Here's a book I have to put on my reading list...

Dr. Lisa Randall, of Harvard University, is one of the world's current experts on particle physics, string theory, and cosmology. In view of some of my posts below regarding dimensionality in our universe, her book Warped Passages: Unraveling the Mysteries of the Universe's Hidden Dimensions looks like a must read.

Thanks to one of my colleagues at work for making me aware of this...

Web 2.0

Before I get to the titular topic, it is worth checking out Marc Andreessen 's blog, which I recently stumbled upon: /http://blog.pmarca.com/. There is lots of interesting information about the Web there, and also about entrepreneur-ship in general. His recent post on personal productivity posted here is at the very least a fascinating read that might challenge some of your own views on achieving high personal productivity.

In any case, the post I wanted to highlight in this post is on Web 2.0 - and exactly what it is. The full text of the post and replies can be found here:

http://blog.pmarca.com/2007/06/why_theres_no_s.html


Here is the salient points I took from the article:

"Web 2.0" is a very fuzzy term originally used as a name for a conference on web technology. Web 2.0 isn't a new set of standards or even strictly a new set of technology. Although we might like to think of a variety of "spaces" in cyberspace - usually to categorize technology - these spaces are conceptual in nature where the reality is "market, product and company". There really isn't a "Web 2.0".

One respondent to the blog post (who helped to chair a recent Web 2.0 Expo) commented:

" - web 2.0 is about rich client-side applications (aka ajax, flash, etc)
- web 2.0 is about tagging & ratings & voting (aka Flickr, Loomia, etc)
- web 2.0 is about user participation & user-generated content (aka Digg)
- web 2.0 is about APIs, RSS, and web services (aka Amazon, Google, Yahoo, eBay, etc)
- web 2.0 is about mashups & remixing content (aka HousingMaps, ChicagoCrime, etc)
- web 2.0 is about web 2.0

anyway, while it's tough to define, and it's broad & overly fuzzy, it still is relevant.

- dave mcclure"

which sounds to me like fairly recent technological developments for the web. Web 2.0? Perhaps...

Oh, and by the way, if you don't know who Mark Andreessen is, shame on you. Look it up.

Happy Birthday, Leonhard Euler

Yesterday was Leonhard Euler's 300th birthday. Happy birthday! to one of the most prominent and prolific mathematicians this world has ever seen. In my mind, one of the most unique characteristics about Euler was that he was how normal he was outside of his genius. Most great mathematicians seemed burdened by their genius in how it affected their lives outside of their profession - indeed, many of them had psychological or social difficulties (ever see the movie "A Beautiful Mind"?). But Euler seemed to escape all of that; his remarkable contributions were all the more unique in how unremarkable the rest of his life was.

From Two to Ten Dimensions

In real life, we know that we live for the moment in three dimensions - we can call them height, width, and depth. (Mathematicians cleverly call them "x, y, and "z", or some other combination of letters.) We can also view time as a dimension as well - so we can say that we live in four dimensions - the three spatial dimensions and time.

In a future post, I'd like to talk about 'time'; it's a topic I've thought about a lot recently. But, to set the stage, we need to delve into the "dimensionality" of our current reality.

Our reality has these three spatial dimensions. We can construct three straight lines, meeting at a single point, which are all ninety degrees away from each other - the x, y, and z axes. Even though it is possible mathematically, we cannot imagine a way to create a new line, intersecting at the same point, that is ninety degrees away from each of the other three lines. If we could do that, then we would have a fourth spatial dimension.

Although four dimensions are impossible to imagine, we can gain some insight by going in the other direction; that is, we can imagine a reality which has less dimensions than the one in our reality. This is the pretext for the book Flatland by Edwin Abbott Abbott. This rather dated book, originally published in 1899, describes what life could be like if lived in only two spatial dimensions. For those interested, this book is in the public domain, and can be downloaded in its entirety here.

