#
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.

Continued...Read More than Infinite.

Nicolas Minutillo, January 22, 2003 (revised February 9, 2003) (Mathematics)

#
The Columbia and the 21^{st} Century

As a young boy growing up in the sixties and seventies,
the 21^{st} 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.

Continued...Read The Columbia and the 21^{st} Century.

Nicolas Minutillo, February 1, 2003 (Memoirs)

#
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.

Continued...Read Language and Grammar.

Nicolas Minutillo, January 12, 2003 (revised January 14, 2003) (Linguistics, Book Review)