Doctor Who and the Regeneration Didactic Science Fiction

By Bobby Neal Winters
I started reading science fiction in the 70s. At that time, I was reading mainly stuff from the 50s and the 60s. Think of Isaac Asimov, Robert Heinlein, and Arthur C. Clarke.  I cut my teeth on Lucky Star, The Foundation Trilogy, 2001: A Space Odyssey, and Space Cadet.
This is didactic science fiction.  If you are worried about that word “didactic,” it means that this science fiction, for the most part, was interested in teaching more than it was about the story. In particular, Lucky Star, was intended as a channel to bring the knowledge of science to young minds. The was so much the case that Asimov was moved to put disclaimers at the beginning of certain of these books after the science became obsolete, e.g. when it was learned that Venus wasn’t covered by oceans.
In didactic science fiction, the story is merely a vehicle for teaching.  You get lines like, “We shall head of the pirates at Mars, which you know is the fourth planet from the sun.”  I’d better be careful, I may have actually read that somewhere.  There is a tendency to make fun of this, and truly some of it was so awful as to be funny, but I did learn a lot of science from it.  Some of it was so good that I didn’t have to study from my textbooks much in high school; I’d gotten the “facts” from science fiction.
I want to come at this from a couple of orthogonal ways today. (Notice how I worked in that mathematical term, orthogonal.  It means at right angles, though it can be generalized to apply to vectors in an arbitrary inner product space.  Look it up.)  The first of these is literary and the second--get ready--is didactic.
As a writer, I am interested in that old didactic science fiction because I want to write some new didactic mathematical fiction. I like teaching mathematics; I like writing fiction; writing mathematical fiction seems like a nice marriage between those two.    One thing we writers do in pursuing our aims is to steal and thieves don’t steal things they don’t like.  I like Doctor Who and Doctor Who provides an excellent example.
It began as good didactic science fiction and remains so through many regenerations.  What it teaches isn’t so much hard science as the social sciences.  The Doctor is a time traveler, and this gives him the opportunity to visit a lot of history.  Someone of a suspicious nature--here, here, pick me--might observe that the Doctor spends a lot of time in Great Britain in general, in England, in particular, and in London to be even more particular.  
As Doctor Who has been around so much, it gives us the opportunity to watch it develop over time.  As with every other science fiction franchise, the special effects have improved, but there is more going on than that.  In the new series there is a lot more emphasis on emotional connection with people. In the olden days, there were actors speaking lines; they were carrying out the motions of an adventure; they lived; they died; but somehow they didn’t make me care.
The most recent regeneration of the series is much different. While the special effects in the old series were...uh...awful which I say to avoid the word horrendous, those in the new are grand.  But that is minor in comparison to the level of emotional connection one has with the characters.  Care is taken in the writing, acting, and direction to create characters that one cares about.
And it still teaches.  The new series continues to show history, but it has also ventured into art.  One episode involving Vincent Van Gogh stands up well against Lust for Life.  It also ventures into philosophy and science.  It does it well. What is the secret?
The secret was in how it engaged our emotions. This is done through good story-telling. Time has been taken to create multi-dimensional characters. (How fitting for Doctor Who!) And--wait this is important--the guardians of the franchise apparently do believe in character.  One is able to put oneself within the character and feel what they are feeling.  The Doctor is not treated by the writers as a puppet.  He is an emotional being who, although he is a Time Lord, reacts emotionally like a human.  When his beloved traveling companions Amy and Rory were sent to the past--and effectively killed to the Doctor--by the Weeping Angels, it dealt him a psychic blow requiring about a century to recover from.
