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

There’s nothing to “solve” necessarily. Euclidean geometry is made up of 5 postulates. Postulates are essentially assumed to be true. So the “problem” is that Euclid has 4 very simple beautiful postulates but the 5th one is a bit wordy and doesn’t “fit in” with the other 4 so mathematicians have tried to deduce the 5th postulate from the first four because if they could do that, then the 5th postulate would be obsolete and could be removed. It’s been attempted many times but never successfully deduced from the first 4.

There are much more meaningful and important problems to solve in mathematics than this but it is what it is.

Euclid’s fifth postulate has not been proven, it has to fall back on the parallel lines postulate for its utility. Because it is possible to create entirely self-consistent, non-Euclidean geometries where the parallel postulate doesn’t hold, that means that it’s possible that the 5th *might* not hold even in the Euclidean geometry. Proving it true or false would mean that the rest of the geometry would be more or less reliable.

they were trying to find a way to show that rule 5 isn’t needed and was a consequence or rules 1 – 4. perhaps it was intuition rule 5 is more complicated than the others maybe they just didn’t like it in the end rule 5 is a axiom but swapping it out is fruitful and is the basis for non euclidean geometries.

A proof is a rigorous logical argument, assuming only certain axioms, and arrive at a specific conclusion. The argument must be rigorous beyond any doubts. Most importantly, you cannot use any facts beyond the axioms given. You cannot draw a diagram and point at it (it will help people understand the proof, but the argument must work even if you don’t have the diagram). You cannot rely on what you mean by a “point” or a “line”. For example, if you replace the words “point” with “gfjgndfk” and “line” with “gfjkgnd”, in both axioms and your proof, your proof is still logically sound.

To understand more details, you need to understand the idea of a formal proof, and the Frege-Hilbert controversy. Not to worry, you’re basically on the side of Frege, so even a philosopher of the 20th century get confused about it.

In modern conception, a proof is an human-readable outline of a *formal proof*; the outline should be detailed enough that the intended reader can infer that a formal proof can, in theory, be produced.

So what is a formal proof? The kind of proof you do in geometry came fairly close to being a formal proof, but there are still more. A *formal* proof, as it name implies, is a string of logical symbols in the correct form. What is means by “correct form” is different depends on which logical framework is used, but in general, they have the following purpose:

– Every statement inside the proof is made out of logical symbols, according to a specific grammar.

– The proof declare some axioms to use, and declare what conclusion it is making.

– Every step of a proof correspond to a basic logical step, as basic as possible.

– The entire proof can be checked by just checking what the proof look like. A machine can blindly check the proof for correctness without understanding what it said.

If you look from this perspective, producing a formal proof is like a game of string manipulation, and the question of whether a proof exist is no differences in principal from asking whether some numbers can be written in a specific form (for example, “can number 1/2 be written as sum and products of integers?”). There are no meaning to words like “point” and “line”, they are just symbols to be moved around. This kind of detachment between words and meaning is the hallmark of modern mathematics. From this perspective you can see why there is something needed to be done to prove Euclid’s 5th postulates (from the rest of the axioms), the question is asking for a sequence of symbols satisfying some properties. You can’t just draw a diagram and point at it.

The next question is why? Why would people care about whether Euclid’s 5th postulate is provable from the rest? Actually, you will talk more generally, why would people care about whether something obviously true be provable for previous axioms? There are 2 cases:

– If the obvious claim is not actually provable from the previous axioms, then it tells us that we are limited by our imagination. Something is there. Something that looks very similar to what we had imagined, and hence have many interesting properties, and yet different in its own way. If you we just give up, we might have never discovered these exotic objects. This is the case with Euclid’s 5th postulates, we have non-Euclidean geometry. We eventually discover elliptic geometry and hyperbolic geometry. Thousands of years ago, we only have flat geometry and spherical geometry (we came *this* close to discover elliptic geometry back then, since spherical geometry and elliptic geometry are very similar). Nowaday, we know exactly what are the possible geometries in 2 dimensions; and in some weaker sense, we also classified 3-dimensional geometries as well (this is the famous story of the guy who reject the 1 million prize and isolate himself from the world, Perelman).

– If the obvious claim is actually provable from previous axioms, then proving it will tell us that we no longer have to worry about exotic objects where the claim is false (like in the previous paragraph), because those thing don’t exist. Not just that, if proving the obvious claim is actually hard, it tells us that our mathematical knowledge is limited, so it’s an impetus to develop more math knowledge. The real math isn’t the final confirmation that the claim is true, it’s the theorems we proved along the way. For example, this is what happened to Fermat’s Last Theorem; mathematician essentially develop a whole bunch of technology that eventually just bulldozer over the claim on the way.

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.

…

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”)

——

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.

It was way more complicates and verbose than thenother 4 postulates, so mathmaticians tried to “prove” it in terms of the other 4 so that they could just remove it and its verbosity.

However these days we tend to accept that the fifth isn’t “provable” and removing it results in a different, completely covalid system of geometry: non euclidian geometry (like geometry done on a curved surface)

I’m no expert so take this with a grain of salt.

The issue is when your creating a theory of how the world works at some point you have to make assumptions. The more assumptions you make the more susceptible your theory is to being proven wrong. As I understand it the attempts to “prove” the fifth postulate was an attempt to describe it in terms of the previous 4 postulates thus removing the fifth and making the theory more robust.