For many of us geometry was our first brush with
mathematics beyond the manipulation of numbers.
Its reliance on a few, seemingly reasonable,
starting points, followed by a consistent derivation of all kinds of
other results was a wonder in our eyes.
What a thing of beauty mathematics was!

Going on to algebra and trigonometry found a
reasonably similar paradigm, with extensive results derived logically
from a small start and with the whole presenting a consistent and
unified picture.
Even calculus, at least in its earlier
stages, appeared to be in the same mold.

However, as we got deeper into more advanced
mathematics, this unity broke down.
Mathematics became a bag of techniques, each
of which stood alone and applied to a limited range of problems.
This range might or might not overlap other
techniques in other ranges.
Each of the techniques was logically derived
from the earlier body of mathematics, but various specializations or
approximations or reformulations were used in order to get successful
results that could be used within the region of validity of these
specializations.

We came from expecting a unity of the
mathematical world, as exemplified by Euclid, to a world where our human
capabilities were insufficient to keep that unity and where success
resulted from making quite human approximations to the more general
mathematical reality.
Thus, Euclid did mislead us in that his
path, his paradigm, only goes so far.
If we wish to proceed further than his path
goes, we must be prepared for piecemeal success and inaccuracies outside
of limited ranges.
In other words, mathematics begins to look
more like physics!

But Euclid, and our Western, Greek intellectual
heritage, also had an influence on other areas outside of mathematics.
Philosophy, for example, debated whether it
could use the tools of logic from Euclidian geometry and thus also show
the exquisite unity and deductive power that mathematics -- and geometry
in particular – showed.
This same question arose in the history of
many other fields in the sciences and humanities, but none of them was
able to find a clear path to some approach that resembled Euclidean
geometry except in severely limited areas.

In physics, as well, Newton showed us a simple
set of tools describing the motion of objects that could be used with
enormous success in a wide variety of applications.
This, like Euclidian geometry, suggested
that other fields would yield to such a paradigm: find the basic laws
and a wide field of application would follow.
Other scientific fields attempted to do
this, but the success was spotty.
Maxwell’s Laws of Electromagnetism proved
another stellar example of success.
But other fields, for example, geology,
attempted to find analogous basic principles without success.
As with Euclid, Newton gave us an example to
follow which proved to be largely impossible to reproduce in other areas
of inquiry.

Because of our early experience in education,
where Euclid and Newton feature prominently, we develop expectations of
what we will find in other fields of exploration and actually feel
cheated when the wisdom developed in other fields proves to be messy,
inelegant and not universal.
Of course, it isn’t Euclid and Newton that
misled us, but rather that our educational experiences use these
exquisite examples of mathematical and scientific beauty without
preparing us for the messiness that characterizes much of the rest of
the real world.
This disillusionment might be avoided if we
were shown some examples of actual useful fields of work, messiness and
all, so that we could better appreciate the beauty of Euclid and Newton
(and Maxwell and …) when we get to them, recognizing that they are not
examples to be generalized but rather exceptions to a messy world to be
appreciated for the rare jewels that they are.