> In summary, it appears that implementing reflection is mathematically
> impossible. I may perhaps still be able to realise my dream of rendering
> depth of field effects realistically. (Although without a nice detailed
> scene to look at, it might be rather disappointing...)

Obviously it's not, because there are plenty of shadertoy examples 
showing it:

https://www.shadertoy.com/view/4ssGWX

Surfaces that reflect *OR* refract are trivial:

for(i=0;i<MAX_TRACE_DEPTH;i++)
{
   intersect = Trace( ray );
   if( REFLECT(intersect) ) ray = ReflectRay(ray , intersect);
   if( REFRACT(intersect) ) ray = RefractRay(ray , intersect);
}

To do surfaces that do both you need to choose randomly at each 
intersection which ray to follow (weighted appropriately depending on 
the material) and then average over a larger number of samples. You can 
either do many samples per frame (but obviously this gets slow for 
complex scenes), or what is commonly done is to average over many frames 
(and reset the average if the camera is moved). Or both.

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