more on this theme     |     more from this text

Single Idea 18210

[filed under theme 6. Mathematics / C. Sources of Mathematics / 9. Fictional Mathematics ]

Full Idea

Why regard the axioms of standard mathematics as truths, rather than as fictions that for a variety of reasons mathematicians have become interested in?

Gist of Idea

Why regard standard mathematics as truths, rather than as interesting fictions?

Source

Hartry Field (Science without Numbers [1980], p.viii)

Book Ref

Field,Hartry: 'Science without Number' [Blackwell 1980], p.-7

The 44 ideas from Hartry Field

9160 Lots of propositions are default reasonable, but the a priori ones are empirically indefeasible [Field,H]
9161 Maybe reasonableness requires circular justifications - that is one coherentist view [Field,H]
9162 Believing nothing, or only logical truths, is very reliable, but we want a lot more than that [Field,H]
9163 If we only use induction to assess induction, it is empirically indefeasible, and hence a priori [Field,H]
9164 We treat basic rules as if they were indefeasible and a priori, with no interest in counter-evidence [Field,H]
9166 People vary in their epistemological standards, and none of them is 'correct' [Field,H]
9165 Reliability only makes a rule reasonable if we place a value on the truth produced by reliable processes [Field,H]
8401 Physical laws are largely time-symmetric, so they make a poor basis for directional causation [Field,H]
8400 Identifying cause and effect is not just conventional; we explain later events by earlier ones [Field,H]
8402 The only reason for adding the notion of 'cause' to fundamental physics is directionality [Field,H]
8404 Explain single events by general rules, or vice versa, or probability explains both, or they are unconnected [Field,H]
9226 If mathematical theories conflict, it may just be that they have different subject matter [Field,H]
8714 Fictionalists say 2+2=4 is true in the way that 'Oliver Twist lived in London' is true [Field,H]
8959 Field presumes properties can be eliminated from science [Field,H, by Szabó]
8958 In Field's version of science, space-time points replace real numbers [Field,H, by Szabó]
10261 The application of mathematics only needs its possibility, not its truth [Field,H, by Shapiro]
9570 In Field's Platonist view, set theory is false because it asserts existence for non-existent things [Field,H, by Chihara]
10260 Logical consequence is defined by the impossibility of P and ¬q [Field,H, by Shapiro]
18212 Nominalists try to only refer to physical objects, or language, or mental constructions [Field,H]
18215 It seems impossible to explain the idea that the conclusion is contained in the premises [Field,H]
18214 Mathematics is only empirical as regards which theory is useful [Field,H]
18213 Abstract objects are only applicable to the world if they are impure, and connect to the physical [Field,H]
18216 Abstractions can form useful counterparts to concrete statements [Field,H]
18218 Hilbert explains geometry, by non-numerical facts about space [Field,H]
18220 Both philosophy and physics now make substantivalism more attractive [Field,H]
18219 Relational space is problematic if you take the idea of a field seriously [Field,H]
18222 Beneath every extrinsic explanation there is an intrinsic explanation [Field,H]
18221 'Metric' axioms uses functions, points and numbers; 'synthetic' axioms give facts about space [Field,H]
9623 Field needs a semantical notion of second-order consequence, and that needs sets [Brown,JR on Field,H]
18223 In theories of fields, space-time points or regions are causal agents [Field,H]
9917 'Abstract' is unclear, but numbers, functions and sets are clearly abstract [Field,H]
8757 The Indispensability Argument is the only serious ground for the existence of mathematical entities [Field,H]
18211 You can reduce ontological commitment by expanding the logic [Field,H]
18210 Why regard standard mathematics as truths, rather than as interesting fictions? [Field,H]
13499 Tarski reduced truth to reference or denotation [Field,H, by Hart,WD]
10818 Tarski really explained truth in terms of denoting, predicating and satisfied functions [Field,H]
7615 Field says reference is a causal physical relation between mental states and objects [Field,H, by Putnam]
10817 Tarski just reduced truth to some other undefined semantic notions [Field,H]
10819 Tarski gives us the account of truth needed to build a group of true sentences in a model [Field,H]
10820 In the early 1930s many philosophers thought truth was not scientific [Field,H]
10826 'Valence' and 'gene' had to be reduced to show their compatibility with physicalism [Field,H]
10825 The notion of truth is to help us make use of the utterances of others [Field,H]
10827 Model theory is unusual in restricting the range of the quantifiers [Field,H]
22244 'Partial reference' is when the subject thinks two objects are one object [Field,H, by Recanati]