Ideas from 'Science without Numbers' by Hartry Field [1980], by Theme Structure

[found in 'Science without Number' by Field,Hartry [Blackwell 1980,0-631-13037-3]].

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4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
In Field's Platonist view, set theory is false because it asserts existence for non-existent things
                        Full Idea: Field commits himself to a Platonic view of mathematics. The theorems of set theory are held to imply or presuppose the existence of things that don't in fact exist. That is why he believes that these theorems are false.
                        From: report of Hartry Field (Science without Numbers [1980]) by Charles Chihara - A Structural Account of Mathematics 11.1
                        A reaction: I am sympathetic to Field, but this sounds wrong. A response that looks appealing is that maths is hypothetical ('if-thenism') - the truth is in the logical consequences, not in the ontological presuppositions.
5. Theory of Logic / B. Logical Consequence / 1. Logical Consequence
Logical consequence is defined by the impossibility of P and ¬q
                        Full Idea: Field defines logical consequence by taking the notion of 'logical possibility' as primitive. Hence q is a consequence of P if the conjunction of the items in P with the negation of q is not possible.
                        From: report of Hartry Field (Science without Numbers [1980]) by Stewart Shapiro - Philosophy of Mathematics 7.2
                        A reaction: The question would then be whether it is plausible to take logical possibility as primitive. Presumably only intuition could support it. But then intuition will equally support natural and metaphysical possibilities.
6. Mathematics / A. Nature of Mathematics / 3. Numbers / a. Numbers
In Field's version of science, space-time points replace real 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.
                        From: report of Hartry Field (Science without Numbers [1980]) by Zoltán Gendler Szabó - Nominalism 5.1
                        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.
6. Mathematics / B. Foundations for Mathematics / 2. Axioms for Geometry
Hilbert's geometry is interesting because it captures Euclid without using real numbers
                        Full Idea: Hilbert's formulation of the Euclidean theory is of special interest because (besides being rigorously axiomatised) it does not employ the real numbers in the axioms.
                        From: Hartry Field (Science without Numbers [1980], 3)
                        A reaction: Notice that this job was done by Hilbert, and not by the fictionalist Hartry Field.
'Metric' axioms uses functions, points and numbers; 'synthetic' axioms give facts about space
                        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.
                        From: Hartry Field (Science without Numbers [1980], 5)
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
The Indispensability Argument is the only serious ground for the existence of mathematical entities
                        Full Idea: There is one and only one serious argument for the existence of mathematical entities, and that is the Indispensability Argument of Putnam and Quine.
                        From: Hartry Field (Science without Numbers [1980], p.5), quoted by Stewart Shapiro - Thinking About Mathematics 9.1
                        A reaction: Personally I don't believe (and nor does Field) that this gives a good enough reason to believe in such things. Quine (who likes 'desert landscapes' in ontology) ends up believing that sets are real because of his argument. Not for me.
6. Mathematics / C. Sources of Mathematics / 3. Mathematical Nominalism
Nominalists try to only refer to physical objects, or language, or mental constructions
                        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.
                        From: Hartry Field (Science without Numbers [1980], Prelim)
                        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.
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / b. Indispensability of mathematics
The application of mathematics only needs its possibility, not its truth
                        Full Idea: Field argues that to account for the applicability of mathematics, we need to assume little more than the possibility of the mathematics, not its truth.
                        From: report of Hartry Field (Science without Numbers [1980]) by Stewart Shapiro - Philosophy of Mathematics 7.2
                        A reaction: Very persuasive. We can apply chess to real military situations, provided that chess isn't self-contradictory (or even naturally impossible?).
Hilbert explains geometry, by non-numerical facts about space
                        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.
                        From: Hartry Field (Science without Numbers [1980], 3)
                        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.
Field needs a semantical notion of second-order consequence, and that needs sets
                        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.
                        From: comment on Hartry Field (Science without Numbers [1980], Ch.4) by James Robert Brown - Philosophy of Mathematics
                        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.
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
It seems impossible to explain the idea that the conclusion is contained in the premises
                        Full Idea: No clear explanation of the idea that the conclusion was 'implicitly contained in' the premises was ever given, and I do not believe that any clear explanation is possible.
                        From: Hartry Field (Science without Numbers [1980], 1)
6. Mathematics / C. Sources of Mathematics / 9. Fictional Mathematics
Abstractions can form useful counterparts to concrete statements
                        Full Idea: Abstract entities are useful because we can use them to formulate abstract counterparts of concrete statements.
