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] |