36 ideas
| 9570 | In Field's Platonist view, set theory is false because it asserts existence for non-existent things [Field,H, by Chihara] |
| 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. |
| 10260 | Logical consequence is defined by the impossibility of P and ¬q [Field,H, by Shapiro] |
| 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. |
| 4568 | If 'Queen of England' does not refer if there is no queen, its meaning can't refer if there is one [Cooper,DE] |
| Full Idea: If 'the Queen of England' is not a referring expression when there is no queen, nor can it be one when there is a queen - since the meaning of the expression is the same in either case. | |
| From: David E. Cooper (Philosophy and the Nature of Language [1973], §4.1) | |
| A reaction: I'm not convinced. Does this mean that since I can point with my finger at nothing, I therefore do not indicate anything when there is an object at which I am pointing. Sounds silly to me. |
| 8958 | In Field's version of science, space-time points replace real numbers [Field,H, by Szabó] |
| 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. |
| 18221 | 'Metric' axioms uses functions, points and numbers; 'synthetic' axioms give facts about space [Field,H] |
| 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) |
| 8757 | The Indispensability Argument is the only serious ground for the existence of mathematical entities [Field,H] |
| 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. |
| 18212 | Nominalists try to only refer to physical objects, or language, or mental constructions [Field,H] |
| 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. |
| 10261 | The application of mathematics only needs its possibility, not its truth [Field,H, by Shapiro] |
| 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?). |
| 18218 | Hilbert explains geometry, by non-numerical facts about space [Field,H] |
| 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. |
| 9623 | Field needs a semantical notion of second-order consequence, and that needs sets [Brown,JR on Field,H] |
| 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. |
| 18215 | It seems impossible to explain the idea that the conclusion is contained in the premises [Field,H] |
| 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) |
| 18216 | Abstractions can form useful counterparts to concrete statements [Field,H] |
| 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. |
| 18214 | Mathematics is only empirical as regards which theory is useful [Field,H] |
| 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. |
| 18210 | Why regard standard mathematics as truths, rather than as interesting fictions? [Field,H] |
| 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) |
| 18211 | You can reduce ontological commitment by expanding the logic [Field,H] |
| 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? |
| 4574 | If some peoples do not have categories like time or cause, they can't be essential features of rationality [Cooper,DE] |
| Full Idea: If our most basic concepts, like time, space, substance or causality, are not shared by some peoples, it puts paid to the cherished ideal of philosophers to discover a set of concepts or categories which any rational human must employ in his thinking. | |
| From: David E. Cooper (Philosophy and the Nature of Language [1973], §5.2) | |
| A reaction: This seems to be a place where a priori philosophy (Aristotle,Kant,Hegel) meets empirical research (Whorf). However, interpreting the research is so fraught with problems it drives you back to the a priori… |
| 8959 | Field presumes properties can be eliminated from science [Field,H, by Szabó] |
| 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. |
| 18213 | Abstract objects are only applicable to the world if they are impure, and connect to the physical [Field,H] |
| 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. |
| 4573 | If it is claimed that language correlates with culture, we must be able to identify the two independently [Cooper,DE] |
| Full Idea: If it is claimed that linguistic differences significantly correlate with cultural differences, it must therefore be possible to identify the linguistic differences independently from the cultural ones. | |
| From: David E. Cooper (Philosophy and the Nature of Language [1973], §5.1) | |
| A reaction: This is a basic objection to any extreme relativist version of the S-P hypothesis. They are part of the conspiracy to overemphasise language in philosophy, and they are wrong. |
| 4575 | A person's language doesn't prove their concepts, but how are concepts deduced apart from language? [Cooper,DE] |
| Full Idea: It would be absurd to say the Hopi lack the concept of time because they lack tensed verbs, ..but how do we find out what a man's concepts are except in terms of his language? | |
| From: David E. Cooper (Philosophy and the Nature of Language [1973], §5.2) | |
| A reaction: Presumably we should look at animals, where concepts must be inferred in order to explain behaviour. I don't see why introspection (scientifically wicked) should not also be employed to detect our own non-verbal concepts. How are new words invented? |
| 18222 | Beneath every extrinsic explanation there is an intrinsic explanation [Field,H] |
| 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. |
| 4561 | Many sentences set up dispositions which are irrelevant to the meanings of the sentences [Cooper,DE] |
| Full Idea: Many sentences set up dispositions which are irrelevant to the meanings of the sentences. | |
| From: David E. Cooper (Philosophy and the Nature of Language [1973], §2.3) | |
| A reaction: Yet another telling objection to behaviourism. When I look at broccoli I may have a disposition to be sick, but that isn't part of the concept of broccoli. |
