25 ideas
| 6299 | Axioms are often affirmed simply because they produce results which have been accepted [Resnik] |
| Full Idea: Many axioms have been proposed, not on the grounds that they can be directly known, but rather because they produce a desired body of previously recognised results. | |
| From: Michael D. Resnik (Maths as a Science of Patterns [1997], One.5.1) | |
| A reaction: This is the perennial problem with axioms - whether we start from them, or whether we deduce them after the event. There is nothing wrong with that, just as we might infer the existence of quarks because of their results. |
| 6304 | Mathematical realism says that maths exists, is largely true, and is independent of proofs [Resnik] |
| Full Idea: Mathematical realism is the doctrine that mathematical objects exist, that much contemporary mathematics is true, and that the existence and truth in question is independent of our constructions, beliefs and proofs. | |
| From: Michael D. Resnik (Maths as a Science of Patterns [1997], Three.12.9) | |
| A reaction: As thus defined, I would call myself a mathematical realist, but everyone must hesitate a little at the word 'exist' and ask, how does it exist? What is it 'made of'? To say that it exists in the way that patterns exist strikes me as very helpful. |
| 6300 | Mathematical constants and quantifiers only exist as locations within structures or patterns [Resnik] |
| Full Idea: In maths the primary subject-matter is not mathematical objects but structures in which they are arranged; our constants and quantifiers denote atoms, structureless points, or positions in structures; they have no identity outside a structure or pattern. | |
| From: Michael D. Resnik (Maths as a Science of Patterns [1997], Three.10.1) | |
| A reaction: This seems to me a very promising idea for the understanding of mathematics. All mathematicians acknowledge that the recognition of patterns is basic to the subject. Even animals recognise patterns. It is natural to invent a language of patterns. |
| 6303 | Sets are positions in patterns [Resnik] |
| Full Idea: On my view, sets are positions in certain patterns. | |
| From: Michael D. Resnik (Maths as a Science of Patterns [1997], Three.10.5) | |
| A reaction: I have always found the ontology of a 'set' puzzling, because they seem to depend on prior reasons why something is a member of a given set, which cannot always be random. It is hard to explain sets without mentioning properties. |
| 6302 | Structuralism must explain why a triangle is a whole, and not a random set of points [Resnik] |
| Full Idea: An objection is that structuralism fails to explain why certain mathematical patterns are unified wholes while others are not; for instance, some think that an ontological account of mathematics must explain why a triangle is not a 'random' set of points. | |
| From: Michael D. Resnik (Maths as a Science of Patterns [1997], Three.10.4) | |
| A reaction: This is an indication that we are not just saying that we recognise patterns in nature, but that we also 'see' various underlying characteristics of the patterns. The obvious suggestion is that we see meta-patterns. |
| 6295 | There are too many mathematical objects for them all to be mental or physical [Resnik] |
| Full Idea: If we take mathematics at its word, there are too many mathematical objects for it to be plausible that they are all mental or physical objects. | |
| From: Michael D. Resnik (Maths as a Science of Patterns [1997], One.1) | |
| A reaction: No one, of course, has ever claimed that they are, but this is a good starting point for assessing the ontology of mathematics. We are going to need 'rules', which can deduce the multitudinous mathematical objects from a small ontology. |
| 6296 | Maths is pattern recognition and representation, and its truth and proofs are based on these [Resnik] |
| Full Idea: I argue that mathematical knowledge has its roots in pattern recognition and representation, and that manipulating representations of patterns provides the connection between the mathematical proof and mathematical truth. | |
| From: Michael D. Resnik (Maths as a Science of Patterns [1997], One.1) | |
| A reaction: The suggestion that patterns are at the basis of the ontology of mathematics is the most illuminating thought I have encountered in the area. It immediately opens up the possibility of maths being an entirely empirical subject. |
| 6301 | Congruence is the strongest relationship of patterns, equivalence comes next, and mutual occurrence is the weakest [Resnik] |
| Full Idea: Of the equivalence relationships which occur between patterns, congruence is the strongest, equivalence the next, and mutual occurrence the weakest. None of these is identity, which would require the same position. | |
| From: Michael D. Resnik (Maths as a Science of Patterns [1997], Three.10.3) | |
| A reaction: This gives some indication of how an account of mathematics as a science of patterns might be built up. Presumably the recognition of these 'degrees of strength' cannot be straightforward observation, but will need an a priori component? |
| 8978 | Events are made of other things, and are not fundamental to ontology [Bennett] |
