On Collisions: Math and Colloquial Speech

So I was remembering this time in, say, fourth grade when the teacher posed the following question:

How many numbers are between ten and twenty?

Of course, this problem is well within the skills of a fourth-grader. Simply subtract ten from twenty and you get ten. In fact, the only appreciable difficulty is in the interpretation of the question. Students, particularly children, struggle with these so-called “word-problems”: problems where the math is relatively simple, but the problem is phrased as a question. You know, using words and stuff.

Being who I am, I completely failed at understanding the question; I answered, “Eleven.”

So I got laughed at… But to this day I maintain that 11 is just as accurate as—indeed more accurate than—10. Why? It all lies in the interpretation of the word “between.” If “between 10 and 20″ means

Then yes, there are 10 such numbers: . But who the hell uses “between” to mean “including the upper limit but excluding the lower limit”? In my opinion, there are only two reasonable ways to interpret “between ten and twenty”:

1. 
2. 

But then the correct answer is either or . I chose option (1) in fourth grade, but option (2) is perfectly reasonable.

The problem is that this becomes no longer strictly a subtraction problem. What the teacher wants is for the students to compute . But this is wrong, so how do we reinterpret the question?

I propose the following: first consider the question, “what are the numbers between ten and twenty?” This question still has the ambiguity of the word “between,” but no person in his right mind would answer:

or

Then we can ask, after the set of numbers has been identified, “how many of them are there?” I think if the problem were posed this way, no one would ever arrive at an answer of eleven (unless they miscount).

Further Rambling

I’ve decided that word problems like the ones we’re given in grade school don’t really count as “math.” I think they’re physics. Consider the topics covered in college- and higher-level mathematics. They’re abstract and almost completely disconnected from the real world. A mathematician doesn’t care if we live in Euclidean space or a Minkowski space; if they can prove something about both, then that is interesting.

A physicist (or statistician maybe) concerns himself with questions like these word problems—problems dealing with possible real-world scenarios. When we answer such questions, we don’t really learn a lot about math; rather, we learn about the connection between math and the world we live in.

That’s not to say word problems aren’t good or helpful, but I think school teachers should be very careful both in posing questions and in receiving answers. What if my teacher had let me explain my logic? It would have been nice to shut up the kids who laughed at me for a “stupid” answer, and they might learn something at the same time.

Borders Is Lame

All summer I’ve been aching for something good to read. A few days ago, when the “aching” began to cause actual physical pain, I caved and began browsing the web for a book I might like.

I decided that I wanted a collection of short stories. I decided this for a number of reasons. First, short stories are short. I have what probably counts as A.D.D., and it can be tough for me to sit through a old, dry book (I’m looking at you, Great Expectations).

The way I see it, reading a few short stories is like watching an episode of The Simpsons, a couple of Family Guy, and maybe some Arrested Development or something. A long novel is like watching Cassablanca.

Also, short stories allow authors to tackle subjects that can’t quite carry themselves through an entire book. In particular, authors often write short math- or science-related stories. One of my favorite stories, The Library of Babel, is one I read years ago. I wanted a collection of stories like that.

My googling eventually brought me to the book Einstein’s Dreams which, although not quite a “collection” of short stories, is pretty close to what I wanted. The only semi-competent bookstore near me is Borders, so I searched their website with my zip code. I was told this:

WTF?

“Likely in store”? What the hell does that mean? Shouldn’t the Borders cashiers be competent enough to scan each book that gets sold, and shouldn’t the computers be competent enough to update the store’s inventory?

I realized, as you probably have, that “likely in store” is their way of telling us that if we can’t find a book in the store then they aren’t responsible. Of course, that big, red button that says “reserve in store” indicates that they have it.

So I went to Borders. Just for la-la’s, I ran the same search on the in-store computer and it told me the same thing. OK, I’ll go find it. Hmm… Literature… Fiction… Fiction/Literature… Aha! Alan Lightman (the author)!

Guess what. No Einstein’s Dreams. I asked a “sales representative” to help me find the book. He heroically took me over to the Fiction/Literature section, failed to find the book, and informed me that the book was not there.

“But the computer says ‘likely in store’!” I protested.

“Yeah, but likely doesn’t mean definitely.“

It took all of my will power to keep myself from telling him to go fuck himself sideways. All the damn “likely in store” label does is give the Borders employees an excuse to not find a book that was either misplaced or somehow lost.

Then again, I guess this is why Amazon is so successful.

Passive Aggression

For one of my final papers, I have the privilege of writing about Immanuel Kant’s Prolegomena to Any Future Metaphysics. Each core humanities class, such as the one I’m taking, comes equipped with a graduate student “writing intern” who reads and sometimes grades our papers.

My writing intern always insists that we send her digital copies of our papers formatted as “.doc” What she doesn’t understand is that there are quite a few different formats that use the .doc extension. Clearly what she wants is something with MIME type application/msword, but that’s not what she asked for. Furthermore, I resent the idea that she is restricting me to a proprietary format.

