The Warped World of a Mathematician

I'm starting to write again...

2006-08-31T22:07:00.000-04:00

... but not here. Check it out at http://www.mathmannick.net

There's not anything new there, right at the moment... but the format's there, and there will be something soon.

Also, it does have an atom feed built in - so if you use Firefox, you can use the Live Bookmarks feature to see if I've posted without loading the page... and if you don't use Firefox, you should probably contemplate starting 8-)

Goodbye, Old Friend

2005-07-21T16:33:00.000-04:00

Rest in Peace, Paws Robert
July 21st, 2005
My faithful friend and ally for well over a decade.
I miss you, buddy.

I just can't believe you're gone.

On Mathematical Method

2005-07-12T22:53:00.000-04:00

In the mathematician's role as a researcher, it is generally the goal to prove as general and powerful of a theorem as possible - it is much more of an accomplishment to have found and proved a formula for the order of the General Linear Group for any n x n matrix, say, than to have calculated the order of the General Linear Group for a 13 x 13 matrix. The second case can be computationally difficult (and in fact it is computationally difficult), but the former reduces it to plugging numbers in to a formula. (I use this example because one of my professors was actually the first to compute and prove the formula for this.) The incentive for powerful results is very powerful - a mathematician's reputation is based largely on the strength of their published results.

However, there are a couple of situations in which general results are either bothersome or impossible. First, an example of the 'bothersome' case:

Much of Topology is designed to link in with intuitive Geometry. Much of elementary Topology is done on topological spaces over the real line, the real plane, etc. The concept of the convergence of a sequence on a topological space matches the concept of convergence of a sequence in general when you are in an Euclidean n-space. However, in general topological spaces, examples can be easily constructed in which this is not true. For example, a topology can easily be defined on the set
{1, 2, 3} for which all sequences converge to 2 - including the sequence 3, 3, 3, ... Obviously, this is still a valid result as defined, but it is not intuitive. As such, many results in Topology are proven under the assumption that the topology forms a special topological space, called a Hausdorff Space (after the mathematician Felix Hausdorff), in which many results are much more intuitive. This clearly makes results less general - there is a restriction on the topological spaces to which it applies, in this case - but it makes it more useful.

Another situation in which generality can be abandoned is the obvious - when it isn't possible. Many examples of this can be found in the field of Real Analysis. Often, when proving results, Analysts choose to make specific assumptions about the function that they have to work with. Very few results can be proven with regard to all possible functions (some of them are quite strange), but more is possible with a function with the property of weak convergence. If the theorem in question still cannot be proved with this assumption, perhaps it can be if given a strongly convergent function, and so on.

So, for mathematicians, it is a constant tightrope-walk - restrictions must be in place to ensure the usefullness, or even the provability, of the theorem in question; but generality must be preserved to ensure its power.

CNN.com - Energy-beam weapons still missing from battlefield - Jul 11, 2005

2005-07-12T09:11:00.000-04:00

Oh, the amusement. I had no idea that energy weapons were this far along in development.

CNN.com - Energy-beam weapons still missing from battlefield - Jul 11, 2005

My First Month as an Independant Adult

2005-06-26T00:53:00.000-04:00

I have now (as it is after midnight) been living in my first apartment for a full month. It is strange to think about, really. The fact that it did not seem like much of a transition is perhaps even stranger. Be that as it may, my last month has been fairly interesting, to me at least; following is a (brief) description of life thus far.

Upon returning to Houghton, I began the next phase of my work in the labs. For those who do not know, I am now responsible for, essentially, all aspects of computing in the math department. I am taking over for both the systems admin and the web admin. In both roles, I have discovered that I really already knew what I would need to know to do my job - I just was not yet aware of it. It has, however, been very rewarding to realize what I'm capable of. The kickstart (automated installation) file for the new Fedora Core 4 installation for the labs is almost complete; it will be done by the time the hardware arrives in the next week or two. Also, my new grading system for the lab courses is being implemented. All grading and submissions will be done through a bar code system. I also wrote scripts that customize Mathematica labs when they are downloaded, allow for administration of the entire system (a full database front-end, basically), and allow for detailed grade reports. We are testing the system for the second summer session, which starts on Monday.

In addition to work, I have been continuing my studies of Mathematics this summer. I am working through Topology by James R. Munkres; essentially, it is the canonical text in the field. Topology, like Group, Ring, and Field Theory, is a branch of Set Theory. However, Topology could more appropriately be called Collection Theory. A collection is a set of sets - a Topology is a special set of subsets of some universal set. As I complete my work, I am posting it at http://www.mathlab.mtu.edu/~nrpeters/munkres/
.I have not been able to work on it for a while, because of a heavy work load. I currently have a PDF up that covers sections 12 and 13, and I hope to have the exercises from sections 14-16 up by Tuesday evening - I only have one more problem to do. It will just be a couple days because I have a physics class which begins on Monday.

