46 ideas
| 18405 | A 'teepee' argument has several mutually supporting planks to it [Cappelen/Dever] |
| Full Idea: In a 'teepee' argument, a number of argumentative planks intersupport each other. No plank is sufficiently strong to establish the position, but each lends credibility to the others because there is the appearance of a unified phenomenon. | |
| From: Cappelen,H/Dever,Josh (The Inessential Indexical [2013], 01.5) | |
| A reaction: To attack it, they say, you have to identify the separate planks of the argument. It is a moot point whether the teepee might be so imprecise that it is better described as 'coherence'. There is a background support, as well as the planks. |
| 17926 | Rejecting double negation elimination undermines reductio proofs [Colyvan] |
| Full Idea: The intuitionist rejection of double negation elimination undermines the important reductio ad absurdum proof in classical mathematics. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.1.3) |
| 17925 | Showing a disproof is impossible is not a proof, so don't eliminate double negation [Colyvan] |
| Full Idea: In intuitionist logic double negation elimination fails. After all, proving that there is no proof that there can't be a proof of S is not the same thing as having a proof of S. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.1.3) | |
| A reaction: I do like people like Colyvan who explain things clearly. All of this difficult stuff is understandable, if only someone makes the effort to explain it properly. |
| 17924 | Excluded middle says P or not-P; bivalence says P is either true or false [Colyvan] |
| Full Idea: The law of excluded middle (for every proposition P, either P or not-P) must be carefully distinguished from its semantic counterpart bivalence, that every proposition is either true or false. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.1.3) | |
| A reaction: So excluded middle makes no reference to the actual truth or falsity of P. It merely says P excludes not-P, and vice versa. |
| 17929 | Löwenheim proved his result for a first-order sentence, and Skolem generalised it [Colyvan] |
| Full Idea: Löwenheim proved that if a first-order sentence has a model at all, it has a countable model. ...Skolem generalised this result to systems of first-order sentences. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 2.1.2) |
| 17930 | Axioms are 'categorical' if all of their models are isomorphic [Colyvan] |
| Full Idea: A set of axioms is said to be 'categorical' if all models of the axioms in question are isomorphic. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 2.1.2) | |
| A reaction: The best example is the Peano Axioms, which are 'true up to isomorphism'. Set theory axioms are only 'quasi-isomorphic'. |
| 17928 | Ordinal numbers represent order relations [Colyvan] |
| Full Idea: Ordinal numbers represent order relations. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.2.3 n17) |
| 17923 | Intuitionists only accept a few safe infinities [Colyvan] |
| Full Idea: For intuitionists, all but the smallest, most well-behaved infinities are rejected. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.1.3) | |
| A reaction: The intuitionist idea is to only accept what can be clearly constructed or proved. |
| 17941 | Infinitesimals were sometimes zero, and sometimes close to zero [Colyvan] |
| Full Idea: The problem with infinitesimals is that in some places they behaved like real numbers close to zero but in other places they behaved like zero. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 7.1.2) | |
| A reaction: Colyvan gives an example, of differentiating a polynomial. |
| 17922 | Reducing real numbers to rationals suggested arithmetic as the foundation of maths [Colyvan] |
| Full Idea: Given Dedekind's reduction of real numbers to sequences of rational numbers, and other known reductions in mathematics, it was tempting to see basic arithmetic as the foundation of mathematics. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.1.1) | |
| A reaction: The reduction is the famous Dedekind 'cut'. Nowadays theorists seem to be more abstract (Category Theory, for example) instead of reductionist. |
| 17936 | Transfinite induction moves from all cases, up to the limit ordinal [Colyvan] |
