Ugly Duckling Theorem

The ugly duckling theorem, in pattern recognition and the philosophy of classification, is an argument showing that classifying objects into similarity based categories is not possible without some prior bias about which properties matter. Under the assumption of finitely many logically definable properties and finitely many objects, the theorem shows that any two distinct objects share exactly the same number of properties in common, a symmetry that defeats any classification scheme built purely on counting shared properties. It is named, following Hans Christian Andersen's story, for the fact that on this measure a duckling is exactly as similar to a swan as two swans are to each other.