Other than that, life is going pretty well right now. I am frustrated by several things, as I suppose is typical of me. I am very lonely, both in the lack of a significant other (which seems to bug me more with each passing day) and in that I don't see people very often in general. Another frustration which people might not expect is more academic in nature.

For an undergraduate, and particularly considering the fact that I am in the summer after my first year, I am fairly accomplished in my science. I am fully capable of doing work on my own at a level most people my age wouldn't consider feasible (Topology, for instance, is typically a first-year graduate school course in a Pure Mathematics program). However, I am still not to the point where I have begun to see areas with research possibilities. Considering the fact that my ultimate goal is to go in to research, its easy to see how this could be disheartening. Additionally, the fact that I prefer not to study on my own topics that I will be studying for courses in the near future limits the amount of work that I can do on my own to get closer to researchable material. I know that I will get there eventually, but that doesn't change the fact that it is frustrating.

Well, seeing as it is now 1:30am in the Upper Penninsula, I think that I will be going to bed. I hope that this finds you well.

DeLay angered by 'Law & Order' mention

2005-05-28T08:31:00.000-04:00

I have to say, I found this so amusing that I just couldn't pass it up. People who know me generally know that I am a huge fan of Law & Order. Well, in the episode this last week, the police were looking for somebody who had murdered a judge. One of the detectives said to put out an APB for "somebody in a Tom DeLay T-shirt." Obviously, of late, Rep. DeLay has been fairly outspoken about his views on the judiciary... And, to be frank, I have considered him to be an idiot the entire time. Well, now he's proven it. He actually wrote an angry letter to NBC and Dick Wolf, the producer of the show.

*dies laughing*
http://www.cnn.com/2005/POLITICS/05/27/delay.law.order/index.html

A Post of a More Personal Nature

2005-05-23T01:53:00.000-04:00

Hello to anybody actually foolish enough to still be looking at this. Due to the craziness of life (between finals, taking over new jobs, goodbyes, moveout, coming home, etc), I have not been able to come up with anything to write about for quite some time. Today, I break from the slight trend I had established, and actually use this as a journal. We'll see how it goes.

I moved back to Bloomington, MN on April 29th, and have been around here ever since. It's been great to see my family, anda great to see my friends (particularly those that I actually still have something in common with)... but I am incredibly ready to get back to my own life. Not that there's any logical reason for it; I get along with my family really well, and I've got a few people around here that I've done quite well at keeping in contact with over the last year. But, it wouldn't suprise me if almost everybody out there understands exactly what I mean.

A brief interjection: Since being back in Bloomington, I've been doing work for the Math Labs at Tech. I've been programming, working on an entirely new grading/information system for the labs. So, I've been doing lots of writing in PHP, HTML, MySQL, and CSS. I've been thoroughly amused, and it has turned out really well. I would post a link, but it requires permission to access it, so it would be an exercise in the worthless. If any of you are employed by the Labs, you'll be introduced soon enough. Also in queue for me for this summer: redesigning the www.mathlab.mtu.edu page, and the www.math.mtu.edu page, in addition to refitting the labs and several other random projects.

Well, back to life. I move in to my apartment (no roommate!) in Houghton this Thursday, after yet another 400-mile stint through the Midwest. I am very excited to get settled in to my apartment. My family (less my brother) is coming up to help me move in, and they'll be in Houghton through the end of the weekend. Then, I'm on my own. I've severely missed Rachel, and Betsy, and Scottie, Steve, Kati, Tim, etc. etc... The first four will even all be in town upon my arrival, which is excellent.

What am I doing this summer? Well, I can only work 20 hours a week for the school (silly regulations), and so I'll have a pretty decent amount of free-time. Hopefully, hanging out with people will be in order. Additionally, I'm teaching myself Topology (out of
Munkres' Topology, a classic text in the field). The second half of the summer, I'm taking a physics course. Lastly, I'll have people up to visit me on occasion, which will be highly amusing... it'll be interesting to see how the crossover between my old life and my new one goes.

All in all, life is going pretty well. I still would like very much to solve the whole issue of being single... and I drive myself crazy on that account. But, It is sadly a solution that I know will not be forthcoming for me, so I can deal with it. I'm just lucky to have good friends, and to have my life progressing along so well. Have a good week, everybody.

Russell's Paradox

2005-04-13T22:49:00.000-04:00

In the beginning of the 20th century, Set Theory was still in its infant stages. At that time, the most commonly accepted version was that of Georg Cantor, known by many as the creator of modern Set Theory. However, Cantor's no-frills, intuitive approach to Set Theory was very susceptible to paradox. The most famous of these was Russell's Paradox, discovered by Bertrand Russell in 1901.

Suppose that you define the set M as follows:
A is a membor of M in case A is not a member of itself.

