Preamble

Conventions

Symbols will often be designated as notation for objects that would otherwise be more tedious to refer to.

Any designated notation will pertain only to the particular heading it’s found under unless otherwise stated.

Unless otherwise clear, in any expression made up of mathematical notation, enclosure of a particular part of that expression within parentheses \(()\), brackets \([]\), or curly braces \(\{\}\) will denote that the enclosed subexpression should be treated as a single unit in the context of the larger expression.

Expressions will sometimes be displayed on separate lines to improve readability.

A comma will sometimes be used as shorthand for “and” when listing multiple objects (but not when stating that things are simultaneously true).

“We” will typically be understood to mean you, the reader(s) and me, the author.

New terminology will be italicized when first introduced.

Math, Intuition, And Rigor

I would define math as the act of deduction and definition by the least questionable means practicable. Thus, the achievement of “perfect math” is most definitely impossible; nothing is absolutely undeniable or able to be defined without reference to anything else. This means intuition is what’s going to have to dominate when we lay our foundations. It’s unfortunate in that it means our foundations will be somewhat wishy-washy, but it’s unavoidable.

Once our foundations are laid, we’ll really hone in on making things minimally questionable by using our foundations to derive everything else. The less questionable something is, the more rigorous it is said to be, so we as mathematicians will be going for maximal rigor. (In the case of deduction, this means every deduction will be accompanied by a thorough proof; once proved, a deduction is called a theorem.) That said, mathematicians are still humans, so we may omit little things if they’re obvious or practically redundant (since perfect rigor is impossible anyway), and we won’t entirely abandon our intuition, but we will restrict ourselves to using it indirectly (i.e., “behind the scenes” as a guide).

Why Math Matters

Math, of course, has plenty of applications to the real world, but it also improves your thinking in all sorts of ways (it’s all about thinking logically, creatively, critically, etc., attention to detail, perseverance, communication, and so on), and it has a rather empowering quality to it—To stick with and struggle through a difficult topic long enough to truly, deeply understand it leaves you with a great feeling. (Just don’t get a big head.)

I, for one, would also say that math is pretty cool.