“The treatise itself, therefore, contains only twenty-four pages — the most extraordinary two dozen pages in the whole history of thought!”Bolyai's non-Euclidean geometry was published in 1832.

This Hungarian postage stamp does not depict János Bolyai. No portraits of him survive. For more information about the "real face of János Bolyai," click the picture to read Tamás Dénes's article on the subject.

“How different with Bolyai János and Lobachévski, who claimed at once, unflinchingly, that their discovery marked an epoch in human thought so momentous as to be unsurpassed by anything recorded in the history of philosophy or of science, demonstrating as had never been proved before the supremacy of pure reason at the very moment of overthrowing what had forever seemed its surest possession, the axioms of geometry.”

— George Bruce Halsted, on János Bolyai’s treatise on non-Euclidean geometry,The Science of Absolute Space.

Another geometry book says:

“The discoverers of noneuclidean geometry fared somewhat like the biblical king Saul. Saul was looking for some donkeys and found a kingdom. The mathematicians wanted merely to pick a hole in old Euclid and show that one of his postulates which he though was not deducible from the others is, in fact, so deducible. In this they failed. But they found a new world, a geometry in which there are infinitely many lines parallel to a given line and passing through a given point; in which the sum of the angles in a triangle is less than two right angles; and which is nevertheless free of contradiction.”The blog concludes:

These quotes all seem a bit dramatic for geometry, but it’s easy not to know how truly revolutionary the discovery of non-Euclidean geometry was. Halsted’s description of Bolyai’s paper as “the most extraordinary two dozen pages in the whole history of thought” certainly sounds hyperbolic, but can you find 24 other pages that compete with it?The creation of non-Euclidean geometry by Bolyai, Gauss, and others was indeed a great vindication of geometry and the axiomatic method. It later became central to 20th century physics when Poincare and Minkowski created a theory of relativity based on it.

Roger,

ReplyDeleteNon-Euclidean geometry is logically dependent upon Euclidean geometry. The converse is not also true. Non-Euclidean geometry does not supplant or replace Euclidean geometry.

As for the statement

" But they found a new world, a geometry in which there are infinitely many lines parallel to a given line and passing through a given point; in which the sum of the angles in a triangle is less than two right angles; and which is nevertheless free of contradiction..."

Sounds about as useful as MWI to me. You can get any answer you like, which what makes it pretty useless too. If you can bend the meanings of words to this point where a line isn't a line and a triangle is no longer a triangle, why yes, you can pretty much make any claim you like re-defining a line as a squiggle and claim a triangle can tap dance I suppose.

"How many legs does a dog have if you call the tail a leg? Four. Calling a tail a leg doesn't make it a leg."

Maybe Lincoln knew something Minkowski and Poincare did not.