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Single Idea 13004

[filed under theme 5. Theory of Logic / K. Features of Logics / 1. Axiomatisation ]

Full Idea

Aristotle's way with axioms, rather than Euclid's, is as assumptions which we are willing to agree on while awaiting an opportunity to prove them

Gist of Idea

Aristotle's axioms (unlike Euclid's) are assumptions awaiting proof

Source

report of Aristotle (Posterior Analytics [c.327 BCE], 76b23-) by Gottfried Leibniz - New Essays on Human Understanding 4.07

Book Ref

Leibniz,Gottfried: 'New Essays on Human Understanding', ed/tr. Remnant/Bennett [CUP 1996], p.419

A Reaction

Euclid's are understood as basic self-evident truths which will be accepted by everyone, though the famous parallel line postulate undermined that. The modern view of axioms is a set of minimum theorems that imply the others. I like Aristotle.

The 31 ideas with the same theme [giving basic truths from which some system is deduced]:

13004 Aristotle's axioms (unlike Euclid's) are assumptions 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]
16871 A truth can be an axiom in one system and not in another [Frege]
16870 Axioms are truths which cannot be doubted, and for which no proof is needed [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]
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]
23443 The axioms of group theory are not assertions, but a definition of a structure [Linnebo]
23444 To investigate axiomatic theories, mathematics needs its own foundational axioms [Linnebo]
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]