27 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) |
| 16007 | I assume existence, rather than reasoning towards it [Kierkegaard] |
| Full Idea: I always reason from existence, not towards existence. | |
| From: Søren Kierkegaard (Philosophical Fragments [1844], p.40) | |
| A reaction: Kierkegaard's important premise to help show that theistic proofs for God's existence don't actually prove existence, but develop the content of a conception. [SY] |
| 16554 | Activities have place, rate, duration, entities, properties, modes, direction, polarity, energy and range [Machamer/Darden/Craver] |
| Full Idea: Activities can be identified spatiotemporally, and individuated by rate, duration, and types of entity and property that engage in them. They also have modes of operation, directionality, polarity, energy requirements and a range. | |
| From: Machamer,P/Darden,L/Craver,C (Thinking About Mechanisms [2000], 3) | |
| A reaction: This is their attempt at making 'activity' one of the two central concepts of ontology, along with 'entity'. A helpful analysis. It just seems to be one way of slicing the cake. |
| 16556 | Penicillin causes nothing; the cause is what penicillin does [Machamer/Darden/Craver] |
| Full Idea: It is not the penicillin that causes the pneumonia to disappear, but what the penicillin does. | |
| From: Machamer,P/Darden,L/Craver,C (Thinking About Mechanisms [2000], 3.1) | |
| A reaction: This is a very neat example for illustrating how we slip into 'entity' talk, when the reality we are addressing actually concerns processes. Without the 'what it does', penicillin can't participate in causation at all. |
| 16013 | Nothing necessary can come into existence, since it already 'is' [Kierkegaard] |
| Full Idea: Can the necessary come into existence? That is a change, and everything that comes into existence demonstrates that it is not necessary. The necessary already 'is'. | |
| From: Søren Kierkegaard (Philosophical Fragments [1844], p.74) | |
| A reaction: [SY] |
| 16562 | We understand something by presenting its low-level entities and activities [Machamer/Darden/Craver] |
| Full Idea: The intelligibility of a phenomenon consists in the mechanisms being portrayed in terms of a field's bottom out entities and activities. | |
| From: Machamer,P/Darden,L/Craver,C (Thinking About Mechanisms [2000], 7) | |
| A reaction: In other words, we understand complex things by reducing them to things we do understand. It would, though, be illuminating to see a nest of interconnected activities, even if we understood none of them. |
| 16563 | The explanation is not the regularity, but the activity sustaining it [Machamer/Darden/Craver] |
| Full Idea: It is not regularities that explain but the activities that sustain the regularities. | |
| From: Machamer,P/Darden,L/Craver,C (Thinking About Mechanisms [2000], 7) | |
| A reaction: Good, but we had better not characterise the 'activities' in terms of regularities. |
| 16555 | Functions are not properties of objects, they are activities contributing to mechanisms [Machamer/Darden/Craver] |
| Full Idea: It is common to speak of functions as properties 'had by' entities, …but they should rather be understood in terms of the activities by virtue of which entities contribute to the workings of a mechanism. | |
| From: Machamer,P/Darden,L/Craver,C (Thinking About Mechanisms [2000], 3) | |
| A reaction: I'm certainly quite passionately in favour of cutting down on describing the world almost entirely in terms of entities which have properties. An 'activity', though, is a bit of an elusive concept. |
| 16528 | Mechanisms are not just push-pull systems [Machamer/Darden/Craver] |
| Full Idea: One should not think of mechanisms as exclusively mechanical (push-pull) systems. | |
| From: Machamer,P/Darden,L/Craver,C (Thinking About Mechanisms [2000], 1) | |
| A reaction: The difficulty seems to be that you could broaden the concept of 'mechanism' indefinitely, so that it covered history, mathematics, populations, cultural change, and even mathematics. Where to stop? |
| 16529 | Mechanisms are systems organised to produce regular change [Machamer/Darden/Craver] |
