Saturday, October 01, 2005

To Infinity and Beyond

I've started reading this book, How the Universe got its Spots: Diary of a Finite Time in a Finite Space by Janna Levin. And the operative word here is started. From the title, the book is about the Universe, but before going there, she discusses mathematics and mathematicians.

The first line of the book perked my interest
"Some of the great mathematicians killed themselves. The lore is that their theories drove them mad, though I suspect they were just lonely, isolated by what they knew."
Of course this line, reminded me of my favorite mathematician, EEE. I hope that he is just a little mad, and I fervently hope that he is not lonely.

In the second chapter, she went into a discussion on infinity and the theory of Ludwig Phillip Cantor called transfinite arithmetic. I thought I was getting along fine with the ideas, countable infinite numbers, infinite rational numbers, infinite irrational numbers, and then the bombshell, some sets of infinite numbers are larger than others. HUH? how can this be? Of course she gave an explanation:
"Cantor realized that the set of irrational numbers was infinite in a way that was so huge as to be uncountable. They cannot be represented by one integer divided by another and some require an infinitely long desciption, such as pi or the golden ratio. An uncountable infinity could never be put into one-to-one correspondence with a countable infinity and so the irrational numbers must be of a larger infinity than the natural numbers"
And this is where I got lost. reminding me why I preferred political science as my undergradute course. I will still read the book, hopefully finish and increase my understanding of mathematics and the universe. Hopefully, this book will not drive me mad. If you can find the book, maybe you can join me in this adventure.


Abe said...

My 15 yr old son, who is in the 4th year of a special 4 year math study, says this is correct. The integers and real numbers are "countable" infinities, since they can be matched up one-to-one with the whole numbers. This cannot be done with the irrational numbers.

In fact, if you throw a dart and hit the number line, the odds are overwhelmingly large that you will hit an irrational number, rather than a rational number.

roy choco said...

Congrtulations for having such a brilliant son, must have taken after his mother :-)

what I actually cannot understand, or my mind just refuses to accept that some infinities are larger than others. They are infinite, by definition they are boundless, so how can one be larger than another?

I don't know, I am still glad I took up a social science course :)

Abe said...

lol. Yes, his mom is pretty sharp, too.

It is hard to accept that "greater infinity" concept. Essentially there is an infinite number of irrational numbers between any pair of rational numbers. Of course there are also an infinite number of rational numbers between any pair of rational numbers.

Anonymous said...

Hi, do you know where I can get the book? Sadly, bookstores near my place have failed me. - Ana from QC

Roy C. Choco, FCD said...

Hi Ana,

I was able to buy the book from the Fully Booked bookstore in Rockwell when they had a sale about a couple of years ago. Maybe you can ask them.

If your really, really want to read the book, I am willing to lend it to you or maybe we can swap books. I liked the book but I barely understood it, you can probably appreciate if more. If you have another book that you think I might enjoy, lets swap.