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On 2005/9/25, PingPongBoy was fascinated by fields. In field theory, the notion of a field may have been motivated by the study of roots of polynomials.

On 2005/9/27, PingPongBoy was fascinated by extensionality. PingPongBoy had always used the practical definition of equality as, loosely speaking, a relationship that holds between two things that are the same in every way. However, the idea of extensionality suggests that equality could be prone to error: what if no one could find a distinguishing test for two inequal things? Things that are equal are and things that aren't aren't, right? The ingenuity of the beholder should have no bearing.

On 2005/9/27, PingPongBoy resumed his interest in algebraic numbers, which caught his attention in field theory. What are algebraic numbers? I mean, why are they so important that they are given such a name? The study of algebra has a far ranging and significant impact, and to call a number algebraic would pretty quickly exclude all other kinds of numbers from being called algebraic, regardless of how deserving they are. How fascinating that it's all related to the roots of polynomials. Well, polynomials are quite desirable for use as mathematical models. We can give them that much.

Who would have thought is transcendental, that is, not a root for a polynomial with rational coefficients?

The import of isomorphism and homomorphism is noted by PingPongBoy. Isomorphism is a strange animal -- how is an isomorphism between Big Ben and a wristwatch expressed in simple terms?