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Hilbert's aim to reduce all mathematics to finite logical system was shown impossible by Goedel. He did mathematical analysis of logic itself (Goedel numbering). Turing defined algorithms, and mathematised (algorithms are objects of study in mathematics/cs) them as well -to solve halting problem, and formalise computation (and there goes the possibility of reducing mathematics to algorithms).

The issue which bothers me is what is "mathematical" analysis (or treatment)? What makes one treatment/analysis mathematical ? Now it is possible to do mathematics in natural language (it will be very long and tedious, but in principle, one only has to follow reasoning as outlined in the write-up). So is mathematical treatment a "style"? Is it the rigour? Is it formalisation?

Edit: I would call Turing's analysis as mathematical -bringing algorithms under the purview of mathematical techniques. This is the sense of "mathematical analysis" in this question (not to be confused with real analysis, etc.).

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  • Please give the original context where the term 'mathematical analysis' is used. Otherwise there are several answers possible about the use of the term.
    – Jo Wehler
    Commented Apr 19, 2022 at 19:38
  • Although Godel showed that the hope of computably deriving all arithmetical truths from a finite set of axioms won't work (at least not if you believe all well-formed formulas of first-order arithmetic must have a truth-value), this doesn't actually rule out the idea of seeing arithmetical in terms of a broader idea of rule-based deductions from axioms, see my answer here.
    – Hypnosifl
    Commented Apr 19, 2022 at 20:19
  • @Ajax In your edit you say that it was you who calls Turing's analysis a mathematical analysis because it brings 'algorithms under the purview of mathematical techniques'. Usually one analyzes algorithms concerning their running time, their need of resources, whether they are runnig sequentially or in parallel. - But these are not typical questions from mathematics. Mathematics proves theorems about the relations between abstract objects in domains like number theory, algebra, complex analysis ... 1/2
    – Jo Wehler
    Commented Apr 19, 2022 at 21:33
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    It is the rigorous treatment of some concept, subject of study with the method and language (using symbols) of mathematics: definitions, axioms, theorems. Commented Apr 20, 2022 at 5:55
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    In the same sense that Euclidean geometry is the mathematical theory of space, so Turing theory is the mathematical theory of algorithm or formal computation and Hilbert's proof system is the mathematical theory of deduction. What is the benefit? we can express rigorously property of the "mathematized" concepts: e.g consistency for formal deductive systems, termination for algorithms, etc. Commented Apr 20, 2022 at 8:19

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