On 9/22/2010 10:56 AM, Alex Shkotin wrote:
> Eliminate from theory all "*closed form* definition" and I do not
> get any idea about practically any theorem. (01)
Of course. It's impossible to state any major innovation with just
closed-form definitions. Any innovation requires something new, and
the new material often requires multiple steps (i.e., a recursive
definition) to construct it. (02)
Peirce observed the following distinction between the way that
Euclid and most mathematicians following him used the terms
'theorem' and 'corollary': (03)
1. Euclid introduces each new theorem with a diagram that
introduces some new construction. (04)
2. Every proof of a corollary uses the same diagram that
was introduced for the theorem. (05)
The diagram always incorporates new structure. Peirce's terms for
the distinction were 'theorematic' vs. 'corollarial' proofs. In effect,
a theorem requires an assumption (abduction), which is justified by
the subsequent deduction. A corollary uses the same assumption that
was justified by the proof of the theorem. (06)
Those misguided mathematicians who tried to eliminate all diagrams
from mathematics were philosophically confused and pedagogically
disastrous. One of the worst offenders was Dieudonné, who was
a hypocrite. (07)
He wrote a textbook on geometry that did not contain a single
diagram. But in one of his lectures, he got stuck in the middle
of a proof. He went to the side of the blackboard, drew a
diagram (which he kept hidden from the audience), erased the
diagram, and continued the proof. (08)
Even though he was a brilliant mathematician, Dieudonné deserved
to be dismissed for pedagogical malpractice. People like that,
when left unchecked by reality, can do enormous harm. (09)
In the US during the 1960s, they started the "New Math"
movement to teach children set theory before they learned
arithmetic. The mathematician Morris Kline documented the
results in a book with a title that summarizes the results:
_Why Johnny Can't Add: the Failure of the New Math_ . (010)
John Sowa (011)
PS: Whenever you read a math paper that starts off by saying
"A whatchamacallit is an n-tuple (A, B, C, D, E)..." you have
found somebody whose mind was warped by people like Dieudonné. (012)
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