15 ideas
5437 | The claim of hermeneutics to give knowledge through understanding is challenged by positivism [Mautner on Dilthey] |
Full Idea: The claim of hermeneutics to give understanding instead of explanation can be seen as part of the theory of knowledge, but it seems to be incompatible with the most accepted aspects of positivism. | |
From: comment on Wilhelm Dilthey (works [1883]) by Thomas Mautner - Penguin Dictionary of Philosophy p.248 | |
A reaction: So much the worse for positivism. The same conflict occurs in modern philosophy of mind. God can be a positivist if he likes, but we must settle for hermeneutics for a lot of our knowledge. We are discussing method, not ontology. |
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) |
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. |
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] |
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. |
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) |
5436 | Natural science seeks explanation; human sciences seek understanding [Dilthey, by Mautner] |
Full Idea: In the natural sciences we seek for causes and ask for explanation (erklären), but in the human or cultural sciences we seek understanding (verstehen) by means of interpretation. | |
From: report of Wilhelm Dilthey (works [1883]) by Thomas Mautner - Penguin Dictionary of Philosophy p.144 | |
A reaction: This seems a nice distinction. The prospects of finding the causes or explanations of Shakespeare's plays don't look good, and when you have explained the causes of a chemical reaction you probably have all you need. |
20653 | Six reduction levels: groups, lives, cells, molecules, atoms, particles [Putnam/Oppenheim, by Watson] |
Full Idea: There are six 'reductive levels' in science: social groups, (multicellular) living things, cells, molecules, atoms, and elementary particles. | |
From: report of H.Putnam/P.Oppenheim (Unity of Science as a Working Hypothesis [1958]) by Peter Watson - Convergence 10 'Intro' | |
A reaction: I have the impression that fields are seen as more fundamental that elementary particles. What is the status of the 'laws' that are supposed to govern these things? What is the status of space and time within this picture? |