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Single Idea 8958
[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
]
Full Idea
Field's nominalist version of science develops a version of Newtonian gravitational theory, where no quantifiers range over mathematical entities, and space-time points and regions play the role of surrogates for real numbers.
Gist of Idea
In Field's version of science, space-time points replace real numbers
Source
report of Hartry Field (Science without Numbers [1980]) by Zoltán Gendler Szabó - Nominalism 5.1
Book Ref
'The Oxford Handbook of Metaphysics', ed/tr. Loux,M /Zimmerman,D [OUP 2005], p.36
A Reaction
This seems to be a very artificial contrivance, but Field has launched a programme for rewriting science so that numbers can be omitted. All of this is Field's rebellion against the Indispensability Argument for mathematics. I sympathise.
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]
|
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]
|
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]
|
10825
|
The notion of truth is to help us make use of the utterances of others
[Field,H]
|
10826
|
'Valence' and 'gene' had to be reduced to show their compatibility with physicalism
[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]
|