In many practical situations, we are faced with a necessity to combine sophisticated mathematical knowledge about the analyzed systems with informal expert knowledge. To make this combination natural, it is desirable to reformulate the abstract mathematical knowledge in understandable intuitive terms. In this paper, we show how this can be done for an abstract metric.
One way to define a metric is to pick certain properties P1, ..., Pn, and to define a similarity between two objects x and y as the degree to which P1(x) is similar to P1(y) and P2(x) is similar to P2(y), etc.
Similarity is naturally described by 1-|d1-d2| (we can use robustness arguments to get this expression). Since we can have infinitely many properties, we should use min for "and". The distance is then 1-this similarity. The resulting metrics are "natural".
It seems, at first glance, that not all metrics are natural in this sense. Interestingly, an arbitrary continuous metric can be thus described.
Similarly, we can thus describe all "kinematic metrics" (space-time analogues of metrics), while probabilistic explanation is difficult.