EQUALITY (added) a=b can be a judgement, observation, assumption, fact, definition, pronouncement, proposition, designation, or axiom.
All mathematics is abstraction from “1”, “+” and “=” by definitions.
Definition: what something is.
+: association between things.
Equality: a=b means a and b represent the same thing, and is true or false depending on what they represent. 1+1=0 and 1=12 are false in Z, but 1+1=0 in Z2 and 1ft=12in are true.
“Advanced” abstract mathematics is a matter of definitions abstracted from familiar concepts and usually presented to students without reference to the familiar concepts.
An Exercise in definition from Birkhoff and McLane, definition of Field assumed, which you can just squeak through comfortably if you are familiar with the concept of a polynomial. Otherwise you can still piece it together, but you won’t feel comfortable with it.
Definition: Let K be any field, and F any subfield of K. An element of c of K will be called algebraic over F if c satisfies a polynomial equation with coefficients not all zero over F.
An element c of K which is not algebraic over F is called transcendental.
Theorem: If c is transcendental over F, the subfield F(c) generated by F and c is isomorphic to the field F(x) of all rational forms in an indeterminate x, with coefficients in F. The isomorphism may be so chosen that c <-> x and a <-> a, for each a in F.
Definition: Polynomial Form in x over F: P(x) = a0+a1x+a2x2+….anxn.
Definition: Rational Polynomial Form: P(x)/Q(x).
Teachers don’t emphasize definitions. That is why many students don’t “get” math. The teacher says something that is not intuitive and the student doesn’t understand it. It’a a definition. It’a a definition. It’a a definition. If intuitive, it should be explained as such.
Let ICC = Ice Cream Cone. There is nothing to understand. IT’S A DEFINITION.