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9226
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If mathematical theories conflict, it may just be that they have different subject matter [Field,H]
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Full Idea:
Unlike logic, in the case of mathematics there may be no genuine conflict between alternative theories: it is natural to think that different theories, if both consistent, are simply about different subjects.
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From:
Hartry Field (Recent Debates on the A Priori [2005], 7)
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A reaction:
For this reason Field places logic at the heart of questions about a priori knowledge, rather than mathematics. My intuitions make me doubt his proposal. Given the very simple basis of, say, arithmetic, I would expect all departments to connect.
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8958
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In Field's version of science, space-time points replace real numbers [Field,H, by Szabó]
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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.
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From:
report of Hartry Field (Science without Numbers [1980]) by Zoltán Gendler Szabó - Nominalism 5.1
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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.
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18212
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Nominalists try to only refer to physical objects, or language, or mental constructions [Field,H]
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Full Idea:
The most popular approach of nominalistically inclined philosophers is to try to reinterpret mathematics, so that its terms and quantifiers only make reference to, say, physical objects, or linguistic expressions, or mental constructions.
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From:
Hartry Field (Science without Numbers [1980], Prelim)
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A reaction:
I am keen on naturalism and empiricism, but only referring to physical objects is a non-starter. I think I favour constructions, derived from the experience of patterns, and abstracted, idealised and generalised. Field says application is the problem.
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18218
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Hilbert explains geometry, by non-numerical facts about space [Field,H]
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Full Idea:
Facts about geometric laws receive satisfying explanations, by the intrinsic facts about physical space, i.e. those laid down without reference to numbers in Hilbert's axioms.
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From:
Hartry Field (Science without Numbers [1980], 3)
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A reaction:
Hilbert's axioms mention points, betweenness, segment-congruence and angle-congruence (Field 25-26). Field cites arithmetic and geometry (as well as Newtonian mechanics) as not being dependent on number.
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9623
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Field needs a semantical notion of second-order consequence, and that needs sets [Brown,JR on Field,H]
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Full Idea:
Field needs the notion of logical consequence in second-order logic, but (since this is not recursively axiomatizable) this is a semantical notion, which involves the idea of 'true in all models', a set-theoretic idea if there ever was one.
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From:
comment on Hartry Field (Science without Numbers [1980], Ch.4) by James Robert Brown - Philosophy of Mathematics
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A reaction:
Brown here summarises a group of critics. Field was arguing for modern nominalism, that actual numbers could (in principle) be written out of the story, as useful fictions. Popper's attempt to dump induction seemed to need induction.
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8714
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Fictionalists say 2+2=4 is true in the way that 'Oliver Twist lived in London' is true [Field,H]
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Full Idea:
The fictionalist can say that the sense in which '2+2=4' is true is pretty much the same as the sense in which 'Oliver Twist lived in London' is true. They are true 'according to a well-known story', or 'according to standard mathematics'.
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From:
Hartry Field (Realism, Mathematics and Modality [1989], 1.1.1), quoted by Michčle Friend - Introducing the Philosophy of Mathematics 6.3
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A reaction:
The roots of this idea are in Carnap. Fictionalism strikes me as brilliant, but poisonous in large doses. Novels can aspire to artistic truth, or to documentary truth. We invent a fiction, and nudge it slowly towards reality.
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