The Black Box

By Bobby Neal Winters

Part 1; The Big Dance

It had been one of those evenings.  I’d sat down at my desk to write a C++ program. I did this because I am trying to convince myself that I can still learn, can still use my mathematical skills to solve problems.  Writing computer programs is a good way to exercise that particular demon, at least for a while.  
Computers don’t care about your feelings.  They don’t want anything from you.  With computers it is either right or wrong, yes or no, 1 or 0.  There is no maybe, kinda sorta.  This goes a long way to explaining why math and science types are they way they are, but I digress.
After an evening of this quasi-mathematical activity, I went to bed with the TV on and set to go to sleep after about a half-hour.  I almost said after about a half hour, but it is run by a computer.  There is no about a half hour.  It either is or it isn’t.
There was a movie on the TV.  I forget which one exactly.  (I am not a computer.) But it was a cowboy movie and a musical kind of like Oklahoma! but a different one.  I remember nothing beyond that because I drifted off into sleep and in my sleep, I dreamed.
In my dream, I was high above the world over a flat plain and it was night.  Before me upon the plain--which was green but not the green associated with grass--was a rectangular black barn.  The barn was completely black except for a non-descript sign on the roof that read “The Black Box.”
I gradually lost altitude, coming closer to the barn, as was I did I noticed people.  There were two lines of people.  In my dream, I was floating with my body parallel to the ground.  The rectangular barn had its long axis running from left to right.  The people in the dream were arranged in two lines, each line running parallel to the left-right axis of the building.  One line was above the barn and the other line was below it.
Drawing closer, I noticed they were all wearing cowboy hats.  Getting even closer, I notice that the ones above the barn had braided ponytails, so they must’ve been girls, and the ones below the barn did not, so they must’ve been boys.  And all of them were carrying candles.  Some of the candles were lit and some weren’t.
Both lines were moving toward the barn with a rhythmic motion; they were dancing.  I then noticed there were doors in the sides of the barns.  I noticed this when each of the rhythmic cowboys and girls danced their way into them.
Then, in the manner of dreams, I was suddenly in the barn without ever having gone through a door.  Once inside, I saw that the cowboys and cowgirls were facing each other, still dancing in place.   In between them were children.  They were also dressed in traditional cowboy accoutrements, but they were androgynous.  While a particular one on the other might have had a ponytail or not, I couldn’t tell.  It wasn’t important to the dream.
Once lined up facing each other, with one little “cow-child” looking up at each couple the following ritual began to take place.  It started on the right end and proceeded in turn to the left.  
Each member of the couples had a candle as did the child and between each of threesome and the next was a pole that held the barn up with a lantern hanging from it.  With some of the cowboys and cowgirls both candles were lit, in some neither, and in some one and not the other, and to begin with, none of the lanterns were lit nor any of the children’s candles.  
When it became a particular couple’s turn they would look at their candles and look at the lantern to their right.  If exactly none were lit, they would do nothing.  If exactly one was lit, they would light the child’s candle.  If exactly two were lit, they would light the lantern to their left and leave the child’s candle unlit.  If all three were lit, they would light the child’s candle and then light the lantern.  Then the next couple took its turn. The first couple didn’t have a lantern to their right to look to, but acted as if they did and as if it were unlit.
At the end, as in the manner of dreams, certain things began to go out of focus while other things came in.  The cowboys, cowgirls , and lanterns faded into the background while the children came into focus, each holding his/her own candle whether lit or unlit.
Then I started awake because a cat was crawling across my face and muttered something about computer programming before bed..
In the cold light of the next morning, I realized this dream was about how binary addition works.  That is to say, how computers add numbers.  For computers, buried deep beyond where most users dare venture, numbers are represented as binary numbers, ones and zeroes. Instead of the place values representing one, ten, one hundred, etc from right to left, the places represent 1, 2, 4, 8, etc.
The binary number 10110 is 0 times one, 1 times two, 1 times four, 0 times eight, and 1 times sixteen; this represents the number twenty-two.  If this number were to be added to the binary 01101--thirteen--we could carry out the operation as below:

Lanterns	1	1	1	0	0	0
Cowboys	0	1	0	1	1	0
Cowgirls	0	0	1	1	0	1
Children	1	0	0	0	1	1

This is a multi-stage process.  Each stage has three inputs: bit (binary digit) from the first summand; bit from the second summand; and the bit that is carried in.  Each stage has two outputs: the bit that goes into the sum and that bit that is carried to the next stage.  So twenty-two plus thirteen is thirty-five.
This turns addition into a mechanical process.  Mechanical operations yield a result that can be interpreted as a number.  When we type numbers, usually base ten, into a computer, this is what goes on behind the mask as it were.  At least this is how I thought about it for years.
But behind this mask, there are other masks.

