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.

John W. Weil