5:30 am

I read the second chapter of "Why Beauty is Truth" by Ian Stewart yesterday evening. I need to have a closer look at it this morning while making a few notes. Last night I had difficulty seeing the relevance of the chapter to the topic of the book as it didn't seem to discuss either equations or symmetry.

2. The Household Name

• "Euclid ... was the author of one of the most famous mathematics texts ever written: the Elements of Geometry. " [p. 19]
  
  
• "This does not mean that Euclid discovered all of the mathematics contained within his book's pages. What he did was to collect and codify a substantial part of ancient Greek mathematical knowledge." [p. 20]
  
  
• "The Greek attitude to mathematics was very different from that of the Babylonians or the Egyptians. Those cultures saw mathematics largely in practical terms ... For some Greek mathematicians, numbers were not tools occasionally employed in support of mystical beliefs, but the very core of those beliefs (e.g. Pythagorus)". [p. 21]
  
  
• "Euclid made two great innovations. The first is the concept of proof. Euclid refuses to accept any mathematical statement as being true unless it is supported by a sequence of logical steps that deduces it from statements already known to be true. The second innovation recognizes that the proof process must start somewhere, and that these initial statements cannot be proved." [p. 22]
  
  
• "Mathematicians may debate whether a given axiomatic system is better than another for some purpose, or whether a given axiomatic system has any intrinsic merit or interest. But these discussions are not about the internal logic of any particular axiomatic game. They are about which games are worthwhile, interesting, or fun." [p. 24]
  
  
• "... the main aspect of The Elements that affects our story is not what it contains but what it doesn't." [p. 26]
  
  
• "By the late 1800s ... mathematicians were learning to prove that no method of this-and-that kind is capable of solving such-and-such problem." [p. 27]
  
  
• "When someone invents a powerful new method to characterize those things that can be constructed using straightedge and compass, and distinguish them from those that cannot, then you have an entirely new way of thinking. And with that come new thoughts, new problems, new solutions - and new mathematical theories and tools. ... So tool-building can be at least as important as problem-solving." [p. 29]
  
  
  • This is the key sentence of this chapter. This is what Galois did - he created a new way of thinking about equations using a new notation, called Group Theory. Slowly rereading this chapter and noticing this sentence, has made the effort well worthwhile.