39 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. |
16185 | Causality indicates which properties are real [Cartwright,N] |
Full Idea: Causality is a clue to what properties are real. | |
From: Nancy Cartwright (How the Laws of Physics Lie [1983], 9.3) | |
A reaction: An interesting variant on the Shoemaker proposal that properties actually are causal. I'm not sure that there is anything more to causality that the expression in action of properties, which I take to be powers. Structures are not properties. |
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] |
16182 | Two main types of explanation are by causes, or by citing a theoretical framework [Cartwright,N] |
Full Idea: In explaining a phenomenon one can cite the causes of that phenomenon; or one can set the phenomenon in a general theoretical framework. | |
From: Nancy Cartwright (How the Laws of Physics Lie [1983], 4.1) | |
A reaction: The thing is, you need to root an explanation in something taken as basic, and theoretical frameworks need further explanation, whereas causes seem to be basic. |
16184 | An explanation is a model that fits a theory and predicts the phenomenological laws [Cartwright,N] |
Full Idea: To explain a phenomenon is to find a model that fits it into the basic framework of the theory and that thus allows us to derive analogues for the messy and complicated phenomenological laws that are true of it. | |
From: Nancy Cartwright (How the Laws of Physics Lie [1983], 8.3) | |
A reaction: This summarises the core of her view in this book. She is after models rather than laws, and the models are based on causes. |
16171 | The covering law view assumes that each phenomenon has a 'right' explanation [Cartwright,N] |
Full Idea: The covering-law account supposes that there is, in principle, one 'right' explanation for each phenomenon. | |
From: Nancy Cartwright (How the Laws of Physics Lie [1983], Intro) | |
A reaction: Presumably the law is held to be 'right', but there must be a bit of flexibility in describing the initial conditions, and the explanandum itself. |
16167 | Laws get the facts wrong, and explanation rests on improvements and qualifications of laws [Cartwright,N] |
Full Idea: We explain by ceteris paribus laws, by composition of causes, and by approximations that improve on what the fundamental laws dictate. In all of these cases the fundamental laws patently do not get the facts right. | |
From: Nancy Cartwright (How the Laws of Physics Lie [1983], Intro) | |
A reaction: It is rather headline-grabbing to say in this case that laws do not get the facts right. If they were actually 'wrong' and 'lied', there wouldn't be much point in building explanations on them. |
16169 | Laws apply to separate domains, but real explanations apply to intersecting domains [Cartwright,N] |
Full Idea: When different kinds of causes compose, we want to explain what happens in the intersection of different domains. But the laws we use are designed only to tell truly what happens in each domain separately. | |
From: Nancy Cartwright (How the Laws of Physics Lie [1983], Intro) | |
A reaction: Since presumably the laws are discovered through experiments which try to separate out a single domain, in those circumstances they actually are true, so they don't 'lie'. |
16176 | Covering-law explanation lets us explain storms by falling barometers [Cartwright,N] |
Full Idea: Much criticism of the original covering-law model objects that it lets in too much. It seems we can explain Henry's failure to get pregnant by his taking birth control pills, and we can explain the storm by the falling barometer. | |
From: Nancy Cartwright (How the Laws of Physics Lie [1983], 2.0) | |
A reaction: I take these examples to show that true explanations must be largely causal in character. The physicality of causation is what matters, not 'laws'. I'd say the same of attempts to account for causation through counterfactuals. |
16177 | I disagree with the covering-law view that there is a law to cover every single case [Cartwright,N] |
Full Idea: Covering-law theorists tend to think that nature is well-regulated; in the extreme, that there is a law to cover every case. I do not. | |
From: Nancy Cartwright (How the Laws of Physics Lie [1983], 2.2) | |
A reaction: The problem of coincidence is somewhere at the back of this thought. Innumerable events have their own explanations, but it is hard to explain their coincidence (see Aristotle's case of bumping into a friend in the market). |
16180 | You can't explain one quail's behaviour by just saying that all quails do it [Cartwright,N] |
Full Idea: 'Why does that quail in the garden bob its head up and down in that funny way whenever it walks?' …'Because they all do'. | |
From: Nancy Cartwright (How the Laws of Physics Lie [1983], 3.5) | |
A reaction: She cites this as an old complaint against the covering-law model of explanation. It captures beautifully the basic error of the approach. We want to know 'why', rather than just have a description of the pattern. 'They all do' is useful information. |
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) |