| Full Idea: Events are not basic items in the universe; they should not be included in any fundamental ontology...all the truths about them are entailed by and explained and made true by truths that do not involve the event concept. | |
| From: Jonathan Bennett (Events and Their Names [1988], p.12), quoted by Peter Simons - Events 3.1 | |
| A reaction: Given the variable time spans of events, their ability to coincide, their ability to contain no motion, their blatantly conventional component, and their recalcitrance to individuation, I say Bennett is right. |
| 23708 | Humeans see properties as having no more essential features and relations than their distinctness [Friend/Kimpton-Nye, by PG] |
| Full Idea: The Humean view says properties are 'quiddities', which individuates properties by nothing more than their distinctness from one another, so that dispositions are not essential to them, and there is no limit to possible property recombination. | |
| From: report of Friend/Kimpton-Nye (Dispositions and Powers [2023], 3.3.1) by PG - Db (ideas) | |
| A reaction: [my summary] All of this is implied by Hume, rather than stated. David Lewis supports this view. The theory of basic powers is the view's main opponent. This quidditist view is not found in physics, where a property's modal profile matters. |
| 23709 | Dispositions are what individuate properties, and they constitute their essence [Friend/Kimpton-Nye] |
| Full Idea: Dispositions constitute the essences of properties, and hence the identity of a property is not primitive ('quidditism'), but is given in terms of its dispositional relations to other properties. | |
| From: Friend/Kimpton-Nye (Dispositions and Powers [2023], 3.3.1) | |
| A reaction: I like the picture that powers are basic, giving rise to dispositions, which combine to produce qualitative and active properties. Powers are precise and relatively few, and properties are ill-defined and very numerous. Being 'influential', for example. |
| 23707 | Powers are properties which necessitate dispositions [Friend/Kimpton-Nye] |
| Full Idea: In broad terms: powers are properties that necessitate dispositions. | |
| From: Friend/Kimpton-Nye (Dispositions and Powers [2023], 3.2) | |
| A reaction: If powers are properties then they must be properties 'of' something, which then seems to be more fundamental than the powers. Maybe our concept of an electron helps, which seems to be a bundle of a few properties, but no one even asks 'of' what. |
| 23714 | Dispositional essentialism (unlike the grounding view) says only fundamental properties are powers [Friend/Kimpton-Nye] |
| Full Idea: Dispositional essentialism yields the view that just fundamental properties and some evolved macro properties are powers. The grounding view, by contrast, seems to yield the result that all properties are powers. | |
| From: Friend/Kimpton-Nye (Dispositions and Powers [2023], 3.7) | |
| A reaction: For the second view, Mumford (for example) claims that the sphericity of a ball is a power, but that seems to miss the whole motivation for the powers ontology, which offers a fairly fundamental explanation of laws and modality. |
| 23711 | A power is a property which consists entirely of dispositions [Friend/Kimpton-Nye] |
| Full Idea: In the 'dispositional essentialist' account (the main view) …what it is to be a power is to be a property whose essence is exhaustively constituted by dispositions. | |
| From: Friend/Kimpton-Nye (Dispositions and Powers [2023], 3.4) | |
| A reaction: [compressed] Sounds wrong to me. A very complex property (such as 'stormy' weather) could be nothing more than a large bundle of dispositions, but that wouldn't make it a 'power', which has to be simpler and more basic. |
| 23712 | Powers are qualitative properties which fully ground dispositions [Friend/Kimpton-Nye] |
| Full Idea: In the 'grounding' view of powers …powers are qualitative, because their essence can be specified independently of any dispositions or relations, but they fully ground dispositions. | |
| From: Friend/Kimpton-Nye (Dispositions and Powers [2023], 3.4) | |
| A reaction: [compressed] They give this as the rival view to dispositional essentialism. It may be a mistake to call a power a property (which needs to be 'of' something). Not sure how powers can be both fundamental and qualitative. Don't they also ground qualities? |
| 23698 | Dispositions have directed behaviour which occurs if triggered [Friend/Kimpton-Nye] |
| Full Idea: The three platitudes about dispositions are that 1) they are directed towards some specific behaviour, 2) they can be triggered under specific conditions, and 3) their directedness is modal, meaning not 'when it is triggered' but 'it it were triggered'. | |
| From: Friend/Kimpton-Nye (Dispositions and Powers [2023], 2.1.1) | |
| A reaction: [PG summary] This is the preliminary to an attempt at a precise formal analysis, covering a number of hypothetical problem cases. 3) is the counterfactual rather than material conditional. Seems accurate. |
| 23699 | 'Masked' dispositions fail to react because something intervenes [Friend/Kimpton-Nye] |
| Full Idea: A disposition is 'masked' when it fails to manifest due to interference, such as a fragile vase packed in bubble wrap, or an antidote taken after some poison. | |
| From: Friend/Kimpton-Nye (Dispositions and Powers [2023], 2.2.1) | |
