After my post yesterday about the mine wars, a friend asked about the Sid Hatfield murder. Sid Hatfield was the Chief of Police in Matewan, WV in 1920. Some mine "detectives" came to town to evict strikers from their company-owned houses, and Hatfield got together a posse, saying he would kill the detectives. There were several confrontations, the last resulting in a shootout that left seven detectives and three townspeople dead, including the mayor, Cab Testerman.

Some said Hatfield boasted about pulling the trigger on the head detective, Albert Felts, while others said they saw him track an injured Felts into the post office and finish him off. Two weeks later, Hatfield married Testerman's widow, leading to speculation that the whole shootout was a pretext to get Testerman killed so Hatfield could have his wife. Hatfield took over Testerman's jewelry store and turned it into a firearm supply shop for union men.

Hatfield was tried, but beat the murder charges by claiming self-defense. He said Felts fired the first shot, at Testerman. Highly unlikely, not because Felts was an especially honorable man, but because he was a professional detective who had no particular reason to shoot a town official.

Soon after, Hatfield was charged with another shooting at Mohawk, for which he was supposed to stand trial in Welch. He had also been accused of blowing up a tipple and rifle-butting the superintendent of the Stone Mountain Coal Company when that company's mines were being struck by the UMWA. This was not all the trouble Hatfield was in, but I want to wrap this up.

When Hatfield went to Welch to stand trial, Everett Lively, a secret agent working for the mine operators, shot him dead on the courthouse steps in front of his new wife, the erstwhile Widow Testerman. Hatfield, who was clearly what we today would refer to as a total dirtbag, became an instant martyr to the UMWA.

Anyway, my friend asked whether Hatfield's family had been killed with him. They were not, but I wondered whether he was thinking of the Jock Yablonski murder in 1969. Yablonski had just been elected president of the UMWA as a "reform" candidate. The former president, Tony Boyle, hand-picked successor to John L. Lewis, hired a hitman using embezzled union funds to kill Yablonski and his wife and daughter. Such a nice group of people.

I wrote yesterday that the mine wars were started by the UMWA on behalf of competing coal companies. They sent organizers, but things quickly spun out of control. After the 1921 "Battle of Blair Mountain," which resulted in a visit from the U.S. Army, including a wing of bombers, the UMWA tried to wash their hands of the situation, but they should have known what they were getting into. (The Army did not actually drop bombs on the union. That was done by planes privately hired by the Sheriff of Logan County.)

The union men tied red bandannas around their necks and called themselves rednecks, although the word already meant exactly what it means now. Are rednecks prone to violence? Not really. The violent crime rate in Mingo County, WV today is lower than the national average. It is 2.07, compared to a national average of 3.8. For comparison, the rate in Chicago is 9.00. But crime seems to come in waves, when there are are rival groups. Rednecks do not shoot each other over shoes or video games; there has to be some kind of group dynamic going on. This was well known when the UMWA went into WV, because the Hatfield-McCoy and other feuds that were mainly an aftermath of the Civil War had just died down. The UMWA going into West Virginia, and playing groups off one another, is like Nike trying to market shoes by getting the Rolling 60's to adopt them as a gang uniform. You don't think personal grudges played a role? Everett Lively and the officers of the UMWA local were childhood friends turned enemies!

Anyway. I wanted to say one last thing about John L. Lewis, the so-called hero of the UMWA. Here is a typical miner's house

and here is John L. Lewis's house

## Tuesday, January 27, 2015

## Monday, January 26, 2015

### Another Book About the Mine Wars

I'm reading a new history of the West Virginia mine wars by historian James Green, called

Prof. Green says he is an "activist scholar," and his career has been one of advocacy for collectivism. It is no surprise, then, that the book paints the union organizers, who were generally associated with the United Mine Workers of America, as heroic defenders of the working man. As for myself, I am no historian, just the descendant of "company men" who were caught in the middle of the so-called mine wars. When UMWA members were stockpiling guns and ammunition, blowing up tipples and killing company officials in 1921, my grandfather was working miles from home, because the mines back near his home in Ottawa, Boone County, were shut down by strikes. Later that year he was called back to Ottawa to help with an armed defense of the mines and to stop the miners' army from getting through to Logan County to continue wreaking havoc. So you can expect that I have a somewhat different take.

Apparently, public TV has bought the rights to this book and plans to make an

Like Green's book, most histories of these events uncritically support the union men. The only really objective one I know of is

You have to realize what a strike meant in the early 1900s. Today, a strike means a picket line. Back then, a strike meant shutting down an operation by beating or killing people on their way to work and by destroying equipment. It is easy to criticize the mine operators for hiring private armies to break strikes if you have the modern kind of strike in mind, instead of what strikes were really like in those days.

By 1912, both the mine operators and the union were using tactics that have no place in decent society. The operators were running company towns like lords over a medieval manor, firing and blacklisting any miners who spoke or acted in ways that displeased management. For their part, the union was using guns and dynamite to shut down nonunion mines.

The natural question is, who started it? Who set off the chain of events that spun out of control until the Army had to be called in?

Contrary to Green's telling, the mine wars were anything but grassroots in origin and anything but ignited by oppression. In 1898, a coal cartel called the "Central Competitive Field" was formed, covering Ohio, Indiana, Illinois and western Pennsylvania. It sought to preserve profits by limiting production, exactly like an OPEC for coal, preventing price competition that would have reduced the cost of coal to consumers. Ultimately, the CCF was willing to pay whatever wage the UMWA demanded and just pass it all onto the consumer. The trouble is, the plan could only work if it were extended over the whole industry, otherwise other mines could undercut the cartel. In exchange for high wages, the UMWA promised the CCF it would unionize West Virginia, which would stop mines there from selling cheap coal. Green claims this is a "conspiracy theory," but alas it was a real conspiracy, not just a theory. Here is a high official of the UMWA, complaining to Congress in 1921 that the CCF operators were not helping them carry out this very promise:

Let me point to the fact that the United Mine Workers of America have diligently and aggressively attempted to carry out the promise made in Chicago in 1898 that they have done everything in their power to redeem any promise they may have made to organize West Virginia. Since 1898 our organization has at various times spent hundreds of thousands of dollars trying to unionize West Virginia. We have also sacrificed human life in the attempt to redeem that promise. In view of the fact that we have spent hundreds of thousands of dollars and that our organizers, our members who have gone there as missionaries in an attempt to redeem that promise have sacrificed their lives and their liberties, we should be given credit for what we have done. I want to ask the operators how much money they have spent and what they have done to aid us to organize West Virginia?

I can guess that the CCF would respond by saying that it was ultimately their mines that were the source of every one of the hundreds of thousands of dollars mentioned above. In effect, the CCF gave the workers a raise, a good portion of which went to the UMWA's strike fund, which had a double function as a sabotage operation against the CCF's competitors. The CCF agreement is a much more convincing explanation of when and why the mine wars happened in West Virginia than any spontaneous uprising of oppressed miners.