Therefore I am not writing my paper in MS Word. I instead am writing it in LaTeX, typsetting it with TeX, and will use the no-longer proprietary application/pdf format. Of course, I don’t want to disobey her request, so I will copy and paste the LaTeX into an MS Word file and send her that. Here’s part of one of the paragraphs she (a divinity school graduate student — not a techie) will have to read:

“All our intuition,” he says, happens only by means of the senses.”\footnote{\S 13 Note II} The basic intuitions we possess are all concepts which relate to the appearances of objects. Kant mentions “extension, place, and more generally space”\footnote{\S 13 Note II} as examples. In fact, he says that representations of space and time exist \emph{only} in intuition,\footnote{\S 12} and this intuition of appearances, or \emph{intuitive sensibility} is the only way through which we can understand or know about an object.

Hopefully in the future she’ll be less of a bitch.

The Thingness Of A Thing

A hamburger -- the object of our scrutiny

In the vein of Martin Heidegger, it is my purpose today to discuss the thingness of a thing. In particular, I wish to examine hamburgerness and hamburgers. In this post I will attempt to enumerate the qualities and characterstics related to hamburgerness, and I will use these components to strictly define a “complete hamburger.”

To begin, let us consider first the properties an object must possess in order to be considered a hamburger at all. There are some obvious ones: extension in at least three physical dimensions (I say at least because there may exist 4-dimensional “hyperburgers” and so forth), it must have mass, and it must be made of meat. We can generalize these ideas by saying a hamburger is a patty of some meat.

This defintion includes the properties of extension and mass because we assume them inherent to the idea of meat. Indeed, this is quite a useful definition for a hamburger: we have not specified the degree to which a hamburger must be cooked, nor have we specified what sort of things ought to be eaten with a hamburger. Therefore, a hamburger is an independant entity that can come in many forms.

The Hamburgerness of a Hamburger

With our definition in hand. let us consider the properties that in some way modify or affect the hamburgerness of a hamburger. For example, if a hamburger is placed between two slices of bread or between two buns, we consider this to be more hamburgerly than a burger lacking these qualities.

Therefore we can declare that the presence of certain things improves the hamburgerness of a hamburger. Similarly, we can conclude that the absence of these things impairs the hamburgerness. I have come up with what I consider to be the complete set of properties which, when absent, impair the hamburgerness of a hamburger:

• Buns
• Lettuce
• Ketchup
• Pickles

Note that the meat patty itself is excluded from this list. It is excluded because if a hamburger is lacking a patty then it is simply not a hamburger. Also note that we now have a full, rigorous definition of a “complete hamburger.” That is, a complete hamburger is a hamburger that possesses each of the elements of the above list.

If a hamburger is lacking, say, pickles, then I assert that it is less of a hamburger, or less complete than a hamburger which possesses pickles.

Superburgers

Imagine, if you will, a hamburger that possesses the additional property of having cheese on it. Such a hamburger (referred to colloquially as a “cheeseburger”) is something that I call a “superburger.” A superburger I define as follows:

A superburger is a hamburger that is complete and posseses an additional property that also lends to its hamburgerness.

A complete superburger

Clearly a cheeseburger is a superburger because a cheeseburger is just like a hamburger but better. However, a complete hamburger smothered with fish oil is not a superburger because the presence of the fish oil does not increase the hamburgerness—it impedes it.

Also worthy of mention is the incomplete-superburger hybrid. Such a hamburger is one that lacks an essential characteristic such as ketchup, but possesses a super characteristic like barbecue sauce.

Additional Considerations

So far we have considered only properties of hamburgers that are inherent to the hamburger itself: composition, mass, extension, &c. Let us now consider properties that may have their impetus in some source external to the hamburger. For example, let us consider the relative velocity of a hamburger.

Is a hamburger that is falling with respect to its observer any more or less of a burger than one that is stationary with respect to its observer? The obvious answer is no, but let’s not jump to conclusions. Certainly a moving hamburger still looks like a hamburger. Indeed, its inherent characteristics (mass, etc.) are in no way changed. But something else is changed: the edibility of the hamburger.

A bad, incomplete hamburger

But edibility is not on the list of inherent hamburger attributes! And it’s not that I have made an egregious omission. Rather, we have identified a new sort of characteristic. That is, there are some characteristics that are not inherent to a hamburger, and therefore not essential to its hamburgerness nor its completeness. These attributes which rely on the relationship and interaction the hamburger has with its observer, like relative velocity, we can call characteristics that are essential to the goodness of a hamburger.

For example, a well-cooked, complete hamburger that is stationary with respect to its observer is a good hamburger. One that is raw, smelly, and moving rapidly towards its observer is a bad hamburger.

A Final Question

What is the relationship, if any, between the completeness of a hamburger and the goodness of a hamburger? I will discuss this more fully in a later work, but my conclusion is as follows:

A hamburger must be complete to be good, but it mustn’t necessarily be good to be complete. That is, there exist bad hamburgers that are nevertheless complete.

The Height Of Materialism

Materialism, thy name is SKINNY LONG LENGTH SCARF. For only £90 ($130 USD) you can be the proud new owner of the same scarf everyone in the entire city of Chicago owns.

