Somewhere on his blog, the indefatigable Cosma Shalizi has written about the disciplinary universe of mathematics – that in addition to pure mathematics itself, pure mathematics is used in (and is essential to) the disciplines of Statistics and Computer Science.  This idea struck a chord, and I began to wonder exactly what particular aspect of pure mathematics was being used in each of these other disciplines and where else such methods or approaches were being used.  Of course, having trained as a pure mathematician who turned to mathematical statistics and then eventually to computer science, I know precisely what parts or theories of  pure math were being used in these two disciplines, so this is not my question.    For example, the theory and practice of mathematical statistics draw on probability theory (which itself draws on measure theory and the theory of integration, which in turn require Cantor’s theory of infinite collections), and, in statistical decision theory,  on the differential geometry of information.    (Indeed, I recall being strongly annoyed in my introductory statistics courses that so often proofs of theorems were postponed until “after you know measure theory.”)  Rather, what interests me is what abstract processes – what we might call, mathematical styles of thinking (mathmind, as distinct from, say, the styles of thinking of anthropology or history or chemistry)  – were being used, and where.
Not for the first time, I considered an input-process-output model.  From this viewpoint, we can view pure mathematics itself as a process of (mostly) deductive reasoning that transforms facts about abstract formal objects into other facts about abstract formal objects.   The abstract formal objects may have a basis in some (apprehension of some manifestation of) some real domain or objects, but such a basis is neither necessary nor important to the mathematics.   Until the mid 20th century, people used to say that mathematics was the theory of number, although why they thought this when the key examplar of this theory was Euclidean geometry, a theory which is mostly number-free and scale-invariant, I don’t know.    Since the mid-20th century, people have tended to say that mathematics is the theory of structure and relationship, which better describes most parts of pure mathematics, including Euclidean geometry, and better describes the potential applications and utility of the subject.
Many of the disciplines in the mathematical universe use the same processes – essentially deductive reasoning and, sometimes, calculation – to transform different inputs to certain outputs.   Here is my list (to be added to, when I think of others).
Pure Mathematics

• Input = Abstract formal structures and objects
• Process = Manipulation based on deductive reasoning (and, occasionally, calculation)
• Output = Knowledge about abstract formal structures and objects

Theoretical Physics

• Input = Mathematical models of physical reality
• Process = Manipulation based on deductive reasoning and calculation
• Output = Knowledge about (mathematical models of) physical reality

Mainstream Economics

• Input = Mathematical models of economic reality
• Process = Manipulation based on deductive reasoning and calculation
• Output = Knowledge about (mathematical models of) economic reality

Computational Economics

• Input = Computational models of economic reality
• Process = Manipulation based on deductive and inductive reasoning, calculation and simulation
• Output = Knowledge about (computational models of) economic reality

Exploratory Statistics:

• Input = Raw data
• Process = Processing and manipulation
• Output = Information

Computer Processing:

• Input = Information
• Process = Processing and manipulation, including operations derived from both deductive and inductive reasoning, and simulation
• Output = Information

Statistical Decision Theory (quantitative decision theory)

• Input = Information
• Process = Processing and manipulation, both inductive and deductive reasoning
• Output = Knowledge, Actions

Computer Science:

• Input = Abstract formal structures and objects, intended as models of computational processes
• Process = Manipulation based on both deductive and inductive reasoning, and simulation
• Output = Knowledge about (abstract formal structures and objects, as models of) computational processes

Engineering:

• Input = Physical objects and materials
• Process = Manipulation based on deductive reasoning and calculation
• Output = Physical objects and materials

(Formal) Logic

• Input = Formal representations of statements and arguments
• Process = Manipulation based on deductive reasoning
• Output = Formal representations of statements and arguments

AI Planning

• Input = Information, actions
• Process = Manipulation based on deductive reasoning and simulation
• Output = Knowledge, actions, plans

Qualitative Decision Theory

• Input = Information, actions
• Process = Processing and manipulation, both inductive and deductive reasoning
• Output = Knowledge, actions, plans.

Musical composition

• Input = Abstract formal structures and objects with a sonic semantics
• Process = Manipulation, based on deductive-like reasoning or simulation-like generation
• Output = (Plans for the production of) sounds

Some of the statements implied by these input-process-output schemas are contested.  I would argue that, for instance, any knowledge gained by mathematical economics is only ever knowledge about the mathematical model being studied, and not about the real world which the model is intended to represent.   But this is not the view of most economists, who seem to think they are talking about reality rather than their model of it.  Perhaps this view explains why economics seems peculiarly immune to the major revision or rejection of models on the basis of their failure to predict or describe actual empirical data.
A word on the last schema above:   The composition, performance and even the auditing of music may involve thinking, as I argue here.  Some of the specific modes of musical thinking involved have much in common with deductive mathematical reasoning, in the sense that they can involve the working out of the logical consequences of musical ideas, where the logic being used is not Modus Ponens or Reductio ad Absurdum (as in pure mathematics), but a logic of sounds, pitches, rhythms, timbre and parts.