It is worse than that. Not only were the "mine wars" sponsored by the CCF, they actually achieved nothing but to rain death and destruction on operators and miners alike. There is no definitive tally of deaths, but in reading these histories, there are one or two killed here, and ten killed there, and a powerhouse blown up here, and children starving in strike colonies there, so it adds up to a lot of suffering.

In the end, the strikes and violence did nothing to unionize southern West Virginia. It was unionized (in 1933) by the federal government, when Roosevelt's NIRA was passed. And once again, the welfare of the workers was incidental to the real goal. The NIRA was drafted by another cartel, this one covering much of American manufacturing. Even liberals came to recognize that the NIRA was crony capitalism at its worst. Most of it was found unconstitutional in 1935, but the labor provisions were retained by subsequent legislation.

The role of the UMWA and its leader, John L. Lewis, is outlined here:

Northern coal-operator associations joined John L. Lewis...in helping to draft Section 7(a) of the NIRA. Because most northern mines had been unionized long before 1933 and could not be deunionized, their owners had long sought the unionization of Appalachia's mines as the ultimate solution to Appalachia's low coal prices...Lewis accepted what amounted to price fixing under the NIRA...in exchange for labor leaders' treasured Section 7(a) of the legislation. Lewis reportedly said at this time that he was only looking out for current miners. Miners' sons, he reportedly said, would have to look for work in the cities. Lewis was manifestly not, however, looking out for all current miners, His initial proposal for a nationwide daily minimum wage of five dollars for all coal miners would have ruined many Appalachian operations, throwing their miners out of work.

Let's think about this. In the depths of the Depression, with people breaking up furniture and burning it in their fireplaces to keep warm, the main concern of the federal government is...that there is too much cheap coal being mined in Appalachia. That is some kind of crazy.

The UMWA, like all unions, promoted the idea that workers have a claim on profits which only the union can help them get. But the CCF and NIRA episodes show that the miners' gains didn't come out of existing profits, they came out of coal prices paid by the public that were inflated by monopolies set up with the connivance of the UMWA. It sort of undermines the whole union narrative.

The UMWA and its allies in the press tried to drum up support for the union by depicting the desperate conditions that existed at times in the mining camps as luridly as possible. It is true that life was desperate at times in the West Virginia mining camps. But whose fault was that? The papers complained that families were living in tent colonies with little to eat. But they failed to point out that those same families had been living in houses with a regular wage

The saddest chapter is that the UMWA sold out the miners partially after the 1921 battle and completely in 1951 when John L. Lewis accepted the mechanization of the mines in exchange for higher wages for existing miners. It fulfilled his threat that miners' sons would have to look for work in the cities. Mechanization was of course inevitable, and beneficial to the consumers of coal, but that argument would not have garnered any support from the miners, had they been asked.

Green acknowledges this repeated betrayal by the UMWA, which started the whole mess back in 1898, but ends the book by describing a utopian scene of union solidarity in which West Virginia miners enjoy safe working conditions, job security, and sunny days forever. In fact, that lasted about ten years, because the UMWA was bleeding the mines dry. Green says he went to West Virginia recently to see Blair Mountain, where the UMWA battled the coal operators and the police. Do you know what he didn't see? Any of my grandfather's seven kids or their descendants, because they had to disperse from West Virginia to five different states in order to make a living.

Somebody will say that West Virginia's problems were due to the decline of the coal industry, which the union couldn't control. Not so. Even with recent moves to restrict coal-burning power plants,

*The Devil Is Here in These Hills: West Virginia's Coal Miners and Their Battle for Freedom*. The story, though told many times, is still exciting.From the front page of the New York Times, August 2, 1921 |

Prof. Green says he is an "activist scholar," and his career has been one of advocacy for collectivism. It is no surprise, then, that the book paints the union organizers, who were generally associated with the United Mine Workers of America, as heroic defenders of the working man. As for myself, I am no historian, just the descendant of "company men" who were caught in the middle of the so-called mine wars. When UMWA members were stockpiling guns and ammunition, blowing up tipples and killing company officials in 1921, my grandfather was working miles from home, because the mines back near his home in Ottawa, Boone County, were shut down by strikes. Later that year he was called back to Ottawa to help with an armed defense of the mines and to stop the miners' army from getting through to Logan County to continue wreaking havoc. So you can expect that I have a somewhat different take.

Apparently, public TV has bought the rights to this book and plans to make an

*American Experience*episode out of it. You probably also know about the movie*Matewan*. A thousand times more people will read the book and see the TV show than read this blog, but still, one must try. It's not my contention that unions are always wrong and companies are always right. In fact, I would never have been born were it not for the UMWA hospital at Welch, West Virginia. But I will make the case that the mine wars were a tragic waste of lives and property that had little to do with workers' rights per se.Like Green's book, most histories of these events uncritically support the union men. The only really objective one I know of is

*Bloodletting in Appalachia*by Howard Lee. Nearly all the others see the conflict as a moralistic good-versus-evil battle pitting a grassroots movement of saintly miners against oppressive mine operators straight out of a Dickens novel. These kinds of fairy tales are not only condescending, they are by now boring.You have to realize what a strike meant in the early 1900s. Today, a strike means a picket line. Back then, a strike meant shutting down an operation by beating or killing people on their way to work and by destroying equipment. It is easy to criticize the mine operators for hiring private armies to break strikes if you have the modern kind of strike in mind, instead of what strikes were really like in those days.

By 1912, both the mine operators and the union were using tactics that have no place in decent society. The operators were running company towns like lords over a medieval manor, firing and blacklisting any miners who spoke or acted in ways that displeased management. For their part, the union was using guns and dynamite to shut down nonunion mines.

The natural question is, who started it? Who set off the chain of events that spun out of control until the Army had to be called in?

Contrary to Green's telling, the mine wars were anything but grassroots in origin and anything but ignited by oppression. In 1898, a coal cartel called the "Central Competitive Field" was formed, covering Ohio, Indiana, Illinois and western Pennsylvania. It sought to preserve profits by limiting production, exactly like an OPEC for coal, preventing price competition that would have reduced the cost of coal to consumers. Ultimately, the CCF was willing to pay whatever wage the UMWA demanded and just pass it all onto the consumer. The trouble is, the plan could only work if it were extended over the whole industry, otherwise other mines could undercut the cartel. In exchange for high wages, the UMWA promised the CCF it would unionize West Virginia, which would stop mines there from selling cheap coal. Green claims this is a "conspiracy theory," but alas it was a real conspiracy, not just a theory. Here is a high official of the UMWA, complaining to Congress in 1921 that the CCF operators were not helping them carry out this very promise:

Let me point to the fact that the United Mine Workers of America have diligently and aggressively attempted to carry out the promise made in Chicago in 1898 that they have done everything in their power to redeem any promise they may have made to organize West Virginia. Since 1898 our organization has at various times spent hundreds of thousands of dollars trying to unionize West Virginia. We have also sacrificed human life in the attempt to redeem that promise. In view of the fact that we have spent hundreds of thousands of dollars and that our organizers, our members who have gone there as missionaries in an attempt to redeem that promise have sacrificed their lives and their liberties, we should be given credit for what we have done. I want to ask the operators how much money they have spent and what they have done to aid us to organize West Virginia?

