[SJC_Hacker]

Unfortunately it seems their proof already had the Pythagorean Theorem embedded within its implicit assumptions - they define measure of an angle through rotation of a circle. They don't explicitly define circle, but from their diagram they hint at the "understood" definition, namely a set of points equidistance from a central point, while using Euclidean distance as the metric.

[lupire]

That's not true at all.

To understand why, read Euclid.

[SJC_Hacker]

Geometry has made a bit of progress since Euclid's time. Its become a bit more rigorous.

Euclidean geometry is based on five axioms, and some other terms left undefined.

The fifth postulate - the parallel postulate - was considered so irksome that for hundreds of years, many attempted to prove it using the other four, but failed to do so, and almost drove some crazy. In the late 19th century it was shown you can generate perfectly valid geometries if you assume it to be false somehow - either no-parallel (spherical geometry) or infinite parallel (hyperbolic)

Euclid's third postulate - "a circle can be drawn with any center and radius - doesn't define how to do it. Like I could draw a "circle with a radius of 1" using taxicab distance, and it would look like a diamond shape.

Conversely, if you take the "conventional" definition, than the Pythagorean theorem falls out almost immediately.

[gowld]

The (non-generalized) Pythagorean theorem is part of Euclidean geometry, so non-Euclidean geometry is irrelevant to this discusion.

> Euclid's third postulate - "a circle can be drawn with any center and radius - doesn't define how to do it.

You do it using an axiomatic compass, a device that copies length in a circular pattern but does not measure it. Lengths are measured using constructable line segments.

Are you implying that nearly all the hundreds of proofs of Pythagorean theorem, which do not use modern rigorous definitions, are not valid proofs?

> Conversely, if you take the "conventional" definition, than the Pythagorean theorem falls out almost immediately.

So? The Pythagorean theorem is very easy to prove. There are hundreds of proofs created by amateurs. That doesn't make them "not proofs" simply because other proofs exist.

[SJC_Hacker]

Strictly speaking, the postulates say nothing about compasses, or even straihhtedges/constructions. Also introducing lengths similarly, involves introducing number which is not a "pure" geometry concept. The third postulate just says that a "circle" exists defined by a point and a radii (which also, not a "pure" geometry construct since it involves a metric - i.e. number.

I would say yes, alot of the fundamental proofs while not striclty "incorrect" or false, are rather informal and contain some hidden axioms/circularities.

Tarski put geometry on a more secure footing using first-order logic.

Similar to how Calculus wasn't on a solid logical foundation until Riemann.