More than Infinite?

You might think that infinity is pretty easy to understand. All you have to do is start counting, 1, 2, 3, 4, … and then imagine that you never stop. The set of numbers so described {1, 2, 3, 4, …} is called the natural numbers and is an infinite set. So far so good. But then you start asking yourself some questions and things seem to quickly go awry.

For example, which set is bigger: {1, 2, 3, 4, …} or {2, 4, 6, 8, …}? How could you even answer the question? You could assume that since the even numbers are a proper subset of the natural numbers that the set of natural numbers is bigger. That makes sense, we took away all the odd numbers. Okay, but, let’s start counting the even numbers, the first (1) even number is 2, the second (2) even number is 4, the third (3) even number is 6, and so on. But doesn’t this counting of the even numbers go on forever just like the natural numbers. As a matter of fact, isn’t this counting of the even number exactly the set of natural numbers? So how can we say that the evens are a smaller set; if to count them you need all of the natural numbers?

Maybe you’re beginning to see that things aren’t as simple as they seem. We need to be careful when thinking about concepts like infinity. This is where mathematics comes to our rescue. It is there that we will find the careful definitions and methods that help us answer questions about infinity in a logical and consistent manner.

The Columbia and the 21st Century

As a young boy growing up in the sixties and seventies, the 21st century was my destination. It was the future and I wanted to be ready. I can remember a strong sense that the entire world was changing. The chaos I would witness on the nightly news (the war, Martin Luther King Jr. and his assassination, the riots in the cities) were clear signs of changing world. A new world was coming, the old order was desperately holding on. But it was dying; the future was inevitable.

Language and Grammar
Language Learnability and Language Development, Steven Pinker

Almost without exception, children are able to learn how to speak the language of the adults around them. They do this using only the example those adults supply by speaking. While adults do simplify their speech when talking to very young children, we rarely have a formal program of language instruction in mind. Rather, we simplify our speech so that we can be understood. From this input children are able to learn to distinguish words, understand the meaning of words and combine them into sentences. In no time a child is speaking his language and speaking it correctly.

How this process occurs is the topic of Steven Pinker’s monograph Language Learnability and Language Development. He focuses on how a child can learn the grammar of his language. His approach is quite formal and technical, as is fitting for a professor of linguistics writing for an audience of professional researchers. The ultimate goal is to define a set of algorithms that processes the input (the sentences heard by the child) and creates a set of rules that define the grammar of the language the child is hearing.

Occasional (May 3)


I‘ve been enjoying my new job, I am already right in the thick of things. No LISP yet, but I am getting closer. Right now I am doing C++.

My new commute to work is very quick. I can either walk over the Mass Ave bridge (about 5 blocks distance) and then go about 6 blocks to the office or I can hop on the number one bus, get off at MIT and have about a 4 block walk. What I do depends on the weather. It’s close enough to home that Sandy and the baby can come by and go to lunch with me if they feel like it.

When I worked in Lexington I would spend about 1.5 hours a day commuting. That gave me plenty of time to read. My new commute does allow much time for that. I will have to remember to make time for reading.

I have new motivation to hurry home from work. Dante has started waving and laughing and otherwise carrying on when I get home. He won't even let me take my coat off; he has to be picked up right and played with. Combine that with Sandy’s pleasure at seeing me home and it’s quite a joy.


Neverwhere, Niel Gaiman

The Culture of Cities, Lewis Mumford


Tonight Sandy went out with her friend Jane. Dante had oatmeal and carrots and I had some ravioli.



[ Language and Grammar ]


[ More than Infinite ]


[ Columbia and the 21st Century ]



[ A.R.T ]

[ Metropolis ]

[ Abitare ]

[ How ]

[ Think Do Think, Blog Do Blog ]

[ The Museum of Fine Arts, Boston ]

[ clio ]

[ Café Scientifique ]

[ Language Learning and Language Development ]

[ Steven Pinker ]

[ Chomsky ]

[ LFG ]


[ Steve Minutillo :: Weblog ]

[ The Julie/Julia Project ]

[Boing Boing]


What’s new

The first few installments of the think do think blog are up. More to come as time allows. So far we’ve decided to learn J2EE technologies, chosen some tools to do the work and installed the first tool (MySQL 4.0) on our Mac. Next time we install JBOSS and make sure everything is working so far.

Also made a minor change to my RDF/RSS template (I still hand code everything) to make sure it validates.

What is Ponderful?

Ponderful is a weblog of sorts created by Nicolas Minutillo.


This weblog uses only standard web technologies. I’ve made pretty lightweight use of the more modern standards so your browser should be able to image this page well enough. I would display the appropriate badges, but they are really ugly and clash with the simple page design I’ve chosen.

[ XHTML 1.0 ]

[ CSS ]

[ Bobby WorldWide Approved (AAA and Section 508) ]

[ RSS ]


Hand coded XHTML and hand coded RSS, edited with BBEdit 7.0.1, on my iMac (400 Mhz G3, 512 MB, 20GB + 80GB LaCie external)