| A reaction: [compressed] The easiest account of these would be to say that the stimulus or trigger of the disposition never completely occurs. Poisons are only disposed to kill when they are fully ingested. Bubble wrapped vases can't be properly struck. |
| 23700 | A disposition is 'altered' when the stimulus reverses the disposition [Friend/Kimpton-Nye] |
| Full Idea: A disposition is subject to 'altering' when the stimulus of the disposition influences whether (and to what degree) an object has that disposition. Either a live wire goes dead when it is touched, or a dead wire has a sensor making it live when touched. | |
| From: Friend/Kimpton-Nye (Dispositions and Powers [2023], 2.2.2) | |
| A reaction: The word 'fink' is used of such interference. Not much of a problem, I would say, because at the moment when the stimulus comes to do its job, there is no longer a disposition for it to trigger. No different from switching off a light. |
| 23701 | A disposition is 'mimicked' if a different cause produces that effect from that stimulus [Friend/Kimpton-Nye] |
| Full Idea: A disposition is 'mimicked' by objects without that disposition which behave as though they do have it. Styrofoam plates are not fragile, but make a horrible sound when stressed, causing some annoyed person to break them. | |
| From: Friend/Kimpton-Nye (Dispositions and Powers [2023], 2.2.3) | |
| A reaction: A rather strained example! It shouldn't be a problem if the same cause (stress) leads to the same effect (breaking), but by a different path which is not the same as fragility. A formal analysis must obviously cover this case. |
| 23702 | A 'trick' can look like a stimulus for a disposition which will happen without it [Friend/Kimpton-Nye] |
| Full Idea: A 'trick' can behave like a disposition, as when someone says 'abracadabra' over a hot cup of coffee, stimulating it (?) to gradually cool down. | |
| From: Friend/Kimpton-Nye (Dispositions and Powers [2023], 2.2.4) | |
| A reaction: This is like Humean constant conjunction which is obviously not a cause, such as night following day. Only a problem is this cup of coffee is seen in isolation from all other cups of coffee. Post hoc propter hoc does not apply to all stimuli! |
| 23703 | Some dispositions manifest themselves without a stimulus [Friend/Kimpton-Nye] |
| Full Idea: Some dispositions, such as loquaciousness or irascibility, are disposed to manifest whether they are provoked to do so. | |
| From: Friend/Kimpton-Nye (Dispositions and Powers [2023], 2.3.3) | |
| A reaction: We might surmise that such people have internal triggers that get them going, rather than overt ones. The Sun has a disposition to shine, without an external stimulus. The theory of powers says nature is active, rather than being disposed to activity. |
| 23704 | We could analyse dispositions as 'possibilities', with no mention of a stimulus [Friend/Kimpton-Nye] |
| Full Idea: We might abandon the relational analysis of dispositions (as stimulus-effect), and just say a disposition is a 'possibility', which simply can manifest, however that manifestation comes about. | |
| From: Friend/Kimpton-Nye (Dispositions and Powers [2023], 2.3.5) | |
| A reaction: [Compressed. He particularly cites Barbara Vetter] A mere 'possibility' seems to cover passive states as well as potentially active ones. A cushion can be dented, but I wouldn't say it was 'disposed' to dent. Radioactive decay is a disposition, though. |
| 23710 | Dispositionalism says modality is in the powers of this world, not outsourced to possible worlds [Friend/Kimpton-Nye] |
| Full Idea: Dispositionalism does not 'outsource' modality to other possible worlds, it roots modality in the powers of concrete individuals in this world. | |
| From: Friend/Kimpton-Nye (Dispositions and Powers [2023], 3.3.3) | |
| A reaction: Possible worlds are to abolish modality, by treating it as the non-modal facts of different worlds. I see the dispositional view as vastly superior, because the world is awash with vivid and undeniable potentialities, and one world is better ontology. |
| 10364 | Facts are about the world, not in it, so they can't cause anything [Bennett] |
| Full Idea: Facts are not the sort of item that can cause anything. A fact is a true proposition (they say); it is not something in the world but is rather something about the world. | |
| From: Jonathan Bennett (Events and Their Names [1988], p.22), quoted by Jonathan Schaffer - The Metaphysics of Causation 1.1 | |
| A reaction: Compare 10361. Good argument, but maybe 'fact' is ambiguous. See Idea 10365. Events are said to be more concrete, and so can do the job, but their individuation also seems to depend on a description (as Davidson has pointed out). |
| 23706 | Hume's Dictum says no connections are necessary - so mass and spacetime warping could separate [Friend/Kimpton-Nye] |
| Full Idea: Hume's Dictum says there are no necessary connections between existences, …and also between the distinct properties that individuals instantiate. …It follows that an object's property of mass and its disposition to warp space-time could come apart. | |
| From: Friend/Kimpton-Nye (Dispositions and Powers [2023], 3.2) | |
| A reaction: [compressed] This nicely pinpoints the heart of the Humean view, to which scientific essentialists and fans of powers in nature object. The objectors include me. |