| 9917 | 'Abstract' is unclear, but numbers, functions and sets are clearly abstract [Field,H] |
| 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. |
| 4564 | I can meaningfully speculate that humans may have experiences currently impossible for us [Cooper,DE] |
| Full Idea: It is not meaningless for me to postulate the potential for humans to sense in a manner which is at present unimaginable and indescribable. There is no reason to believe me, but I might be right. | |
| From: David E. Cooper (Philosophy and the Nature of Language [1973], §3.1) | |
| A reaction: The key counterexample to verificationist theories of meaning is wild speculations, which are clearly meaningful, though frequently far beyond any likely human experience. Logical positivists are allergic to imagination. |
| 4565 | The verification principle itself seems neither analytic nor verifiable [Cooper,DE] |
| Full Idea: It seems that the positivists must admit that there is at least one statement which is meaningful, but which is neither verifiable nor analytic - namely, the statement of the principle of verification itself. | |
| From: David E. Cooper (Philosophy and the Nature of Language [1973], §3.1) | |
| A reaction: Some people think this objection is decisive, but I think any theory must be permitted a few metatheoretic assertions or axioms which are beyond discussion. Ayer thought the VP might be treated as analytic. Everyone has to start somewhere. |
| 4563 | 'How now brown cow?' is used for elocution, but this says nothing about its meaning [Cooper,DE] |
| Full Idea: The sentence 'How now brown cow?' has its use in elocutions classes, yet this aspect of its use tells us nothing about its meaning. | |
| From: David E. Cooper (Philosophy and the Nature of Language [1973], §2.4) | |
| A reaction: Indeed, and also there are weird sentence of which we can assemble a meaning, but cannot think of any conceivable use ('rats swim in purple marmalade'). |
| 4562 | Most people know how to use the word "Amen", but they do not know what it means [Cooper,DE] |
| Full Idea: Most people know how to use the word "Amen", but they do not know what it means. | |
| From: David E. Cooper (Philosophy and the Nature of Language [1973], §2.4) | |
| A reaction: Personally I find examples like this decisive against the 'use' theory of meaning. Maybe the defence is that the theory works for sentences, and individual words (like passwords) are peripheral. |
| 4571 | Reference need not be a hit-or-miss affair [Cooper,DE] |
| Full Idea: Reference need not be a hit-or-miss affair. | |
| From: David E. Cooper (Philosophy and the Nature of Language [1973], §4.2) | |
| A reaction: Sounds right. If the basic scenario is picking someone out in a crowd, your listener may think they know which person you are talking about, with a high degree of probability. |
| 4566 | Any thesis about reference is also a thesis about what exists to be referred to [Cooper,DE] |
| Full Idea: Any thesis about reference is also going to be a thesis about what there is in existence to refer to. | |
| From: David E. Cooper (Philosophy and the Nature of Language [1973], §4) | |
| A reaction: I see the point, but we must not put the cart before the horse. I may have an intuition that something exists, but not know how to refer to it (because of my small vocabulary). |
| 4572 | If predicates name things, that reduces every sentence to a mere list of names [Cooper,DE] |
| Full Idea: If predicates are names of entities, then subject/predicate sentences are pairs of names, since subjects are names (or referring expressions). But a pair of names is not a sentence at all, it is a mere list. | |
| From: David E. Cooper (Philosophy and the Nature of Language [1973], §4.4) | |
| A reaction: If that is meant to demolish universals it is too quick. Concatenating names is not the same as listing them. A relationship is asserted. There is a (mysterious) Platonic 'partaking' between form and particular. Perhaps. |
| 4576 | An analytic truth is one which becomes a logical truth when some synonyms have been replaced [Cooper,DE] |
| Full Idea: The definition of analytic truth which has, I believe, the most chance of success is one in terms of synonymy; ..an analytic truth is one which can be transformed into a logical truth once synonyms are replaced by synonyms. | |
| From: David E. Cooper (Philosophy and the Nature of Language [1973], §7.1) | |
| A reaction: Sounds promising, though there is Quine's notorious problem of circularity in all these concepts. If synonymy is conventional, then so is analyticity. I personally feel that the circle can be broken. |
| 1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
| Full Idea: Archelaus was the first person to say that the universe is boundless. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.Ar.3 |
| 18223 | In theories of fields, space-time points or regions are causal agents [Field,H] |
| 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) |
| 18220 | Both philosophy and physics now make substantivalism more attractive [Field,H] |
| 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. |
| 18219 | Relational space is problematic if you take the idea of a field seriously [Field,H] |
| 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. |
| 5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |
| Full Idea: Archelaus wrote that life on Earth began in a primeval slime. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Malcolm Schofield - Archelaus | |
| A reaction: This sounds like a fairly clearcut assertion of the production of life by evolution. Darwin's contribution was to propose the mechanism for achieving it. We should honour the name of Archelaus for this idea. |