I would draw the lesson that one puts good, well-drawn characters into an engaging story, and then work in any didactic elements that one can make comfortably fit.  The comfortable fit is important.
I look back on that and see that although I was talking about writing I was, at the same time being didactic.  I hope it was a comfortable fit. Let me now be didactic and I will see if I can work something else in comfortably at the same time.
T. S. Eliot wrote in The Hollow Men “Between the conception/ And the creation/ Between the emotion/ And the response/ Falls the Shadow.” Well, between being didactic and doing product placement lies social engineering.
Oops! There I’ve said it. The phrase “social engineering” evokes dark images of evil old men in back rooms moving around people like pawns and of raving, drooling conspiracy theorists clawing the furniture and howling at the moon.  
I am okay with both of those.
We don’t necessarily need science fiction for social engineering, but it has a comfortable place there.  We can look at Star Trek for example.  Captain Kirk had a Vulcan first officer and an African Communication officer.  This made people more comfortable with the idea of working with people who were different from them in general.  In particular, it was aimed at making whites more comfortable working with blacks as equals.  
It was a good cause. Was it education, social engineering, or the spirit of the times?  Not an easy question.  An easier question would be what’s the difference.  I would say the difference is the level of organization.  Social engineering would be need to be organized at some level higher than those involved in any particular step and emerging at multiple venues.  As for the spirit of the times, let me remark that any effort at social engineering will be more successful if it can work in concert with something of the spirit of the times.  It is easier to grow an oak from an acorn than it is to create the acorn.
Let me mention briefly also the topic of Anthropogenic Global Warming.  I saw this referenced in Robert Silverberg’s Hot Sky at Midnight back in 1999.  This was five years before the release of The Day After Tomorrow in 2004 to use the disaster movie tropes to bring the message to a much broader audience.  When An Inconvenient Truth came out in 2006, a broad audience was ready.
One could create a conspiracy theory about a group of people who wish to shift the world to the use of nuclear energy by creating a fake crisis.  One could spin a story about concerned scientists harnessing sympathetic ears in the media to get their message out.  One could imagine simply an idea emerging into a culture.  
Take your pick.
It does strike me that if you want to get a message out and you can plan and are patient--oh, and if you have the money--then this is a smart way to do it.
To get back to Doctor Who as an example, I will say that it does a bit of social engineering itself.  These days that a program will provide examples of “different” people working together is a given and it does plenty of that, but other topics appear that I find charming:  It is a patriotic show.  I find the pride displayed in being British to be delightful even when it is at the expense of Americans. (They think we are too violent and gun crazy.  Go figure.) The most delightful and effective example of patriotism occurred when the Doctor’s companion Rose wore a Union Jack and hung from a barrage balloon during the Battle of Britain.
The show also fosters a notion that Church of England will continue into the future to fight evil and provide comfort.  Those who follow the travails of the C of E might find these the most daring science fictional elements of all, but I digress.
In the end, I do believe they’ve got something that works, and something others might want to try regardless of their agenda:  Good characters and engaging stories.  Once you’ve got those you can work in whatever is comfortable.

