By Pfeffer, Riemannian

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Symmetry) D3: d(a, b) + d(b, c) Â³ d(a, c). (triangle inequality) A space that has a distance function is called a metric space. For example, in the two-dimensional space satisfies D1-D3. Also a finite graph where R 2 , the Euclidean distance the distance between points a and b is defined as the number of steps on the shortest path between a and b is a metric space. In a metric space, one can define a betweenness relation B and an equidistance relation E in the following way: Def B: B(a, b, c) if and only if d(a, b) + d(b, c) = d(a, c).

Humans and other animals that did not have a sufficiently adequate representation of the spatial structure of the external world were disadvantaged by natural selection. 5 for more connections to neuroscience). The structuring principles of these mappings are basically innate, even if the fine tuning is established during the development of the human or animal. 34 The same principles appear to govern most of the animal kingdom. Gallistel (1990, 105) argues: [T]he intuitive belief that the cognitive maps of "lower" animals are weaker than our own is not well founded.

On this topic, Aisbett and Gibbon (1994, 143) write: People are willing to rank simple objects of different shape and colour on the basis of "similarity". If machines are to reason about structure, this comparison process must be formalized. That is, a distance measure between formal object representations must be defined. If the machine is reasoning with information to be presented to a human, the distance measure needs to accord with human notions of object similarity. Since our perception of similarity is subjective and strongly influenced by situation, the measure should be tunable to particular users and contexts.