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### 5. Theory of Logic / K. Features of Logics / 1. Axiomatisation

#### [giving basic truths from which some system is deduced]

30 ideas
 13004 Aristotle's axioms (unlike Euclid's) are assumption awaiting proof [Aristotle, by Leibniz]
 13002 It is always good to reduce the number of axioms [Leibniz]
 17624 To understand axioms you must grasp their logical power and priority [Frege, by Burge]
 16866 Tracing inference backwards closes in on a small set of axioms and postulates [Frege]
 16868 The essence of mathematics is the kernel of primitive truths on which it rests [Frege]
 16870 Axioms are truths which cannot be doubted, and for which no proof is needed [Frege]
 16871 A truth can be an axiom in one system and not in another [Frege]
 16886 The truth of an axiom must be independently recognisable [Frege]
 17963 The facts of geometry, arithmetic or statics order themselves into theories [Hilbert]
 17966 Axioms must reveal their dependence (or not), and must be consistent [Hilbert]
 6109 Some axioms may only become accepted when they lead to obvious conclusions [Russell]
 17629 Which premises are ultimate varies with context [Russell]
 17630 The sources of a proof are the reasons why we believe its conclusion [Russell]
 17640 Finding the axioms may be the only route to some new results [Russell]
 17613 We should judge principles by the science, not science by some fixed principles [Zermelo]
 17886 The limitations of axiomatisation were revealed by the incompleteness theorems [Gödel, by Koellner]
 19292 Logic doesn't split into primitive and derived propositions; they all have the same status [Wittgenstein]
 19632 An axiom has no more authority than a frenzy [Cioran]
 17606 Axioms reveal the underlying assumptions, and reveal relationships between different areas [Kline]
 17622 We come to believe mathematical propositions via their grounding in the structure [Burge]
 17779 'Classificatory' axioms aim at revealing similarity in morphology of structures [Mayberry]
 17778 Axiomatiation relies on isomorphic structures being essentially the same [Mayberry]
 17780 'Eliminatory' axioms get rid of traditional ideal and abstract objects [Mayberry]
 13416 Mathematics must be based on axioms, which are true because they are axioms, not vice versa [Tait, by Parsons,C]
 17605 Hilbert's geometry and Dedekind's real numbers were role models for axiomatization [Maddy]
 17625 If two mathematical themes coincide, that suggest a single deep truth [Maddy]
 9649 Axioms are either self-evident, or stipulations, or fallible attempts [Brown,JR]
 17761 A compact axiomatisation makes it possible to understand a field as a whole [Walicki]
 17763 Axiomatic systems are purely syntactic, and do not presuppose any interpretation [Walicki]
 17930 Axioms are 'categorical' if all of their models are isomorphic [Colyvan]