Sum Numbers: More Than I know

By Bobby Neal Winters
I am teaching analysis once again out of A Radical Approach to Real Analysis by David Bressoud. It has done a lot over the years to broaden my understanding of the subject.  It has good examples, classical examples.  While many threads of the subject are explored, very good attention is paid to summing series.
We teach our students to do things without a second thought in introductory calculus that the ancient Greeks would’ve balked at, e.g. the summing of infinite series.  For example, consider the series 1+ 1/2 + 1/4 + 1/8+ ..., where the ellipsis indicates the terms go on forever. Note the pattern that each term is half of the previous term.  
This is an example of a geometric series.  Geometric series are distinguished by the fact that there are a common ratio between every term and the previous term. Such a series is determined by the first term and the common ratio.  The ratio in this case is 1/2. This particular example sums to 2.  Do the following thought experiment.  The number 1 of course is 1 less than 2; 1+1/2 is 1/2 less than 2; 1+1/2+1/4 is 1/4 less than 2.  The pattern emerges that, with each additional term, we are are splitting the difference between the current sum and 2.  In this way, any finite sum can be made arbitrarily close to 2.  We say that the series converges to 2.
But, as I said, this is just one example of a more general type of series.  We can consider the series 1+x+x2+x3+... .  Again, this is a geometric series because there is a common ratio between the current term and the previous is x.  This series sums to 1/(1-x).  there are careful ways to show this and less careful ones.  For the sake of ease, let s=1+x+x2+x3+... .  Note that we can write s=1+x+x2+x3+...= 1+(x+x2+x3+...)=1+x(1+x+x2+x3+...)=1+xs.  That is to say,  s=1+xs.  Solving for s in terms of x gives us, s=1/(1-x).  Therefore, 1/(1-x)=1+x+x2+x3+... .
If you are in a mood to do algebra, you can let x=1/2 in that formula, and observe the sum is 2, exactly as we argued in the first example.   I think that’s pretty cool, but then look at my life choices.
This is a keen little formula, but there are problems.  It won’t work for every choice of x.  If we let x=1, for example, then 1/(1-x) is one divided by zero, and as we have been taught from a very early age, dividing by zero is a no no.  It’s bad.  Don’t do it.  The 1+x+x2+x3+...  becomes 1+1+1+1+... which becomes larger with each additional summand and will, in fact, increase without bound.  We mathematicians say it “diverges to infinity.”
More of a problem is when we allow x=-1.  In this case, 1/(1-x) becomes 1/2, and 1+x+x2+x3+...  becomes 1-1+1-1+1-... .Here let s=1-1+1-1+1-... .  If we group the terms of the series  as s=(1-1)+(1-1)+(1-1)+...=0+0+0+...=0, so it appears we are done.  However, looks can be deceiving as we can also group them like s=1-(1-1)+(1-1)+(1-1)+...=1-0-0-0-...=1.  So we’ve proven that 1=0 which, though it might be useful at tax time, doesn’t really strike us at true.  
To muddy the waters even further, Gottfried Leibniz, made the observation that 1/2=1(1+1)=1-1/(1+1)=1-1+1/(1+1)=1-1+1-1+... and so forth, agreeing that the sum should be 1/2. This is halfway between 0 and 1, so in an age where we seem to seek compromise, 1/2 would represent meeting in the middle, making everybody happy.
In case you haven’t notice, though, mathematicians don’t put a high premium on making people happy.  
The lesson drawn from this is that one can’t simply deal with infinite sums the way one would deal with finite some.  Infinity makes a difference.  Mathematicians, again unconcerned about making people happy but deeply concerned about making sense, ferreted out conditions in which one could treat infinite sums more or less like finite ones.  They distilled the concept of convergent series.  The series 1+x+x2+x3+...  converges to 1/(1-x) only when |x|<1 .="." nbsp="nbsp" span="span">
For many, this closed the book on series that don’t converge, the so-called divergent series.  Yet such great lights as Leonhard Euler--which is pronounced Oiler not Yooler.  You sound like a hick when you say Yooler--used divergent series to get results which were, in fact, true.  This is insane. It is reminiscent of the joke about the woman whose husband thought he was a chicken.  She didn’t have him put away because they needed the eggs.
This is an area where my reach exceeds my grasp. Suffice it to say there are ways where this sort of thing can be made precise and consistent results can be obtained, but it comes at a price of having a different understanding of what the sum means.
The formula 1/(1-x)=1+x+x2+x3+...  gives and gives.  One can use it to prove that 1/(1-x)2 = 1+2x+3x2+4x3+... .  If we let x=-1, this yields 1/4=1-2+3-4+5-... .  The technical term for this sort of behavior is “nuts.” But it gets worse.  Let t=1-2+3-4+5-....  and let r=1+2+3+4+5+.... While one might harbor some sort of hope for the fate of t because the terms bounce back and forth between being positive and negative, r must be positive.  But wait.  r-1/4=r-t=2(2+4+6+...)=4(1+2+3+...)=4r.  By algebra r=-1/12.  So 1+2+3+4+...=-1/12.
There are ways of making this all precise, but it requires a different interpretation of the symbols involved than is usually given them.
And I’ve told you more than I know about this.

