Cross-ratio can be defined intrinsically in projective ge... — Carmelics

History Edit See Inverse

Cross-ratio can be defined intrinsically in projective geometry using quadruples of collinear points

Truth & Knowledge

?Rate how convincing each reason is below to see the overall strength.

1 reason for

2 reasons against

Reasons For

1 perspective

Reason for

?

1.The projective group preserves straight lines
?
Think about whether this reason is strong or weak
2.Any ordered triple of distinct collinear points can be mapped uniquely to any other ordered triple of distinct collinear points
?
Think about whether this reason is strong or weak
3.No transformation in the projective group can map an arbitrary quadruple of collinear points onto an arbitrary such quadruple
?
Think about whether this reason is strong or weak

Reasons Against

2 perspectives

Reason against 1 of 2

?

1.The cross-ratio requires coordinatization by real numbers, and this numerical structure is not derivable from purely projective incidence axioms alone.
?
Think about whether this reason is strong or weak
2.Pappus's theorem must be assumed as an independent axiom to guarantee commutativity of the coordinate field, revealing an algebraic presupposition external to projective geometry.
?
Think about whether this reason is strong or weak
3.Therefore the cross-ratio is not intrinsic to projective geometry but depends on importing Euclidean or arithmetic structure through the back door.
?
Think about whether this reason is strong or weak

Reason against 2 of 2

?

Sign in or register to share your perspective on this statement.

Next step

Based on where you are in your exploration

Strongest counterpoint

Explore the most compelling reason on the other side.