Absolute versus relative measurements in geometry | Rational Geometry Math Foundations 134

In science and ordinary life, the distinction between absolute and relative measurements is very useful. It turns out that in mathematics this is also an important distinction. We must be prepared that some aspects of mathematics are more naturally measured relatively, rather than absolutely.

This is very much in the spirit of the ancient Greeks, who did not impose an arbitrary measuring unit into the plane to define lengths, but preferred to consider the ratios, or proportions, formed by segments and other quantities.

We will see that while signed lengths of segments are not generally defined in the plane, the proportions along any line of two segments contained in that line is well-defined. Similar considerations apply to signed areas in three dimensions etc.

Video Content:
00:00 Introduction
2:53 Ancient Greeks held proportions in high regard
9:49 Euclid's theory of proportions
14:00 Segments
15:11 Natural numbers are absolute
19:18 Affine points
23:27 Distinguishing points and vectors
25:50 Affine plane

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[Royvan]

i don't see a reason to make such arbitrary decision as to say that particular values cannot exist. also, seeing as vector spaces are brought up, so long as a metric is defined the square of the magnitude can be found. seems like an arbitrary decision to me.