Subject: Re: [HM] What is good math history?
From: Milo Gardner (milo.gardner@24stex.com)
Date: Sat Jan 15 2000 - 08:14:33 EST
Dear HM list members:
Jan Mycielski's recent post covering certain 20th century foundations
of mathematics was very well written. Jan detailed aspects of Logicism,
Formalism and Intuitionism in ways that are mathematically true in
the 20th century. Clearly, modern foundations of calculus does require
the use of the good ideas of axiomatic set theory, and other related
mathematical tools, as outlined by Jan.
However, I am intrigued by the historical and intellectual basis for Jan's
introduction of Plato as one difference between imaginary and real objects,
as he suspected are unique to his point of view. Clearly from a 19th
century point of view, the foundation of algebra itself (and its fundamental
theorem) would not have taken place without Gauss's addition of imaginary
numbers to the mathematicians' tool kit. Plato and his view of the
imaginary and real objects are not needed to open up the domains of the
additional numbers needed by our now 21th century mathematicians. Yet, in
a math history course, Plato needs to be discussed for several reasons,
one being the range of dead ends it has produced.
That is to suggest, Gauss may have complained of Jan's 20th century
view of mathematics, if he was alive today. Gauss showed that Newton's
foundations of calculus was based on much older ideas, such as the work
of Archimedes. Stated another way, modern mathematicians need to be taught
to be respectful to many 'giants' on which they must stand to achieve a
mathematician's point of view.
In closing, I would like to cover two final points, the role of pedagogy
in a history of mathematics class, and the role of all the numbers,
as each was 'discovered' along the way. Math history is filled
with wide ranging pedagogical methods, such as Jan outlined in the 20th
century under the titles of logicism, formalism and intuitionism.
Any class of pedagogies can be used as 'blinders', or as eye openers.
Considering the eye opening aspects, a fair review of pedagogies
should cite the range of numbers that the giants of the past 'discovered'
and applied within their selected discipline.
For example, Pythagoras can be given credit for introducing aspects
of irrational numbers to the history of mathematics. Greeks like
Archimedes generalized the use of any number, including irrational and
higher order numbers like pi, in terms of the oldest foundations of
calculus, one being the use of the 1/4th geometric series to find exact
areas of certain geometric shapes. Plato would have accepted aspects of
Pythagorean views, and discarded other aspects, based on his view of the
role of rational numbers, a step backwards from Archimedes' generalized
acceptance of all known numbers.
Regards,
Milo Gardner
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