## Re: [SCA Newcomers] Education in the Middle Ages

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• ... The knowledge of mathematics also depended on cultural context. For example, the Arab cultures invented much of our numbering system and higher
Message 1 of 10 , Aug 28, 2009
On Thu, 2009-08-27 at 19:50 -0400, bronwynmgn@... wrote:
> Village officials who couldn't write or read could none the less do
> enough math, and use marks on a tally stick to keep track of the grain
> and animals produced on the manor and render an exact account to the
> lord several times a year - and be able to figure money well enough to
> determine whether he owed the lord money or the other way around, and
> exactly how much money.

The knowledge of mathematics also depended on cultural context. For
example, the Arab cultures invented much of our numbering system and
higher mathematics, including a new mathematic called "al-gabr" (named
after the treatise "Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr
wa’l-muqābala" (Arabic for "The Compendious Book on Calculation by
Completion and Balancing"). This was written by Muhammad ibn Mūsā
al-Khwārizmī, a Persian mathematician circa 820 CE.

The "al-gabr" is not, as I once thought, part of the mathematician's
name, but rather is the name of the mathematical operation of moving an
equation term across the equal sign while negating it, for instance:

x + 20 = 3x (original equation)
x = 3x - 20 (al-gabr step)
-2x = -20 (al-muqabala step)
x = 10 (dividing both sides by -2 to get the answer)

The notion of symbolic logic is an elegant leap of intellect. Early
mathematical systems could perform concrete calculations but could not
express abstract relationships between quantities. According to
Wikipedia, the roots of algebra go back to the Babylonians. However,
during the intervening times not all cultures had the abstraction
concepts. Even today, there are people who can do computations very well
but whose brains just don't grasp abstract symbolic logic -- in the same
way that *my* brain doesn't grasp music or artistic creativity. :-)
We're all wired differently.

(I have an interest in this topic because my persona traded his
inheritance rights to his father's estate for tuition to study al-jabr
and astronomy in the Arab lands during one of the brief intervals when
we Byzantines weren't at war with them.)

Another important mathematical innovation that was not culturally
universal was the notion of place value. The Roman numeral system, for
example, has only a primitive left-or-right concept of place value. VI
means six, and IV means four, but they didn't have the concept of a base
number such as our decimal system. Computations of large quantities are
extremely cumbersome without a place-value (radix) system. Again, the
Babylonians had this, but later cultures like the Romans often did not.

The other often-overlooked mathematical breakthrough was the concept of
zero, as a number like any other rather than as the absence of a number.
The notion that you could use a symbol for "nothing" as part of a
calculation dates back to 9th century India, though earlier cultures
(including, once again, the Babylonians) had concepts that *almost* got
there. The key concept is that zero is a number like any other, that can
be included in calculations to generalize mathematical rules.

As an interesting side note, although modern computers treat zero as
just another number, we have had to go back to the medieval concept of a
different symbol to represent a placeholder for "something that should
have been a number but isn't". For example, when a computer program
tries to calculate 35/0, this produces an error. However, even though an
error message might be issued to a log or displayed to the user, you
still have to put *something* into the memory slot for the answer. The
Institute of Electrical and Electronic Engineers (IEEE) defined a
standard that includes special "NaN" (Not a Number) symbols that can be
used for situations like this. Essentially, they mark a memory location
as containing "invalid data" so that later calculations won't rely on
these data items as being real numbers.

The difference between modern and early medieval thought is that we
treat zero as a number but retain the concept of a placeholder for
things that truly *aren't* numbers, such as the result of division by
zero. In early medieval times, zero was thought of as being somehow not
a real number, because you couldn't have "something" that represented
"nothing". Again, it is a leap in intellect to understand the difference
between the symbol for a number and the abstract concept of "number".

Very interesting thread -- thanks to our resident historical scholars
for some very enlightening posts!

Justin

--
()xxxx[]::::::::::::::::::> <::::::::::::::::::[]xxxx()
Maistor Justinos Tekton called Justin (Scott Courtney)
Gules, on a bezant a fleam sable and on a chief dovetailed Or two keys
fesswise reversed sable.

justin@... http://4th.com/sca/justin/
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