our wings; we are lifted so high we are never seen again.

There is a quote I’ve heard attributed to RH Bing, a Texas mathematician who is a personal hero of mine.  When asked about a visiting topologist, he is said to have replied, “He studies spaces of which there are only one example and only in England.”

Mathematics, especially abstract mathematics, is best when it is equipped with numerous examples. Examples give breadth and richness.  Examples guarantee you aren’t just proving theorems about the empty set.  But I digress.

General topology is alive with examples.  It is wide and it is deep.

There was a time in my career, and I will say this without shame, that I taught subjects simply because I wanted to learn them myself, without regard to the student.  I say it without shame because the students can still get a lot of value from that provided they are motivated themselves and their needs are being regarded other places.  Time has dealt with me in any case. I find myself singing along with Bob Seger:

Well those drifter's days are past me now

I've got so much more to think about

Deadlines and commitments

What to leave in, what to leave out

As I teach my courses now, I try to focus on what I think the student needs.  One great need that students have as they enter into graduate mathematics is to have their pre-assumptions stripped away. The abstractness of general topology is the best method I know.  That having been said, there is so much of it. What do I leave in?  What do I leave out?

In the end, my prejudice is to choose topics that will lead my students toward areas where mathematics is growing, places where many branches come together, places where there is structure--much structure. Then they will be able to choose.

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.