Sound and Fury

Signifying nothing

What is a straight line? Pt. II

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Euclid’s definition of a straight line is actually a line which ‘lies evenly with the points on itself.’ A line is a ‘breadthless length.’ So whatever a straight line is, it isn’t necessarily ‘the shortest distance between two points.’ In Geometry you can define lines in terms of a vector (a direction) and a point. Or as the intersection of two planes (if we have a 3rd dimension). But all this talk of vectors and planes seems to rest on some premathematical conception of what it means to say that a line is ‘straight’ Straightness is in a sense more basic than talk of vectors or planes or whatever. So even if it isn’t dependent on a distance function, there is something difficult about the concept of a straight line.

The definition of a straight line in spherical geometry can be done in terms of the intersection of a the sphere with plane in the ambient 3-space (which is Euclidean) which passes through the centre of the sphere. But a plane in the ambient space is defined in terms of lines and points. So that isn’t helpful. Similarly other non-Euclidean spaces can in some sense be embedded in a higher dimensional Euclidean space and have their straight lines defined in the same way. (I don’t know about spaces you can’t embed in E-space. But never mind. The point is that you still can’t define ‘straightness’ this way.) Russell wrote a book which suggests that projective geometry is somehow more basic, more foundational. But he later changed his mind on that. But I don’t know enough about P space to say any more on that topic. But it might be worth looking at.

Essentially, the way out is just to take the concept of a straight line as a primitive. You just pretend you know what you’re talking about. Well, that’s fair enough really. Then you can do stuff like add postulates which increase the content of the concept (ie there is exactly one line which goes through a point not on a line which never crosses the line. Or there are no such lines, or there are infinitely many…) But at the basis is some foundational primitive ‘straight line’ concept.

OK, but this is so basic it is common to the whole family of Riemannian geometries. However you want to negate the parallel postulate (or keep it, whatever) all the resultant geometries share this basal concept. Which is weird. Lines in S2 (the sphere) that are straight are curved when considered as lines in the ambient Euclidean space. So what is it that is shared?


Written by Seamus

November 22, 2007 at 4:57 pm

Posted in maths, philosophy

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