| Full Idea: Mechanisms are entities and activities organized such that they are productive of regular change from start or set-up to finish or termination conditions. | |
| From: Machamer,P/Darden,L/Craver,C (Thinking About Mechanisms [2000], 1) | |
| A reaction: This is their initial formal definition of a mechanism. Note that a mere 'activity' can be included. Presumably the mechanism might have an outcome that was not the intended outcome. Does a random element disqualify it? Are hands mechanisms? |
| 16530 | A mechanism explains a phenomenon by showing how it was produced [Machamer/Darden/Craver] |
| Full Idea: To give a description of a mechanism for a phenomenon is to explain that phenomenon, i.e. to explain how it was produced. | |
| From: Machamer,P/Darden,L/Craver,C (Thinking About Mechanisms [2000], 1) | |
| A reaction: To 'show how' something happens needs a bit of precisification. It is probably analytic that 'showing how' means 'revealing the mechanism', though 'mechanism' then becomes the tricky concept. |
| 16553 | Our account of mechanism combines both entities and activities [Machamer/Darden/Craver] |
| Full Idea: We emphasise the activities in mechanisms. This is explicitly dualist. Substantivalists speak of entities with dispositions to act. Process ontologists reify activities and try to reduce entities to processes. We try to capture both intuitions. | |
| From: Machamer,P/Darden,L/Craver,C (Thinking About Mechanisms [2000], 3) | |
| A reaction: [A quotation of selected fragments] The problem here seems to be the raising of an 'activity' to a central role in ontology, when it doesn't seem to be primitive, and will typically be analysed in a variety of ways. |
| 16559 | Descriptions of explanatory mechanisms have a bottom level, where going further is irrelevant [Machamer/Darden/Craver] |
| Full Idea: Nested hierachical descriptions of mechanisms typically bottom out in lowest level mechanisms. …Bottoming out is relative …the explanation comes to an end, and description of lower-level mechanisms would be irrelevant. | |
| From: Machamer,P/Darden,L/Craver,C (Thinking About Mechanisms [2000], 5.1) | |
| A reaction: This seems to me exactly the right story about mechanism, and it is a story I am associating with essentialism. The relevance is ties to understanding. The lower level is either fully understood, or totally baffling. |
| 16564 | There are four types of bottom-level activities which will explain phenomena [Machamer/Darden/Craver] |
| Full Idea: There are four bottom-out kinds of activities: geometrico-mechanical, electro-chemical, electro-magnetic and energetic. These are abstract means of production that can be fruitfully applied in particular cases to explain phenomena. | |
| From: Machamer,P/Darden,L/Craver,C (Thinking About Mechanisms [2000], 7) | |
| A reaction: I like that. It gives a nice core for a metaphysics for physicalists. I suspect that 'mechanical' can be reduced to something else, and that 'energetic' will disappear in the final story. |
| 16561 | We can abstract by taking an exemplary case and ignoring the detail [Machamer/Darden/Craver] |
| Full Idea: Abstractions may be constructed by taking an exemplary case or instance and removing detail. | |
| From: Machamer,P/Darden,L/Craver,C (Thinking About Mechanisms [2000], 5.3) | |
| A reaction: I love 'removing detail'. That's it. Simple. I think this process is the basis of our whole capacity to formulate abstract concepts. Forget Frege - he's just describing the results of the process. How do we decide what is 'detail'? Essentialism! |
| 16558 | Laws of nature have very little application in biology [Machamer/Darden/Craver] |
| Full Idea: The traditional notion of a law of nature has few, if any, applications in neurobiology or molecular biology. | |
| From: Machamer,P/Darden,L/Craver,C (Thinking About Mechanisms [2000], 3.2) | |
| A reaction: This is a simple and self-evident fact, and bad news for anyone who want to build their entire ontology around laws of nature. I take such a notion to be fairly empty, except as a convenient heuristic device. |