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
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.
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