Learning:
The Journey of a Lifetime
A Cloud Chamber of the Mind

February 2006 Mathematics Notebook

An Example of a "Learning Process" Journal

I completed reading "Coincidences, Chaos, and All That Math Jazz" yesterday and will now make my notes for Part IV Transcending Reality which discusses the fourth dimension and infinity.

Chapter 10 The Universe Next Door.

• This material was familiar to me. However the explanation for unfolding a cube into 6 squares and then extending this idea to unfolding a 4-D cube into a cross of 8 cubes is one of the clearest I have seen.

  I would have liked a short section where the 4th (and higher) dimensions are only expressed algebraically. This is a straight-forward extension which requires no special visual imagination.

  
  

Chapter 11 Moving Beyond the Confines of Our Nutshell

• "Numbers with a million digits are essentially meaningless, yet most numbers are far larger still." [p. 233]
  
  
• "How can we hope to understand infinity? As Woody Allen once lamented, ' I'm astounded by people who want to 'know' the universe when it's hard enough to find your way around Chinatown.' " [p. 233]
  
  
• One way to understand numbers is by counting. Another way is to view them as a 1-1 pairing. Using this latter idea, two collections of things have the same size if their elements can by paired up evenly. This is a totally different idea of number. It corresponds with our conventional idea when one is thinking about finite numbers but it provides new ways of thinking about infinity.
• "Thus, adding any finite number of elements to an infinite collection does not increase the size of that collection." [p. 238]
  
  
• The mathematical term cardinality is used to refer to size. ... 'Two collections have the same cardinality' means that the elements of the two collections can be put in a one-to-one pairing."
  
  
• "Mathematically speaking, we have thus shown that the collection of even counting numbers has the same cardinality as the collection of all the counting numbers. Basically, half of infinity is no smaller than infinity." [p. 239]

Chapter 12 In Search of Something Still Larger

• "Asking whether all infinities have the same size - or, mathematically speaking, whether they have the same cardinality - really means, 'Suppose someone gives us two infinite collections. Is it always possible to pair them up in a one-to-one manner?' " [p. 247]
  
  
• There is more than one kind of infinity since one can imagine scenarios where a one-to-one pairing is impossible.
  
  
• There are more real numbers than there are counting numbers.
  
  
• There are an infinity of infinities.