32 ideas
| 5438 | Hermeneutics of tradition is sympathetic, hermeneutics of suspicion is hostile [Ricoeur, by Mautner] |
| Full Idea: Ricoeur distinguishes a hermeneutics of tradition (e.g. Gadamar), which interprets sympathetically looking for hidden messages, and a hermeneutics of suspicion (e.g. Nietzsche, Freud) which sees hidden drives and interests. | |
| From: report of Paul Ricoeur (works [1970]) by Thomas Mautner - Penguin Dictionary of Philosophy p.249 | |
| A reaction: Obviously the answer is somewhere between the two. Nietzsche's suspicion can be wonderful, but Freud's can seem silly (e.g. on Leonardo). On the whole I am on the 'tradition' side, because great thinkers can rise above their culture (on a good day). |
| 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. |
| 19566 | Epistemology does not just concern knowledge; all aspects of cognitive activity are involved [Kvanvig] |
| Full Idea: Epistemology is not just knowledge. There is enquiring, reasoning, changes of view, beliefs, assumptions, presuppositions, hypotheses, true beliefs, making sense, adequacy, understanding, wisdom, responsible enquiry, and so on. | |
| From: Jonathan Kvanvig (Truth is not the Primary Epistemic Goal [2005], 'What') | |
| A reaction: [abridged] Stop! I give in. His topic is whether truth is central to epistemology. Rivals seem to be knowledge-first, belief-first, and justification-first. I'm inclined to take justification as the central issue. Does it matter? |
| 19261 | Understanding is seeing coherent relationships in the relevant information [Kvanvig] |
| Full Idea: What is distinctive about understanding (after truth is satisfied) is the internal seeing or appreciating of explanatory and other coherence-inducing relationships in a body of information that is crucial for understanding. | |
| From: Jonathan Kvanvig (The Value of Knowledge and the Pursuit of Understanding [2003], 198), quoted by Anand Vaidya - Understanding and Essence 'Distinction' | |
| A reaction: For me this ticks exactly the right boxes. Coherent explanations are what we want. The hardest part is the ensure their truth. Kvanvig claims this is internal, so we can understand even if, Gettier-style, our external connections are lucky. |
| 19568 | Making sense of things, or finding a good theory, are non-truth-related cognitive successes [Kvanvig] |
| Full Idea: There are cognitive successes that are not obviously truth related, such as the concepts of making sense of the course of experience, and having found an empirically adequate theory. | |
| From: Jonathan Kvanvig (Truth is not the Primary Epistemic Goal [2005], 'Epistemic') | |
| A reaction: He is claiming that truth is not the main aim of epistemology. He quotes Marian David for the rival view. Personally I doubt whether the concepts of 'making sense' or 'empirical adequacy' can be explicated without mentioning truth. |
| 19567 | The 'defeasibility' approach says true justified belief is knowledge if no undermining facts could be known [Kvanvig] |
| Full Idea: The 'defeasibility' approach says that having knowledge requires, in addition to justified true belief, there being no true information which, if learned, would result in the person in question no longer being justified in believing the claim. | |
| From: Jonathan Kvanvig (Truth is not the Primary Epistemic Goal [2005], 'Epistemic') | |
| A reaction: I take this to be an externalist view, since it depends on information of which the cognizer may be unaware. A defeater may yet have an undiscovered counter-defeater. The only real defeater is the falsehood of the proposition. |
| 19679 | 'Access' internalism says responsibility needs access; weaker 'mentalism' needs mental justification [Kvanvig] |
| Full Idea: Strong 'access' internalism says the justification must be accessible to the person holding the belief (for cognitive duty, or blame), and weaker 'mentalist' internalism just says the justification must supervene on mental features of the individual. | |
| From: Jonathan Kvanvig (Epistemic Justification [2011], III) | |
| A reaction: [compressed] I think I'm a strong access internalist. I doubt whether there is a correct answer to any of this, but my conception of someone knowing something involves being able to invoke their reasons for it. Even if they forget the source. |
| 19730 | Epistemic virtues: love of knowledge, courage, caution, autonomy, practical wisdom... [Kvanvig] |
| Full Idea: Virtue theorists may focus on the particular habits or virtues of successful cognizers, such as love of knowledge, firmness, courage and caution, humility, autonomy, generosity, and practical wisdom. | |
| From: Jonathan Kvanvig (Virtue Epistemology [2011], III) | |
| A reaction: [He cites Roberts and Wood 2007] It is interesting that most of these virtues do not merely concern cognition. How about diligence, self-criticism, flexibility...? |
| 19731 | If epistemic virtues are faculties or powers, that doesn't explain propositional knowledge [Kvanvig] |
| Full Idea: Conceiving of the virtues in terms of faculties or powers doesn't help at all with the problem of accounting for propositional knowledge. | |
| From: Jonathan Kvanvig (Virtue Epistemology [2011], IV B) | |
| A reaction: It always looks as if epistemic virtues are a little peripheral to the main business of knowledge, which is getting beliefs to be correct and well-founded. Given that epistemic saints make occasional mistakes, talk of virtues can't be enough. |
| 19732 | The value of good means of attaining truth are swamped by the value of the truth itself [Kvanvig] |
| Full Idea: The Swamping Problem is that the value of truth swamps the value of additional features of true beliefs which are only instrumentally related to them. True belief is no more valuable if one adds a feature valuable for getting one to the truth. | |
| From: Jonathan Kvanvig (Virtue Epistemology [2011], IV B) | |
| A reaction: His targets here are reliabilism and epistemic virtues. Kvanvig's implication is that the key to understanding the nature of knowledge is to pinpoint why we value it so much. |
| 19678 | Strong foundationalism needs strict inferences; weak version has induction, explanation, probability [Kvanvig] |
| Full Idea: Strong foundationalists require truth-preserving inferential links between the foundations and what the foundations support, while weaker versions allow weaker connections, such as inductive support, or best explanation, or probabilistic support. | |
| From: Jonathan Kvanvig (Epistemic Justification [2011], II) | |
| A reaction: [He cites Alston 1989] Personally I'm a coherentist about justification, but I'm a fan of best explanation, so I'd vote for that. It's just that best explanation is not a very foundationalist sort of concept. Actually, the strong version is absurd. |
| 19570 | Reliabilism cannot assess the justification for propositions we don't believe [Kvanvig] |
| Full Idea: The most serious problem for reliabilism is that it cannot explain adequately the concept of propositional justification, the kind of justification one might have for a proposition one does not believe, or which one disbelieves. | |
| From: Jonathan Kvanvig (Truth is not the Primary Epistemic Goal [2005], Notes 2) | |
| A reaction: I don't understand this (though I pass it on anyway). Why can't the reliabilist just offer a critique of the reliability of the justification available for the dubious proposition? |
| 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) |