The course that was and is...

There are many challenges involved in providing a web based class, and this one is no exception. While overall the class was delivered effectively, there are a few things which could be done better or perhaps omitted in the future.

One of the things I used most often in this course was the group wiki. While it provides an excellent platform to discuss ideas, it is at times difficult to use due to bugs in its programming. The benefits of the wiki outweighed its problems, since for collaborative efforts it works quite well. The calendar was also helpful, but it could be refined. Specifically events which spanned multiple days sometimes would overlap other events and it was difficult to see that multiple assignments were going on at the same time. I liked the announcements via email and I think that an option to notify a user by email with deadlines would be advantageous to this sort of class.

There were other parts of this course which were less than satisfactory, but did not interfere too much with accomplishing the assigned tasks. The "podcasts", although novel, did not add significantly to the course, as all the material in the podcast is available in other portions of the website. The forum is an important part of the feedback portion of this course, however it would have been more effective had there been more possibilities of keeping the information on one page. The calibrated peer review is also a good part of the course, but in order for it to reach its full potential, it needs slight adjustment. The text submission portion at turnitin.com and then transferring this to CPR is a hassle which could be avoided by choosing to either do all work though CPR or turnitin.com. Choosing both is overkill.

The worst thing about this course was the initial setup. The sheer amount of login information required was daunting. To make things worse, the usernames on some of the websites were pre-chosen and so keeping up with more than one username may pose a problem for some people. Unfortunately I see no way to fix this totally given the diversity of assignments.


Although there are many facets of this course which could be improved, it is a good course. I would certainly take another course like this and suggest that others take web based courses as well. The assignments were interesting and informative. My favorite part of the course was the group portion as it broke pace with the other business oriented assignments. As web courses in general become more developed, they will increasingly become the place for students to take this course.

The Problem of Zeno: Part II

Last time I promised a formulation of Zeno's Paradox which would make the word "paradox" inadequate to describe the story. The most important things to consider are the definitions and assumptions. Some of these concepts are not easy to understand, so looking them up at Mathworld or on Google may prove to be useful. The key is the difference between "infinitely divisible" and "discrete".

There are many equivalent definitions of discrete, but the one I will use is that given two points in a subset of the real numbers, there are only a finite number of points between them.

Infinitely divisible is nearly the opposite of discrete, that is between any two points in a subset of the real numbers, there are an infinite number of points between them.

Now what does this have to do with Zeno's Paradox? Well everything, in my view. There is one very important concept one needs to be familiar with in order to understand this discussion and that is the idea of a function or map. This is simply an association of two points in a set, for instance if you have a map from time into space, you associate a time, like 2:00 PM Tuesday, to a place, such as a grocery store.

The tortoise implicitly assumes a few things, first that he is in one spot at any given point in time, so in terms of his mapping from time into space this means that the map is "well-defined". Since this is a race its also assumed that the participants are always moving forward, but this can be done without losing generality since even in a complicated path there will be periods where motion is only in one direction for a period of time. What this means for our hero and tortoise is that they have two maps which describe their locations during the race which are injective, that is one point in space is associated to one point in time. The tortoise also assumes that every time Achilles moves a certain period of time passes. From his argument it seems that he assumes that there is a smallest period of time which cannot be divided, which translates into the story as Achilles never catching up to him. Therefore this can be viewed as the assumption that time is discrete. With this in mind, the tortoise assumes that space is infinitely divisible since every point Achilles has to catch up to lies between his previous point and the tortoise. Now we are ready to discuss the proof of Zeno's Paradox. The fact he can always pinpoint the time Achilles reaches these points indicates the assupmtion that the map from space into time is also injective, that is there is a point in space associated to every point in time. This makes the map surjective. Note the major things revealed so far:

Time is discrete.

Space is infinitely divisible.

The map will be well-defined, injective and surjective and hence bijective.

One more definition is necessary, that of "common motion". We said nothing about our map from time into space other than it is well-defined and injective, but this definition would include all sorts of things like bouncing around the universe from one galaxy to another instantaneously. This will not do, so in order to "keep it real" so to speak, call the set of maps which represent a finite distance traversed in a finite time "common motion", i.e. something that we may see in real life. Now this includes superluminal travel, which is certainly uncommon nowadays, but for our discussion this will be a non-issue. Zeno's Paradox is now:

If space is infinitely divisible and time is discrete the set of common motion is empty, that is there are no paths which traverse a finite distance in a finite time.

