29 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. |
| 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. |
| 5163 | Basic propositions refer to a single experience, are incorrigible, and conclusively verifiable [Ayer] |
| Full Idea: There is a class of empirical propositions, which I call 'basic propositions', which can be verified conclusively, since they refer solely to the contents of a single experience, which are incorrigible. | |
| From: A.J. Ayer (Introduction to 'Language Truth and Logic' [1946], p.13) | |
| A reaction: A classic statement of empirical foundationalism. I sort of agree that 'single experiences' are a 'given' for philosophy, but is questionable whether there is anything which could both be a single experience AND give rise to a 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) |
| 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) |
| 5167 | The argument from analogy fails, so the best account of other minds is behaviouristic [Ayer] |
| Full Idea: There are too many objections to the argument from analogy, so I am inclined to revert to a 'behaviouristic' interpretation of propositions about other people's experiences. | |
| From: A.J. Ayer (Introduction to 'Language Truth and Logic' [1946], p.26) | |
| A reaction: It seems odd to vote for behaviourism on one issue, if you aren't a general subscriber. It is one thing to say that behaviour is the best evidence for your explanation, quite another to equate the other mind with its behaviour. |
| 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) |
| 5164 | A statement is meaningful if observation statements can be deduced from it [Ayer] |
| Full Idea: In the improved version, a statement was verifiable, and consequently meaningful, if 'some observation-statement can be deduced from it in conjunction with certain other premises, without being deducible from those other premises alone'. | |
| From: A.J. Ayer (Introduction to 'Language Truth and Logic' [1946], p.15) | |
| A reaction: I.Berlin showed that any statement S could pass this test, because if you assert 'S' and 'If S then O', these two statements entail O, which could be some random observation. Hence a 1946 revised version had to be produced. |
| 5165 | Directly verifiable statements must entail at least one new observation statement [Ayer] |
| Full Idea: A statement is directly verifiable if it is either itself an observation-statement,or is such that in conjunction with one or more observation-statements it entails at least one observation-statement which is not deducible from these other premises alone. | |
| From: A.J. Ayer (Introduction to 'Language Truth and Logic' [1946], p.17) | |
| A reaction: This is the 1946 revised version of the Verification Principle, which was then torpedoed by an elaborate counterexample from Alonzo Church. Ayer thereafter abandoned attempts to find a precise statement of it. |
| 5166 | The principle of verification is not an empirical hypothesis, but a definition [Ayer] |
| Full Idea: I wish the principle of verification to be regarded, not as an empirical hypothesis, but as a definition. | |
| From: A.J. Ayer (Introduction to 'Language Truth and Logic' [1946], p.21) | |
| A reaction: This is Ayer's attempt to meet the well known objection of 'turning the tables' on his theory (by asking whether it is tautological or empirically verifiable). However, if it is just a definition, then presumably it is completely arbitrary… |
| 5162 | Sentences only express propositions if they are meaningful; otherwise they are 'statements' [Ayer] |
| Full Idea: I suggest that every grammatically significant indicative sentence expresses a 'statement', but the word 'proposition' will be reserved for what is expressed by sentences that are literally meaningful. | |
| From: A.J. Ayer (Introduction to 'Language Truth and Logic' [1946], p.10) | |
| A reaction: We don't have to accept Ayer's over-fussy requirements for what is meaningful to accept that this is a good distinction. Every day we hear statements from people (e.g. politicians) in which we can fish in vain for the underlying proposition. |
| 7667 | There are two sides to men - the pleasantly social, and the violent and creative [Diderot, by Berlin] |
| Full Idea: Diderot is among the first to preach that there are two men: the artificial man, who belongs in society and seeks to please, and the violent, bold, criminal instinct of a man who wishes to break out (and, if controlled, is responsible for works of genius. | |
| From: report of Denis Diderot (works [1769], Ch.3) by Isaiah Berlin - The Roots of Romanticism | |
| A reaction: This has an obvious ancestor in Plato's picture (esp. in 'Phaedrus') of the two conflicting sides to the psuché, which seem to be reason and emotion. In Diderot, though, the suppressed man has virtues, which Plato would deny. |
| 5168 | Moral approval and disapproval concerns classes of actions, rather than particular actions [Ayer] |
| Full Idea: The common objects of moral approval and disapproval are not particular actions so much as classes of actions. | |
| From: A.J. Ayer (Introduction to 'Language Truth and Logic' [1946], p.27) | |
| A reaction: This 1946 revision of his pure emotivism looks like a move towards Hare's prescriptivism, where classes, rules and principles are seen as the window-dressing of emotivism. It's still a bad theory. |