| Full Idea: Transfinite inductions are inductive proofs that include an extra step to show that if the statement holds for all cases less than some limit ordinal, the statement also holds for the limit ordinal. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 5.2.1 n11) |
| 17940 | Most mathematical proofs are using set theory, but without saying so [Colyvan] |
| Full Idea: Most mathematical proofs, outside of set theory, do not explicitly state the set theory being employed. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 7.1.1) |
| 17931 | Structuralism say only 'up to isomorphism' matters because that is all there is to it [Colyvan] |
| Full Idea: Structuralism is able to explain why mathematicians are typically only interested in describing the objects they study up to isomorphism - for that is all there is to describe. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 3.1.2) |
| 17932 | If 'in re' structures relies on the world, does the world contain rich enough structures? [Colyvan] |
| Full Idea: In re structuralism does not posit anything other than the kinds of structures that are in fact found in the world. ...The problem is that the world may not provide rich enough structures for the mathematics. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 3.1.2) | |
| A reaction: You can perceive a repeating pattern in the world, without any interest in how far the repetitions extend. |
| 19542 | It is nonsense that understanding does not involve knowledge; to understand, you must know [Dougherty/Rysiew] |
| Full Idea: The proposition that understanding does not involve knowledge is widespread (for example, in discussions of what philosophy aims at), but hardly withstands scrutiny. If you do not know how a jet engine works, you do not understand how it works. | |
| From: Dougherty,T/Rysiew,P (Experience First (and reply) [2014], p.24) | |
| A reaction: This seems a bit disingenuous. As in 'Theaetetus', knowing the million parts of a jet engine is not to understand it. More strongly - how could knowledge of an infinity of separate propositional truths amount to understanding on their own? |
| 19543 | To grasp understanding, we should be more explicit about what needs to be known [Dougherty/Rysiew] |
| Full Idea: An essential prerequisite for useful discussion of the relation between knowledge and understanding is systematic explicitness about what is to be known or understood. | |
| From: Dougherty,T/Rysiew,P (Experience First (and reply) [2014], p.25) | |
| A reaction: This is better. I say what needs to be known for understanding is the essence of the item under discussion (my PhD thesis!). Obviously understanding needs some knowledge, but I take it that epistemology should be understanding-first. That is the main aim. |
| 19541 | Rather than knowledge, our epistemic aim may be mere true belief, or else understanding and wisdom [Dougherty/Rysiew] |
| Full Idea: If we say our cognitive aim is to get knowledge, the opposing views are the naturalistic view that what matters is just true belief (or just 'getting by'), or that there are rival epistemic goods such as understanding and wisdom. | |
| From: Dougherty,T/Rysiew,P (Experience First (and reply) [2014], p.17) | |
| A reaction: [compressed summary] I'm a fan of understanding. The accumulation of propositional knowledge would relish knowing the mass of every grain of sand on a beach. If you say the propositions should be 'important', other values are invoked. |
| 19540 | Don't confuse justified belief with justified believers [Dougherty/Rysiew] |
| Full Idea: Much theorizing about justification conflates issues of justified belief with issues of justified/blameless believers. | |
| From: Dougherty,T/Rysiew,P (What is Knowledge-First Epistemology? [2014], p.12) | |
| A reaction: [They cite Kent Bach 1985] Presumably the only thing that really justifies a belief is the truth, or the actual facts. You could then say 'p is a justified belief, though no one actually believes it'. E.g. the number of stars is odd. |
| 19539 | If knowledge is unanalysable, that makes justification more important [Dougherty/Rysiew] |
| Full Idea: If knowledge is indeed unanalyzable, that could be seen as a liberation of justification to assume importance in its own right. | |
| From: Dougherty,T/Rysiew,P (What is Knowledge-First Epistemology? [2014], p.11) | |
| A reaction: [They cite Kvanvig 2003:192 and Greco 2010:9-] See Scruton's Idea 3897. I suspect that we should just give up discussing 'knowledge', which is a woolly and uninformative term, and focus on where the real epistemological action is. |