Making Those Distinctions: 2 is not 3  

By Bobby Neal Winters
As I’ve mentioned before, I am a low-dimensional topologist.  At least I was before I began spending my time shuffling papers.  Doe the quality of being a low-dimensional topologist persist over time?  Perhaps it can be proven or perhaps I am an example that it does not.  
But I digress.
My time away from doing mathematics has given me time to think about mathematics.  Does a eunuch think more about women than other men?  I am not sure I want to research this. In any case, when I was doing mathematical research, I didn’t take the time to reflect on it. Perhaps that’s why I wasn’t more successful.  Now that I am viewing it from a distance there are things I ponder.  One of the things I ponder is the methods by which mathematicians make distinctions.  
In low dimensional topology this comes to the forefront because so much of the subject deals with pictures and mathematicians distrust pictures.  Pictures can be deceiving either purposefully or accidentally.  As a result of this, we make a practice of translating things from pictures to something that is closer to words.  We create artificial languages in which we can carry on our arcane conversations.  Sometimes differences which are easy to see (but difficult to be sure of)  in the picture are difficult to see (but absolutely certain) in our symbolic language.
Consider the following two objects:

2-holed torus

3-holed torus

These are called the two-holed torus and the three-holed torus, respectively.  They are generalizations of the torus.  A torus is a surface that looks like a donut and I will omit the picture, assuming everyone has seen one.
These two surfaces are different.  I mean, look at them.  One has two holes and the other has three.  As an old professor of mine used to say, “A blind cow could tell them apart.”  Indeed, once it has been proven, we can say they are different, but such are the standards of the subject that we can’t just assume this a priori; it has to be proven.
Okay, how?
There are a number of ways.  Consider something called homology theory. This is complex (and that is a great pun for the in-group, but unless you want to spend a couple of semesters working at it, let it go). You begin by creating models of your surfaces using triangles.  One can then obtain abelian groups from these triangles.
Okay, I’ve just slipped in the concept of “group” along with the notion that there are special groups that are referred to as “abelian.”  Let it go.  Let it go, I say.
I can’t.  
A group is a mathematical object that has a binary operation that is modeled on multiplication.  The positive real numbers with multiplication are a group.  Those of you who have been abused by being taught about matrices will take comfort in the fact that two by two (n by n, actually) matrices with a nonzero determinant form a group.
Now, those of you who have studied two by two matrices may recall that they differ from real numbers because A times B is not necessarily B times A.   Groups where A times B is always equal to B times A are called abelian.  If you didn’t know that before, now you do.  Among those abelian groups there are free abelian groups.  To describe them would make this too technical and you might rightly fear that prospect.
One can use homology theory to associate a different abelian group to each of the surfaces shown above.  The two-holed torus can be associated to a free abelian group of order two and the three-holed torus can be associated to a free abelian group of order three.  
And, yes, the number of holes of the torus will always be the same as the order of the free abelian group that it is associated with.  It’s kind of happy (or ironic depending on your mood) that it turns out that way.  We are allowed to know that two does not equal three when we are in the world of free abelian groups, but we are not when we are just looking at the pictures.

What It Is

By Bobby Neal Winters

We use science to study nature because we want to be able to predict; we want to be able to control.  We try to translate the language of nature to the language of man so that we can name the creatures of nature and thereby control them.

This, I am told, comes from an ancient tradition that we can see examples of in the Bible.  When Jacob wrestles with the unnamed entity in the night he asks the entity’s name and the entity is cagey about it.

When Moses is talking to God in the burning bush, he asks God for his name.  One would be reckless to try to even number the pages of commentary on the answer God gives.  In the Ancient tradition, it is left four consonants and is only pronounced once a year and only in the Holy of Holies.  Others have been bolder to translate it as “I am that I am” or “I am He who is.” Still others have said “I am all that is and was and will be.”  Perhaps one could even be so rash as to say “Being Itself.”