Part 2; The Logical Mask

Behind the mask of binary arithmetic is the mask of symbolic logic.  The symbols 1 and 0 can be interpreted to have the truth values of true and false, respectively.  These symbols can be combined using the logical operators of & (and), OR (or), and ~(not), and the statements obtained from these various operations can be evaluated as being either true or false. Consider the following:
“Red is a color” and “Dogs are mammals.”  Each of the these statements is true and the combined statement is true.  However, “Red is a color” and “Dogs are reptiles” is a false statement because the second statement is false.  One can put this into symbols as 1&1=1 and 1&0=0, respectively.  We can summarize these in tables as below:

&	0	1
0	0	0
1	0	1

OR	0	1
0	0	1
1	1	1

x	~x
0	1
1	0

We see the 1s and the 0s are transformed in different ways with each of these operations and this might make us think about how they interact with each other in binary addition. Using this to capture addition adds additional (HA!) complexity which is why it is hidden behind a mask.
It is at this point where those of you who get motion sick in the movies might want to skim for a while.
The addition of one bit to another actually requires two stages because of the having to deal with whatever is being carried in from the previous stage.  The first stage is called a half adder and it requires two pieces as well: sum and carry.
Think about how we add numbers when we do it by hand. We place our numbers one above the other with the digits having the same place value lined up.  We proceed column by column.  We add together the digits from the current column and we add in the carry from the previous column.
To describe this, I am going to have to label various bits, and to keep it from looking too much like algebra class, I will refer to the names in the story I began with.
When we add together those two bits in the column under consideration, the cowboy bit and the cowgirl bit, the only time there will be a carry is when both bits are 1.  So the carry to the next column--at least before we add in the carry from the previous column--is (cowboy bit)&(cowgirl bit). We will call this the left lantern bit.  Don’t go to thinking we are done with the left lantern bit; it will change before this is over.
Let us now worry about what we give to the child. It’s not so straightforward. If exactly one bit from the cowboy or cowgirl is 1, then the child bit is 1.  Here it looks like OR. However, when both bits are 1 then the child bit is 0.  It doesn’t look like anything we’ve seen before from out logical operations.  However, we can create a new operation by compounding the others to take care of this.  It is called exclusive or and it is denoted XOR.  We can define it by the formula:

a XOR b= (a OR b) & (~(a & b)).

There is a reason for the mask, no?
We set the child bit as (cowboy bit) XOR (cowgirl bit).  Again, don’t grow too attached because child bit will change once before we are done. This is because we must now account for the possibility of a carry from the previous stage.  
(Breath, just breath.  In and out, that’s okay.)
We simply add the carry from the previous stage--right lantern bit-- to the child bit.  This means the final bit that we write down for the some will be (child bit) XOR (right lantern bit).  What happens to the carry to the next column--the left lantern bit--is a bit more subtle.  When we sum the child bit (not the new child bit but the old child bit) with right lantern bit, this creates a carry of its own that is to be added to the carry that we originally generated by adding the cowboy bit to the cowgirl bit. (Did you get that bit?) That carry is:

(child bit) & (right lantern bit).

While it might seem that summing this with the left lantern bit would be,

[(child bit) & (right lantern bit)] XOR (left lantern bit),

The fact is that because of the small number of possibilities for right lantern bit, cowboy bit, and cowgirl bit, there will never be a time when the quantities on the left hand side and the right hand side of this XOR will ever be the same.  As a consequence it is:

[(child bit) & (right lantern bit)]  OR (left lantern bit) .

If you have been paying attention to this, you should be just a little dizzy. Air sickness bags are available from the seat back in front of you.

Part 3; Beneath the Mask of Logic, Transistors

Let us now recall that all of this is to get a computer to add numbers together.  In the computer, the numbers are represented by electricity.  One represents where there is current and zero represents where there is not.  This is a mask too, but I will not remove it until the last section. 