| 17943 | Probability supports Bayesianism better as degrees of belief than as ratios of frequencies [Colyvan] |
| Full Idea: Those who see probabilities as ratios of frequencies can't use Bayes's Theorem if there is no objective prior probability. Those who accept prior probabilities tend to opt for a subjectivist account, where probabilities are degrees of belief. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 9.1.8) | |
| A reaction: [compressed] |
| 17939 | Mathematics can reveal structural similarities in diverse systems [Colyvan] |
| Full Idea: Mathematics can demonstrate structural similarities between systems (e.g. missing population periods and the gaps in the rings of Saturn). | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 6.3.2) | |
| A reaction: [Colyvan expounds the details of his two examples] It is these sorts of results that get people enthusiastic about the mathematics embedded in nature. A misunderstanding, I think. |
| 17938 | Mathematics can show why some surprising events have to occur [Colyvan] |
| Full Idea: Mathematics can show that under a broad range of conditions, something initially surprising must occur (e.g. the hexagonal structure of honeycomb). | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 6.3.2) |
| 17934 | Proof by cases (by 'exhaustion') is said to be unexplanatory [Colyvan] |
| Full Idea: Another style of proof often cited as unexplanatory are brute-force methods such as proof by cases (or proof by exhaustion). | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 5.2.1) |
| 17933 | Reductio proofs do not seem to be very explanatory [Colyvan] |
| Full Idea: One kind of proof that is thought to be unexplanatory is the 'reductio' proof. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 5.2.1) | |
| A reaction: Presumably you generate a contradiction, but are given no indication of why the contradiction has arisen? Tracking back might reveal the source of the problem? Colyvan thinks reductio can be explanatory. |
| 17935 | If inductive proofs hold because of the structure of natural numbers, they may explain theorems [Colyvan] |
| Full Idea: It might be argued that any proof by induction is revealing the explanation of the theorem, namely, that it holds by virtue of the structure of the natural numbers. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 5.2.1) | |
| A reaction: This is because induction characterises the natural numbers, in the Peano Axioms. |
| 17942 | Can a proof that no one understands (of the four-colour theorem) really be a proof? [Colyvan] |
| Full Idea: The proof of the four-colour theorem raises questions about whether a 'proof' that no one understands is a proof. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 9.1.6) | |
| A reaction: The point is that the theorem (that you can colour countries on a map with just four colours) was proved with the help of a computer. |
| 17937 | Mathematical generalisation is by extending a system, or by abstracting away from it [Colyvan] |
| Full Idea: One type of generalisation in mathematics extends a system to go beyond what is was originally set up for; another kind involves abstracting away from some details in order to capture similarities between different systems. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 5.2.2) |
| 18422 | Prioprioception focuses on your body parts, not on your self, or indexicality [Cappelen/Dever] |
| Full Idea: Proprioception is not focused single-mindedly on the self, but is focused on a number of objects - the component bodily parts that belong to the self. There is no obvious need for a concept of the self, or of indexicality. | |
| From: Cappelen,H/Dever,Josh (The Inessential Indexical [2013], 07.2) |
| 18425 | We can acquire self-knowledge with mirrors, not just with proprioception and introspection [Cappelen/Dever] |
| Full Idea: Imagine a being that learns everything about itself by watching itself in mirrors, rather than by proprioception and introspection. Surely it can get wet in a storm, even though allegedly distinctive routes of self-knowledge are not available to it? | |
| From: Cappelen,H/Dever,Josh (The Inessential Indexical [2013], 09.3) | |
| A reaction: [compressed] |
| 18421 | Proprioception is only immune from error if you are certain that it represents the agent [Cappelen/Dever] |
