stoutfellow

[My apologies for the long delay in posting this. There were several directions I could have gone at this point, and it took me a while to find one that I liked.]

The discovery of hyperbolic and elliptic geometry was the tip of the iceberg. In the second half of the nineteenth century, a wide variety of new geometries sprouted, with far-ranging consequences in and out of mathematics. For example:

Jean Poncelet resurrected Desargues' projective geometry.

Like Euclidean geometry, the new geometries - hyperbolic, elliptic, and projective - were "homogeneous"; that is, every point looks like every other. Gauss' inquiries into the geometry of surfaces introduced the possibility of inhomogeneous geometries. (Consider a "torus", the surface of a doughnut. The immediate vicinity of a point on the inner rim is almost-but-not-quite hyperbolic; that around a point on the outer rim is almost-but-not-quite elliptic, and points in between have in-between properties.)

Gauss' great student Bernhard Riemann extended these ideas, by considering spaces of three, four, and more dimensions and introducing the notion of a "Riemannian metric", describing how the geometry varies from point to point.

Projective geometry, unlike the others mentioned, does not involve any notion of distance, and new geometries, lacking any metric structure, followed in its wake; the field of topology emerged.

And on and on. Faced with this bewildering variety of geometries, mathematicians had several options. It was possible to restrict one's attention to one geometry: the study of hyperbolic geometry, or of projective geometry, or (to a lesser extent) venerable old Euclidean geometry still proved fruitful. Otherwise, one could take another step up the ladder of abstraction. Just as the passage from arithmetic to algebra involves moving from talking about specific numbers to talking about properties of all numbers, or of classes of numbers (squares, primes), so many geometers began to study entire classes of geometries: metric spaces, Riemannian spaces, Hausdorff spaces, and so on. Yet again, it was possible to choose a particular viewpoint from which to study geometries; differential geometry, algebraic geometry, and combinatorial geometry, to name three, came to be.

Most important was the philosophical change. No longer did geometers proclaim, "This is what the universe is like, and here is the truth that can be deduced about it"; increasingly, the view became, "If the universe were like this, here is what would be true about it." And when the time came that mathematical physicists like Hermann Minkowski and Albert Einstein began to suspect that the universe was not, in fact, what Euclid had thought it to be, other options were ready to hand. Elsewhere - in number theory, in quantum theory, in any number of fields pure and applied - the new geometries found use. By abandoning Euclid's claim to universal truth, geometers became able to find particular truths in previously undreamed-of areas. The reach of mathematics extended further than ever before.

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