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The article numbers (such as 0.0 for this article) are for troubleshooting problems with my writing. When you find that something is unclear, inconsistent, stupid, or wrong any other way, the numbers can help you tell me about it. Don't expect this structured number space to be densely filled-in or have other formal consistensies.
I may skip numbers arbitrarily.
I'll keep them in graduated order, but their
purpose is naming, not doing arithmetic.
They are numbers only up to
unique naming and ordering.
They should merely be unique in their space,
the space of this work.
0.2 The Rhythm of Reading Math
0.2.1 Eating
Many people read faster than they can eat. You may be used to devouring a chapter in about the same time as a snack, or a short book in the same time as a long meal with friends.
Math reading is slower than eating. Because mathematics is itself an unnatural language, it uses natural language in a way different from everyday talk. For example, "in general" means "in every case imaginable" in mathematical talk, but "usually" in casual talk. This makes a special demand on the person reading. Either you slow down and let your mind shape around some new thought, or you refuse to try.
In the second instance, you might tell yourself you can't do it and quit.
The only thing wrong in the second instance is telling yourself you can't. It's better to tell yourself you're not yet ready for a different learning rhythm.
0.2.2 Spurts
It's okay to read math in uneven spurts. That's in keeping with how it is discovered and with how it is written down. A mathematical philosopher, Russell, spent eight years sitting in his garden thinking about how to avoid writing a particular chapter of the book he and Whitehead were writing, hoping instead to write only a single sentence. He never found that sentence. Perhaps he did that chapter of Principia Mathematica well enough you could read it somewhat speedily, but it's more likely you would feel the need to take some time at it.
I remember a week in which I memorized two short statements from the book on which this work is based, just so I could think about them as I walked to work and back. I didn't see how they could both be true at the same time. What it took me a week to figure out was that Spencer-Brown was not claiming that the two conditions they describe can be made to hold simultaneously. Indeed they can't always, but that doesn't prevent both statements from being true independently.
Maybe if I had been a more experienced reader
of math I would have noticed right away.
As it was, it took me a week.
When "light dawned over Marblehead" I was
walking on a bridge over the South Branch of
the Chicago River. I stopped walking,
to digest the new thought.
0.3 Let's Say
In this work I will be let's-saying things rather carefully. The purpose is to blaze an unbroken trail of results all the way back to the foundations in the early part. Those let's-say statements are technically called axioms and postulates. They are what the whole enterprise of axiomatic development and proof rests on.
We're following a blazed axe-trail -- a trail of axioms, definitions and postulates. I'll change the original blazes a little bit, just to keep us out of a thicket I found confusing. You can read about that in a footnote if you are venturesome.
Footnote: Value from
Reversing Operator or Vice-Versa
Students of Spencer-Brown's original work will notice a great many differences in presentation, even in essentials. The greatest material difference I am aware of is that he defines his "cross" (our "pancake") as a reversing operator, while at the same time avoiding postulating value or any values, carefully saying instead, "the simple expressions." That made for the greatest difficulty I had with his exposition: what in tarnation gets reversed if not value, a perfectly serviceable, standard mathematical concept? A "marked state," fastidiously never called a value, then abandoned after the definition.
Here we do the reverse: postulate that the two simple expressions are indeed values -- not necessarily opposites until proven so -- and derive the operator character of the pancake from these values and one of the two arithmetic primitives, showing the reversing property of the operator as a further consequence.
That proof depends on first proving everywhereness for the invisible symbol.