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.