23 ideas
| 21753 | If we look at the world rationally, the world assumes a rational aspect [Hegel] |
| Full Idea: Whoever looks at the world rationally will find that it in turn assumes a rational aspect; the two exist in a reciprocal relationship. | |
| From: Georg W.F.Hegel (Lectures on the Philosophy of (World) History [1837], p.29), quoted by Stephen Houlgate - An Introduction to Hegel 01 | |
| A reaction: What happens when I look at irrationality rationally? |
| 10571 | Concern for rigour can get in the way of understanding phenomena [Fine,K] |
| Full Idea: It is often the case that the concern for rigor gets in the way of a true understanding of the phenomena to be explained. | |
| From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 2) | |
| A reaction: This is a counter to Timothy Williamson's love affair with rigour in philosophy. It strikes me as the big current question for analytical philosophy - of whether the intense pursuit of 'rigour' will actually deliver the wisdom we all seek. |
| 21974 | The world seems rational to those who look at it rationally [Hegel] |
| Full Idea: To him who looks at the world rationally, the world looks rationally back; the two exist in reciprocal relationship. | |
| From: Georg W.F.Hegel (Lectures on the Philosophy of (World) History [1837], Intro p.29), quoted by A.W. Moore - The Evolution of Modern Metaphysics 07.4 | |
| A reaction: This is a nice variation on the stoic idea that nature is essentially rational. If we are capable of rationality, then nature has made us that way. Romantics seem to prefer looking at nature less rationally, so what do they see in nature? |
| 15375 | If terms change their designations in different states, they are functions from states to objects [Fitting] |
| Full Idea: The common feature of every designating term is that designation may change from state to state - thus it can be formalized by a function from states to objects. | |
| From: Melvin Fitting (Intensional Logic [2007], 3) | |
| A reaction: Specifying the objects sounds OK, but specifying states sounds rather tough. |
| 15376 | Intensional logic adds a second type of quantification, over intensional objects, or individual concepts [Fitting] |
| Full Idea: To first order modal logic (with quantification over objects) we can add a second kind of quantification, over intensions. An intensional object, or individual concept, will be modelled by a function from states to objects. | |
| From: Melvin Fitting (Intensional Logic [2007], 3.3) |
| 15378 | Awareness logic adds the restriction of an awareness function to epistemic logic [Fitting] |
| Full Idea: Awareness logic enriched Hintikka's epistemic models with an awareness function, mapping each state to the set of formulas we are aware of at that state. This reflects some bound on the resources we can bring to bear. | |
| From: Melvin Fitting (Intensional Logic [2007], 3.6.1) | |
| A reaction: [He cites Fagin and Halpern 1988 for this] |
| 15379 | Justication logics make explicit the reasons for mathematical truth in proofs [Fitting] |
| Full Idea: In justification logics, the logics of knowledge are extended by making reasons explicit. A logic of proof terms was created, with a semantics. In this, mathematical truths are known for explicit reasons, and these provide a measure of complexity. | |
| From: Melvin Fitting (Intensional Logic [2007], 3.6.1) |
| 10565 | There is no stage at which we can take all the sets to have been generated [Fine,K] |
| Full Idea: There is no stage at which we can take all the sets to have been generated, since the set of all those sets which have been generated at a given stage will itself give us something new. | |
| From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 1) |
| 10564 | We might combine the axioms of set theory with the axioms of mereology [Fine,K] |
| Full Idea: We might combine the standard axioms of set theory with the standard axioms of mereology. | |
| From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 1) |
| 11026 | Classical logic is deliberately extensional, in order to model mathematics [Fitting] |
| Full Idea: Mathematics is typically extensional throughout (we write 3+2=2+3 despite the two terms having different meanings). ..Classical first-order logic is extensional by design since it primarily evolved to model the reasoning of mathematics. | |
| From: Melvin Fitting (Intensional Logic [2007], §1) |
| 11028 | λ-abstraction disambiguates the scope of modal operators [Fitting] |
| Full Idea: λ-abstraction can be used to abstract and disambiguate a predicate. De re is [λx◊P(x)](f) - f has the possible-P property - and de dicto is ◊[λxP(x)](f) - possibly f has the P-property. Also applies to □. | |
| From: Melvin Fitting (Intensional Logic [2007], §3.3) | |
