45 ideas
12452 | Our dislike of contradiction in logic is a matter of psychology, not mathematics [Brouwer] |
Full Idea: Not to the mathematician, but to the psychologist, belongs the task of explaining why ...we are averse to so-called contradictory systems in which the negative as well as the positive of certain propositions are valid. | |
From: Luitzen E.J. Brouwer (Intuitionism and Formalism [1912], p.79) | |
A reaction: Was the turning point of Graham Priest's life the day he read this sentence? I don't agree. I take the principle of non-contradiction to be a highly generalised observation of how the world works (and Russell agrees with me). |
15941 | For intuitionists excluded middle is an outdated historical convention [Brouwer] |
Full Idea: From the intuitionist standpoint the dogma of the universal validity of the principle of excluded third in mathematics can only be considered as a phenomenon of history of civilization, like the rationality of pi or rotation of the sky about the earth. | |
From: Luitzen E.J. Brouwer (works [1930]), quoted by Shaughan Lavine - Understanding the Infinite VI.2 | |
A reaction: [Brouwer 1952:510-11] |
18119 | Mathematics is a mental activity which does not use language [Brouwer, by Bostock] |
Full Idea: Brouwer made the rather extraordinary claim that mathematics is a mental activity which uses no language. | |
From: report of Luitzen E.J. Brouwer (Mathematics, Science and Language [1928]) by David Bostock - Philosophy of Mathematics 7.1 | |
A reaction: Since I take language to have far less of a role in thought than is commonly believed, I don't think this idea is absurd. I would say that we don't use language much when we are talking! |
18247 | Brouwer saw reals as potential, not actual, and produced by a rule, or a choice [Brouwer, by Shapiro] |
Full Idea: In his early writing, Brouwer took a real number to be a Cauchy sequence determined by a rule. Later he augmented rule-governed sequences with free-choice sequences, but even then the attitude is that Cauchy sequences are potential, not actual infinities. | |
From: report of Luitzen E.J. Brouwer (works [1930]) by Stewart Shapiro - Philosophy of Mathematics 6.6 | |
A reaction: This is the 'constructivist' view of numbers, as espoused by intuitionists like Brouwer. |
12451 | Scientific laws largely rest on the results of counting and measuring [Brouwer] |
Full Idea: A large part of the natural laws introduced by science treat only of the mutual relations between the results of counting and measuring. | |
From: Luitzen E.J. Brouwer (Intuitionism and Formalism [1912], p.77) | |
A reaction: His point, I take it, is that the higher reaches of numbers have lost touch with the original point of the system. I now see the whole issue as just depending on conventions about the agreed extension of the word 'number'. |
18118 | Brouwer regards the application of mathematics to the world as somehow 'wicked' [Brouwer, by Bostock] |
Full Idea: Brouwer regards as somehow 'wicked' the idea that mathematics can be applied to a non-mental subject matter, the physical world, and that it might develop in response to the needs which that application reveals. | |
From: report of Luitzen E.J. Brouwer (Mathematics, Science and Language [1928]) by David Bostock - Philosophy of Mathematics 7.1 | |
A reaction: The idea is that mathematics only concerns creations of the human mind. It presumably has no more application than, say, noughts-and-crosses. |
12454 | Intuitionists only accept denumerable sets [Brouwer] |
Full Idea: The intuitionist recognises only the existence of denumerable sets. | |
From: Luitzen E.J. Brouwer (Intuitionism and Formalism [1912], p.80) | |
A reaction: That takes you up to omega, but not beyond, presumably because it then loses sight of the original intuition of 'bare two-oneness' (Idea 12453). I sympathise, but the word 'number' has shifted its meaning a lot these days. |
12453 | Neo-intuitionism abstracts from the reuniting of moments, to intuit bare two-oneness [Brouwer] |
Full Idea: Neo-intuitionism sees the falling apart of moments, reunited while remaining separated in time, as the fundamental phenomenon of human intellect, passing by abstracting to mathematical thinking, the intuition of bare two-oneness. | |
From: Luitzen E.J. Brouwer (Intuitionism and Formalism [1912], p.80) | |
A reaction: [compressed] A famous and somewhat obscure idea. He goes on to say that this creates one and two, and all the finite ordinals. |
8728 | Intuitionist mathematics deduces by introspective construction, and rejects unknown truths [Brouwer] |
Full Idea: Mathematics rigorously treated from the point of view of deducing theorems exclusively by means of introspective construction, is called intuitionistic mathematics. It deviates from classical mathematics, which believes in unknown truths. | |
From: Luitzen E.J. Brouwer (Consciousness, Philosophy and Mathematics [1948]), quoted by Stewart Shapiro - Thinking About Mathematics 1.2 | |
