As someone who failed his algebra 1 final twice and geometry once, can someone explain to me what is meant be “proving” Euclid’s postulate five? Like, the point of the postulate is two lines that cross another line will, at some point, meet if they’re angled toward each other. I get that.
What I don’t understand is why that needs to be ‘solved’ or ‘proven’. What were so many mathematicians trying to do? How would they go about ‘completing’ it? Why did it need to be completed?
In: Mathematics
Euclid was an ancient Greek who formalized the math of geometry. In the USA, you usually encounter him second year in high school, I think? He did so using a few “axioms” – things that were self-evident, like “between any two points, you can draw a straight line”. There were five of these, along with five statements on the order of “If equals are added to equals, the reults are equal” which basically said “hey, this is basic arithmetic we’re using”.
From these axioms, he developed a whole lot of simple proofs of things in plane geometry, using unassailable logic. The first 28 proofs just used the first four axioms. See if maybe you can see why:
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1. Between any two points, you can draw a straight line.
2. If you have a straight line that ends in a point, you can continue the straight line past that point, without limit.
3. Given a point and a distance, you can make a circle centered on that point, whose radius is that distance.
4. Any right angle (90-degree angle, or square corner) is equal to any other right angle. No matter how it’s turned on the paper.
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5. If a straight line crosses two other straight lines, and the ‘interior angles’ – the two angles facing each other inside the crossing – on one side add up to less than two right angles do, then extending those two other straight lines indefinitely? Causes them to meet & cross, on the side that had the interior angles less than two right angles. (“The Parallel Postulate”)
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Those are the five axioms. FOR SOME REASON, the ancients didn’t think the fifth one was as pretty or intuitive as the first four. You can reword it in a good many ways, one of which is “Two lines are parallel ONLY if a line crossing them has its interior angles add up to two right angles exactly, on both sides.”
They felt SURE it wasn’t independent of the first four, and could be deduced, or proven, from them somehow. But nobody EVER managed to do so, for thousands of years. Some people thought they had, but they’d usually just flexed it into a different form which stated something they didn’t think was true (but was). It _frustrated_ mathematicians greatly, not least because it WASN’T beautiful and “self-evident” the way the first four were.
Problem is … Euclid didn’t actually DEFINE what a “point” or “line” or “angle” WERE. He had hidden assumptions in there that he didn’t realize were there. (And a few others, like “a line will have more than one point on it” – simple, right? but you CANNOT prove it from what’s given.) Including what a “plane” was – which is gonna be important in “plane geometry”, right?
It turns out there are at least two forms of “non-Euclidean” geometry which use the first four axioms, and a different fifth axiom. Instead of “you can always draw one, and only one, line through a point parallel to another given line the point isn’t on”, one of them says “No such parallel line can be drawn” (!). The other says “An unlimited number of such parallel lines, each different, can be drawn” (!!).
And you can prove the same sorts of things, some analogous things, and some completely different things, for each of these alternates. The first 28 of Euclid’s proofs carry right over, because they didn’t even use the parallel postulate. But what these are clearly not is “ordinary flat geometry on a flat plane that extends off infinitely in every direction”. Each time someone had thought they’d “proven” the parallel postulate from the others, they’d actually proved something that worked for one of these other two, and said “wait. that doesn’t make sense. that’s wrong. you can clearly see (blah blah yadda yadda)”.
One of them I can even ELY5: it’s the geometry on a SPHERE, a ball, a globe of the world. Here, a “point” is a PAIR of points on opposite sides of the sphere, exactly, and a “line” is a great circle. Any two different “lines” meet at one “point”, one pair of points; any two different “points” determine exactly one “line”. You can draw smaller circles, all the way up to a great circle; for a radius bigger than that, you end up drawing a smaller and smaller circle on the other side of the sphere, and this repeates periodically. You can draw triangles. Etc.
But … NO two “lines” are ever parallel; two “lines” ALWAYS meet if you go far enough. The sum of a triangle’s inside angles is always BIGGER than two right angles, though if it’s a very small triangle the sum’s only a little bigger. And you can extend a line indefinitely … except that eventually it wraps back around and starts tracing itself again. But it’s a perfectly good, perfectly consistent geometry – that isn’t on a flat infinite plane.
–Dave, the other one is called “hyperbolic geometry”, and ends up being on a kind of surface that’s curved like a saddle, or a Pringle, at every point; this one’s almost impossible to visualize, because it doesn’t fit into three-space correctly. Here “lines” are also curved, but “away” from each other instead of towards; you can have infinitely many other lines, all different, that never meet a given line in either direction; and a triangle’s interior angles always sum to LESS than two right angles.
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