The difference between mathematics and logic is that logical truths are true model-independently, while mathematical truths are merely true model-dependently.

Of course, logic is used in mathematics, but the domains are distinguished. The core of logic is classical first-order logic (FOL), which is the logic of well-formed, classical, first-order sentences.

  • well-formed: are valid according to the language definition
  • first-order: include quantifiers over a fixed and not self-referential domain
  • classical: are either true or false

The formalization of logics such as classical FOL is the study of model theory, which is a branch of mathematics. However, can only ever study the truth of sentences (which are used to indicate propositions) within the context of a model. The study of logic itself is concerned only with model-independent truths. In this way, model theory is like a strictly cross-sectional study of logical truth.

In mathematics more broadly, model-dependency is the norm. Every branch of mathematics has a model that includes an enumeration of axioms and, usually, inherits from a model of classical FOL or some similar logic. For example, field theory is the study of fields which is a mathematics object that obeys a selection of axioms called the field axioms. The existence of these mathematical objects is posited. Only under all of these conditions, and within a chosen model (that is typically beyond first-order), does field theory contain truths. Nearly all mathematical theories have this form:

  • posit a new mathematical object
  • specify axioms about the new object
  • choose a model within which to reason about the object with the axioms

In pure logic, there is no underlying structure of the truths. The rule often called “modus ponens” which states “if A and A implies B, then B” is not an axiom given for reasoning about sentences that correspond to propositions. Modus ponens is just a description of what it means for A to imply B. The phrase “A implies B” indicates a proposition that we understand to have that meaning. And when one writes “A implies B” they are indending to indicate the proposition, without necessarily specifying a framework for interpreting it. Implicly, the intepretation framework is the “common interpretation framework” which is the way in which people normally understand each other. It is impossible to completely formalize this framework because, in doing so, it must be communicated by appealing to the framework itself.