20 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. |
| 22246 | A train of reasoning must be treated as all happening simultaneously [Recanati] |
| Full Idea: For logic purposes, a train of reasoning has to be construed as synchronic. | |
| From: François Recanati (Mental Files in Flux [2016], 5.2) | |
| A reaction: If we are looking for a gulf between logic and the real world this is a factor to be considered, along with Nietzsche's observation about necessary simplification. [ref to Kaplan 'Afterthoughts' 1989, 584-5] |
| 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? |
| 19553 | Commitment to 'I have a hand' only makes sense in a context where it has been doubted [Hawthorne] |
| Full Idea: If I utter 'I know I have a hand' then I can only be reckoned a cooperative conversant by my interlocutors on the assumption that there was a real question as to whether I have a hand. | |
| From: John Hawthorne (The Case for Closure [2005], 2) | |
| A reaction: This seems to point to the contextualist approach to global scepticism, which concerns whether we are setting the bar high or low for 'knowledge'. |
| 19551 | How can we know the heavyweight implications of normal knowledge? Must we distort 'knowledge'? [Hawthorne] |
| Full Idea: Those who deny skepticism but accept closure will have to explain how we know the various 'heavyweight' skeptical hypotheses to be false. Do we then twist the concept of knowledge to fit the twin desiderata of closue and anti-skepticism? | |
| From: John Hawthorne (The Case for Closure [2005], Intro) | |
| A reaction: [He is giving Dretske's view; Dretske says we do twist knowledge] Thus if I remember yesterday, that has the heavyweight implication that the past is real. Hawthorne nicely summarises why closure produces a philosophical problem. |
| 19552 | We wouldn't know the logical implications of our knowledge if small risks added up to big risks [Hawthorne] |
| Full Idea: Maybe one cannot know the logical consequences of the proposition that one knows, on account of the fact that small risks add up to big risks. | |
| From: John Hawthorne (The Case for Closure [2005], 1) | |
| A reaction: The idea of closure is that the new knowledge has the certainty of logic, and each step is accepted. An array of receding propositions can lose reliability, but that shouldn't apply to logic implications. Assuming monotonic logic, of course. |
| 19554 | Denying closure is denying we know P when we know P and Q, which is absurd in simple cases [Hawthorne] |
| Full Idea: How could we know that P and Q but not be in a position to know that P (as deniers of closure must say)? If my glass is full of wine, we know 'g is full of wine, and not full of non-wine'. How can we deny that we know it is not full of non-wine? | |
| From: John Hawthorne (The Case for Closure [2005], 2) | |
| A reaction: Hawthorne merely raises this doubt. Dretske is concerned with heavyweight implications, but how do you accept lightweight implications like this one, and then suddenly reject them when they become too heavy? [see p.49] |
| 22247 | Indexicality is not just a feature of language; examples show it also occurs in thought [Recanati] |
| Full Idea: People once took indexicality to be exclusively a property of language, ....but a series of examples seemed to establish that the thought expressed by uttering an indexical sentence is itself indexical (and is thus 'essential'). | |
| From: François Recanati (Mental Files in Flux [2016], 6.1) | |
| A reaction: Perry's example of not realising it is him leaking the sugar in a supermarket is the best known example. Was this a key moment for realising that philosophy of thought is (pace Dummett) more important than philosophy of language? |
| 22248 | How can we communicate indexical thoughts to people not in the right context? [Recanati] |
| Full Idea: Indexical thoughts create an obvious problem with regard to communication. How can we manage to communicate such thoughts to those who are not in the right context? | |
| From: François Recanati (Mental Files in Flux [2016], 7.1) | |
| A reaction: One answer is that you often cannot communicate them. If I write on a wall 'I am here now', that doesn't tell the next passer-by very much. But 'it's raining here' said in a telephone call works fine - if you know the location of the caller. |
| 22242 | Mental files are concepts, which are either collections or (better) containers [Recanati] |
| Full Idea: Mental files are entries in the mental encyclopedia, that is, concepts. Some, following Grice, say they are information collections, but I think of them as containers. Collections are determined by their elements, but containers have independent identity. | |
| From: François Recanati (Mental Files in Flux [2016], Pref) | |
| A reaction: [compressed] [Grice reference is 'Vacuous Names' (1969)] I agree with Recanati. The point is that you can invoke a file by a label, even when you don't know what the content is. |
| 22243 | The Frege case of believing a thing is both F and not-F is explained by separate mental files [Recanati] |
| Full Idea: Frege's Constraint says if a subject believes an object is both F and not-F (as in 'Frege cases'), then the subject thinks of that object under distinct modes of presentation. Having distinct mental files of the object is sufficient to generate this. | |
| From: François Recanati (Mental Files in Flux [2016], Pref) | |
| A reaction: [compressed] When you look at how many semantic puzzles (notably from Frege and Kripke) are solved by the existence of labelled mental files, the case for them is overwhelming. |
| 22245 | A linguistic expression refers to what its associated mental file refers to [Recanati] |
| Full Idea: Mental files determine the reference of linguistic expressions: an expression refers to what the mental file associated with it refers to (at the time of tokening). | |
| From: François Recanati (Mental Files in Flux [2016], 5) | |
| A reaction: Invites the question of how mental files manage to refer, prior to the arrival of a linguistic expression. A mental file is usually fully of descriptions, but it might be no more than a label. |
| 22250 | There are speakers' thoughts and hearers' thoughts, but no further thought attached to the utterance [Recanati] |
| Full Idea: There is the speaker's thought and the thought formed by the hearer. That is all there is. We don't need an additional entity, the thought expressed by the utterance. | |
| From: François Recanati (Mental Files in Flux [2016], 7.2) | |
| A reaction: This fits my view of propositions nicely. They are the two 'thoughts'. The notion of some further abstract 'proposition' with its own mode of independent existence strikes me as ontologically absurd. |
| 22249 | The Naive view of communication is that hearers acquire exactly the thoughts of the speaker [Recanati] |
| Full Idea: The Naive Conception of Communication rests on the idea that communication is the replication of thoughts: the thought the hearer entertains when he understands what the speaker is saying is the very thought which the speaker expressed. | |
| From: François Recanati (Mental Files in Flux [2016], 7.1) | |
| A reaction: It is hard to believe that any modern thinker would believe such a view, given holistic views of language etc. |