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Re: [ontolog-forum] Grand Unified Theories

To: ontolog-forum@xxxxxxxxxxxxxxxx
From: John F Sowa <sowa@xxxxxxxxxxx>
Date: Tue, 03 Mar 2015 23:30:36 -0500
Message-id: <54F68A6C.9080407@xxxxxxxxxxx>
Pat, David, Bruce, and Ed L,    (01)

>> Double entry bookkeeping was in use in the 13th century. Calculus was
>> developed in the 17th century. Four hundred years is a pretty long
>> ferment.    (02)

> To a certain extent I'd argue that things moved way slower back then.    (03)

The two centuries from the 13th c to the 15th c were a very rapid
period of development.  When the first European universities were
established in the 12th c, Europe was an underdeveloped backwater
compared to the East (everything from Greece and Egypt eastward to
Persia, India, and China).    (04)

But by the 13th c, an educated class was creating new science,
technology, and finance.  By the 15th c, Europe had the Renaissance
in the arts, and their economies were strong enough to send ships
on the high seas -- both eastward and westward.  By the 17th c,
they were setting up colonies around the world.    (05)

As just one example, in 1275 there were no mechanical clocks in
Europe.  But by 1300, nearly every town either had a church with
a clock tower or was planning to build one.  The first clocks
just rang bells on the hour, then every 15 minutes, then an hour
hand, then a minute hand, then all sorts of dials for phases of
the moon, mechanical things popping out, playing melodies, etc.    (06)

> "reality is continuous, concepts are discrete."  (does that
> include calculus?)    (07)

> I expect they will come to realize that the teaching of mathematics
> and other technical subjects is best redeveloped using well-designed
> information building blocks as foundational.    (08)

This gets into a huge swamp of issues -- about what to teach,
how to teach it, and where to get/find/train teachers.    (09)

High-school teaching declined significantly in the past half
century -- partly because talented women can now get better jobs.    (010)

I credit my high-school English teacher, Miss Elliott, with teaching
me how to write.  Today, women with her talents would be teaching
at the university level.    (011)

re calculus:  Yes, it can represent and analyze continuous
spaces and functions on them.  And yes, many teachers, especially
at the high-school level, don't know how to teach calculus.    (012)

Furthermore, most textbooks still use the epsilon-delta methods,
which can be hopelessly confusing to a beginner (or a poorly trained
teacher).  I was fortunate to learn calculus from my father's old
textbook, which used the (then disparaged) method of infinitesimals.    (013)

But in the 1960s, Abraham Robinson proved that infinitesimals
are perfectly respectable.  They are the original way that
Leibniz thought about the calculus, and they are the best way
to teach it beginners.    (014)

John    (015)

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