There, that project is done. Well mostly done. I finally finished the first draft of A Tale of Two Paradoxes. I suppose that I will have to someday look at it again and correct all the spelling and grammar mistakes. I wish I could afford an editor. I find I can do a passable editing job if I leave and article alone for a few months, then revisit it.

Anyway, this is the latest effort to address the paradoxes of transfinite theory. It has a few modest improvements over the previous efforts. It strongly emphasize that the diagonal method is simply a form of the liar's paradox. The two paradoxes are Galileo's paradox and the liar's paradox. Transfinite theory uses Galileo's paradox to assert that the set of rationals is the same "size" as the integers, then uses the liar's paradox to say that the real numbers are a different size. This is the denumerable/nondenumerable dichotomy. Mathematics is supposed to somehow arise from the opposition of these two terms.

The purpose of this set of articles is to address the use of paradoxes as the foundations of mathematics. It is not about the conclusions of theory, but about the foundations. So I added a few words on the difference between discrete and continuous mathematics. Discrete mathematics works only with terms of finite length. Continuous math uses terms of infinite length. The differences between these branches arise from definitions, not paradoxes.

We can create only a finite number of unique strings from a finite number of characters. Expressing a totality, like the set of real numbers, requires an infinite number things: Discrete mathematics is content to work with finite entities. Continuous mathematics must allow for the use of infinite terms. The difference between the subjects does not arise from a fundamental dichotomy created by paradoxes.

The article does not deny the existence of paradoxes. It simply says that they sould be treated as a side dish. For example, we should not use Galileo's paradox as the definition of the infinite!!!!! Anyway, I hate this work on transfinite theory. I will not let the site fester for a few months and work on things I enjoy.

I can't believe twenty days past this quickly. I am finally getting around to doing the hour or so program writing and debugging needed to finish the comment program. I think it was the dread of javascript that made me put writing off. Once I finish the comments, I will add more stuff.

BTW, I just added myself to Blogarama.com.

The comment section is still in test mode. That is why you see a test on all of the blog entries on this page. I didn't get a chance to finish the code. Of course, having an unfinished project in production has inspired me to work on other things.

I finally figured out what to do with the site map problem. The site map for the Salt Lake City community directory had close to 300 links. Google suggests limiting the page to 100 links. So, I made a buffer that lets me select which pages I block from the site map. I also now display it in alphabetical order.

Kids are coming over. That means I will not be able to finish the comments for another day. You can try the program. It is in a half-assed state.

In the time honored spirit of cowboy programming. I will be doing some testing in production.

I decided to write a program so people could leave comments. I want to see what Blogger passes to my web host...so I added a test button which does absolutely nothing of interest except show the unique ID of each post on this page. In a couple of hours, I will probably turn the "test" link into a comment link. If you are interested in programming. You could read my intro to PHP.

I was linking through the many different art sites that I enjoyed. Sadly many seem to have disappeared. One of my big hopes was that web would help revitalize the arts community. During the dot com boom, many artists were able to supplement their income with design.

Anyway, I've always liked the texture used on the pages of Salt Clay City. This is a site for a Utah potter. She did a great job of capturing dimension with just a few simple textures. I am sad to see the site doesn't have that much more information from several years again when I added it to my collection of favorites.

That was embarassing. I almost lost a site. I went to check my email and I couldn't get to my main address on rgreetings.com. I discovered not only had the site expired, but that it was still registered with Network Solutions. I thought I had rid myself of all Network Solution domain names ages ago. Fortunately, while Network Solutions makes just about every activity as hard as possible. They've streamline the renewal process. As for domains, I retiring infosuprhwy.com. This was a domain I inherited.

This just in:

Landmines are Okay!!!!!!!

For some reason unknown to the sane, the United States is now unilaterally pursuing an aggressive landmine campaign (See DoD.mil). The United States is now firmly standing against Vietnam Vets, the United Nations and other liberal organizations, and is pursuing the development of safer landmines. These new mines will be easier to detect...decreasing the number of MILITARY casualties, and will have deactivation devices that will cause them to self destruct after a preset time.

I suspect that in Rumsfeld mind, anyone opposing the proliferation of landmines is un-American.

I have no doubt that the landmines created by defense contractors will be technological wonders. A good contractor will have an answer on paper for every criticism leveled at the use of mines.

This is bad news because the world had a very workable landmine ban on the table.

By actively thwarting the ban, the US has just given defacto persmission for every rogue state to follow the United State's example and start an aggressive campaign of building and stockpiling *cough* *cough* "smart" landmines.

The really odd thing is that the current landmine ban gave the US an decided edge in military campaigns. Landmines are a low tech device that rogue nations can mass produce. It is the one weapon that poor nations can use against the United States. Saddam Hussein had destroyed a good portion of his stockpiles. If he had not, then he could have planted his stock piled mines in Sh'ite and Kurdish areas in the final days of the battle.