According to Cantor, M is well-defined. However, one question muddies the waters: Is M a member of itself?

Suppose that it is. Because M is a member of itself, it does not belong to M - a serious contradiction. Alternatively, suppose that M is not a member of itself. Then, it must be a member of the set of all such sets - meaning that it must be a member of M. Another contradiction.

The moral of the story? This is one sure example of why modern mathematics is axiomatic. By defining a strict set of axioms, we can exclude cases that we don't feel should be part of the problem. One of the most famous sets of axioms is the Peano Axioms - the axioms used to generate the Natural number system and guarantee induction.

After several flaws, such as Russell's Paradox, were discovered in Cantor's so-called Naive Set Theory, it was clear that either it or traditional logic had to be revised, and logic was not likely to go anywhere. So, Ernst Zermelo made the first excursions in to Axiomatic (Rigorous) Set Theory in 1908, and it was later modified by Adolf Frankel and Thoralf Skolem. Modern Set Theory uses several axioms, such as the Axiom of Extensionality, the Axiom of Infinity, and the Axiom Schema of Comprehension. For more information on Rigorous Set Theory, visit this Wiki page.

Ponderings of the Infinite

2005-03-31T13:04:00.001-05:00

First off, I'd like to say that I wish I would have updated more commonly (i.e. at all) in recent history. However, I picked a bad time to try to get in to the habit of doing so. We'll see what I can do from here on out. However, I would now like to write out the first mathematical article to be posted in this space. Without further ado, here it is.

∞. Almost everybody believes that they understand infinity, and yet the vast majority are completely and totally incorrect. So what is ∞? There are two different number systems that incorporate the infinite: The "Extended Reals" and the HyperReals.

The Extended Real number system is essentially the standard Real number system, with two symbols attached: ∞ and -∞. The infinite values do not represent numbers; rather, they represent any number greater (or more negative, respectively) than any Real number. Of course, we can make Real numbers arbitrarily large, so it cannot represent any number. In fact, this is the very weakness of the Extended Real system. Because ∞ is not a number, we can not do algebra with it. So, the ordering properties of the Real numbers remain intact.

The Hyper Reals are another extension of the Real Number system that includes the infinite. It begins by establishing the infinitesimal numbers - numbers smaller than any given real number. Essentially, the infinitesimals fill in 'gaps' around the real numbers. Then, we establish the multiplicative inverses of the infinitesimals. The inverse of something smaller than any Real number must be larger than any Real number - infinite. Defining the infinite and infinitesimal numbers in this way gives us one extreme benefit over the Extended Reals - because there are multiple different infinities (in fact, as many as there are Real numbers to begin with) we can perform algebra.

Why would I bother writing this short introduction? Well, the simple reason is this: On a forum a few weeks ago, I saw somebody make the comment that "anything divided by 0 is ∞," and nobody questioned it. Yes, you could say that the limit as the divisor approaches 0 will be infinite (though positive or negative depends on the particular problem).

Dagnabbit.

2005-03-14T21:37:00.000-05:00

Yes, the title about says it all. I had a decently sized post prepared (about Minesweeper, actually... yes, I know) but it and all of my visuals for it are on my laptop...

The very same laptop whose motherboard fried today.

I am only going to be able to access the internet sporadically over the next couple of weeks, as in only when I am in the labs. So, it will be a while.

My appologies, yet again.

An Explanation

2005-03-04T10:35:00.000-05:00

I appologize for my recent silence - the past week has been absolutely insane. I am now actually at home, on Spring Break. I'm working on doing some writing, so there should be something up within the next few days.

Enjoy your breaks, MTU, and everybody else enjoy whatever you may be doing.

Having Secured My Method of Posting...

2005-02-25T19:25:00.000-05:00

I deleted the last post - it was merely a medium in which to experiment with my methods of posting math here. That being done, you can expect that I will make use of it in the near future.

For those who are interested in these methods:
I will be typsetting Mathematical posts (or at least the portions that require it) using LaTeX. In order to achieve the integration of these documents in to my blog, I will use the color package to set the background color to match, and adjust the layout using the textwidth option. I will then convert the result to a JPEG image, and post that image here.

If anybody who reads this has any questions regarding the use of LaTeX for typesetting Mathematics, feel free to ask. That being said, I will be taking part in a presentation on LaTeX at Michigan Tech sometime in the next few weeks. My portion will be specifically about Mathematics documents. More details on that to follow.

On the Questions of Readership and Presentation

2005-02-24T09:11:00.000-05:00

It occurs to me, as I propose to write points of Mathematical interest in this space, that I have a very... widely varied readership with regard to Mathematical knowledge, ranging from clueless, to math minors, to a grad student or two. So, here I shall address what must be the solution, should I wish to post anything of interest.

