While in popular literature and philosophical circles the discovery of non-Euclidean geometry may seem like an earth-shattering discovery, subverting our confidence in the absolute truth and nature of mathematics and reality, its actual study may seem like a trick or word play.

Consider Spherical Geometry, an example of non-Euclidean Geometry which I think should be the easiest to understand and the most empirically familiar. To understand how Non-Euclidean geometry works you have to understand how geometricians understand a "line" and a "point". We normally think that a "line" is just a "straight line", but if you restrict the referent of "line" to just the great circles on a sphere, and "point" to any points *on* the surface of the sphere (thus there are no "points" inside the sphere), you get a very different and interesting geometry. (Roughly, a great circle is a line on the surface of the sphere generated by cutting a plane passing through the center of the sphere.)

Thus, given that on the sphere the only "lines", or line segments, are these great circles on the sphere, or parts thereof, and the only "points" are points on the surface of the sphere, it is easy to see why core axioms of Euclidean Geometry fail to obtain here. Consider for example the parallel postulate (using Playfair's Version): Given any "line" CD, and any "point" S not on the line CD, there exists one and only one "line" QR passing through the "point" S that does not meet the CD.

But in Spherical Geometry, as *we* have restricted the referent of "line" and "point", there are no parallel lines because every "line" (great circle), passing through a point P not on another "line" (great circle) l, must meet the latter line/great circle somewhere, either at the equator or at the poles.

At this point you'll complain, well, that's cheating, you have just *redefined* the meaning of "line" and restricted it to the surface of this sphere, but that's not what people usually mean by "line" in "the real world", they mean a "straight line". But what is the "real world"? We're literally talking about a sphere which is as close an approximation to the literal world/earth (technically more like an ellipsoid). It seems like doing geometry *on the actual spherical/elliptical planet *is as "real world" an activity as one can get.

Even if we want to discuss the usual Euclidean geometry of a "straight line" and the parallel postulate, the problem is that our "common sense" understanding of this isn't very "real" either. Think again about Playfair's parallel postulate, there is one and only one line on a point not on another line that will never meet the latter line. But has anyone ever seen an *infinitely long line* that has never met another line? "In the real world"? Or even in our universe? (Which according to some physicists is merely finite in extent, etc) This brings us to an important insight: what we imagine to be a "common sense" "real world" axiom is actually itself an idealisation, dare I even say, a construct, an idealised infinitely long 'straight' line which no one has ever seen anywhere in the "real world". If we are allowed to idealise infinitely long straight lines, which no one has ever seen "in the real world", why can't we also idealise other lines on spheres and other exotic geometries?

After all, if the "real world" is a cosmos constituted by a vast elliptical or spherical space (maybe due to General Relativity or whatever), then *locally* it looks like the parallel postulate holds, given a "line" or "great circle" on this cosmic sphere, that great circle wouldn't meet another great circle for millions and millions and millions of miles. So, "for all practical purposes", it *looks* parallel, but eventually it may still meet somewhere halfway across the galaxy.

The point of this little excursion into Non-Euclidean Geometry is a two-fold: it is not as earth shattering a discovery as we might be lead to believe, it may even feel a little disappointing because it feels like "cheating" by redefining the ordinary meaning of words, but it is also an important insight into how much *unconscious* expectations goes into our definition and referent of words. Yes, "everyone knows" that when we use the word "line", we "really" mean a straight line, in that sense Spherical Geometry is "cheating", but then we need to ask ourselves, why do we privilege "the straight line" as the universal standard definition of line? Who is to say that this definition is always applicable everywhere? Non-Euclidean geometry, by interrogating the "ordinary" meaning of words, at the same time deepens our understanding of these ordinary words, expands our minds to other possibilities, while also introducing a new rigor in being more precise about what we mean or refer to by our words.