The landmine ban gave the US a decided military edge. This absurd game where the Bush Administration automatically rejects all work by the international community is insane.

It is sad, all of the effort that people have invested in removing landmines is now lost and worthless. Without the landmine ban, then landmine removal is just a circular game where armies plant mines and charities run themselves ragged removing the mines. Landmine removal is now a round robbin game. The economics are simple. It costs a country $3.00 per mine to mine villiage. The international community will sink $300.00 per mine in the country to remove it. Planting mines means big business.

Yep, creating community sites means a lot of work. The categories on Salt Lake Sites have grown too large. So, I've been trying to figure out how to break things into subcategories. Today, I've been working on the photo section. I've decided to break the section into art photos, scenic photos, fashion and photographers. I have a feeling the process will backfire. I put the best galleries in the Salt Lake City Art Photo gallery. I am worried, however, that everyone will end up visiting the lower quality galleries.

In the culture department, I loved the PBS Series on the Medici. My work on mathematics starts with Brunelleschi's rediscovery of visual perspective. The Medici documentary does an excellent job showing the important role that patronage played in the development of Renaissance arts, and I was delighted from start to finish with the production.

The goal of the production was to give the world view of the Medici's. As such, it tends to overrate their contributions. The documentary gives the impression that the Medici's created the Renaissance. I see them more as a product of their time. For example, the Medici's were able to rise from obscurity to power because Florence was a republic. In allowing more freedom and active participation in the government and economy, Florence opened its society to merchants and bankers.

As for the patronage of the arts, again the Medici were very much a creation of their times. The patronage of the arts was not simply the Medici's being wonderfully generous. With the Catholic Church in a major cathedral building mode, art creation was an extremely vibrant part of the economy. Creating public art was like film making today. The Medici's came into the art and architecture world because that is where the money was. Buying and trading art work was the pinnacle of high finance in the 1400s. The documentary seems to imply that the Medicis created the republic in Florence and that they created the grand masters of the Renaissance.

I suspect that the real dynamics of the economy was much as it is today with the creative agents needing capital and bankers finding clever ways to snag the lion's share of the rewards from the creations. In other words, the Medici's were a leach with a symbiotic relation to the creators. In all likelihood any other banking family would have become rich. The fact that Florence was an open society that allowed Brunelleschi, Da Vinci, Giotto, ..., and the Medici's to pursue their own intellectual thoughts was the cause of the Renaissance. Add to that the work of Roger Bacon and St Augustine that helped promote the sciences of the ancients, and you have a formula for success.

In some ways, I dislike how the Catholic Church was portrayed as reactionaries throughout the work. I personally think a better description of the day is that both the CC and the people at large were finally in a position to recover from the invasions and plagues of the dark ages. The CC was the primary funder for the arts. It was trying to find a way to progress.

It is interesting to note that the conservative elements that persecuted Galileo got into power in raction to the corruption of the Medici popes. When you look at that ugly little nest of politics, you see a lot more going on than the rift between science and religion.

BTW, I was very pleased that the documentary did not go into the Machiavelli worship that is common in many modern universities. My personal take is that Machiavelli wrote the prince just as Scott Adams draws caricatures of pointy haired bosses. Machiavelli saw the collapse of the Republic that brought so much to the world. Machiavelli's The prince was not the recipe for subjegation that Mansfield falls over to worship, but a description of the horrible things that the powerful were doing to gain power.

The Medici's destroyed the republic that allowed the Medici's to rise to power. Just as Bill Gates seems to be doing everything in his power to destroy the economic foundations that allowed businesses like Microsoft to be born and flourish. It is called kicking down the ladder behind you. Nothing new there.

Anyway, the Medici, Godfathers of the Renaissance is an absolutely supurb documentary.

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In the burning up time department, I took a trial of the There.com community. Wow, they've got a really cool site in the works. Basically, you are an avator and run around communing with others on a tropical island. I prefer this model to the dark shoot up worlds of unreal.

About every six months, I actually do add a feature to the web sites. I added a new page that shows any new entries for the community directories. For example, here is the new listing for Grand Junction, Co.

Realizing that I was getting a penalty for using excessively long site maps, I cut the full directory listing from the site map, then put a link back to the original map. I exclude the original map from the search engines: Here is the new site map for Salt Lake City. Here is the old map.

It is always strange when I find sites that I've added to the Internet. While trying to track down some old friends, I happened on my Back Flip site. I am really happy to see that Backflip is still up and running. Like Blogger, BackFlip is a large public site that lets people build their own personal link list. It is like the favorites on your web browser, but shared publicly.

What you can do with blogger is add links to all of the sites that you think have good content (or links to all of your friends). The search engines read BackFlip, and your little directory becomes your vote for the sites that you think merit note. Unfortunately, its been a while since I had visited my BackFlip folder. I had to remove about a dozen broken links.

Although I didn't find what I was looking for this morning, it was fun finding my backflip account again. I hope the site sticks around.