I can guess that the CCF would respond by saying that it was ultimately their mines that were the source of every one of the hundreds of thousands of dollars mentioned above. In effect, the CCF gave the workers a raise, a good portion of which went to the UMWA's strike fund, which had a double function as a sabotage operation against the CCF's competitors. The CCF agreement is a much more convincing explanation of when and why the mine wars happened in West Virginia than any spontaneous uprising of oppressed miners.

It is worse than that. Not only were the "mine wars" sponsored by the CCF, they actually achieved nothing but to rain death and destruction on operators and miners alike. There is no definitive tally of deaths, but in reading these histories, there are one or two killed here, and ten killed there, and a powerhouse blown up here, and children starving in strike colonies there, so it adds up to a lot of suffering.

In the end, the strikes and violence did nothing to unionize southern West Virginia. It was unionized (in 1933) by the federal government, when Roosevelt's NIRA was passed. And once again, the welfare of the workers was incidental to the real goal. The NIRA was drafted by another cartel, this one covering much of American manufacturing. Even liberals came to recognize that the NIRA was crony capitalism at its worst. Most of it was found unconstitutional in 1935, but the labor provisions were retained by subsequent legislation.

The role of the UMWA and its leader, John L. Lewis, is outlined here:

Northern coal-operator associations joined John L. Lewis...in helping to draft Section 7(a) of the NIRA. Because most northern mines had been unionized long before 1933 and could not be deunionized, their owners had long sought the unionization of Appalachia's mines as the ultimate solution to Appalachia's low coal prices...Lewis accepted what amounted to price fixing under the NIRA...in exchange for labor leaders' treasured Section 7(a) of the legislation. Lewis reportedly said at this time that he was only looking out for current miners. Miners' sons, he reportedly said, would have to look for work in the cities. Lewis was manifestly not, however, looking out for all current miners, His initial proposal for a nationwide daily minimum wage of five dollars for all coal miners would have ruined many Appalachian operations, throwing their miners out of work.

Let's think about this. In the depths of the Depression, with people breaking up furniture and burning it in their fireplaces to keep warm, the main concern of the federal government is...that there is too much cheap coal being mined in Appalachia. That is some kind of crazy.

The UMWA, like all unions, promoted the idea that workers have a claim on profits which only the union can help them get. But the CCF and NIRA episodes show that the miners' gains didn't come out of existing profits, they came out of coal prices paid by the public that were inflated by monopolies set up with the connivance of the UMWA. It sort of undermines the whole union narrative.

The UMWA and its allies in the press tried to drum up support for the union by depicting the desperate conditions that existed at times in the mining camps as luridly as possible. It is true that life was desperate at times in the West Virginia mining camps. But whose fault was that? The papers complained that families were living in tent colonies with little to eat. But they failed to point out that those same families had been living in houses with a regular wage

*until the UMWA induced them to strike.*And the very worst times in the coalfields happened in the late 1930s,*after*unionization*.*The saddest chapter is that the UMWA sold out the miners partially after the 1921 battle and completely in 1951 when John L. Lewis accepted the mechanization of the mines in exchange for higher wages for existing miners. It fulfilled his threat that miners' sons would have to look for work in the cities. Mechanization was of course inevitable, and beneficial to the consumers of coal, but that argument would not have garnered any support from the miners, had they been asked.

Green acknowledges this repeated betrayal by the UMWA, which started the whole mess back in 1898, but ends the book by describing a utopian scene of union solidarity in which West Virginia miners enjoy safe working conditions, job security, and sunny days forever. In fact, that lasted about ten years, because the UMWA was bleeding the mines dry. Green says he went to West Virginia recently to see Blair Mountain, where the UMWA battled the coal operators and the police. Do you know what he didn't see? Any of my grandfather's seven kids or their descendants, because they had to disperse from West Virginia to five different states in order to make a living.

Somebody will say that West Virginia's problems were due to the decline of the coal industry, which the union couldn't control. Not so. Even with recent moves to restrict coal-burning power plants,

*more bituminous coal is being mined now than before unionization*, but the industry supports only a small fraction of the families it used to. The UMWA didn't stop the process or even try to slow it down. So what did the violence achieve? Do we really need to hold up this blood-soaked history as something to be proud of, or as an inspiration to today's workers? Blair Mountain should be protected from mountaintop removal and preserved in the interest of history, but as caution, not inspiration.## Saturday, January 17, 2015

### Math Extra #3: Functions

Ten months ago I posted Math Extras #1 and 2 which I wrote for my son. At long last, here is Math Extra #3. This is for kids taking algebra. By the way, I solved the problem of how to typeset math in Blogger by using MathJax. It isn't perfect, but it's a whole lot better than the alternatives.

The following shows the difference between writing a function explicitly and referring it to by name:

Explicitly: ``The function $y=x^2+x-6$ has the value zero for $x=2$ and $x=-3$."

By name

The explicit statement of the function above uses the dependent variable $y$. Don't confuse the name of the function with the dependent variable. We can use any dependent variable we want; for example, we could write $y=f(x)$, meaning $y=x^2+x-6$, or we could write $z=f(x)$, meaning $z=x^2+x-6$. You say $f(x)$ like this: ``f of x."

Naming a function makes it easy to talk about different arguments. We can write $f(5)$ when we mean $5^2+5-6$. We can also write $f(x+2)$ when we mean $(x+2)^2+(x+2)-6$. Eventually, you will study functions that have more than one argument, such as $f(x,y,z)$. You would say this as, ``f of x, y and z."

Did you notice that when I named the function, I used the symbol $\equiv$ instead of $=$? The three lines are on purpose. They indicate that this is a definition, just giving something a name. That's a completely different idea than stating two things are equal to each other. Not everyone is careful about this. They might give the definition as $f(x)= x^2+x-6$. But I think you can see that $f(x)=2$ is an equation, and doesn't have anything to do with the definition $f(x)\equiv x^2+x-6$. An explicit way to write that equation is $x^2+x-6=2$.

