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

August 2005 Mathematics Learning Log

An Example of a "Learning Process" Journal

Sunday August 28, 2005
Learning Log Number 3

8:20 am Lethbridge, Alberta

I am listening to the 4th day of the 4th match of the ashes series. Australia have just lost their last wicket and only have a lead of 128 runs. Now to do some maths.


My content notes from the previous session cover material from chapter 1 of the book "The Art of the Infinite: Our Lost Language of Numbers" by Robert & Ellen Kaplan (2003), pages 13 - 28. The chapter discusses rational numbers and irrational numbers giving a couple of examples of why there are an infinite number of each. There is then mention of the real and complex numbers. I read the second chapter a few day ago and yellow highlighted a number of sentences.

My next step will be to make a few notes about chapter two "How Do We Hold These Truths?" [p. 29 - 55].



The authors give an example of a "proof" of the Theorem of Pythagorus that simply involves looking at two diagrams.

They follow this with a proof that the sum of n integers is by simply pairing the first and last numbers of the sequence and showing that there are n/2 such pairs each of which adds to (n+1). These are both examples of a geometric approach to proof. The latter may also be proven using mathematical induction, which is an algebraic approach to proof.

The Greeks then came up with the idea of proof, to use as tight a system of logic as possible to substantiate the various relationships that they had noticed.

Descartes (1619) continued this tradition by stating that one must "begin with what we can intuit clearly and immediately". The Englishman John Locke took the opposite approach by beginning with reality and looking for "self-evident" truth. The German Immanuel Kant then elaborated on this idea by stating that we begin with perception, but that such perceptions are initially shaped by our intuitions of space and time. Thus mathematics is fundamentally about the logical conclusions that follow from our apriori ideas of space and time.

The axioms of mathematics were then refined during the eighteenth and nineteenth centuries as we became more adept at clarifying our ideas and concepts. The basic axioms for the natural numbers included the ideas of associativity and commutativity. Formalism, where the focus was on the relationships among the symbols, and where the symbols need not refer to anything in particular, became popular. However this approach eventually led, in the case of geometry, to a number of different geometries (depending on a basic assumption of parallel lines). However in the case of the natural numbers such formalism appeared to hold and provide a clear logical basis for the theorems about these numbers.

Eventually numbers became to be subsumed under the more general idea of sets, and set theory became the fundamental foundation for arithmetic. Richard Dedekind (1858) developed the idea that an irrational number could be conceptualized as what one has when one divided the numbers into two distinct sets where all of the members of one set are less than all the members of the other set. Such a cut is now referred to a Dedekind cut.

The Italian mathematician Giuseppe Peano developed the five key axioms that define the natural numbers which includes the idea of mathematical induction.

The German David Hilbert focused on trying to show that Peano's axioms were both consistent and complete. But in 1931 an Austrian mathematician Kurt Godel showed that no proof of consistency and completeness could ever be made within the system to which it referred.

At the moment there is less interest in pursuing the foundations of mathematics. Instead we are focusing on developing new theorems that simply show stunning patterns, relationships and results that are both useful and aesthetically pleasing.

End of chapter 2. [p. 55]

Here are a few quotes from "The Art of the Infinite: Our Lost Language of Numbers" that I enjoyed:
"Some people relish the geometric approach, some the symbolic. This tells you at once that personality plays as central a role in mathematics as in any of the arts." [p. 30] I lean toward geometric approaches as they seem to me to be more intuitive, while a symbolic approach seems to me to be more mechanistic.
"But the Garden of Eden is famous for its snake, and the snake is the desire for more precise knowledge." [p. 41] I first saw this metaphor in a book by John Ralston Saul, "The Unconscious Civilization" (1995). It is one of my favorite metaphors.
"Induction has confirmed the truth of many an important mathematical insight, but that insight had to have come from some other source." [p. 44] The role of insight is fascinating in its own right.

This chapter gives a good overview of the historical and logical foundations of mathematics.

9:35 am

Total elapsed time: 1 hr 15 min.

I am now ready to read and yellow highlight chapter 3 "Designs on a Locked Chest" which is about prime numbers.

Now to check on the cricket score. England has lost 2 wickets but only need 77 more runs for victory.

Total elapsed time for the day: 1 hr 15 min