| A reaction: Compare the Barcan formula. Originated with Church in the 1930s, and Carnap 1947, but revived by Stalnaker and Thomason 1968. Because it refers to the predicate, it has a role in intensional versions of logic, especially modal logic. |
| 10569 | If you ask what F the second-order quantifier quantifies over, you treat it as first-order [Fine,K] |
| Full Idea: We are tempted to ask of second-order quantifiers 'what are you quantifying over?', or 'when you say "for some F" then what is the F?', but these questions already presuppose that the quantifiers are first-order. | |
| From: Kit Fine (Replies on 'Limits of Abstraction' [2005]) |
| 10570 | Assigning an entity to each predicate in semantics is largely a technical convenience [Fine,K] |
| Full Idea: In doing semantics we normally assign some appropriate entity to each predicate, but this is largely for technical convenience. | |
| From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 2) |
| 10573 | Dedekind cuts lead to the bizarre idea that there are many different number 1's [Fine,K] |
| Full Idea: Because of Dedekind's definition of reals by cuts, there is a bizarre modern doctrine that there are many 1's - the natural number 1, the rational number 1, the real number 1, and even the complex number 1. | |
| From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 2) | |
| A reaction: See Idea 10572. |
| 10575 | Why should a Dedekind cut correspond to a number? [Fine,K] |
| Full Idea: By what right can Dedekind suppose that there is a number corresponding to any pair of irrationals that constitute an irrational cut? | |
| From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 2) |
| 10574 | Unless we know whether 0 is identical with the null set, we create confusions [Fine,K] |
| Full Idea: What is the union of the singleton {0}, of zero, and the singleton {φ}, of the null set? Is it the one-element set {0}, or the two-element set {0, φ}? Unless the question of identity between 0 and φ is resolved, we cannot say. | |
| From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 2) |
| 10560 | Set-theoretic imperialists think sets can represent every mathematical object [Fine,K] |
| Full Idea: Set-theoretic imperialists think that it must be possible to represent every mathematical object as a set. | |
| From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 1) |
| 10568 | Logicists say mathematics can be derived from definitions, and can be known that way [Fine,K] |
| Full Idea: Logicists traditionally claim that the theorems of mathematics can be derived by logical means from the relevant definitions of the terms, and that these theorems are epistemically innocent (knowable without Kantian intuition or empirical confirmation). | |
| From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 2) |
| 10563 | A generative conception of abstracts proposes stages, based on concepts of previous objects [Fine,K] |
| Full Idea: It is natural to have a generative conception of abstracts (like the iterative conception of sets). The abstracts are formed at stages, with the abstracts formed at any given stage being the abstracts of those concepts of objects formed at prior stages. | |
| From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 1) | |
| A reaction: See 10567 for Fine's later modification. This may not guarantee 'levels', but it implies some sort of conceptual priority between abstract entities. |
| 15377 | Definite descriptions pick out different objects in different possible worlds [Fitting] |
| Full Idea: Definite descriptions pick out different objects in different possible worlds quite naturally. | |
| From: Melvin Fitting (Intensional Logic [2007], 3.4) | |
| A reaction: A definite description can pick out the same object in another possible world, or a very similar one, or an object which has almost nothing in common with the others. |
| 10561 | Abstraction-theoretic imperialists think Fregean abstracts can represent every mathematical object [Fine,K] |
| Full Idea: Abstraction-theoretic imperialists think that it must be possible to represent every mathematical object as a Fregean abstract. | |
| From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 1) |
| 10562 | We can combine ZF sets with abstracts as urelements [Fine,K] |
| Full Idea: I propose a unified theory which is a version of ZF or ZFC with urelements, where the urelements are taken to be the abstracts. | |
| From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 1) |
| 10567 | We can create objects from conditions, rather than from concepts [Fine,K] |
| Full Idea: Instead of viewing the abstracts (or sums) as being generated from objects, via the concepts from which they are defined, we can take them to be generated from conditions. The number of the universe ∞ is the number of self-identical objects. | |
| From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 1) | |
| A reaction: The point is that no particular object is now required to make the abstraction. |