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We want to explore some ideas
expressed in Keith Devlin's book The Math Gene. (He's "The
Math Guy" frequently heard on NPR.). Devlin explores the roots of
mathematics that are entangled in our ability to use language.
As we humans evolved, we developed perceptions that let us see many kinds of regularities and patterns which give us knowledge that lets us interact effectively with our surroundings. But we also developed "offline" thinking which lets us imagine things not currently being perceived, not currently directing our behavior. Our brain also developed some ability to trigger itself in ways that led to metaphor (like the metaphor we call "energy"), and that led to language, representations of the world around us, and that language has certain organizing structures. metaphor
(Lakoff Principle)
We can sometimes recognize some of that organization. We "see" certain
patterns within our language, within our representations of the world.
We recognize these patterns of patterns. Here lies mathematics.
We tend to feel that what we perceive and imagine is the world, in spite of the fact that we constantly discover new things. The world is what we comprehend...or so it seems. But some of those new things of modern science—like the mysteries of quarks and the mysteries of quantum mechanics—just don't make sense to us: we don't comprehend. It's magic. It's beyond those evolutionary developments by which we "see." However, Devlin warns us not to shy away from mathematics, not to think it's "too hard a subject."?it's not math we hate, but rather some boring drudgery inflicted on us in math classes. Mathematicians, too, hate that drudgery. Real mathematics is a beautiful experience that comes naturally to us, as does language. Mathematics is a search for patterns. It's a delightful discovery of patterns. And we all can do it. The above in fable form: HERE 
Patterns
? easy This dimension is abstraction > difficult > (the human evolutionary path)





even plants do it (a bit) LOOK!
PERCEPTION 
MAYBE... ????? SENSING WHAT MIGHT BE
IMAGINATION 
LANGUAGE
METAPHOR 
MEAN DEEP DOWN ABSTRACT RELATIONSHIPS
MATHEMATICS 
HOW
DID
HE DO THAT
MAGIC 

Imagine that you are wandering in these three forests of dots but can see only about as far as the average distance between dots. 
Positions of more distant dots can be accurately determined by sampling a small cluster of nearby dots 
Information
content is minimal. These patterns can also represent
entropy, but only at a very deep level of abstraction.
The pattern is a square array of dots extending indefinitely in two dimension. 


How could you become able to predict where the more distant dots are located? 
Positions of distant dots cannot be estimated by examining nearby dots. (There is some uniformity of density of dots.) 
Information
content is maximal.
The dots are randomly placed. 

This forest of dots has almost as much pattern as the regular array. You can actually see the pattern...if you discover how. 
(but the order is hidden) 
The pattern is a twodimensional space group of point groups which are squares of nine dots in a symmetrical square array. 
Cross or spread your eyes slightly to fuse any adjacent pair of those four dots across the bottom of the array. The squares of nine are then seen stereoscopically. 
The above might be seen as a metaphor for life.
We look at the world (or listen, or feel, or...)
and, according to what we discover, decide how to act on that world.
We need to anticipate the outcome of our action so we can gain some kind
of advantage. It's through patterns, regularities, and constancies
that we can anticipate.
(The above concept is a good place to start to understand thermodynamics and, especially, entropy. Part of Erwin Schr?dinger's answer to the question "What is Life?" was the observation that the essence of "energy," the thing in food and fuel that is necessary for life and for the operations of our engines, is not the energy of physics but rather is the removal of unavoidable entropy buildups. Entropy is a measure of the predictability of what a system of particles—atoms, molecules, electrons, etc—will do next. CLICK HERE to clear up your misconceptions about entropy.) 
It is also a metaphor for science: . Knowledge starts with PERCEPTION, proceeds to speculation
then to definitions of terms
Science leaves ordinary human thinking by going to the next step:

Teachers love the magic. Want to share the magic. Teach by starting with the magic. BIG mistake! We cannot skip a step on the path to science. "Pseudoscience" skips that crucial step.
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