by David Zethmayr
Contents
ForewordTopic 0 How To Read Math
Topic 1 Let's Say We Make A Calculus
Topic 2 Revealing an Operator
Topic 3 The Pancake Algebra
Topic 4 Implication, Root of Logic
Topic 4b Meta-implication: modus ponens, sorites
Topic 5 Antilogisms
Topic 6 Syllogisms
Topic 7 Predicate Logic in Pancake
Topic 8 Boolean in PancakeAppendix A Summary of Algebraic Consequences
This is a set of traps, laid out carefully for
your early success in doing axiomatic proofs.
The activities laid out here can be done
quite independently of the regular curriculum,
beginning at no particular stage of it.
It is for the bright, the bored, the ADD
(like me and most of my family) and
the dyslexic.
It is for the teacher wanting a sustainable,
perennial, coherent enrichment activity for
them without worry over prerequisites.
This branch of math makes success in proof
accessible to anyone. Less than ever before is there
excuse for the perennial complaint heard in graduate
programs in mathematics: students who fear proof because
of too little experience with it. If you want strong math
majors, push the teachers to use "pancake"
in the lower grades.
Don't wait until plane geometry to introduce proof --
there's too much other careful abstraction to geometry.
Proof, clearly a more fundamental skill than geometry,
should be already in hand.
In what follows it will seem that I talk to students and
teachers without making much difference. That's because
I think the best teachers are still students and
don't fear to be known to their students as
fellow-students.
And the best students are teachers: self-teachers,
future teachers, or any professionals generous
with what they learn.
I hope the teachers reading this will
draw the inference
that I assume they are among the best.
To keep the main thread short, there are links to notes,
marked [*]. At the end of each note there is a backlink
to its context.
What is a formal system, anyway, and why does it matter? Our species has been building formal systems as a matter of play and other work for a long time.
Most games, and spectator sports, are formal systems.
Calculation and resolution seem to be a pervasive notion in formal systems. What is calculated, what is resolved, by playing a game? A game formally sets up a conflict to be resolved. Sometimes, in the simpler games, it's a single winner. A single winner! What an abstraction! And what an illusion.
In other games it's a team that wins or loses. In the game of planetary hygiene, with its set of rules we are learning, forgetting, re-learning, ignoring -- it is species and sometimes whole phyla that that win or lose, and life itself that thrives or languishes.
The syllogistic logic completely investigated long ago by Aristotle and company is a formal system. Euclid's geometry is a formal system. Number arithmetic and algebra, trigonometry, analytic geometry and the infinitesimal calculus all stem from the same formal system. The pancake calculus we start with here is a formal system.
Who can deal with formal systems?
You can.
Did you and another child ever
see a patterned floor and agree, "let's say you can't
step on any of the green squares, okay?" You noticed
a formal regularity and made a formal system on it.
You defined an illegal move, agreeing
that all other moves were legal.
[*]
What you did was in the spirit of pure mathematics.
You did this without any
worry about what use your formal system
might have. Your immediate motivation was fun,
or a kind of interest related to beauty,
and that was enough to make it worthwhile. You agreed
on a statement requiring no proof
--an axiom--a "let's say"--and played with it just to see
what might happen.
Do symbols on the page jump around and rearrange themselves
on you? That's okay too; it doesn't mean you can't think
straight. Gather some of the
concrete materials
suggested below and have someone read this with you
and play/work along.
Stop playing around and DO something.
I was only playing.
We are, most of us, quite adept at abstracting --artificially isolating-- some aspect of a problem in order to get understanding and control. To get the best use of this tool we must recognize what it is not. An abstraction is not the problem itself, but only an incomplete model of the problem. An artifice.
You have been using
abstraction ever since you learned to say "no."
Abstraction is the most powerful intellectual tool
ever. It made language possible
(or vice versa, maybe). Part of your work as
a learner is to distinguish between abstractions and
the problems that spawn them.
We are not at first accustomed to examining the roots of our knowledge. I guess that's because it's not regarded as a basic survival skill. We accept what we think we know because it mostly seems to "work." In those formal systems labeled mathematics, however, the notion of axiomatic proof is central. The infinitesimal calculus invented by Newton and Leibnitz was used for many decades by astronomers and other scientists and engineers before being firmly set on its axiomatic foundation in a complete formal system. It "worked;" it gave useful, credible results. But nobody could say it was proven until, by careful, close thinking, Dedekind [*] made the crucial breakthrough to proof. After that, it became possible to say exactly in what circumstances the infinitesimal calculus can be validly applied.
I say again, the math presented here makes
success in proof accessible to anyone.
.
[*] Notes, with backlinks to context
next
contents
LawsOfForm.org where to meet others exploring this mathematics
David Zethmayr
Manteno, Illinois, US(A)