Total tangential point: I noticed something while reading this otherwise good explanation. I can't speak for others, but for me it explains why I often had a tough time with math in school and possibly also why pure functional languages have had slow uptake.

The thing I hate about math (and mathematical functional coding by proxy) is how it is explained.

This article contains an absolutely perfect example. Go to the heading entitled "Church Numerals," which explains how numerals are declared as functions. Above we've been introduced to lambda function notation, which is fine. Now he throws this at you:

0 = λs.λz. z the function s is applied to the argument z zero times

1 = λs.λz. s z the function s is applied once

2 = λs.λz. s (s z) the function s is applied twice

What the bleep is 's'?!?!?

I spent several minutes trying to figure that out. What is s? (bang head on table) What is 's'? (bang head on table more)

OOOOOOOHHHH... there it is in tiny f'ing print:

"(the names s and z are short for successor and zero)"

It's in tiny print so you won't see it, because knowing what symbols mean evidently is not important to understanding them.

Mathematicians do this habitually. Every math class I attended in college would consist of a mathematician who threw symbols and notations at you, proceeded to do stuff with them, and then maybe got around to defining them after he'd already expected you to get it. Maybe. If he feels like it.

It's like teaching a foreign language by beginning with poetry reading, progressing to literary theory, and then oh by the way -- as an unimportant aside -- visiting the topic of words' meanings and their grammar.

Why do mathematicians do this? Are they working to make their subject impossible to explain? Or is it some weird artifact of how the mathematically-inclined think-- do they think nonlinearly and so do not grasp that people learn things by building up ideas hierarchically?

If math were taught by beginning with symbols and operators and their conceptual meaning and then progressing to their use, I think people would have far less of a problem with it. It would be like learning any other language, followed by learning the "literature" that is expressed in that language.

Edit: I sometimes see programming languages explained this way too. Some code will be thrown at you, followed by some higher-order language concepts, and then oh by the way the symbols % means ____ in this language. The example, of course, was utterly incomprehensible prior to that little detail.