This naming may seem like too much trouble, but functions can become very lengthy and we may want to refer to them again and again. For example,

``The gamma function is defined as $\Gamma(t)\equiv\int_0^\infty x^{t-1} \text{e}^{-x} \, \text{d}x.$ It satisfies the identity $\Gamma(t+1)=t\Gamma(t)$. It is closely related to the factorial, because for integer arguments, $\Gamma(n)=(n-1)!$"

It's not important to understand what the preceding sentence means, just to appreciate that it would have been a lot longer and more confusing if we had to refer to the gamma function explicitly instead of as just $\Gamma(t)$.

Giving a function a name enables

\begin{align}

1 + 2 &= 2 + 1 \\

1 + 3 &= 3 + 1 \\

1 + 4 &= 4 + 1 \ldots

\end{align}

and so on, covering all possible sums of two numbers. Without using variables, it would literally require an infinite number of specific examples to completely define the commutative property of addition.

Using named functions enables a similar kind of abstraction. For example, a fact about functions from calculus is

\begin{equation}

\frac{\text{d}}{\text{d}t}f(x(t))=\frac{\text{d}}{\text{d}x}f(x)\frac{\text{d}}{\text{d}t}x(t)

\end{equation}

This is true no matter what the functions $f$ and $x$ are. For now, you don't need to know what all the d's are doing, but you can see that this would be impossible to write without being able to refer to functions by name instead of explicitly. This example also shows that the argument of a function can be a function itself.

Examples

Let $f(x)\equiv x^2+2x-3$. If $x=0$, then $f(x)=-3$. Another way to write this is $f(0)=-3$. On the other hand, $f(-1)=-4$. The values of $x$ for which $f(x)=0$ are called the zeros or roots of $f(x)$. Note that these are

A function that has the same value for both signs of an argument is called an even function. $g(x)=x^2$ is an example of an even function. Its value is 4 for both $x=2$ and $x=-2$. If a function has the same value except with the sign flipped, it is called an odd function. $h(x)=x^3$ has the value $h=27$ for $x=3$ and $h=-27$ for $x=-3$. Therefore, it is an odd function. The function $f(x)$ as defined above is neither even nor odd.

It's important to understand the difference between the value of a function and the function itself. Let $f_1(x)\equiv 4x-4$ and $f_2(x)\equiv 3x-3$. It is true that $f_1(1)=f_2(1)=0$. However, $f_1(x)\neq f_2(x)$ otherwise. They are two different functions.

Sometimes we can draw conclusions about a function even if we don't know exactly what it is. ``Forget" the definition of $f(x)$ we stated above, so that we can re-use the name $f$. Now, if $f(x)=3f(2x)$, and $f(6)=-5$, then $f(3)=-15$. How do we know this? Let $x=3$ and evaluate $f(x)$ and $f(2x)$. In this paragraph, we have discarded the old definition of $f(x)$. You have to be careful to understand when a function has been redefined. Math books or exams will typically be very careful about this. When they talk about $f(x)$ in a single problem or explanation, they usually mean the same definition of $f(x)$. Science books might not be so careful about it. When they say ``If $f(x)=\ldots$", they probably only intend that definition of $f(x)$ to hold in that particular statement.

The name we give to the argument doesn't matter in some situations. If $x$ and $z$ are two unknown numbers, then writing $f(x)$ is pretty much the same as writing $f(z)$ unless we know something about $x$ and $z$. The only thing to be careful of is not to use the same name for the argument as for the value of the function, unless you really mean it. If you write $y=f(x)$, there is a value of $y$ corresponding to each possible value of $x$. But if you write $x=f(x)$, you've written an equation that may be solvable for $x$. For example, if $f(x)\equiv 5x-3$, then the equation $x=f(x)$ has one solution, $x=3/4$.

Because the important thing is the form of the function, and not the names of the variables, sometimes people write functions as $f(\bullet)$. That means any name could be used for the argument. This is because, for example, $f(x)\equiv 4x^2-x+1$ and $f(z)\equiv 4z^2-z+1$ are really the same function. Everything that is true of $f(x)$ is also true of $f(z)$, as long as there is nothing special about $x$ and $z$.

When the argument of a function is itself a function, the overall function is called the composition of two functions. It can just be written as $f(g(x))$, or it can be written more formally as $f \circ g$. For example, if $f(x)\equiv 2x-1$ and $g(x)\equiv -x+8$, then $f(g(x))=2(-x+8)-1=-2x+15.$

When a function is raised to a power, say the power of 2, it's written $f^2(x)$, not $f(x)^2$. Note that $f^2(x)$ is not the same thing as $f(x^2)$.

Problems

1. If $f(x)\equiv x+3$ and $g(x)\equiv 2x-3$, for what value or values of $x$ does $f(x)=g(x)$?

2. Is $g(x)\equiv x^4+x^2+3$ even, odd or neither?

3. What are the roots of $f(x)\equiv x^2-36$?

4. Suppose $f(x)$ is odd and $g(x)=3f(x)$. If $f(2)=-4$, then what is $g(-2)$?

5. If $\phi(y)\equiv 3y$, what is $\phi^3(z)$?

6. If $f(x)\equiv 4x+9$ and $g(x)\equiv x^2+3x+7$, what is $f \circ g$ for $x=0$?

7. If $f(x)$ and $g(x)$ are both even, and $f(5)=1$ and $g(-1)=5$, then what is $f \circ g$ if $x=1$?

8. Is composition of functions commutative? Hint: Let $g(x)\equiv -x$ and $f(x)\equiv x^2$. What is $f \circ g$? What is $g \circ f$? If you have trouble keeping track of the signs, substitute in a specific value like $x=1$.

Extra Info

You have probably seen things like $\sin(x)$ and $\log(x)$ on your calculator or elsewhere. These are named functions. They have fixed definitions (not given here) that everyone in math accepts. Whenever anyone writes $\sin(x)$, everyone knows it is the sine function from trigonometry. So don't choose names for your functions that are the same as the standard ones. But aside from the fact that these are standard functions, they work just like the functions you name yourself.

One thing about the standard functions is that their names are always typeset in Roman font, whereas the variables are always written in italics. This helps people not get confused into thinking that $\sin(x)$ means $s$ times $i$ times $n$ times $x$. One can argue that functions we name ourselves should also be in Roman font: $\text{f}(x)$ instead of $f(x)$. But nobody ever does this.

Usually, people omit the parentheses from standard functions. They write $\sin x$ instead of $\sin(x)$. They only use the parentheses in cases where there could be some confusion, such as $\tan (3x+6)$. That clarifies that the argument is $3x+6$. If you wrote $\tan 3x+6$ someone might think you meant $\tan(3x)+6$. But you never write $f x$. It always has to be $f(x)$ or else people will think you mean $f$ times $x$.

