Posted by: Dave Richeson | October 13, 2010

The hitchhiker’s guide to infinity

I’ve had a few encounters with large numbers and the infinite lately.

First, I came across this interesting blog post that asks how many times you would have to fold a piece of paper so that its thickness would be the same as the distance from the earth to the moon. It is a really nice illustration of exponential growth.

Speaking of 42

I also received a nice email from David Freeman, whom I’ve gotten to know (electronically speaking) since he first contacted me after I wrote a blog post that mentioned him (in my 13th Division by Zero blog post). David’s email contained three of his favorite mathematical quotes from The Hitchhiker’s Guide to the Galaxy. (I especially like the second one. The first is funny because it so true.)

The startling truth finally became apparent, and it was this: Numbers written on restaurant checks within the confines of restaurants do not follow the same mathematical laws as numbers written on any other pieces of paper in any other parts of the Universe.

“The Hitchhiker’s Guide to the Galaxy” offers this definition of the word “Infinite.” INFINITE: Bigger than the biggest thing ever and then some. Much bigger than that in fact, really amazingly immense, a totally stunning size, real “wow, that’s big,” time. Infinity is just so big that by comparison, bigness itself looks real titchy. Gigantic multiplied by colossal multiplied by staggeringly huge is the sort of concept we’re trying to get across here.”

And finally this humorous quote that contains some shady mathematical reasoning:

It is known that there is an infinite number of worlds, simply because there is an infinite amount of space for them to be in. However, not every one of them is inhabited. Therefore, there must be a finite number of inhabited worlds. Any finite number divided by infinity is as near to nothing as makes no odds, so the average population of all the planets in the Universe can be said to be zero. From this it follows that the population of the whole Universe is also zero, and that any people you meet from time to time are merely the products of a deranged imagination.

Today I read a short blog post by John D. Cook called There isn’t a googol of anything, which is a nice discussion a very large number. I think it interesting to put it up against the second of the Hitchhiker quotes.

Finally, here’s a quote that one of my colleagues sent to me the other day—it is also related to the infinite and the infinitesimally small. It is St. Thomas Aquinas’ solution to Zeno’s arrow paradox:

Instants are not parts of time, for time is not made up of instants any more than a magnitude is made of points, as we have already proved. Hence it does not follow that a thing is not in motion in a given time, just because it is not in motion in any instant of that time.

Responses

1. It’s not about infinity, but there’s a great article by Scott Aaronson at: http://www.scottaaronson.com/writings/bignumbers.html

It’s about such things as Ackermann sequences and Busy Beaver numbers and defining large numbers.

The thing that fascinates me is that, regardless of what number is named or defined in such an exercise in logic, computing, and set theory – as utterly fascinating as that is – is that almost every number is larger than that number.

I have yet to find anyone else who is as transfixed by that as I am, so maybe it’s just me.

2. I’d like to recommend another excellent illustration of exponential growth.

Dr. David Suzuki has a video called “The Test Tube” at:
http://interactive.nfb.ca/#/testtube

It won’t be anything new to a mathematician but it is an excellent teaching tool.

3. My favorite take on exponentials and the moon is this one:

http://abstrusegoose.com/218