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Single Idea 16866
[filed under theme 5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
]
Full Idea
We can trace the chains of inference backwards, …and the circle of theorems closes in more and more. ..We must eventually come to an end by arriving at truths can cannot be inferred, …which are the axioms and postulates.
Gist of Idea
Tracing inference backwards closes in on a small set of axioms and postulates
Source
Gottlob Frege (Logic in Mathematics [1914], p.204)
Book Ref
Frege,Gottlob: 'Posthumous Writings', ed/tr. Hermes/Long/White etc [Blackwell 1979], p.204
A Reaction
The rival (more modern) view is that that all theorems are equal in status, and axioms are selected for convenience.
The
31 ideas
with the same theme
[giving basic truths from which some system is deduced]:
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13004
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Aristotle's axioms (unlike Euclid's) are assumptions awaiting proof
[Aristotle, by Leibniz]
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13002
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It is always good to reduce the number of axioms
[Leibniz]
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17624
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To understand axioms you must grasp their logical power and priority
[Frege, by Burge]
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16866
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Tracing inference backwards closes in on a small set of axioms and postulates
[Frege]
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16868
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The essence of mathematics is the kernel of primitive truths on which it rests
[Frege]
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16871
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A truth can be an axiom in one system and not in another
[Frege]
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16870
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Axioms are truths which cannot be doubted, and for which no proof is needed
[Frege]
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16886
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The truth of an axiom must be independently recognisable
[Frege]
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17963
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The facts of geometry, arithmetic or statics order themselves into theories
[Hilbert]
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17966
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Axioms must reveal their dependence (or not), and must be consistent
[Hilbert]
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6109
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Some axioms may only become accepted when they lead to obvious conclusions
[Russell]
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17629
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Which premises are ultimate varies with context
[Russell]
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17630
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The sources of a proof are the reasons why we believe its conclusion
[Russell]
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17640
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Finding the axioms may be the only route to some new results
[Russell]
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17886
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The limitations of axiomatisation were revealed by the incompleteness theorems
[Gödel, by Koellner]
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19292
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Logic doesn't split into primitive and derived propositions; they all have the same status
[Wittgenstein]
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19632
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An axiom has no more authority than a frenzy
[Cioran]
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17606
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Axioms reveal the underlying assumptions, and reveal relationships between different areas
[Kline]
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17622
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We come to believe mathematical propositions via their grounding in the structure
[Burge]
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17779
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'Classificatory' axioms aim at revealing similarity in morphology of structures
[Mayberry]
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17778
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Axiomatiation relies on isomorphic structures being essentially the same
[Mayberry]
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17780
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'Eliminatory' axioms get rid of traditional ideal and abstract objects
[Mayberry]
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13416
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Mathematics must be based on axioms, which are true because they are axioms, not vice versa
[Tait, by Parsons,C]
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17605
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Hilbert's geometry and Dedekind's real numbers were role models for axiomatization
[Maddy]
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17625
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If two mathematical themes coincide, that suggest a single deep truth
[Maddy]
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9649
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Axioms are either self-evident, or stipulations, or fallible attempts
[Brown,JR]
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23443
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The axioms of group theory are not assertions, but a definition of a structure
[Linnebo]
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23444
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To investigate axiomatic theories, mathematics needs its own foundational axioms
[Linnebo]
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17761
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A compact axiomatisation makes it possible to understand a field as a whole
[Walicki]
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17763
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Axiomatic systems are purely syntactic, and do not presuppose any interpretation
[Walicki]
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17930
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Axioms are 'categorical' if all of their models are isomorphic
[Colyvan]
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