We represent the wiring in symbols.  For example, NOT is represented by:

 

& is represented by:

 
OR is represented by

:
and XOR is denoted by:

 
To spare you further suspense, the wiring the denotes addition is represented by:
For any of you readers who are still there, we are already well past any reasonable threshold of complication.  However, this diagrams still represents a mask because none of those symbols actually stands for a single component.  Rather, they stand for a collection of electronic components that are wired together in a particular way.
One way to make them work is to use something we’ve all heard of but very few think much about: the transistor.
A transistor is a switch, and there are a few different kinds.  The kind I am going to tell you about is the NPN.  An NPN transistor has three legs pictured as below:
Notice the legs are labeled E, B, and C.  This is for emitter, base, and collector.  Think of electricity as coming in C (the collector--they are working with us) and going out E (the emitter--it works). Think of B as an on-off switch (base? on-off switch? they were doing so nicely).  We send in one current through C to E and when we send another current to B, it will turn the switch on and allow it to pass.  When we turn that switch off, it will not.  
This is represented in a wiring diagram by the figure  below:

 

By wiring transistors together in various imaginative ways, we can create the logical gates as pictured above. For example the NOT gate can be wired as:

 

Part 4; Burning Your Fingers

A good question at this point would be how does this circuit yield NOT as an output.  The answer goes as follows.  There is current coming in from the top, all the time.  When there is no signal sent in through the input, i.e. when it is false or 0, the current cannot go through the transistor because it is off.  Therefore it puts out a current, i.e. a true or a 1.  That is to say 0 input gives 1 as an output.  
By way of contrast, if there is a current coming in through the base, i.e. it is true or 1, then current can flow through the transistor from C to E.  This will be the path of least resistance so the current will not go out through the output.  Therefore, there will be no output current.  Therefore, it will be false or 1.  So 1 input gives 0 output.  This is what the NOT gate is supposed to do.
Except its NOT that simple. How do I know?  I wired it on a breadboard with a transistor, resistors, batteries, the whole shooting match.  It took me awhile to get it to work.  Why?  I didn’t use resistors at first.  Resistors are simple--but handy--little things.  They are cheap.  Their purpose is to slow electricity down.  When you don’t have them, the transistor gets hot and doesn’t work well anymore.
How do I know it gets hot?  I burned my finger on it, that’s how.  I got a little transistor-sized blister.
Oh, and the path of least resistance thing.  That’s not exactly true.  It will go through the transistor, but you need to be careful about the amount of resistance in the output direction.  Regardless of that, there will still be a little current.  I never got the resistance high enough so that, although the LED I hooked up to check for current did dim, it never went entirely out.  I suspect you can calibrate things so that you can’t see the LED light but there will still be a tiny amount of current.  
So the 1 and the 0, true and false, are simply masks themselves.  True means the current is above a certain threshold and false means the current is below it.

Conclusion

So what is the point?  There is a collection of masks going downward like a Matryoshka doll.  I was satisfied I’d gotten close enough to the bottom of it when I burned my fingers.  The masks are there for a purpose.  The complexity of this endeavor is just too massive to take in all at once.  We must organize it by putting the masks on so that it can be learned one step at a time.
Consider now the fact that I’ve only talked about addition and only shown the wiring for a small part of even that.  It is all still inside a Black Box, as it were.  The relative number of people who can understand this is vanishingly small, and, as a wise friend of mine says, it is becoming increasingly like magic to the average person.
As a teacher, this motivates me.

Battlestar Galactica: All along the Watchtower

By Bobby Neal Winters
"No reason to get excited", the thief he kindly spoke
"There are many here among us who feel that life is but a joke
But you and I, we've been through that, and this is not our fate
So let us not talk falsely now, the hour is getting late".
--Bob Dylan