16183 | In science, best explanations have regularly turned out to be false [Cartwright,N] |
Full Idea: There are a huge number of cases in the history of science where we now know our best explanations were false. | |
From: Nancy Cartwright (How the Laws of Physics Lie [1983], 5.3) | |
A reaction: [She cites Laudan 1981 for this] The Ptolemaic system and aether are the standard example cited for this. I believe strongly in the importance of best explanation. Only a fool would just accept the best explanation available. Coherence is needed. |
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) |
3400 | Things must have parts to intermingle [Gassendi] |
Full Idea: If you are no larger than a point, how are you joined to the whole body, which is so large? …and there can be no intermingling between things unless the parts of them can be intermingled. | |
From: Pierre Gassendi (Objections to 'Meditations' (Fifth) [1641]), quoted by Jaegwon Kim - Philosophy of Mind p.131 | |
A reaction: As Descartes says that mind is distinct from body because it is non-spatial, it doesn't seem quite right to describe it as a 'point', but the second half is a real problem. Being non-spatial is a real impediment to intermingling with spatial objects. |
16175 | A cause won't increase the effect frequency if other causes keep interfering [Cartwright,N] |
Full Idea: A cause ought to increase the frequency of the effect, but this fact may not show up in the probabilities if other causes are at work. | |
From: Nancy Cartwright (How the Laws of Physics Lie [1983], 1.1) | |
A reaction: [She cites Patrick Suppes for this one] Presumably in experimental situations you can weed out the interference, but that threatens to eliminate mere 'probability' entirely. |
6781 | There are fundamental explanatory laws (false!), and phenomenological laws (regularities) [Cartwright,N, by Bird] |
Full Idea: Nancy Cartwright distinguishes between 'fundamental explanatory laws', which we should not believe, and 'phenomenological laws', which are regularities established on the basis of observation. | |
From: report of Nancy Cartwright (How the Laws of Physics Lie [1983]) by Alexander Bird - Philosophy of Science Ch.4 | |
A reaction: The distinction is helpful, so that we can be clearer about what everyone is claiming. We can probably all agree on the phenomenological laws, which are epistemological. Personally I claim truth for the best fundamental explanatory laws. |
16166 | Laws of appearances are 'phenomenological'; laws of reality are 'theoretical' [Cartwright,N] |
Full Idea: Philosophers distinguish phenomenological from theoretical laws. Phenomenological laws are about appearances; theoretical ones are about the reality behind the appearances. | |
From: Nancy Cartwright (How the Laws of Physics Lie [1983], Intro) | |
A reaction: I'm suspecting that Humeans only really believe in the phenomenological kind. I'm only interested in the theoretical kind, and I take inference to the best explanation to be the bridge between the two. Cartwright rejects the theoretical laws. |
16179 | Good organisation may not be true, and the truth may not organise very much [Cartwright,N] |
Full Idea: There is no reason to think that the principles that best organise will be true, nor that the principles that are true will organise much. | |
From: Nancy Cartwright (How the Laws of Physics Lie [1983], 2.5) | |
A reaction: This is aimed at the Mill-Ramsey-Lewis account of laws, as axiomatisations of the observed patterns in nature. |
16178 | There are few laws for when one theory meets another [Cartwright,N] |
Full Idea: Where theories intersect, laws are usually hard to come by. | |
From: Nancy Cartwright (How the Laws of Physics Lie [1983], 2.3) | |
A reaction: There are attempts at so-called 'bridge laws', to get from complex theories to simple ones, but her point is well made about theories on the same 'level'. |
16170 | To get from facts to equations, we need a prepared descriptions suited to mathematics [Cartwright,N] |
Full Idea: To get from a detailed factual knowledge of a situation to an equation, we must prepare the description of the situation to meet the mathematical needs of the theory. | |
From: Nancy Cartwright (How the Laws of Physics Lie [1983], Intro) | |
A reaction: She is clearly on to something here, as Galileo is blatantly wrong in his claim that the book of nature is written in mathematics. Mathematics is the best we can manage in getting a grip on the chaos. |
16181 | Simple laws have quite different outcomes when they act in combinations [Cartwright,N] |
Full Idea: For explanation simple laws must have the same form when they act together as when they act singly. ..But then what the law states cannot literally be true, for the consequences that occur if it acts alone are not what occurs when they act in combination. | |
From: Nancy Cartwright (How the Laws of Physics Lie [1983], 3.6) | |
A reaction: This is Cartwright's basic thesis. Her point is that the laws 'lie', because they claim to predict a particular outcome which never ever actually occurs. She says we could know all the laws, and still not be able to explain anything. |