more on this theme
|
more from this thinker
Single Idea 18221
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
]
Full Idea
There are two approaches to axiomatising geometry. The 'metric' approach uses a function which maps a pair of points into the real numbers. The 'synthetic' approach is that of Euclid and Hilbert, which does without real numbers and functions.
Gist of Idea
'Metric' axioms uses functions, points and numbers; 'synthetic' axioms give facts about space
Source
Hartry Field (Science without Numbers [1980], 5)
Book Ref
Field,Hartry: 'Science without Number' [Blackwell 1980], p.42
Related Idea
Idea 18218
Hilbert explains geometry, by non-numerical facts about space [Field,H]
The
21 ideas
from 'Science without Numbers'
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]
|
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]
|
8959
|
Field presumes properties can be eliminated from science
[Field,H, by Szabó]
|
18212
|
Nominalists try to only refer to physical objects, or language, or mental constructions
[Field,H]
|
18213
|
Abstract objects are only applicable to the world if they are impure, and connect to the physical
[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]
|
18216
|
Abstractions can form useful counterparts to concrete statements
[Field,H]
|
18218
|
Hilbert explains geometry, by non-numerical facts about space
[Field,H]
|
18219
|
Relational space is problematic if you take the idea of a field seriously
[Field,H]
|
18220
|
Both philosophy and physics now make substantivalism more attractive
[Field,H]
|
18221
|
'Metric' axioms uses functions, points and numbers; 'synthetic' axioms give facts about space
[Field,H]
|
18222
|
Beneath every extrinsic explanation there is an intrinsic explanation
[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]
|