When picking names for functions, people usually start with $f$ (for ``function"), then go to $g$ and $h$. After that, practices vary. If you run out of Roman letters, you can use Greek letters like $\Gamma$ (capital gamma), or Gothic letters like $\mathfrak{F}$ or whatever. Avoid using calligraphic letters like $\mathcal{F}$, script letters like $\mathscr{F}$ or bold letters like $\mathbb{F}$. Those are usually used for other kinds of mathematical objects.

Other ways to write the definition of a function are $f(x) \triangleq 2x-7$ and $f(x)$ ${\scriptsize{\text{def}}}\atop{=}$ $ 2x-7$. A really formal way to do it is

\begin{equation}

f(x): x \rightarrow 2x-7.

\end{equation}

This formal way is called a ``mapping". It means that the function $f(x)$ ``maps" any argument $x$ to a different number $2x-7$.

In the beginning, I said that a function is usually an algebraic expression, but doesn't have to be. It could just as well be defined by a table, like this:

$x$ $f(x)$

$-2$ $2$

$-1$ $0$

$0$ $-2$

$1$ $-4$

$2$ $-6$

The function defined by this table is only defined for the specific arguments listed; that is, the integers between $-2$ and 2. $f(10)$ has no meaning, and neither does $f(0.5)$.

A function can also be defined in words, like this: ``For any real number $x$, let $f(x)$ be the next biggest integer." For example, $f(4.66)=5$. You can then use $f(x)$ in discussions.

Some algebra books say that a function can have only one value for any given argument, or else it technically isn't a function. For example, if $f(x) \equiv \sqrt{x}$, then for $x=9$, $f(x)=3$ and $-3$. They would say $\sqrt{x}$ is an expression, but not a function. This is a matter of words, not mathematics. Almost every book after algebra doesn't bother with this kind of hairsplitting. You might say that $f(x)$ is multi-valued, but you'd still call it a function. If you want to get technical, you can say that $f(x)=\sqrt{x}$ is not a one-to-one mapping.

## Functions

In the later stages of algebra, and on into calculus, we start to study functions a lot. A function is just a rule for generating a value from an input. The input is called an argument. Usually, the function is an algebraic expression, like $x^2-10$. For an input or argument of $x=1$, the value is $1^2-10$, which is $-9$. Sometimes we want to refer to functions by name instead of explicitly. First I'll talk about*why*we would want to do this. Then I'll give some examples and problems.The following shows the difference between writing a function explicitly and referring it to by name:

Explicitly: ``The function $y=x^2+x-6$ has the value zero for $x=2$ and $x=-3$."

By name

- First we give the function a name: $f(x) \equiv x^2+x-6$
- Then we can refer to it by name: ``$f(x)=0$ for $x=2$ and $x=-3$."

The explicit statement of the function above uses the dependent variable $y$. Don't confuse the name of the function with the dependent variable. We can use any dependent variable we want; for example, we could write $y=f(x)$, meaning $y=x^2+x-6$, or we could write $z=f(x)$, meaning $z=x^2+x-6$. You say $f(x)$ like this: ``f of x."

Naming a function makes it easy to talk about different arguments. We can write $f(5)$ when we mean $5^2+5-6$. We can also write $f(x+2)$ when we mean $(x+2)^2+(x+2)-6$. Eventually, you will study functions that have more than one argument, such as $f(x,y,z)$. You would say this as, ``f of x, y and z."

Did you notice that when I named the function, I used the symbol $\equiv$ instead of $=$? The three lines are on purpose. They indicate that this is a definition, just giving something a name. That's a completely different idea than stating two things are equal to each other. Not everyone is careful about this. They might give the definition as $f(x)= x^2+x-6$. But I think you can see that $f(x)=2$ is an equation, and doesn't have anything to do with the definition $f(x)\equiv x^2+x-6$. An explicit way to write that equation is $x^2+x-6=2$.

This naming may seem like too much trouble, but functions can become very lengthy and we may want to refer to them again and again. For example,

``The gamma function is defined as $\Gamma(t)\equiv\int_0^\infty x^{t-1} \text{e}^{-x} \, \text{d}x.$ It satisfies the identity $\Gamma(t+1)=t\Gamma(t)$. It is closely related to the factorial, because for integer arguments, $\Gamma(n)=(n-1)!$"

It's not important to understand what the preceding sentence means, just to appreciate that it would have been a lot longer and more confusing if we had to refer to the gamma function explicitly instead of as just $\Gamma(t)$.

Giving a function a name enables

*abstraction*, which is what math is all about. By abstraction, I mean talking about the general properties of something instead of just specific examples. You already know that a variable can be used to represent a number, as in $x=5$. And you also know that we can make statements such as $x+y=y+x$. That equation expresses the commutative property of addition. It's true no matter what the specific values of $x$ and $y$ are. If you think about it, this simple statement contains a tremendous, in fact infinite, amount of information. To write it without using variables, you would have to write\begin{align}

1 + 2 &= 2 + 1 \\

1 + 3 &= 3 + 1 \\

1 + 4 &= 4 + 1 \ldots

\end{align}

and so on, covering all possible sums of two numbers. Without using variables, it would literally require an infinite number of specific examples to completely define the commutative property of addition.

Using named functions enables a similar kind of abstraction. For example, a fact about functions from calculus is

\begin{equation}

\frac{\text{d}}{\text{d}t}f(x(t))=\frac{\text{d}}{\text{d}x}f(x)\frac{\text{d}}{\text{d}t}x(t)

\end{equation}

This is true no matter what the functions $f$ and $x$ are. For now, you don't need to know what all the d's are doing, but you can see that this would be impossible to write without being able to refer to functions by name instead of explicitly. This example also shows that the argument of a function can be a function itself.

Examples

Let $f(x)\equiv x^2+2x-3$. If $x=0$, then $f(x)=-3$. Another way to write this is $f(0)=-3$. On the other hand, $f(-1)=-4$. The values of $x$ for which $f(x)=0$ are called the zeros or roots of $f(x)$. Note that these are

*not*$f(0)$. They are the solutions of the equation $f(x)=0$. The roots of this $f(x)$ are $x=1$ and $x=-3$. Not every function has roots.A function that has the same value for both signs of an argument is called an even function. $g(x)=x^2$ is an example of an even function. Its value is 4 for both $x=2$ and $x=-2$. If a function has the same value except with the sign flipped, it is called an odd function. $h(x)=x^3$ has the value $h=27$ for $x=3$ and $h=-27$ for $x=-3$. Therefore, it is an odd function. The function $f(x)$ as defined above is neither even nor odd.

It's important to understand the difference between the value of a function and the function itself. Let $f_1(x)\equiv 4x-4$ and $f_2(x)\equiv 3x-3$. It is true that $f_1(1)=f_2(1)=0$. However, $f_1(x)\neq f_2(x)$ otherwise. They are two different functions.