When I post something that is Mathematical in nature, I will endeavor to supply the background necessary to at least understand what I am saying, if not necessarily all of the nuances of the subject. I hope in this manner to strike a good balance - I will not bore the grad students with background that has become second nature, and I will not (I hope) confuse the other readers.
Additionally, if there is a topic that I have already supplied the background for, I will most likely just provide a link to the post in which I discussed it previously.

That being said, if there is anything that requires further explanation, or even if you just have any interesting questions or propositions, please post them. I am always happy to help.

Additionally, a fairly serious issue with presenting Mathematics in a medium such as this is the limitation of symbolic manipulation. Most likely, to avoid further confusion due to trying to decipher what I mean by things such as S_n as opposed to Sn, I will generate the necessary code in LaTeX and convert it to a JPEG format, and use the images to present my points. It will take a bit of work on my part, but few things that are worthwhile are otherwise.

Adieu for now.

An Interesting Possibility

2005-02-23T12:10:00.000-05:00

There was an email sent to all of the math majors today regarding two possible Undergraduate research positions in Combinatorics and Coding Theory this summer. I've sent an email asking for more information; The position includes a $2400 stipend for 8 weeks of work, plus living expenses apparently. This would be perfect; there is nothing that could be closer to an internship for me. I will undoubtedly post more details as I learn more.

Mid-term TA evaluations were today. Hopefully my students were all nice... I don't have much doubt, though. I would like to think that I do a good job. I believe that I don't get the results of that until the end of the semester, though, so I will not know for quite a while.

Well, that is about everything that I have for news at the moment. I hope that everybody is having a good week thus far.

An Introduction to My World

2005-02-22T00:13:00.000-05:00

As this is something that is, theoretically, accessible to people all around the world, I feel that it is only appropriate that I start it out by saying a few words about myself, my world, and perhaps even somewhat of my intentions for this space. I have little doubt that anybody and everybody who reads this will do so because they already know me... However, the world is ripe with possibilities, and one never knows. So, without further ado, here is a "brief" introduction to the Warped World... I appologize ahead of time if it gets a bit long-winded.

My name is Nicholas Peterson. As of the time of this writing, I am 19 years old. I am currently an undergraduate student in the Department of Mathematical Sciences at Michigan Technological University, Houghton, MI. This is my first year residing in the Upper Penninsula, and my first year as a traditional college student. However, I am a junior by credits, I already TA for Multivariable Calculus labs, and I will be graduating with a degree in General and Discrete Mathematics, with minors in Physics and Computer Science, in Spring 2007.

Not suprisingly, I tend to spend my free time pondering many of the same topics that I discuss in class. Mathematics is my passion; My ultimate dream is to go in to mathematical research. Of course, this means that I will need to attend graduate school, and I intend to do so at either Massachusetts Institute of Technology or at Princeton, depending upon the area of mathematics in which I end up focusing. In addition to doing math in my free time, I do a fair amount of work with computers. My computer-related project for the last several months has been familiarization with, and mastery of, Linux. Along with my more academic pursuits, I enjoy reading, hunting, hiking, music, movies, and several other such activities.

My world is something of an odd place. The Upper Penninsula is not exactly the entertainment capital of the world. I originally hail from Bloomington, MN, home of the Mall of America and several other attractions that are actually interesting and/or entertaining. So, as is the case for most people, the lack of entertainment that I found in Houghton was something of a shock. However, where most people around here turn to alcohol to relieve boredom, I've always had a nack for entertaining myself. So, Houghton is a magnificent environment in which to learn, for those who enjoy their studies.

Admittedly, I spend a considerable amount more time on my own than most. However, I am not exactly a recluse. I have friends around campus that I see on a fairly regular basis, when I have time... after all, I am enrolled in 19 credits of courses this semester, and I am definitely a person who needs time to myself. In some ways, this allows me to observe the ways in which people behave in something of a 'detached' capacity, and I find that I often understand social dynamics incredibly well, provided that they don't directly affect me. When they do directly affect me, I fear that I am not nearly that fortunate. But, we all have our weaknesses.

Lastly, I will say something regarding my plans for this blog. I do not have any one particular purpose in mind, nor have I really ruled many out. There are a few things that I have decided, however. I intend to post in here frequently. As is to be expected, I will be posting events in my life that I feel are important, and will probably be posting opinions and views that are of timely relevance to me. Additionally, as I am a scientist at heart, I will post things in here that I find to be interesting; the majority of these will undoubtedly be mathematical in origin, with a fair amount of physics and computer science, and a small amount of whatever else comes to mind.

Well, those of you who have actually managed to read through all of that, I thank you for your patience. If you didn't... well, I wouldn't exactly say that I could blame you! I look forward to writing in this, and please feel free to comment. I would love to hear from you! And I promise, I will attempt to not be nearly so long-winded in the future.