Convention (3) – why we need conventions

by Neil Rickert

In this post, I shall give a couple of examples to illustrate why we use and need conventions.

3D Space

For the first example, consider 3-dimensional space as discussed by mathematicians and scientists.  We usually represent objects in space in terms of coordinate axes.  We typically use an x-axis in the left-right direction, a y-axis in the forward (away from me) direction, and a z-axis which is vertical.

The first thing to notice about this choice of axes,  is that the choice is rather arbitrary.  When I am at home, the x-axis is in a north-south direction, because my desk happens to face east.  And at work, where my desk faces roughly north, the x-axis is close to an east-west orientation.  The direction of the vertical axis also changes with location, due to the curvature (the spheroidal shape) of the earth.

If we look at the mathematics, we see that we could choose any three independent directions for our axes.  However, it works out better if we choose three directions that are mutually at right angles to one another.  Such a set of axes is called a set of orthogonal coordinates or orthogonal axes.

The mathematics also tells us that two coordinate axes are not sufficient for all of space.  And it tells us that three are always sufficient, regardless of their mutually orthogonal directions.

What the mathematics cannot tell us, is what directions to choose.  Any set of three mutually orthogonal axes will do just as well.  And there are infinitely many possible choices.

Mathematicians typically describe this situation, by saying that there is no canonical choice of axes.  The nature of the mathematics does not uniquely determine the choice.  So we are left with having to make our own choice.  And if we wish to use the coordinate system for communication with others, then we need to make it a social convention, or at least a temporary convention, among those with whom we will communicate.


For a second example, look at driving.  In fact we have already looked at it.  The problem is that if people drive all over the road, there are likely to be head on collisions.  If everyone drives on the left (as in Australia) we can avoid that.  Likewise, if everyone drives on the right (as in America), we can avoid the problem.  But there is no canonical choice of whether to select the right or the left side of the road as the side to drive on.  And we want people to all follow the same decision.  So we need a social convention.


There are often situations where a choice is needed, but the nature of the problem does not dictate which choice to make.  Or, as mathematicians describe it, there is no canonical answer available.  Under these circumstances, we have little alternative but to establish a convention.


5 Comments to “Convention (3) – why we need conventions”

  1. Like you said, some choice is required, some “canon” needs established to allow meaningful navigation in relationships. In space, there even seems to be constraint that only >2-axis choices will work, even if those three or more are not canonical.

    Mathematical question: can one have a workable Triangulation method (love that word for some reason — nature seems to love Triangulations), which uses axises other that straight lines: spirals, hyperbole … ?


    • can one have a workable Triangulation method (love that word for some reason — nature seems to love Triangulations), which uses axises other that straight lines: spirals, hyperbole … ?

      Navigation (as by sailor and pilots) uses spherical trigonometry which is based on circles. Einstein’s General Relativity uses “curvilinear coordinates”.


  2. Polar coordinates are still lines: rays and angles — if I am not mistaken.

    Does Gereral Relativity use “curvilinear coordinates” as in latitude – longitude or as in non-Euclidean “parallel lines meet” mathematics?


    • Here’s a way to look at it. Instead of using x and y as coordinates, we can use F(x,y) and G(x,y), where F and G are two continuous functions of two variables. It’s a bit related to the stuff you learned on the chain rule, back in Calculus 101.


  3. You obviously don’t understand the word “learned”, but instead use it in the “conventional” sense.


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