19 ideas
22664 | I do not care if my trivial beliefs are false, and I have no interest in many truths [Nozick] |
Full Idea: I find that I do not mind at all the thought that I have some false beliefs (of US state capitals), and there are many truths I do not care to know at all (total grains of sand on the beach). | |
From: Robert Nozick (The Nature of Rationality [1993], p.67) | |
A reaction: A useful corrective to anyone who blindly asserts that truth is the supreme human value. I would still be annoyed if someone taught me lies about these two types of truth. |
22665 | Maybe James was depicting the value of truth, and not its nature [Nozick] |
Full Idea: We might see William James's pragmatic view that truth is what works as depicting the value of truth, and not its nature. | |
From: Robert Nozick (The Nature of Rationality [1993], p.68) | |
A reaction: James didn't think that he was doing this. He firmly says that this IS truth, not just the advantages of truth. Another view is that pragmatists are giving a test for truth. |
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) |
20618 | Persons must be conscious, reasoning, motivated, communicative, self-aware [Warren, by Tuckness/Wolf] |
Full Idea: Suggested characteristics of personhood: consciousness (esp. of pain); reasoning and problem solving; self-motivated activity; varied communication on many topics; self-concepts and self-awareness. | |
From: report of Mary Anne Warren (On the Moral and Legal State of Abortion [1973], p.55) by Tuckness,A/Wolf,C - This is Political Philosophy 8 'Standing' | |
A reaction: [a 'famous' article] A number of non-human animals come very close to passing these tests. I suspect the complex communication is only in there to disqualify them from getting the full certificate. (But she wrote on animal rights). |
22662 | In the instrumental view of rationality it only concerns means, and not ends [Nozick] |
Full Idea: On the instrumental conception of rationality, it consists in the effective and efficient achievement of goals, ends, and desires. About the goals themselves it has little to say. | |
From: Robert Nozick (The Nature of Rationality [1993], p.64) | |
A reaction: [He quotes Russell 1954 p.viii as expressing this view] A long way from Greek logos, which obviously concerns the rational selection of right ends (for which, presumably, reasons can be given). In practice our ends may never be rational, of course. |
22666 | Is it rational to believe a truth which leads to permanent misery? [Nozick] |
Full Idea: If a mother is presented with convincing evidence that her son has committed a grave crime, but were she to believe it that would make her life thereafter miserable, is it rational for her to believe her son is guilty? | |
From: Robert Nozick (The Nature of Rationality [1993], p.69) | |
A reaction: I assume there is a conflict of rationalities, because there are conflicting ends. Presumably most mothers love the truth, but most of us also aim for happy lives. It is perfectly rational to avoid discovering a horrible family truth. |
22667 | Rationality needs some self-consciousness, to also evaluate how we acquired our reasons [Nozick] |
Full Idea: Rationality involves some degree of self-consciousness. Not only reasons are evaluated, but also the processes by which information arrives, is stored, and recalled. | |
From: Robert Nozick (The Nature of Rationality [1993], p.74) | |
A reaction: I defend the idea that animals have a degree of rationality, because they can make sensible judgements, but I cannot deny this idea. Rationality comes in degrees, and second-level thought is a huge leap forward in degree. |
22663 | Rationality is normally said to concern either giving reasons, or reliability [Nozick] |
Full Idea: The two themes permeating the philosophical literature are that rationality is a matter of reasons, or that rationality is a matter of reliability. | |
From: Robert Nozick (The Nature of Rationality [1993], p.64) | |
A reaction: Since a clock can be reliable, I would have thought it concerns reasons. Or an unthinking person could reliably recite truths from memory. There is also the instrumental view of rationality. |