| Full Idea: The guarantee of immunity from error in prioprioception is only as strong as the guarantee that proprioception only ever represents the proprioceiving agent. | |
| From: Cappelen,H/Dever,Josh (The Inessential Indexical [2013], 07.1) | |
| A reaction: This is part of an interesting and sustained attack on the idea that self-knowledge is immune from error. They are thinking of science-fictiony situations where I am wired up to experience your leg movement. My experiences usually track me, that's all. |
| 18419 | Folk Functionalism is a Ramsification of our folk psychology [Cappelen/Dever] |
| Full Idea: According to Folk Functionalism, mental states are theoretically defined by Ramsifying on our folk-psychological theory. | |
| From: Cappelen,H/Dever,Josh (The Inessential Indexical [2013], 06.2) |
| 18404 | It is assumed that indexical content is needed to represent the perspective of perception [Cappelen/Dever] |
| Full Idea: Because our perceptual states typically represent the world as seen from a perspective, it is sometimes thought that some distinctively indexical kind of content is needed to characterise those states. | |
| From: Cappelen,H/Dever,Josh (The Inessential Indexical [2013], 01.4) | |
| A reaction: They are summarising this view precisely so that they can oppose it, and I think they are right. |
| 18426 | All information is objective, and purely indexical information is not much use [Cappelen/Dever] |
| Full Idea: Fundamentally, all information is objective information. ...[176] What we want is fully portable information, and information that co-ordinates on the world, rather than on us, is best suited for the task. | |
| From: Cappelen,H/Dever,Josh (The Inessential Indexical [2013], 10) | |
| A reaction: I agree entirely with their thesis. We just pick up information about ourselves, such as who and where we are, which is just like equivalent information about other people. It is isn't a special type of information. |
| 18427 | If some of our thought is tied to its context, it will be hard to communicate it [Cappelen/Dever] |
| Full Idea: It is bad news if some of our contents are essentially tied to particular contexts. ...If information needs to be assessed relative to some ur-context, later recipients won't know what to do with it. | |
| From: Cappelen,H/Dever,Josh (The Inessential Indexical [2013], 10) |
| 18428 | You don't remember your house interior just from an experienced viewpoint [Cappelen/Dever] |
| Full Idea: When you recall the look of the inside of your house ....where things are relative to one another is what persists in memory, not where they were relative to you when seen. | |
| From: Cappelen,H/Dever,Josh (The Inessential Indexical [2013], 10) | |
| A reaction: This seems to be a very telling example, though you could postulate some system which converts perspectival input into objective information. But why bother? We seek objective information, not perspectives. |
| 18429 | Our beliefs and desires are not organised around ourselves, but around the world [Cappelen/Dever] |
| Full Idea: Our view on the world is not primarily a view from a perspective. Our beliefs and desires are not organized around us. They are instead organized around the world itself. Our view is a view from everywhere. | |
| From: Cappelen,H/Dever,Josh (The Inessential Indexical [2013], 10) | |
| A reaction: Slipping in the claim that our desires are also organised around the world is not quite as persuasive as the claim about beliefs. If you want to draw a freehand straight line, focus on the far end of it. The world will guide your hand. |
| 18407 | Indexicality is not significantly connected to agency [Cappelen/Dever] |
| Full Idea: There are no interesting or distinctive explanatory connections between indexicality and agency. | |
| From: Cappelen,H/Dever,Josh (The Inessential Indexical [2013], 01.8) |
| 19538 | Entailment is modelled in formal semantics as set inclusion (where 'mammals' contains 'cats') [Dougherty/Rysiew] |
| Full Idea: Entailment is modelled in formal semantics as set inclusion. 'Cat' entails 'mammal' because the cats are a subset of the mammals. | |
| From: Dougherty,T/Rysiew,P (What is Knowledge-First Epistemology? [2014], p.10) | |
| A reaction: I would have thought that this was only one type of entailment. 'Travelling to Iceland entails flying'. Travelling includes flying, the reverse of cats/mammals, to a very complex set-theoretic account is needed. Interesting. |
