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Single Idea 17761
[filed under theme 5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
]
Full Idea
Having such a compact [axiomatic] presentation of a complicated field [such as Euclid's], makes it possible to relate not only to particular theorems but also to the whole field as such.
Gist of Idea
A compact axiomatisation makes it possible to understand a field as a whole
Source
Michal Walicki (Introduction to Mathematical Logic [2012], 4.1)
Book Ref
Walicki,Michal: 'Introduction to Mathematical Logic' [World Scientific 2012], p.118
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]
|