After reading this text, we then wonder if there really is a fourth spatial dimension, and if so, how would we living in three dimensions interact with it? Fourth dimensional beings (and perhaps we ourselves have a fourth dimensional component?) would view our existence quite differently than we who are trapped in it do, much like we would view the world of the flatlanders.

Interestingly, today's physicists and mathematicians now theorize that there are indeed more than three dimensions plus time in reality; the current thought is that there is ten dimensions. Ten may seem like a rather arbitrary number, but you can visit the site Imagining the Tenth Dimension by Rob Brayton to see an interesting flash production which attempts to explain the current thinking.

I personally would not be surprised if there are indeed more than 10 dimensions. In fact, I expect that there are infinitely many dimensions (there certainly are mathematically). But, what is reality?

Infinity

When I was young, I remember sitting on top of the steps on the outside of my house, lying back and gazing up into the night sky. It all seemed so vast, and made me feel small, and perhaps insignificant. I thought, "where does this end - how far is up?" I thought that somewhere out there, if you kept going far enough, you would hit a brick wall (I'm not sure why I thought it would be brick, but it just seemed right) and that would be the end. But only a few moments later, I wondered what would be on the other side of that brick wall, and how far past that. At that, I sat up, when into the house and stopped thinking about it, because I knew it was something beyond my ability to comprehend.

But one thing we do know - infinity is more than a mathematical concept, it is real, and it does exist. The concept of reality we cannot understand is a theme I'd like to delve into later.

Anyway, where I wanted to go with this post takes me to graduate school, where I studied math. My favorite courses there were in "Real Analysis", and in that course we did talk about the infinite in several different ways. We all know that the counting numbers - the positive integers - are infinite. There is no "last number". So, consider the collection ("set") of all counting numbers. There is a term called 'cardinality' which reflects the number of elements in any set. A finite set - say a set of all the members of your family - would have cardinality of seven if you had a mom, dad, and four other siblings (plus yourself). The cardinality of the set of counting numbers is, of course, infinite, since there are infinitely many numbers in the set.

Now, let's look at other infinite sets. Say, first, the set of both positive and negative whole numbers (the integers). If you can (and you can, take my word for it) create a one-to-one mapping between the set of integers and the set of counting numbers, then we can say the set of integers is 'countably' infinite. Interestingly, you can also create such a one-to-one mapping between the set of counting numbers and the set of 'rational' numbers (rational numbers are those that can be expressed as a fraction with integral numerators and denominators). So, the set of rational numbers have the same cardinality as the set of counting numbers.

So far, so good. You might know that there are numbers that are not rational - numbers that cannot be expressed as a fraction with integral numerators and denominators. Examples are numbers like 'pi' and some square roots (like the square root of 2). Numbers like this are called 'irrational' and the set of all numbers both rational and irrational are called the 'real' numbers. The curious thing is that you can prove mathematically that it is impossible to create a one to one mapping between the set of counting numbers and the set of real numbers. So, amazingly, we have two sets of infinite cardinality - yet one set is 'bigger' than the other! Thus, there is more than one infinity. In math, we call the cardinality of the counting numbers 'aleph null' and the cardinality of the reals 'aleph 1'. If we use the constructive nature of the aforementioned proof of the existence of 'aleph 1', we can construct infinite sets which have cardinality greater than 'aleph 1' (call that one 'aleph 2') - repeat and you have 'aleph 3' and so on. Thus, there are infinitely many infinities. From the way I have described it, perhaps the number of infinities is countably infinite (going from 0 to 1 to 2 ...) but I bet they aren't. At this, I think I should just get up from my stoop and go into the house and stop thinking about it...

welcome to my blog

So why am I starting a blog, anyway. Well, it just seems the thing to do. This blog will be my own reflections on computer science (I teach a few courses for a well known university), net-centric technology (my current area of interest), and mathematics (the discipline of my degree). So it will be an mix of topics. More later...