Occam's razor, as it is popularly known, states that "the simplest answer is most likely to be correct"1. It has been noted in other discussion threads that the phrase "simplest description" is somewhat misleading, and that it actually means something along the lines of "description that is easiest to express concisely using natural language". Occam's razor typically comes into play when we are trying to explain some observed phenomenon, or, in terms of model-building, when we are trying to come up with a model for our observations. The verbal complexity of a new model will depend on the models that already exist in the observer's mind, since, as humans, we express new... (read 757 more words →)
Two sheep plus three sheep equals five sheep. Two apples plus three apples equals five apples. Two Discrete ObjecTs plus three Discrete ObjecTs equals five Discrete ObjecTs.
Arithmetic is a formal system, consisting of a syntax and semantics. The formal syntax specifies which statements are grammatical: "2 + 3 = 5" is fine, while "2 3 5 + =" is meaningless. The formal semantics provides a mapping from grammatical statements to truth values: "2 + 3 = 5" is true, while "2 + 3 = 6" is false. This mapping relies on axioms; that is, when we say "statement X in formal system Y is true", we mean X is consistent with... (read more)