I leave it to you to ponder in your time alone when you are in the right mood how one might control Being Itself.

As we study nature and translate it into our own language, we create models.  The Ancients in studying health talked about the humors. I don’t mean to make fun of them.  When you look at a human, we are largely liquid.  They chose to classify that liquid as black bile, yellow bile, phlegm, and blood.  They used this language in order to make predictions and to attempt to exert some sort of control.

As I said, I don’t mean to make fun of this model.  Indeed, I think that, from a certain point of view, we still use this model.  Our body has various substances--chemicals--within it that can get out of balance.  Blood is still there and having the correct amount of it is still an issue, but the model has become more refined.  The blood has other substances within it that can become out of balance:  hormones, sugar, salt, etc.  There is a huge industry devoted to trying to keep these modern humors in order. This is useful.  It does allow us to make predictions and to exercise a certain amount of control.

The danger we danger we face is in confusing the model--our name for the thing--with the thing itself.  I am thinking specifically here of the photon, the fundamental particle of light.  Indeed, thinking of the photon was the impetus of this essay.

The question goes back again to the ancients, way before Isaac Newton, but I will mention Newton because he did a lot of research on light.  He thought that light was made of corpuscles, often imagined as little billiard balls, and he used his ideas and mathematics to make accurate predictions and to exert control.  He invented the reflector telescope.  

There were contemporaries of Newton, Robert Hooke and Christian Huygens, who proposed a wave theory of light and imagined light a wave in the ether, whatever ether might be.  The wave theory didn’t gain wide acceptance until the 1800s when other scientists were able to make accurate mathematical predictions themselves.  

Students in introductory science classes might rightly be confused because so much of the time the teacher doesn’t bring it to closure.  So is the photon a little billiard or a wave in the ether.  Inquiring minds want to know.

Here’s the answer: A photon is a photon.

It has some properties that can be modeled by thinking of it as a tiny, little, teeninsy billiard ball and others that can be modeled by thinking of it as a wave in the non-existent luminiferous ether.  It is what it is.

These models, these “names” we give the thing in our own language enable us to devise mathematical gadgets that we can manipulate to make predictions and exert control.  These are useful, sometimes shockingly so.

But we ought not confuse them with the thing itself which might very well be unknown and unknowable; uncontrolled and uncontrollable.

Trefoils, Borromean Rings, and Athanasius

By Bobby Neal Winters

You can look up in church windows and see them.  They are the Trefoil Knot and the Borromean rings.  I learned about them first as mathematical objects rather than religious ones because I grew up in a religious tradition that did not truck with such abstract notions.  

*	*
Trefoil Knot	Borromean Rings

But as abstract as they are, the Trefoil Knot and the Borromean Rings are more concrete than the religious concepts they were trying to capture.  I learned about them as I was working on my doctorate in the area of low-dimensional geometric topology.  Specifically, I was beginning the study of knot theory.  My two main sources on knot theory in those days were an article “A Quick Trip through Knot Theory” by Ralph Fox and a wonderful book called Knots and Links by Dale Rolfsen. These were my Old and New Testaments, but not necessarily in that order.

Once while I was working on my doctorate, I had a conversation with one of my cousins with regard to what I was studying him.  I told him about knot theory.  His intelligent middle school aged daughter was listening and interrupted with an indignant tone.

“That isn’t math!” she said.

It leaves that impression with a lot of people.

Knot theory, like her great great grandmother Geometry, attempts to describe the visual world with words. She sees and then she describes.

Let us take the Trefoil as an example.  From the topological point of view, it is a circle. Put your finger on it and trace around it.  It comes back around on itself and starts over.  You can keep going around and around forever just like a circle.  There is a difference, though.  When you draw a standard circle, the lines never do, but in drawing a Trefoil there will be places where the lines cross.