It is rare to get any sort of depiction of religion in the popular media wherein religion is taken seriously by the characters and where it is an integral part of the story.  Whatever its failings, Battlestar Galactica does that.  Religion is taken seriously by the characters and it is integral to the story.  There are hairs to be split, important hairs, but for those who are interested and have the patience, there is something of a reward waiting.  
I am not very learned in the way drama is usually classified, so please be patient with my characterization. Battlestar Galactica is a post-911, science fiction, war, soap opera.  The soap opera is not just tacked on at the end.  The creators of BS-G have embraced the concept and use many of its motifs: amnesia, false deaths, confused sex, adultery, troubled pregnancy, and stolen babies.  This is off my main theme, but I don’t want any of you getting into BS-G without being aware of its soap-operatic nature.  I’ve watched General Hospital, All My Children, and Days of Our Lives back in the day (as well as Hill Street Blues and the like) and have periodically sworn off of them, saying never again.  If you’ve made a similar vow, I’ve now warned you.
The post-911 characterization is important to my theme, however.  Battlestar Galactica is set within a society in which quasi-Greco-Roman polytheism is the established religion.  The society has at some point in the past created robots to be servants and soldiers.  The robots--called Cylons--were sentient and rebelled.  After a war, there was an armistice. Following an extended period of peace, the Cylons returned and attempted the genocide of the human race.  
The Cylons are monotheists.
There monotheism and the nature of the attack on the prevailing human society is heavily influenced by the 911 experience.  The Cylons are portrayed as worshipping One True God. Having a moral arbiter gives them a very black and white view of right and wrong.  While my ignorance of Islam prevents me from comparing the religion of the Cylons with Islam in its multitudinous varieties, I believe it was the intent of the creators of BS-G to invite comparison.  
By way of contrast, while there are pious folks among the polytheistic humans, most have a faith that is shallow or superstitious or both.  The elite are portrayed as either being atheists or practical atheists for the most part. It is here, of course, we come closest to a true representation of religion in modern western civilization.
Beyond the religion of the humans, BS-G also presents a religion/metaphysical framework (the language is escaping me) that is true within the framework of the story.  While it is not made fully explicit, it is undeniably there.  There is a God within the story’s framework.  The question is left open whether this God is natural or supernatural.  There were things said which give me the suspicion that the God of BS-G exist within the natural universe as opposed to being above the natural universe.  (This is one of those fine hairs I promised you.)
The True Religion within the reality of the series appears to be an amalgam of liberal Judeo-Christianity, Buddhism, and Islam.
Having mentioned the natural vs Supernatural dichotomy, the religion of BS-G tends towards those elements that appear to be more supernatural, perhaps with an eye toward giving them a naturalistic interpretation.  That having been said, what is notably missing is much of a connection between religion--whether of the humans, the Cylons, or the True Religion of the series--and sexual morality.
Everybody on the show has sex with everybody else without much input from their religion.  The putative-supernatural entities are no exception to this.  When individuals are untrue, there is concern about possible repercussions because of jealousy; there are concerns with breaking oaths; but there are no expressions that this is simply wrong, not even by the Cylons who supposedly view morality in black and white.
There is an honest moment in the series with regard to abortion.  After the attempted genocide by the Cylons, the human population is down to about 50 thousand.  It has been stated that if the human race is to continue to exist then there must be babies.  A pregnant member of a strict fundamentalist sect then comes seeking abortion.  Leaders within that sect are seeking to block it.  
Then an interesting sequence of events occurs.  The president, who is brilliantly portrayed by Mary McConnell, vows her support for a woman’s right to choose and allows the abortion.  It is then pointed out to her the dire need for babies within their dying civilization and that she will lose political support from her constituents if she sticks with her position; she therefore switches her position. Her opposition immediately takes the opposite side and uses it, and other issues, to win the election.
To me, that particular pericope might be the most honest of the show.
BS-G likes to present black and white lines between good and evil and then gray them out in subsequent episodes.  This is one place where we see the post-911 world we live in and the worldview of the Old Testament come into an odd alignment.
BS-G appears to take pains to get things right for the eyes of our current veterans of Iraq and Afghanistan. We are shown scenes of egregious torture and we are shown “enhanced interrogation.”  We are given the opportunity to ask the question of whether there is a difference and what the difference would be.  We are given situations wherein the only choices are between bad alternatives and worse ones.
BS-G and its prequel, Caprica, which only lasted a season, also raise philosophical questions on the nature of existence, identity, and personhood.  The Cylons are persons but not humans.  They are persons because they are sentient and self-aware.  They are not humans even though they have DNA which cannot be easily distinguished from human DNA and can interbreed with humans. They are not human because they are artificially created and not initially born of humans. Yes, indeed, I did promise you a lot of fine hairs to split.
This series is not for everyone. If any of this seems interesting, I would suggest Netflixing the first five episodes or so just to see.  One or two is not a large enough sample.
And, it is a soap opera.  Don’t say you weren’t warned.

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.