                        From: Hartry Field (Science without Numbers [1980], 3)
                        A reaction: He defends the abstract statements as short cuts. If the concrete statements were 'true', then it seems likely that the abstract counterparts will also be true, which is not what fictionalism claims.
Mathematics is only empirical as regards which theory is useful
                        Full Idea: Mathematics is in a sense empirical, but only in the rather Pickwickian sense that is an empirical question as to which mathematical theory is useful.
                        From: Hartry Field (Science without Numbers [1980], 1)
                        A reaction: Field wants mathematics to be fictions, and not to be truths. But can he give an account of 'useful' that does not imply truth? Only in a rather dubiously pragmatist way. A novel is not useful.
Why regard standard mathematics as truths, rather than as interesting fictions?
                        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?
                        From: Hartry Field (Science without Numbers [1980], p.viii)
7. Existence / D. Theories of Reality / 10. Ontological Commitment / a. Ontological commitment
You can reduce ontological commitment by expanding the logic
                        Full Idea: One can often reduce one's ontological commitments by expanding one's logic.
                        From: Hartry Field (Science without Numbers [1980], p.ix)
                        A reaction: I don't actually understand this idea, but that's never stopped me before. Clearly, this sounds like an extremely interesting thought, and hence I should aspire to understand it. So I do aspire to understand it. First, how do you 'expand' a logic?
8. Modes of Existence / B. Properties / 12. Denial of Properties
Field presumes properties can be eliminated from science
                        Full Idea: Field regards the eliminability of apparent reference to properties from the language of science as a foregone result.
                        From: report of Hartry Field (Science without Numbers [1980]) by Zoltán Gendler Szabó - Nominalism 5.1 n50
                        A reaction: Field is a nominalist who also denies the existence of mathematics as part of science. He has a taste for ontological 'desert landscapes'. I have no idea what a property really is, so I think he is on to something.
9. Objects / A. Existence of Objects / 2. Abstract Objects / d. Problems with abstracta
Abstract objects are only applicable to the world if they are impure, and connect to the physical
                        Full Idea: To be able to apply any postulated abstract entities to the physical world, we need impure abstact entities, e.g. functions that map physical objects into pure abstract objects.
                        From: Hartry Field (Science without Numbers [1980], 1)
                        A reaction: I am a fan of 'impure metaphysics', and this pinpoints my reason very nicely.
14. Science / D. Explanation / 2. Types of Explanation / a. Types of explanation
Beneath every extrinsic explanation there is an intrinsic explanation
                        Full Idea: A plausible methodological principle is that underlying every good extrinsic explanation there is an intrinsic explanation.
                        From: Hartry Field (Science without Numbers [1980], 5)
                        A reaction: I'm thinking that Hartry Field is an Aristotelian essentialist, though I bet he would never admit it.
18. Thought / E. Abstraction / 4. Abstracta by Example
'Abstract' is unclear, but numbers, functions and sets are clearly abstract
                        Full Idea: The term 'abstract entities' may not be entirely clear, but one thing that does seem clear is that such alleged entities as numbers, functions and sets are abstract.
                        From: Hartry Field (Science without Numbers [1980], p.1), quoted by JP Burgess / G Rosen - A Subject with No Object I.A.1.a
                        A reaction: Field firmly denies the existence of such things. Sets don't seem a great problem, if the set is a herd of elephants, but the null and singleton sets show up the difficulties.
26. Natural Theory / B. Concepts of Nature / 3. Space / b. Points in space
In theories of fields, space-time points or regions are causal agents
                        Full Idea: According to theories that take the notion of a field seriously, space-time points or regions are fully-fledge causal agents.
                        From: Hartry Field (Science without Numbers [1980], n 23)
26. Natural Theory / B. Concepts of Nature / 3. Space / c. Substantival space
Both philosophy and physics now make substantivalism more attractive
                        Full Idea: In general, it seems to me that recent developments in both philosophy and physics have made substantivalism a much more attractive position than it once was.
                        From: Hartry Field (Science without Numbers [1980], 4)
                        A reaction: I'm intrigued as to what philosophical developments are involved in this. The arrival of fields is the development in physics.
26. Natural Theory / B. Concepts of Nature / 3. Space / d. Relational space
Relational space is problematic if you take the idea of a field seriously
                        Full Idea: The problem of the relational view of space is especially acute in the context of physical theories that take the notion of a field seriously, e.g. classical electromagnetic theory.
                        From: Hartry Field (Science without Numbers [1980], 4)
                        A reaction: In the Leibniz-Clarke debate I sided with the Newtonian Clarke (defending absolute space), and it looks like modern science agrees with me. Nothing exists purely as relations.