How to Model The Continuum; Or, What is Continuity?

This month we will explore The Continuum, mathematical representations of continuity, and the number systems that have been used to model it. What is the continuum? How can it be represented or modeled? Do rational or real numbers better model the continuum? Can Stevin numbers or infinitesimals or Dedekind cuts model continuous phenomena? How can we represent a continuum in a computer?

• Elementary videos: one and a half hour (37 videos) Michael Starbird series on Rational and Irrational Numbers and Decimal Representation (https://www.youtube.com/playlist?list=PLkCiNL_gZp2ci1mVKaTXt37qe6shAZBcP). Note: advanced members might want to skip these videos, but they provide an excellent review of the basics about decimals, rational and real numbers. They are an excellent exercise for anyone wanting to improve their thinking skills (pause the video often and work out a precise answer to each question before Starbird and his student give hints or solutions, if you get stuck watch a bit more but challenge yourself to answer the questions before the video does). They are also worth studying from the perspective of the value of "understanding simple things deeply": how deeply do you understanding decimals? Educators looking to better understand how elementary concepts can be taught from an advanced perspective will also find these videos useful.

• 44 minute Norman Wildberger video introducing the most fundamental and important problem in mathematics, namely, How To Model The Continuum.

http://www.youtube.com/watch?v=Nu-YPJSNFpE

• 37 minute Wildberger video exploring the rational number line and irrationalities.

http://www.youtube.com/watch?v=t0aHtXud9r4

• 43 minute Wildberger video exploring Stevin numbers, infinitesimals, and complex numbers

http://www.youtube.com/watch?v=Y3-wqjV6z5E

• 70 minute Wildberger video exploring Dedekind cuts and problems with the real numbers

http://www.youtube.com/watch?v=4DNlEq0ZrTo

• John Lane Bell's long essay "Continuity and Infinitesimals" in the Stanford Encyclopedia of Philosophy (http://plato.stanford.edu/entries/continuity/)