Proof: Since the set common motion must consist of injective maps, we should be able to construct at least one. Assume there is a bijective map from time into space. The problem with this is that there are not enough points in time to put all the places from space. There are an infinite number of spatial coordinates which we are trying to fit in a finite number of time coordinates. This is what is known in mathematics as the "Pigeonhole principle", that is with five boxes and six oranges, if one decides to put all the oranges in the boxes, there will be at least one box with two oranges. Therefore there is no bijective map in the set of common motion. Q.E.D.

That's the argument, I hope it convinces you that under this formulation, Zeno's Paradox is indeed a theorem.

The Problem of Zeno: Part I

There is an argument called Zeno's Paradox which shows unequivocally that motion cannot exist. Hold on, you say, I can move about quite nicely, so how can it be that motion does not exist? Consider this post and see for yourself. Achilles the hero and the tortoise were very good friends in Zeno's day. Achilles always had the upper hand when it came to physical toughness, but the tortoise had a very crafty mind. One day, Achilles challenged the tortoise to a race. Achilles was so sure that he could beat the tortoise, he offered to give the tortoise a head start. The tortoise then politely let Achilles in on our paradox. He said that if Achilles were to give the tortoise a head start, he would never pass the tortoise and thus the tortoise would win the race. Achilles thought this was foolish and demanded an explanation from the tortoise. His explanation may have gone something like this.

"Now Achilles, is it not true that when the race begins, I will be some distance ahead of you?"

"Yes that is true, for I will give you a head start."

"Indeed, but before you can pass me, you must first pass the point where I start the race, correct?"

"Yes, this is true."

"But in that time, I will have moved forward."

"If you intend on participating in the race!"

"Quite. However, in order to pass me, you will then have to reach my new position."

"Agreed."

"But by that time, I again will have moved forward some distance."

(It is at this point that Achilles begins to realize that he will not win the race.)

"But-"

"And so you see Achilles, you will never pass me, because you will have to reach a point I was at previously before passing me."

Begrudgingly, Achilles responded, "Tortoise, you are correct, I cannot win if I give you a head start. I concede the race."

Examine this argument, try to understand what is going on here. See how the tortoise wins. Is there a flaw in his argument? If so, what is it? And if there is a flaw in his argument, why did Achilles miss it? I will publish my interpretation in my next post two weeks from now. In the interim, I am presenting a talk for Student Research Week at Texas A&M discussing this subject. It will be Wednesday the 29th around 9:00 AM in Rudder tower. I will present my interpretation in the hopes of showing, under a careful formulation, that this is not a paradox after all, but a genuine theorem of mathematics.

The Peculiarities of Podcasting

Podcasting is a new buzzword which has been floating around the internet since 2004. To many, or at least to me, this word may initially sound like a marketing tool for Apple's iPod. While this is not totally false, this use of the word does not aptly describe what podcasting has become. In Wikipedia a podcast is described as a web feed of audio or video files placed on the Internet for anyone to download or subscribe to, and also the content of that feed. (The feed portion generally refers to RSS which is an XML file format which allows computer programs to read sites which change often and synchronize with them accordingly.) A few misconceptions about podcasting are that you can only listen to a podcast on an iPod and podcasting is just a new word for streaming audio. I was duped by the first one until quite recently, but it is true that an iPod is NOT required to listen to podcasts. The other misconception will be addressed below. The popularity of podcasting has grown considerably in the last year which many people attribute to the freedom podcasts offer. One major difference with podcasting and older forms of audio/visual media served on the Internet is that a user is not required to be online while enjoying a podcast. Podcasts in many ways are more dynamic than these other forms which include streaming internet radio and static, prerecorded files. Podcasts fill a niche that lies somewhere between radio, television and the always accessible internet and for this reason a person can listen to content on his schedule or lack thereof. For instance, on a recent three hour trip I decided to listen to a few episodes of the FrenchPodClass which worked out quite well as it was educational during a time I would normally be doing nothing productive. Clearly, podcasts fit quite well into my schedule, as I'm sure they will fit into yours in the future.

The power of CSS

Cascading Style Sheets is a specification for adding style to HTML documents for viewing on the internet. This standard has been around since 1994 and was one of the first steps in separating data and display in documents, including this one. As a result, CSS is used for the display and arrangement of data on webpages. One of the most impressive websites which displays the power of CSS is CSS Zen Garden. This website has many examples of designs which use the same html file with different sytlesheets. Unfortunately, CSS is not XML and therefore is mostly limited to the web. Other style formats such as XSL which are XML based can be used on many devices more easily than CSS. This is a problem since the recent development of XHTML is moving more towards XML based data structures. While CSS is currently the only option for style sheets on the internet, in the future it may prove to be incompatible with the XML based languages.