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