Sometimes we can draw conclusions about a function even if we don't know exactly what it is. ``Forget" the definition of $f(x)$ we stated above, so that we can re-use the name $f$. Now, if $f(x)=3f(2x)$, and $f(6)=-5$, then $f(3)=-15$. How do we know this? Let $x=3$ and evaluate $f(x)$ and $f(2x)$. In this paragraph, we have discarded the old definition of $f(x)$. You have to be careful to understand when a function has been redefined. Math books or exams will typically be very careful about this. When they talk about $f(x)$ in a single problem or explanation, they usually mean the same definition of $f(x)$. Science books might not be so careful about it. When they say ``If $f(x)=\ldots$", they probably only intend that definition of $f(x)$ to hold in that particular statement.

The name we give to the argument doesn't matter in some situations. If $x$ and $z$ are two unknown numbers, then writing $f(x)$ is pretty much the same as writing $f(z)$ unless we know something about $x$ and $z$. The only thing to be careful of is not to use the same name for the argument as for the value of the function, unless you really mean it. If you write $y=f(x)$, there is a value of $y$ corresponding to each possible value of $x$. But if you write $x=f(x)$, you've written an equation that may be solvable for $x$. For example, if $f(x)\equiv 5x-3$, then the equation $x=f(x)$ has one solution, $x=3/4$.

Because the important thing is the form of the function, and not the names of the variables, sometimes people write functions as $f(\bullet)$. That means any name could be used for the argument. This is because, for example, $f(x)\equiv 4x^2-x+1$ and $f(z)\equiv 4z^2-z+1$ are really the same function. Everything that is true of $f(x)$ is also true of $f(z)$, as long as there is nothing special about $x$ and $z$.

When the argument of a function is itself a function, the overall function is called the composition of two functions. It can just be written as $f(g(x))$, or it can be written more formally as $f \circ g$. For example, if $f(x)\equiv 2x-1$ and $g(x)\equiv -x+8$, then $f(g(x))=2(-x+8)-1=-2x+15.$

When a function is raised to a power, say the power of 2, it's written $f^2(x)$, not $f(x)^2$. Note that $f^2(x)$ is not the same thing as $f(x^2)$.

Problems

1. If $f(x)\equiv x+3$ and $g(x)\equiv 2x-3$, for what value or values of $x$ does $f(x)=g(x)$?

2. Is $g(x)\equiv x^4+x^2+3$ even, odd or neither?

3. What are the roots of $f(x)\equiv x^2-36$?

4. Suppose $f(x)$ is odd and $g(x)=3f(x)$. If $f(2)=-4$, then what is $g(-2)$?

5. If $\phi(y)\equiv 3y$, what is $\phi^3(z)$?

6. If $f(x)\equiv 4x+9$ and $g(x)\equiv x^2+3x+7$, what is $f \circ g$ for $x=0$?

7. If $f(x)$ and $g(x)$ are both even, and $f(5)=1$ and $g(-1)=5$, then what is $f \circ g$ if $x=1$?

8. Is composition of functions commutative? Hint: Let $g(x)\equiv -x$ and $f(x)\equiv x^2$. What is $f \circ g$? What is $g \circ f$? If you have trouble keeping track of the signs, substitute in a specific value like $x=1$.

Extra Info

You have probably seen things like $\sin(x)$ and $\log(x)$ on your calculator or elsewhere. These are named functions. They have fixed definitions (not given here) that everyone in math accepts. Whenever anyone writes $\sin(x)$, everyone knows it is the sine function from trigonometry. So don't choose names for your functions that are the same as the standard ones. But aside from the fact that these are standard functions, they work just like the functions you name yourself.

One thing about the standard functions is that their names are always typeset in Roman font, whereas the variables are always written in italics. This helps people not get confused into thinking that $\sin(x)$ means $s$ times $i$ times $n$ times $x$. One can argue that functions we name ourselves should also be in Roman font: $\text{f}(x)$ instead of $f(x)$. But nobody ever does this.

Usually, people omit the parentheses from standard functions. They write $\sin x$ instead of $\sin(x)$. They only use the parentheses in cases where there could be some confusion, such as $\tan (3x+6)$. That clarifies that the argument is $3x+6$. If you wrote $\tan 3x+6$ someone might think you meant $\tan(3x)+6$. But you never write $f x$. It always has to be $f(x)$ or else people will think you mean $f$ times $x$.

When picking names for functions, people usually start with $f$ (for ``function"), then go to $g$ and $h$. After that, practices vary. If you run out of Roman letters, you can use Greek letters like $\Gamma$ (capital gamma), or Gothic letters like $\mathfrak{F}$ or whatever. Avoid using calligraphic letters like $\mathcal{F}$, script letters like $\mathscr{F}$ or bold letters like $\mathbb{F}$. Those are usually used for other kinds of mathematical objects.

Other ways to write the definition of a function are $f(x) \triangleq 2x-7$ and $f(x)$ ${\scriptsize{\text{def}}}\atop{=}$ $ 2x-7$. A really formal way to do it is

\begin{equation}

f(x): x \rightarrow 2x-7.

\end{equation}

This formal way is called a ``mapping". It means that the function $f(x)$ ``maps" any argument $x$ to a different number $2x-7$.

In the beginning, I said that a function is usually an algebraic expression, but doesn't have to be. It could just as well be defined by a table, like this:

$x$ $f(x)$

$-2$ $2$

$-1$ $0$

$0$ $-2$

$1$ $-4$

$2$ $-6$

The function defined by this table is only defined for the specific arguments listed; that is, the integers between $-2$ and 2. $f(10)$ has no meaning, and neither does $f(0.5)$.

A function can also be defined in words, like this: ``For any real number $x$, let $f(x)$ be the next biggest integer." For example, $f(4.66)=5$. You can then use $f(x)$ in discussions.

Some algebra books say that a function can have only one value for any given argument, or else it technically isn't a function. For example, if $f(x) \equiv \sqrt{x}$, then for $x=9$, $f(x)=3$ and $-3$. They would say $\sqrt{x}$ is an expression, but not a function. This is a matter of words, not mathematics. Almost every book after algebra doesn't bother with this kind of hairsplitting. You might say that $f(x)$ is multi-valued, but you'd still call it a function. If you want to get technical, you can say that $f(x)=\sqrt{x}$ is not a one-to-one mapping.

## Saturday, January 10, 2015

## Friday, January 9, 2015

### The Dope Filter

Mario Cuomo died this week, so he was in the news. Something that caught my eye was the following thing he said: "You're telling me that the Mafia is an organization, and I'm telling you that's a lot of baloney."

Now, Mario Cuomo was obviously a very able and intelligent person, so I can't imagine he really believed that. But was he lying? For normal people the answer would be yes. For politicians, though, the answer is more complicated.

