more on this theme | more from this thinker
Full Idea
Mathematics investigates the deductive consequences of axiomatic theories, but it also needs its own foundational axioms in order to provide models for its various axiomatic theories.
Gist of Idea
To investigate axiomatic theories, mathematics needs its own foundational axioms
Source
Øystein Linnebo (Philosophy of Mathematics [2017], 4.1)
Book Ref
Linnebo,Øystein: 'Philosophy of Mathematics' [Princeton 2017], p.56
A Reaction
This is a problem which faces the deductivist (if-then) approach. The deductive process needs its own grounds.
| 23448 | Mathematics is the study of all possible patterns, and is thus bound to describe the world [Linnebo] |
| 23441 | Logical truth is true in all models, so mathematical objects can't be purely logical [Linnebo] |
| 23442 | Game Formalism has no semantics, and Term Formalism reduces the semantics [Linnebo] |
| 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] |
| 23445 | Naïve set theory says any formula defines a set, and coextensive sets are identical [Linnebo] |
| 23446 | You can't prove consistency using a weaker theory, but you can use a consistent theory [Linnebo] |
| 23447 | In classical semantics singular terms refer, and quantifiers range over domains [Linnebo] |