When knot theorists draw the trefoil, we avoid letting the lines touch.  We imagine the Trefoil as being in space and the crossing as being a place where the trefoil passes behind itself.  We draw the part that passes behind as broken.

The Trefoil Knot is the simplest (non-trivial) knot.  Knot theorists can draw it many different ways, but the way one sees it in the church window is probably the best.

*

The Trefoil Knot

It exhibits a three-fold  symmetry.  Note that there are four finite regions that the knot bounds: One is in the exact center and the other three are distributed around it. If you rotate the picture around the center of the middle piece by one third of a rotation, each of the other three pieces will lay exactly on one of the others.  They are completely equal in size and shape.

This three-fold symmetry is something the Trefoil has in common with the Borromean Rings.

*

The Borromean Rings

The Borromean Rings are an example of a link.  Links and knots are inextricably interrelated.  (This is to avoid saying they are linked. While I don’t object to puns in general, this one is too cute.)   Links are groups of knots that are...uh..linked to each other.  The Borromean Link is special in that each of its pieces (components they are called) is a simple unknotted circle and if any of the circles is removed the rest falls apart, i.e. becomes unlinked.

The question one might ask, because of how I began this article, is why are these religious symbols.  Why do they appear in church windows? They are, of course, symbols of the Trinity.

I once asked a dear friend of mine to explain the Trinity.  I asked him because: one, he has a master’s degree in theology so he is learned in these things, and, two, he is Catholic and they just know this stuff.  He talked to me for ten minutes.  Every sentence he said made sense; they piled one upon another in coherent paragraphs.  At the end of the discourse he said, “And if you understood what I just said, then you weren’t paying attention.”

The Trinity is not something those explain the faith would’ve made-up simply help sell it better.  One can easily infer oneself to misunderstands such as Jesus is his own father.  One can find the concept in the Athanasian Creed or one can refer to the diagram below:

*

The Relationships among the Persons of the Trinity

The Trefoil and the Borromean Rings provide a comparatively concrete example of how something can have three parts and still be one thing. They provide a way to defuse the easy quasi-numerical arguments against Trinitarian faith.

The strength of mathematics lies in its exactness.  There is a quote by Roger Bacon that I know because I play Civilization IV, not because I read. It is: "If in other sciences we should arrive at certainty without doubt and truth without error, it behooves us to place the foundations of knowledge in mathematics."  This certainty comes because in mathematics we are able to create an artificial world that is an approximation in some way of the real one.  We may then extrapolate from the artificial one in order to approximate the real one. Nature leaves a footprint; scientists take a cast of it; mathematicians draw a sketch of the cast.

Mathematics is Man’s extension of his language in an attempt to capture the language of Nature.  I almost said “Reality” instead of “Nature,” but that would carry the implicit assumption that Nature and Reality are the same.  I find that assumption to be farther and farther from obvious as I grow older.  Maybe it’s Alzheimer’s.

As a person of faith, I believe things exist beyond those I can touch.  I suppose this is part of being a mathematician.  We create worlds that at first seem to exist only in our own heads, but, as we interact with other of our kind, we discover they have created those very same worlds in their heads as well.  

Christian theologians found themselves heirs to millennia of tradition and scripture.  From that they distilled the concept of the Trinity. We might at this point recall the reaction my cousin’s daughter had to Knot Theory: That is not Mathematics.

While theology is certainly not mathematics, we might have even more sympathy for the theologian than for the knot theorist.  While the knot theorist is attempting to use language to capture the visual world, i.e. what we can see, the theologian is trying to use language to capture what we can’t see. Optimistic might be one adjective to describe such people; insane might be another.

But I don’t want to dismiss them.

I find something good in staring at a stained glass window being reminded that there are things I don’t understand and never will and that no one else will either, but that we still desire to strive toward understanding.

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.