I believe Cuomo was employing what you might call the "dope filter". Good politicians know how to use it. The dope filter is what you pass your real beliefs through before stating them for public consumption. Cuomo certainly knew that the Mafia is a real organization, but that most Italians are law-abiding. But he also knew that many voters can't understand that both of those statements can be true at the same time. For them, it's one or the other.

Given the limited ability of many people to deal with subtlety, Cuomo chose the path of least harm. He said the Mafia wasn't an organization not because it was true, but to preserve the fact that most Italians are law-abiding.

Most average people don't really care enough about the Mafia to get upset if some politician pretends it doesn't exist. The ones who do can be divided into two groups: Italians and "law-and-order" types. There is of course some overlap between the two groups, but only the "law-and-order" types would take issue with what Cuomo said. And when they did, he could easily deflect it by constructing some weird definition of an organization that the Mafia doesn't meet, or say that he really meant they have five organizations in New York and others in other cities, and they aren't all masterminded by one guy, so therefore it isn't "an organization", or other weaselings.

Don't get the idea that I am looking down on dopey voters. (Well, I am a little.) If you ask me why the Columbia burned up, I'm just going to say the heat shield was damaged by a piece of ice. I'm not going to get into the fact that this had happened many times previously, so the real cause was a flawed means of assessing the damage, blah blah blah, because it's just going to obscure the basic facts.

Sometimes I worry that people are using the "dope filter" on me. I'm a very literal person, so I don't catch on to body language and tone of voice as well as other people. I can imagine that people have to adopt a special, unusually direct way of talking when they talk to me, for fear of not communicating. They have to turn down the bandwidth a little, so there are things they can't communicate to me that they might be able to communicate to others.

Now, Mario Cuomo was obviously a very able and intelligent person, so I can't imagine he really believed that. But was he lying? For normal people the answer would be yes. For politicians, though, the answer is more complicated.

I believe Cuomo was employing what you might call the "dope filter". Good politicians know how to use it. The dope filter is what you pass your real beliefs through before stating them for public consumption. Cuomo certainly knew that the Mafia is a real organization, but that most Italians are law-abiding. But he also knew that many voters can't understand that both of those statements can be true at the same time. For them, it's one or the other.

Given the limited ability of many people to deal with subtlety, Cuomo chose the path of least harm. He said the Mafia wasn't an organization not because it was true, but to preserve the fact that most Italians are law-abiding.

Most average people don't really care enough about the Mafia to get upset if some politician pretends it doesn't exist. The ones who do can be divided into two groups: Italians and "law-and-order" types. There is of course some overlap between the two groups, but only the "law-and-order" types would take issue with what Cuomo said. And when they did, he could easily deflect it by constructing some weird definition of an organization that the Mafia doesn't meet, or say that he really meant they have five organizations in New York and others in other cities, and they aren't all masterminded by one guy, so therefore it isn't "an organization", or other weaselings.

Don't get the idea that I am looking down on dopey voters. (Well, I am a little.) If you ask me why the Columbia burned up, I'm just going to say the heat shield was damaged by a piece of ice. I'm not going to get into the fact that this had happened many times previously, so the real cause was a flawed means of assessing the damage, blah blah blah, because it's just going to obscure the basic facts.

Sometimes I worry that people are using the "dope filter" on me. I'm a very literal person, so I don't catch on to body language and tone of voice as well as other people. I can imagine that people have to adopt a special, unusually direct way of talking when they talk to me, for fear of not communicating. They have to turn down the bandwidth a little, so there are things they can't communicate to me that they might be able to communicate to others.

## Monday, January 5, 2015

### We Who Write Bad Books

I read some interesting quotes by George Kennan, the diplomat and historian, so on impulse I bought his diary and a short book he wrote called

Well, I figured wrong. The diary was so whiny that I wished I hadn't ordered the book, but it was already done. I understand that a lot of diaries are whiny, but you shouldn't publish the ones that are. One thing I did learn was why the military has always been uncomfortable with the State Department. Guys like Kennan, Dulles (both of them) and James Jesus Angleton were so literate and glib that they could say one thing and then convince you they'd meant the exact opposite and that it was your fault for misunderstanding.

The book was better than the diary, but man, was it long-winded. He takes about four pages to say that nationalism is good as long as you don't take it too far. You know how Strunk and White say to omit needless words, like "at this juncture it is perhaps not inappropriate to point out that"? He must have skipped that chapter. I've come to dislike that sort of excessively precise writing, like "He had had a serious operation" instead of just "He had a serious operation". If you need to support your ideas with so many props and ornaments, maybe you had better consider whether they are good ideas. I think I mostly agreed with Kennan's politics. I think.

Come to that, I've lost interest in the "magnum opus". I used to think it would be cool to write a magnum opus; you know, put down on paper the sum total of your views on some topic of importance. But why? The theme of a magnum opus isn't the ideas, it's the author. And let's face it, most of us aren't that interesting, except to people who know us personally. If you want to write a magnum opus, you should give it away free to your friends and family, not charge a stranger money for it.

The technical equivalent of the magnum opus is the textbook or handbook. When I see a textbook with a grandiose title like

I got drafted into writing a chapter of one of these magisterial handbooks about five years ago. The trouble with handbooks is that they usually can't get the best people to contribute, because the best people are too busy to write handbook chapters. I happened to be in a slow spot in my career where I had some free time and was looking for something to do, so I finished my chapter right away, and the editors used it to get the whole project greenlighted for publication. You can guess that all the big shots who were supposed to write the other chapters took a lot longer, and in fact the book is still not done. At this rate, the material will be obsolete before it ever hits the presses. If you think you need a handbook, my advice is to find a good review article written by a bona fide expert.

*Around the Cragged Peak*, which was sort of a grab bag of his personal musings on politics, foreign relations, culture and whatnot. I figured anything he wrote would be worth reading, because he lived so long (101 years) and was involved in so many momentous events of the Cold War, an era I've always taken an interest in.Well, I figured wrong. The diary was so whiny that I wished I hadn't ordered the book, but it was already done. I understand that a lot of diaries are whiny, but you shouldn't publish the ones that are. One thing I did learn was why the military has always been uncomfortable with the State Department. Guys like Kennan, Dulles (both of them) and James Jesus Angleton were so literate and glib that they could say one thing and then convince you they'd meant the exact opposite and that it was your fault for misunderstanding.

The book was better than the diary, but man, was it long-winded. He takes about four pages to say that nationalism is good as long as you don't take it too far. You know how Strunk and White say to omit needless words, like "at this juncture it is perhaps not inappropriate to point out that"? He must have skipped that chapter. I've come to dislike that sort of excessively precise writing, like "He had had a serious operation" instead of just "He had a serious operation". If you need to support your ideas with so many props and ornaments, maybe you had better consider whether they are good ideas. I think I mostly agreed with Kennan's politics. I think.

