{"id":4788,"date":"2012-11-08T13:26:58","date_gmt":"2012-11-08T13:26:58","guid":{"rendered":"http:\/\/meeseeks:5080\/blog\/?p=4788"},"modified":"2012-11-08T13:26:58","modified_gmt":"2012-11-08T13:26:58","slug":"the-pure-mathematical-universe","status":"publish","type":"post","link":"https:\/\/vukutu.com\/blog\/2012\/11\/the-pure-mathematical-universe\/","title":{"rendered":"The pure mathematical universe"},"content":{"rendered":"

Somewhere on his blog<\/a>, 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.\u00a0 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.\u00a0 Of course, having trained as a pure mathematician who turned to mathematical statistics and then eventually to computer science, I know precisely what parts<\/em> or theories of\u00a0 pure math were being used in these two disciplines, so this is not my question.\u00a0\u00a0\u00a0 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,\u00a0 on the differential geometry of information.\u00a0\u00a0\u00a0 (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.”<\/em>)\u00a0 Rather, what interests me is what abstract processes – what we might call, mathematical styles of thinking (mathmind<\/a>, as distinct from, say, the styles of thinking of anthropology or history or chemistry)\u00a0 – were being used, and where.
\nNot for the first time, I considered an input-process-output model.\u00a0 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.\u00a0\u00a0 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. \u00a0 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.\u00a0\u00a0\u00a0 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.
\nMany of the disciplines in the mathematical universe use the same processes – essentially deductive reasoning and, sometimes, calculation – to transform different inputs to certain outputs.\u00a0\u00a0 Here is my list (to be added to, when I think of others).
\nPure Mathematics
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