A reaction: Clearly intuitionist mathematics is a close cousin of logical positivism and the verification principle. This view would be anathema to Frege, because it is psychological. Personally I believe in the existence of unknown truths, big time! |
17377 | All descriptive language is classificatory [Dupré] |
Full Idea: Classification pervades any descriptive use of language whatever. | |
From: John Dupré (The Disorder of Things [1993], 1) | |
A reaction: This is because, as Aristotle well knew, language consists almost entirely of universals (apart from the proper names). Language just is classification. |
17376 | We should aim for a classification which tells us as much as possible about the object [Dupré] |
Full Idea: The most important desideratum of a classificatory scheme is that assigning an object to a particular classification tell us as much as possible about that object. | |
From: John Dupré (The Disorder of Things [1993], Ch 1) | |
A reaction: We should probably say that the aim is a successful explanation, rather than a heap of information. If we are totally baffled by a particular type of object, it is presumably important to group the instances together, to focus the bafflement. |
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. |
17390 | Natural kinds don't need essentialism to be explanatory [Dupré] |
Full Idea: The importance of natural kinds for explanation does not depend on a doctrine of essences. | |
From: John Dupré (The Disorder of Things [1993], 3) | |
A reaction: He suggest as the alternative that laws do the explaining, employing natural kinds. He allows that individual essences might be explanatory. |
17388 | It seems that species lack essential properties, so they can't be natural kinds [Dupré] |
Full Idea: It is widely agreed among biologists that no essential property can be found to demarcate species, so that if an essential property is necessary for a natural kind, species are not natural kinds. | |
From: John Dupré (The Disorder of Things [1993], 2) | |
A reaction: This uses 'essential' to mean 'necessary', but I would use 'essential' to mean 'deeply explanatory'. Biological species are, nevertheless, dubious members of an ontological system. Vegetables are the problem. |
17389 | A species might have its essential genetic mechanism replaced by a new one [Dupré] |
Full Idea: Contradicting one of the main points of essentialism, there is no reason in principle why a species should not survive the demise of its current genetic mechanisms (some other species coherence gradually taking over). | |
From: John Dupré (The Disorder of Things [1993], 2) | |
A reaction: I would say that this meant that the species had a new essence, because I don't take what is essential to be the same as what is necessary. The new genetics would replace the old as the basic explanation of the species. |
17374 | The possibility of prediction rests on determinism [Dupré] |
Full Idea: Determinism is the metaphysical underlay of the possibility of prediction. | |
From: John Dupré (The Disorder of Things [1993], Intro) | |
A reaction: Not convinced. There might be micro-indeterminacies which iron out into macro-regularities. |
17503 | Theories can never represent accurately, because their components are abstract [Cartwright,N, by Portides] |
Full Idea: Cartwright objects that the claim that theories represent what happens in actual situations is to overlook that the concepts used in them (such as 'force functions' and 'Hamiltonians') are abstract. | |
From: report of Nancy Cartwright (The Dappled World [1999]) by Demetris Portides - Models 'Current' | |
A reaction: I'm not convinced by this. The term 'abstract' is too loose. In a sense most words are abstract because they are universals. If I say 'that's a cat', that is a very accurate remark, despite the generality of 'cat'. |
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. |
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. |
17378 | Presumably molecular structure seems important because we never have the Twin Earth experience [Dupré] |
Full Idea: It is surely the absence of experiences like the one Putnam describes that makes it reasonable to attach to molecular structure at least most of the importance that Putnam ascribes to it. | |
From: John Dupré (The Disorder of Things [1993], 1) | |
A reaction: That is, whenever we experience water-like stuff, it always turns out to have the same molecular structure. Twin Earth is a nice thought experiment, except that XZY is virtually inconceivable. |
10117 | Intuitonists in mathematics worried about unjustified assertion, as well as contradiction [Brouwer, by George/Velleman] |
Full Idea: The concern of mathematical intuitionists was that the use of certain forms of inference generates, not contradiction, but unjustified assertions. | |
From: report of Luitzen E.J. Brouwer (Intuitionism and Formalism [1912]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.6 | |
