31 ideas
| 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. |
| 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) |
| 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. |
| 7306 | If the only property of a name was its reference, we couldn't explain bearerless names [Miller,A] |
| Full Idea: If having a reference were the only semantic property in terms of which we could explain the functioning of names, we would be in trouble with respect to names that simply have no bearer. | |
| From: Alexander Miller (Philosophy of Language [1998], 2.1.1) | |
| A reaction: (Miller is discussing Frege) 'Odysseus' is given as an example. Instead of switching to a bundle of descriptions, we could say that we just imagine an object which is stamped with the name. Names always try to refer. |
| 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. |
| 3061 | Anaxarchus said that he was not even sure that he knew nothing [Anaxarchus, by Diog. Laertius] |
| Full Idea: Anaxarchus said that he was not even sure that he knew nothing. | |
| From: report of Anaxarchus (fragments/reports [c.340 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 09.10.1 |
| 7322 | Constitutive scepticism is about facts, and epistemological scepticism about our ability to know them [Miller,A] |
| Full Idea: We should distinguish 'constitutive scepticism' (about the existence of certain sorts of facts) from the traditional 'epistemological scepticism' (which concedes that the sort of fact in question exists, but questions our right to claim knowledge of it). | |
| From: Alexander Miller (Philosophy of Language [1998], 4.7) | |
| A reaction: I would be inclined to call the first type 'ontological scepticism'. Miller is discussing Quine's scepticism about meaning. Atheists fall into the first group, and agnostics into the second. An important, and nicely simple, distinction. |
| 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) |
| 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. |
| 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) |
| 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) |
| 7325 | Dispositions say what we will do, not what we ought to do, so can't explain normativity [Miller,A] |
| Full Idea: Dispositional facts are facts about what we will do, not about what we ought to do, and as such cannot capture the normativity of meaning. | |
| From: Alexander Miller (Philosophy of Language [1998], 6.2) | |
| A reaction: Miller is discussing language, but this raises a nice question for all behaviourist accounts of mental events. Perhaps there is a disposition to behave in a guilty way if you do something you think you shouldn't do. (Er, isn't 'guilt' a mental event?) |
| 7324 | Explain meaning by propositional attitudes, or vice versa, or together? [Miller,A] |
| Full Idea: Grice wants to explain linguistic meaning in terms of the content of propositional attitudes, Dummett has championed the view that propositional attitudes must be explained by linguistic meaning, while Davidson says they must be explained together. | |
| From: Alexander Miller (Philosophy of Language [1998], 6.1) | |
| A reaction: A useful map. My intuition says propositional attitudes come first, for evolutionary reasons. We are animals first, and speakers second. Thought precedes language. A highly social animal flourishes if it can communicate. |
| 7323 | If truth is deflationary, sentence truth-conditions just need good declarative syntax [Miller,A] |
| Full Idea: On a deflationary concept of truth, for a sentence to possess truth-conditions it is sufficient that it be disciplined by norms of correct usage, and that it possess the syntax distinctive of declarative sentences. | |
| From: Alexander Miller (Philosophy of Language [1998], 5.3) | |
| A reaction: Idea 6337 gives the basic deflationary claim. He mentions Boghossian as source of this point. So much the worse for the deflationary concept of truth, say I. What are the truth-conditions of "Truth rotates"? |
| 7315 | 'Jones is a married bachelor' does not have the logical form of a contradiction [Miller,A] |
| Full Idea: The syntactic notion of contradiction (p and not-p) is well understood, but is no help in explaining analyticity, since "Jones is a married bachelor" is not of that syntactic form. | |
| From: Alexander Miller (Philosophy of Language [1998], 4.2) | |
| A reaction: This point is based on Quine. This means we cannot define analytic sentences as those whose denial is a contradiction, even though that seems to be true of them. Both the Kantian and the modern logical versions of analyticity are in trouble. |
| 7328 | The principle of charity is holistic, saying we must hold most of someone's system of beliefs to be true [Miller,A] |
| Full Idea: Properly construed, the principle of charity is a holistic constraint applying, not to individual beliefs, but rather to systems of belief: we must interpret a speaker so that most of the beliefs in his system are, by our lights, true. | |
| From: Alexander Miller (Philosophy of Language [1998], 8.7) | |
| A reaction: This is a lot more plausible than applying the principle to individual sentences, particularly if you are in the company of habitual ironists or constitutional liars. |
| 7329 | Maybe we should interpret speakers as intelligible, rather than speaking truth [Miller,A] |
| Full Idea: A more sophisticated version of the principle of charity holds that we interpret speakers not as necessarily having beliefs that are true by our own lights, but as having beliefs that are intelligible by our own lights. | |
| From: Alexander Miller (Philosophy of Language [1998], 8.7) | |
| A reaction: Consider Idea 4161 in the light of this. Presumably this means that we treat them as having a coherent set of beliefs, even if they seem to us to fail to correspond to reality. I prefer the stronger version that there has to be some proper truth in there. |
| 7333 | The Frege-Geach problem is that I can discuss the wrongness of murder without disapproval [Miller,A] |
| Full Idea: The main problem faced by non-cognitivism is known as the Frege-Geach problem: if I say "If murder is wrong, then getting your brother to murder people is wrong", that is an unasserted context, and I don't necessarily express disapproval of murder. | |
| From: Alexander Miller (Philosophy of Language [1998], 9.2) | |
| A reaction: The emotivist or non-cognitivist might mount a defence by saying there is some second-order or deep-buried emotion involved. Could a robot without feelings even understand what humans meant when they said "It is morally wrong"? |