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Single Idea 13416

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

Full Idea

The axiomatic conception of mathematics is the only viable one. ...But they are true because they are axioms, in contrast to the view advanced by Frege (to Hilbert) that to be a candidate for axiomhood a statement must be true.

Gist of Idea

Mathematics must be based on axioms, which are true because they are axioms, not vice versa

Source

report of William W. Tait (Intro to 'Provenance of Pure Reason' [2005], p.4) by Charles Parsons - Review of Tait 'Provenance of Pure Reason' §2

Book Ref

-: 'Philosophia Mathematica' [-], p.222

A Reaction

This looks like the classic twentieth century shift in the attitude to axioms. The Greek idea is that they must be self-evident truths, but the Tait-style view is that they are just the first steps in establishing a logical structure. I prefer the Greeks.

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]
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]
17630 The sources of a proof are the reasons why we believe its conclusion [Russell]
17629 Which premises are ultimate varies with context [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]
17778 Axiomatiation relies on isomorphic structures being essentially the same [Mayberry]
17779 'Classificatory' axioms aim at revealing similarity in morphology of structures [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]