Mathematics as the Science of Patterns - Conclusion

None of these theorems which Steiner demonstrates is immediate without the idea of the ‘power of a point’, but all are almost obvious with it.   How it makes these things obvious is not by supplying some previously unknown property of some geometrical object, but by supplying a kind of organizational principle, a pattern to look for, something providing “scientific unity and coherence,” as Steiner says in another context.  Thus, the comparison between Euclid and Steiner makes it clear that the difference between them is not so much knowledge as it is perspective and how they perceive what it is they are doing when they do mathematics.  Both seem to be concerned with circles, but, in fact, while Euclid looks at circles as objects with properties, Steiner looks at circles as the carriers of patterns.  The ability to take a pattern as a starting point, even if one has a definite object in view, placed Steiner and moderns like him in a conceptual camp quite different from that of Euclid—in fact, one might say that if truth is a great ocean, as Newton put it, surely Euclid and Steiner stood on opposite shores.  But where should we mathematics teachers stand, especially those of us who consider the history of mathematics important?  I am afraid we have no other choice—we are all moderns.  Of course there is no shame in that: the approach to mathematics via patterns has proven very powerful, and mathematics teachers, therefore, do well by helping students see this strength of modern mathematics. But they ought to make it clear also how far the search for pattern is indeed distinct of modern mathematics, that mathematicians have not always approached their subject in this way.  By making the effort—and it is not an easy one—to see how mathematics has been different in the past, teachers will not only show that mathematics can change its way of seeing things, but also that their students may be the ones to change it.