I don’t actually mind people all owning the same thing. In fact, it’s a pretty nice scarf. It’s cashmere, it’s good-looking, and it’s Burberry. My problem with it is that it’s one hundred and thirty dollars.

Sure, cashmere’s a nice, expensive fabric, but there’s so little cloth in a scarf that you’re basically paying $50 just to have it made by Burberry. And actually, I don’t even have a problem with that. Everyone, including me, spends a little extra money for brand name recognition.

The reason I’m angry is that people will by the $130 scarf, and that’s all. If you’re fantastically wealthy, you don’t mind dropping $130 for a scarf and another $400 for a sweatshirt. But most people simply can’t afford to do that. So what do common folk do? They buy the $130 scarf, then walk around wearing $30 jeans from American Eagle and a $25 polo shirt from Gap.

The message you send when you look like this is not “I have a nice scarf.” There is absolutely no reason to wear that scarf except for the brand, so the message you send instead is “I am shallow and materialistic.”  Rich people are excluded here because it is cashmere, so if you can afford such things, then whatever.

Basically, if you can’t afford to buy Burberry clothing, then don’t. If all you can afford at Burberry is a $130 scarf, then go somewhere else and buy a similar one for $10.

Scratch That

I do have something to say. Did you watch the Superbowl? Did you hear that public service announcement? The one that went like this:

One in three kids are obese

Are you kidding me? How the fuck do you get an advertisement during the Superbowl without being able to pay for someone to check your grammar? One in three kids is obese. Jesus Christ, our Lord and Savior, save us now.

I Don’t Have Anything To Say

…But I’ve noticed that frequently updating my site yields more page views, and I’m a sucker for statistics.

Kuviasungnerk/Kangeiko

I’d write something witty and interesting, but I’m too tired.

This is a bad way to get a free t-shirt.

Ender In Exile

I’m not exactly a die-hard Orson Scott Card fan. I’ve read a few of his books, but none of them really made much of an impression on me except for Ender’s Game. Of course, sci-fi addict that I am, I always look forward to any excuse that allows me to spend a couple hours wrapped up in blankets and reading something about aliens. So I picked up Ender In Exile.

I wouldn’t go so far as to say Ender In Exile is bad, but I get the feeling that Card has exhausted his literary creativity and is now using fiction as an outlet for social commentary. I’m reminded somewhat of Ayn Rand: the plot and characters are disposable. Only the political and social philosophy really matters.

This is an irritating trend that I’ve seen in other authors recently. Most notably, Neal Stephenson’s Anathem was, though interesting, basically an 800-page explanation of his theory that there are multiple universes. If I want to learn about the universe, I’d rather go to class.

Characterizing Ender In Exile in this way is a little unfair, though. Card’s stories are never really disposable, and his social commentary, blatant as it is, is held pretty well in check. That and his social commentary is actually pretty interesting. But just as a story, Ender In Exile is no more than an expansion of the last 10-or-so pages of Ender’s Game. There aren’t any striking revelations.

I suppose if you actually read the series in the proper order — not the order in which they were written — Ender In Exile’s existence is more justifiable. In short, I’d say it’s a fun read, but it’s not worth overnighting it from Amazon.

The Worst Thing About Identity Theft

Identity theft seems to be the number-one thing to be scared of in this modern world. Terrorism-shmerrorism, everybody feels safe enough to fly on airplanes, but everyone and his mother seems to be a victim of identity theft.

Not to say that I personally am a victim (my mother isn’t either). What really bothers me is not that there’s a chance some Finnish guy might steal my credit card number, but that it takes me like six and half hours to buy anything worth more than $30.

I was out shopping today, and I tried to, you know, buy stuff with a credit card. The limit on said card is like $25,000 and I was spending a couple hundred so I figured I’d be OK. I wasn’t. The card wasn’t exactly rejected: the cashier just had to call the card company and tell them I was who I said I was. Of course they didn’t believe the cashier, so they had to talk me. Our conversation went something like this:

Credit Card Guy: Hi, are you <my full name>?

Me: Uh-huh

CCG: Yeah, I want to confirm some past purchases. Did you spend twenty dollars at a gas station yesterday?

Me: Uh-huh

CCG: Ok, two weeks ago we rejected your card three times because it was being used in Ukraine.

Me: I was in Ukraine two weeks ago.

CCG: And two days ago we rejected it for being used out of state.

Me: Yeah, that was me trying to buy food on the way back from the airport.

CCG: So this isn’t a concern to you?

Me: My concern is that I can’t buy anything with this card. Can you approve this transaction so I can get my pants and go home? Please?

CCG: <mumbles something>

Basically, people are so paranoid of having their identities stolen that they’re willing to put up with the most annoying crap when they legitimately want to buy something. “Oh come on,” you might say, “it’s a lot worse having your identity stolen than having to deal with credit card protection.” You may very well be right. I’m not really angry at the credit card companies and banks for trying to protect their customers. I’m mad at the stupid thieves who get their ha-ha’s from stealing other people’s private information.

So I say to you so-called identity thieves: fuck off and find a less annoying way of stealing money.