more on this theme | more from this thinker
Full Idea
Many axioms have been proposed, not on the grounds that they can be directly known, but rather because they produce a desired body of previously recognised results.
Gist of Idea
Axioms are often affirmed simply because they produce results which have been accepted
Source
Michael D. Resnik (Maths as a Science of Patterns [1997], One.5.1)
Book Ref
Resnik,Michael D.: 'Mathematics as a Science of Patterns' [OUP 1999], p.84
A Reaction
This is the perennial problem with axioms - whether we start from them, or whether we deduce them after the event. There is nothing wrong with that, just as we might infer the existence of quarks because of their results.
| 6295 | There are too many mathematical objects for them all to be mental or physical [Resnik] |
| 6296 | Maths is pattern recognition and representation, and its truth and proofs are based on these [Resnik] |
| 6299 | Axioms are often affirmed simply because they produce results which have been accepted [Resnik] |
| 6300 | Mathematical constants and quantifiers only exist as locations within structures or patterns [Resnik] |
| 6301 | Congruence is the strongest relationship of patterns, equivalence comes next, and mutual occurrence is the weakest [Resnik] |
| 6302 | Structuralism must explain why a triangle is a whole, and not a random set of points [Resnik] |
| 6303 | Sets are positions in patterns [Resnik] |
| 6304 | Mathematical realism says that maths exists, is largely true, and is independent of proofs [Resnik] |