| 18413 | Fregeans can't agree on what 'senses' are [Cappelen/Dever] |
| Full Idea: There is little agreement among Fregeans about what senses are. | |
| From: Cappelen,H/Dever,Josh (The Inessential Indexical [2013], 04.5) | |
| A reaction: I don't take this to be sufficient grounds for dismissing Fregean senses. When we look into the workings of the linguistic mind, there seems little prospect of clarity or agreement. |
| 18417 | Possible worlds accounts of content are notoriously coarse-grained [Cappelen/Dever] |
| Full Idea: Possible worlds accounts of content are notoriously coarse-grained. They fail to distinguish between logical or mathematical truths, ..between metaphysical equivalences, ..between coreferentials, ..and between indexicals and non-indexicals. | |
| From: Cappelen,H/Dever,Josh (The Inessential Indexical [2013], 05.5) | |
| A reaction: [A nice summary, very compressed] |
| 18408 | Indexicals are just non-constant in meaning, and don't involve any special concepts [Cappelen/Dever] |
| Full Idea: Once the non-constant characters of expressions has been characterised, there is no further need for additional devices like 'first-person concepts' or 'demonstrative concepts'. | |
| From: Cappelen,H/Dever,Josh (The Inessential Indexical [2013], 01.7) | |
| A reaction: This seems to me to be a wonderfully liberating attack on this issue. There is a kind of creepy mysticism that has been allowed to accrue around indexicals, and it's nonsense. |
| 18414 | Fregeans say 'I' differs in reference, so it must also differ in sense [Cappelen/Dever] |
| Full Idea: Fregeans tend to treat as a fundamental tenet that sense determines reference; same sense, same reference. From that it follow trivially that indexicals don't have the same sense: different uses of 'I' have different referents, so sense must differ. | |
| From: Cappelen,H/Dever,Josh (The Inessential Indexical [2013], 04.6) | |
| A reaction: Interesting. Since it seems implausible that 'I' is profoundly different when two people use it, this seems to be a strong argument against Frege's distinction. But I rather like Frege's distinction, while being sceptical about 'I', so I'm baffled.... |
| 18423 | All indexicals can be expressed non-indexically [Cappelen/Dever] |
| Full Idea: Whatever can be expressed indexically could be expressed by non-indexical means. | |
| From: Cappelen,H/Dever,Josh (The Inessential Indexical [2013], 08.1) | |
| A reaction: This is the best summary of the thesis of their book. Indexicality in non-essential. |
| 18406 | The basic Kaplan view is that there is truth-conditional content, and contextual character [Cappelen/Dever] |
| Full Idea: In what we label 'Basic Kaplanianism', each of the sentences 'Smith is happy' and 'I am happy', as uttered by Smith, has two levels of meaning. The 'content' is a truth-conditional representation. The 'character' is a function from contexts to contents. | |
| From: Cappelen,H/Dever,Josh (The Inessential Indexical [2013], 01.6) | |
| A reaction: They give this as a minimal and plausible account of the situation, without reading huge significance into the indexical. I'm inclined to see the situation in terms of the underlying proposition containing both ingredients. |
| 18411 | It is proposed that a huge range of linguistic items are context-sensitive [Cappelen/Dever] |
| Full Idea: An enormous amount has been written about whether 'all', 'know', 'might', 'delicious', 'good', 'if, then', 'and', 'red', 'just', 'justified', 'probable', 'local', 'ready', and 'left-right' are context-sensitive. | |
| From: Cappelen,H/Dever,Josh (The Inessential Indexical [2013], 02.3) | |
| A reaction: The clearest way to approach these things is ask what the (informal) domain of quantification is for that particular context. The domain can shift in the course of a sentence. |
| 18420 | We deny that action involves some special class of beliefs [Cappelen/Dever] |
| Full Idea: Maybe there is a class of beliefs that plays a special role in the explanation of action. We have argued against the existence of such a class (or at least any interesting such class). | |
| From: Cappelen,H/Dever,Josh (The Inessential Indexical [2013], 06.2) | |
| A reaction: The main class which has been proposed is the one that involves indexical beliefs. I agree with this idea. |