A reaction: This seems to be the real origin of the verificationist idea in the theory of meaning. It is a hugely revolutionary idea - that ideas are not only ruled out of court by contradiction, but that there are other criteria which should also be met. |
17381 | Phylogenetics involves history, and cladism rests species on splits in lineage [Dupré] |
Full Idea: The phylogenetic conception of classification reflects the facts of evolutionary history. Cladism insists that every taxonomic distinction should reflect an evolutionary event of lineage bifurcation. | |
From: John Dupré (The Disorder of Things [1993], 1) | |
A reaction: Devitt attacks cladism nicely. It rules out species change without bifurcation, and it insists on species change even in a line which remains unchanged after a split. |
17385 | Kinds don't do anything (including evolve) because they are abstract [Dupré] |
Full Idea: A kind, being an abstract object, cannot do anything, including evolve. | |
From: John Dupré (The Disorder of Things [1993], 2) | |
A reaction: Maybe. We might have an extensional view of the kind, so that 'gold' is the set of extant gold atoms. But possible gold atoms are also gold, and defunct ones too. Virtually every word in English is abtract if you think about it long enough. |
17380 | Wales may count as fish [Dupré] |
Full Idea: The claim that whales are not fish is a debatable one | |
From: John Dupré (The Disorder of Things [1993], 1) | |
A reaction: A very nice challenge to an almost unquestioned orthodoxy. |
17375 | Natural kinds are decided entirely by the intentions of our classification [Dupré] |
Full Idea: The question of which natural kind a thing belongs to ....can be answered only in relation to some specification of the goal underlying the intent to classify the object. | |
From: John Dupré (The Disorder of Things [1993], Intro) | |
A reaction: I don't think I believe this. The situation is complex, and our intents are relevant, but to find an intent which no longer classifies tigers into the same category is wilful silliness. |
17379 | Borders between species are much less clear in vegetables than among animals [Dupré] |
Full Idea: The richest source of illustrations is the vegetable kingdom, where specific differences tend to be much less clear than among animals, and considerable developmental plasticity is the rule. | |
From: John Dupré (The Disorder of Things [1993], 1) | |
A reaction: Nice. Just as the idea that laws of nature are mathematical suits physics, but founders on biology, so natural kinds founder in an area of biology to which we pay less attention. He cites prickly pears and lilies. I'm thinking oranges, satsumas etc. |
17382 | Cooks, unlike scientists, distinguish garlic from onions [Dupré] |
Full Idea: It would be a severe culinary misfortune if no distinction were drawn between garlic and onions, a distinction that is not reflected in scientific taxonomy. | |
From: John Dupré (The Disorder of Things [1993], 1) | |
A reaction: Not every persuasive. We distinguish some cows from others because they taste better, but no one thinks that is a serious way in which to classify cows. |
17384 | Even atoms of an element differ, in the energy levels of their electrons [Dupré] |
Full Idea: Even if we claim that it is really isotopes not atoms that are the natural kinds (thus divorcing chemistry from ordinary language), atoms are said to differ with respect to such features as energy levels of the electrons. | |
From: John Dupré (The Disorder of Things [1993], 2) | |
A reaction: So we can't just pick out the features of one atom, and say that is the essence. Essence always involves some selection. I say the essence arises from the explanation of the atom's behaviour. |
17387 | Ecologists favour classifying by niche, even though that can clash with genealogy [Dupré] |
Full Idea: To the extent that the occupants of a particular niche do not coincide with the members of a particular genealogical line, a possibility widely acknowledged to occur, ecologists must favour a method of classification lacking genealogical grounding. | |
From: John Dupré (The Disorder of Things [1993], 2) | |
A reaction: Zoo keepers probably classify by cages, or which zoo owns what, but that doesn't mean that they reject genealogy. Don't assume ecologists are rejecting any underlying classification that differs from theirs. Compare classification by economists. |
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. |
17386 | The theory of evolution is mainly about species [Dupré] |
Full Idea: Species are what the theory of evolution is centrally about. | |
From: John Dupré (The Disorder of Things [1993], 2) |
17383 | Species are the lowest-level classification in biology [Dupré] |
Full Idea: Species are, by definition, the lowest-level classificatory unit, or basal taxonomic unit, for biological organisms. | |
From: John Dupré (The Disorder of Things [1993], 2) | |
A reaction: I think this is the 'infima species' for Aristotelians. What about 'male' and 'female' in each species? |