On 10/27/09, Jerry wrote:
> Does anyone have a definition of algebra that they like and
> would use for say a college algebra course?
>
> Is it possible to define algebra in a way that makes sense
> in college algebra, and yet would still encompass things
> like linear algebra and the algebra of groups and rings?
>
> Also, I've heard it said that probability does not have an
> algebra. What does this mean?
>
> Thank you.

I have not heard that expression about probability. Without
knowing the context, I wouldn't want to guess what it means.

As for a definition of algebra, I am sorry that I don't have a
clean or all encompassing one but I have a sort of background
to it if that would do? I always think of algebra in terms of
WHO invented it and who developed it. I know that people
usually think of math as something that exists independently
of humans but that humans discover piece by piece. I don't
agree with this entirely because in many ways, math is also a
human construct and it is developed based on the limitations,
experiences, cultures, and biological/genetic characteristics
of the humans who create it. Algebra began in the fertile
crescent, was further developed in the middle east and
mediterranean, and finally has reached its latest stage of
development mostly by western European peoples. What all these
peoples have in common is that they are caucasians. Studies
have shown that caucasians tend to think about or practice
math in a way that is called "linguistic." In other words,
the math developed by caucasians has been of a linguistic
type in that it uses letters to represent numbers and it uses
what you could characterize as math 'sentences' to explore
mathematics. For example A + B = C is a sort of math
sentence. Asians on the other hand tend on the whole to have
an affinity to a sort of math that I would call 'spatial'
rather than 'linguistic.'

I know that we take it for granted that this is the only way
that math could have developed but I think that is not so. I
think that math developed in an algebraic manner because this
fit the mental capacities and propensities of the jews, arabs,
phoenicians, sumerians, Europeans, Babylonians, Egyptians,
Greeks and other caucasians who have the largest hand in
developing it. Perhaps, if some other race had had the most
influence in the body of math that is predominant, then it
would look completely different. I don't mean that the laws of
math would be different. I just mean that we would study it
in an entirely different way.

For example, we take Calculus for granted. But without the
development of the coordinate plane by DesCartes and other
French mathematicians, we would not have the concept of slope
and we would not have Calculus. Calculus is a direct
development from Algebra and in a way, is just an advanced
form of 'linguistic' mathematics. But for slopes and the
coordinate plane, we would not have calculus. We would solve
our math and engineering problems in some entirely different
way.

The coordinate plane itself would probably not be so important
in math if it were not for the mass production of paper
allowing middle class people to have paper to create their
math on. So you could also consider math now to be in many
ways a 2 dimensional field more comfortably than 3
dimensional. Students learn to work 2 dimensional calculus
problems far sooner and more easily than 3 dimensional ones.
That is because we are all used to thinking on paper, which is
a 2 dimensional medium.

Before mathematicians and students all had paper, math was
much more 3 dimensional as in geometry.

I don't know if I am making any sense but I hope so. I guess
what I mean is that math is a product of the people and
times. We are mainly a people who are linguistically-oriented
in our math thinking whose main technology for studying and
producing math is a 2 dimensional medium (paper). When
computers can create a 3 dimensional holograph around people,
then we will see a huge shift in mathematics as children
spend their lives thinking in a 3 dimensional medium instead
of 2 dimensional.

This will produce entire new fields of mathematics like what
happened after paper became widely available. Before paper,
geometry was king in math. After paper, algebra was king to
the degree that people even discuss eliminating geometry as a
high school course. The seeds of algebra were always there in
caucasian cultures but could not bloom until paper came
cheaply in the 1400's.

Because asians tend to be more spatial thinkers, it will be
very interesting to see whether a 3 dimensional medium like a
computer hologram that surrounds the learner/mathematician
might not favor them in making more advances in math in the
future. I think the 2 dimensional medium favored caucasians
which is why it was in the mediterranean and Europe that math
exploded after paper became easy to mass produce.

Did I say anything that made sense to your original question

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