Come to that, I've lost interest in the "magnum opus". I used to think it would be cool to write a magnum opus; you know, put down on paper the sum total of your views on some topic of importance. But why? The theme of a magnum opus isn't the ideas, it's the author. And let's face it, most of us aren't that interesting, except to people who know us personally. If you want to write a magnum opus, you should give it away free to your friends and family, not charge a stranger money for it.

The technical equivalent of the magnum opus is the textbook or handbook. When I see a textbook with a grandiose title like

*Fracture: An Advanced Treatise*, I just want to run in the other direction. Geezus, a treatise? The worst one is*The Structure of Evolutionary Theory*by Gould. The fact that he didn't just call it*Evolution*should be a warning to you.I got drafted into writing a chapter of one of these magisterial handbooks about five years ago. The trouble with handbooks is that they usually can't get the best people to contribute, because the best people are too busy to write handbook chapters. I happened to be in a slow spot in my career where I had some free time and was looking for something to do, so I finished my chapter right away, and the editors used it to get the whole project greenlighted for publication. You can guess that all the big shots who were supposed to write the other chapters took a lot longer, and in fact the book is still not done. At this rate, the material will be obsolete before it ever hits the presses. If you think you need a handbook, my advice is to find a good review article written by a bona fide expert.

## Saturday, January 3, 2015

### I Stole A Bike

Once upon a time, I stole a bike. Sort of.

It was a weekend morning in the spring of 1990 and I was walking from my apartment on East Norwich Avenue in Columbus to the Ohio State football ticket office, which I believe was in French Field House at the time. Spring is when you buy your discounted student football ticket at Ohio State, and in those days, you had to do it in person. I was almost there, around some bike racks near some dormitories, in the area now occupied by the College of Business.

That's when I saw it - a bike I had to have. It wasn't even locked to the rack, and it was so early there was hardly anyone around. You certainly didn't have to worry about security cameras in those days. So I hopped on the bike and took off. I had it back at my apartment in just a few minutes. Then I walked back to the ticket office and bought my ticket. I figured it was a little too soon to be riding around that same area on the bike.

You know there's going to be more to the story. The reason I had to have that particular bike was that it was in fact my bike. Or, it had been my bike, depending on your point of view. It was a beat-up 10-speed I'd had since high school and had brought down to campus the previous spring. I'd stored it, chained and locked, on the bike rack in front of Taylor Tower where I lived.

During the two- or three-week break between spring quarter and summer quarter, I'd left the bike on the rack in front of Taylor, but when I'd come back to campus for summer classes, it was gone. I had heard there was a campus rule against leaving bikes on the racks during breaks, so maybe the campus police cut the chain and took it away. But it's not like there was a sign to that effect. It's also possible that someone just stole it. At that time all over campus you'd see front wheels chained to racks but with the rest of the bike stolen, because people had run the chain only through the wheel and not the frame of the bike. Or, at the kind of rack where you set the front wheel into a big slot, some idiot would lean on the bike until the wheel bent, so there were these old, rusty bent front wheels all over the place.

I also seem to remember that the campus police would sell bikes they'd confiscated for not having a sticker or whatever, so maybe someone had paid the police for my bike. But I might be remembering incorrectly about that.

Anyway, someone had my bike for almost a whole year. The day I was going to the ticket office, I spotted the bike and knew it was mine, not just someone's similar bike, because there was a rip in the seat that I'd fixed with some of that sneaker repair goo that comes in a tube. It was definitely mine. I had the bike all that summer, but not long after, it got stolen again, this time for good. I didn't have another bike until spring of the next year, when my girlfriend (wife-to-be) bought me a nice mountain bike for my birthday. Wisely, I chained the bike frame itself, not the wheel, to the rack outside my apartment, but they just stole the wheel. Eventually someone came with a bolt cutter and took the rest of the bike, too. It is as if people considered bikes to be communal property.

So, overall I left Ohio State down two bikes. But it felt good getting the first bike back for a while. It halfway felt like stealing, so it was kind of exciting when I took off on it, but how can you steal your own bike? Just because I hadn't seen it in a year didn't make it any less my bike, did it? Who the hell knows.

It was a weekend morning in the spring of 1990 and I was walking from my apartment on East Norwich Avenue in Columbus to the Ohio State football ticket office, which I believe was in French Field House at the time. Spring is when you buy your discounted student football ticket at Ohio State, and in those days, you had to do it in person. I was almost there, around some bike racks near some dormitories, in the area now occupied by the College of Business.

That's when I saw it - a bike I had to have. It wasn't even locked to the rack, and it was so early there was hardly anyone around. You certainly didn't have to worry about security cameras in those days. So I hopped on the bike and took off. I had it back at my apartment in just a few minutes. Then I walked back to the ticket office and bought my ticket. I figured it was a little too soon to be riding around that same area on the bike.

You know there's going to be more to the story. The reason I had to have that particular bike was that it was in fact my bike. Or, it had been my bike, depending on your point of view. It was a beat-up 10-speed I'd had since high school and had brought down to campus the previous spring. I'd stored it, chained and locked, on the bike rack in front of Taylor Tower where I lived.

During the two- or three-week break between spring quarter and summer quarter, I'd left the bike on the rack in front of Taylor, but when I'd come back to campus for summer classes, it was gone. I had heard there was a campus rule against leaving bikes on the racks during breaks, so maybe the campus police cut the chain and took it away. But it's not like there was a sign to that effect. It's also possible that someone just stole it. At that time all over campus you'd see front wheels chained to racks but with the rest of the bike stolen, because people had run the chain only through the wheel and not the frame of the bike. Or, at the kind of rack where you set the front wheel into a big slot, some idiot would lean on the bike until the wheel bent, so there were these old, rusty bent front wheels all over the place.

I also seem to remember that the campus police would sell bikes they'd confiscated for not having a sticker or whatever, so maybe someone had paid the police for my bike. But I might be remembering incorrectly about that.

Anyway, someone had my bike for almost a whole year. The day I was going to the ticket office, I spotted the bike and knew it was mine, not just someone's similar bike, because there was a rip in the seat that I'd fixed with some of that sneaker repair goo that comes in a tube. It was definitely mine. I had the bike all that summer, but not long after, it got stolen again, this time for good. I didn't have another bike until spring of the next year, when my girlfriend (wife-to-be) bought me a nice mountain bike for my birthday. Wisely, I chained the bike frame itself, not the wheel, to the rack outside my apartment, but they just stole the wheel. Eventually someone came with a bolt cutter and took the rest of the bike, too. It is as if people considered bikes to be communal property.

So, overall I left Ohio State down two bikes. But it felt good getting the first bike back for a while. It halfway felt like stealing, so it was kind of exciting when I took off on it, but how can you steal your own bike? Just because I hadn't seen it in a year didn't make it any less my bike, did it? Who the hell knows.

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