30 ideas
| 10147 | The Axiom of Choice is consistent with the other axioms of set theory [Feferman/Feferman] |
| Full Idea: In 1938 Gödel proved that the Axiom of Choice is consistent with the other axioms of set theory. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) | |
| A reaction: Hence people now standardly accept ZFC, rather than just ZF. |
| 10148 | Axiom of Choice: a set exists which chooses just one element each of any set of sets [Feferman/Feferman] |
| Full Idea: Zermelo's Axiom of Choice asserts that for any set of non-empty sets that (pairwise) have no elements in common, then there is a set that 'simultaneously chooses' exactly one element from each set. Note that this is an existential claim. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) | |
| A reaction: The Axiom is now widely accepted, after much debate in the early years. Even critics of the Axiom turn out to be relying on it. |
| 10149 | Platonist will accept the Axiom of Choice, but others want criteria of selection or definition [Feferman/Feferman] |
| Full Idea: The Axiom of Choice seems clearly true from the Platonistic point of view, independently of how sets may be defined, but is rejected by those who think such existential claims must show how to pick out or define the object claimed to exist. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) | |
| A reaction: The typical critics are likely to be intuitionists or formalists, who seek for both rigour and a plausible epistemology in our theory. |
| 10150 | The Trichotomy Principle is equivalent to the Axiom of Choice [Feferman/Feferman] |
| Full Idea: The Trichotomy Principle (any number is less, equal to, or greater than, another number) turned out to be equivalent to the Axiom of Choice. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) | |
| A reaction: [He credits Sierpinski (1918) with this discovery] |
| 10146 | Cantor's theories needed the Axiom of Choice, but it has led to great controversy [Feferman/Feferman] |
| Full Idea: The Axiom of Choice is a pure existence statement, without defining conditions. It was necessary to provide a foundation for Cantor's theory of transfinite cardinals and ordinal numbers, but its nonconstructive character engendered heated controversy. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) |
| 10158 | A structure is a 'model' when the axioms are true. So which of the structures are models? [Feferman/Feferman] |
| Full Idea: A structure is said to be a 'model' of an axiom system if each of its axioms is true in the structure (e.g. Euclidean or non-Euclidean geometry). 'Model theory' concerns which structures are models of a given language and axiom system. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) | |
| A reaction: This strikes me as the most interesting aspect of mathematical logic, since it concerns the ways in which syntactic proof-systems actually connect with reality. Tarski is the central theoretician here, and his theory of truth is the key. |
| 10162 | Tarski and Vaught established the equivalence relations between first-order structures [Feferman/Feferman] |
| Full Idea: In the late 1950s Tarski and Vaught defined and established basic properties of the relation of elementary equivalence between two structures, which holds when they make true exactly the same first-order sentences. This is fundamental to model theory. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) | |
| A reaction: This is isomorphism, which clarifies what a model is by giving identity conditions between two models. Note that it is 'first-order', and presumably founded on classical logic. |
| 10160 | Löwenheim-Skolem says if the sentences are countable, so is the model [Feferman/Feferman] |
| Full Idea: The Löwenheim-Skolem Theorem, the earliest in model theory, states that if a countable set of sentences in a first-order language has a model, then it has a countable model. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) | |
| A reaction: There are 'upward' (sentences-to-model) and 'downward' (model-to-sentences) versions of the theory. |
| 10159 | Löwenheim-Skolem Theorem, and Gödel's completeness of first-order logic, the earliest model theory [Feferman/Feferman] |
| Full Idea: Before Tarski's work in the 1930s, the main results in model theory were the Löwenheim-Skolem Theorem, and Gödel's establishment in 1929 of the completeness of the axioms and rules for the classical first-order predicate (or quantificational) calculus. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) |
| 10161 | If a sentence holds in every model of a theory, then it is logically derivable from the theory [Feferman/Feferman] |
| Full Idea: Completeness is when, if a sentences holds in every model of a theory, then it is logically derivable from that theory. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) |
| 10156 | 'Recursion theory' concerns what can be solved by computing machines [Feferman/Feferman] |
| Full Idea: 'Recursion theory' is the subject of what can and cannot be solved by computing machines | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Ch.9) | |
| A reaction: This because 'recursion' will grind out a result step-by-step, as long as the steps will 'halt' eventually. |
| 10155 | Both Principia Mathematica and Peano Arithmetic are undecidable [Feferman/Feferman] |
| Full Idea: In 1936 Church showed that Principia Mathematica is undecidable if it is ω-consistent, and a year later Rosser showed that Peano Arithmetic is undecidable, and any consistent extension of it. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int IV) |
| 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. |
| 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. |
| 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. |
| 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. |
| 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. |
| 20329 | A work of art is an artifact created for the artworld [Dickie] |
| Full Idea: A work of art is an artifact of a kind created to be presented to an artworld public. | |
| From: George Dickie (The New Institutional Theory of Art [1983], p.53) | |
| A reaction: This is the culminating definition in his paper, deriving originally from Danto, and an improvement of his earlier more complex definition. Since this definition amounts to 'this is art if I say it is art', it doesn't seem to reveal much. |
| 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. |
| 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. |
| 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'. |