Ryan Robertson

Answered

2022-07-05

So I'm reading about the history of hyperbolic geometry and something like this came up: "two thousand years later, people gave up on trying to derive the fifth postulate from the other 4 and begun studying the consequences of the mathematical structure without the fifth postulate. The result is a coherent theory". My question: were there any possibility of incoherence? I understand some postulates are actually definition of objects, then, clearly, other postulates might be dependent on this other postulate, but it doesn't seem to be the case.

Answer & Explanation

Zichetti4b

Expert

2022-07-06Added 13 answers

When people say that noneuclidean geometry is equiconsistent with Euclidean geometry, and therefore all those old Renaissance proofs of the parallel postulate were incorrect, actually they're oversimplifying quite a bit, and taking things out of historical context.

Here is a sketch of a proof that elliptic geometry is inconsistent. In elliptic geometry, we have an axiom that says that for a given line L and a point p not on the line, no line through p exists that is parallel to L. From Euclid's first four postulates plus this non-parallelism postulate, we can prove that there is an upper limit on the area of any figure. But then that contradicts the third postulate, which says that we can construct a circle with any given center and radius, since according to the second postulate the radius can be made as big as desired.

Is this proof fallacious? Well, that depends completely on your interpretations of the second and third postulates. For some perfectly reasonable interpretations of them, it's a valid proof. Euclid wrote the postulates in language that was a model of rigor for its time, but before the consistency of noneuclidean geometry could really be addressed, people needed to decide on a disambiguation of them.

So, my answer to the question "was there any possibility of incoherence?" is that yes, there certainly was. It depended completely on the interpretatation of the postulates, which was not at all fixed to a sufficient level of rigor.

There is also the question of whether the first four postulates were consistent, and although nobody has ever believed that they weren't, it was only relatively recently in history that anything was really done to address this question. Tarski proved that in first-order logic, Euclidean geometry was consistent. (This is not a violation of Godel's theorems, because Godel's theorems only apply to theories that can describe a certain amount of arithmetic, which first-order Euclidean geometry can't do.) Furthermore, we now know that if Euclidean geometry is inconsistent, the real number system must also be inconsistent, which seems pretty unlikely; but this is an idea that would not have occurred to anyone before the invention of Cartesian geometry and the 19th-century formalization of the real number system.

Here is a sketch of a proof that elliptic geometry is inconsistent. In elliptic geometry, we have an axiom that says that for a given line L and a point p not on the line, no line through p exists that is parallel to L. From Euclid's first four postulates plus this non-parallelism postulate, we can prove that there is an upper limit on the area of any figure. But then that contradicts the third postulate, which says that we can construct a circle with any given center and radius, since according to the second postulate the radius can be made as big as desired.

Is this proof fallacious? Well, that depends completely on your interpretations of the second and third postulates. For some perfectly reasonable interpretations of them, it's a valid proof. Euclid wrote the postulates in language that was a model of rigor for its time, but before the consistency of noneuclidean geometry could really be addressed, people needed to decide on a disambiguation of them.

So, my answer to the question "was there any possibility of incoherence?" is that yes, there certainly was. It depended completely on the interpretatation of the postulates, which was not at all fixed to a sufficient level of rigor.

There is also the question of whether the first four postulates were consistent, and although nobody has ever believed that they weren't, it was only relatively recently in history that anything was really done to address this question. Tarski proved that in first-order logic, Euclidean geometry was consistent. (This is not a violation of Godel's theorems, because Godel's theorems only apply to theories that can describe a certain amount of arithmetic, which first-order Euclidean geometry can't do.) Furthermore, we now know that if Euclidean geometry is inconsistent, the real number system must also be inconsistent, which seems pretty unlikely; but this is an idea that would not have occurred to anyone before the invention of Cartesian geometry and the 19th-century formalization of the real number system.

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