formal semantics
Once we scribble down the mathematical formulation of an intuitive idea, an essential part of it may get lost in translation. I had touched upon this theme in an old post where I wrote about the mathematical formulation of randomness. Now I would like to give another example.
Model Theory does not use the word "semantics" in the usual sense.
Everything in mathematics is syntactical. There can be no semantics confined within mathematics. Symbols can not donate meanings to other symbols. Meaning is an interpretor-dependent concept. Although formal semantics does not completely banish the interpretor from the picture, it certainly pushes him as far away as possible. For instance, axiomatic formulations of Set Theory effectively reduce all semantic questions to one big semantic question: What does the symbol ∈ mean? If you are asked to teach this subject, you will have to tell your students that "a∈A means that a belongs A." At that moment you will be doing real semantics, namely asking your interpretors to interpret a symbol in the way you intend it to be interpreted. Since the notion of "belonging" is so primitive and intuitive, you will probably have no difficulties in conveying the correct interpretation.
Model Theory does not use the word "semantics" in the usual sense.
Everything in mathematics is syntactical. There can be no semantics confined within mathematics. Symbols can not donate meanings to other symbols. Meaning is an interpretor-dependent concept. Although formal semantics does not completely banish the interpretor from the picture, it certainly pushes him as far away as possible. For instance, axiomatic formulations of Set Theory effectively reduce all semantic questions to one big semantic question: What does the symbol ∈ mean? If you are asked to teach this subject, you will have to tell your students that "a∈A means that a belongs A." At that moment you will be doing real semantics, namely asking your interpretors to interpret a symbol in the way you intend it to be interpreted. Since the notion of "belonging" is so primitive and intuitive, you will probably have no difficulties in conveying the correct interpretation.
