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Logical and semantic purity. (English) Zbl 1159.03004
Preyer, Gerhard (ed.) et al., Philosophy of mathematics. Set theory, measuring theories, and nominalism. Frankfurt: Ontos Verlag (ISBN 978-3-86838-009-5/hbk). LOGOS. Studien zur Logik, Sprachphilosophie und Metaphysik 13, 40-52 (2008).
Starting with Hilbert’s formulation of the concern for the purity of the method (“one strives to use in the proof of a theorem as far as possible only those auxiliary means that are required by the content of the theorem”), the author distinguishes between two kinds of pure proofs of theorems: (i) logically pure proofs, which are proofs carried out from a minimal subset of axioms of a given axiom system, and (ii) semantically pure proofs, which “draw only on what must be understood and accepted in order to understand that theorem”.
He then shows that: (1) “Some results require more concepts and/or propositions to be proved than to be understood”, and (2) “Some results require more concepts and/or propositions to be understood than to be proved”. To establish (1), he uses the example of the casus irreducibilis for cubic polynomials (which requires complex numbers for its solution, but not for its understanding) and that of Gödel sentences in Peano Arithmetic (which can be understood as arithmetical statements, but not proved as such statements), whereas for (2) he uses the theorem stating that there are infinitely many primes, which can pe proved in fragments of arithmetic, but requires a more generous axiom system to be understood.
For the entire collection see [Zbl 1149.03003].

03A05 Philosophical and critical aspects of logic and foundations
00A30 Philosophy of mathematics
03B30 Foundations of classical theories (including reverse mathematics)
03F07 Structure of proofs