29 ideas
| 13520 | A 'tautology' must include connectives [Wolf,RS] |
| Full Idea: 'For every number x, x = x' is not a tautology, because it includes no connectives. | |
| From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 1.2) |
| 13524 | Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof [Wolf,RS] |
| Full Idea: Deduction Theorem: If T ∪ {P} |- Q, then T |- (P → Q). This is the formal justification of the method of conditional proof (CPP). Its converse holds, and is essentially modus ponens. | |
| From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 1.3) |
| 13522 | Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x) [Wolf,RS] |
| Full Idea: Universal Generalization: If we can prove P(x), only assuming what sort of object x is, we may conclude ∀xP(x) for the same x. | |
| From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 1.3) | |
| A reaction: This principle needs watching closely. If you pick one person in London, with no presuppositions, and it happens to be a woman, can you conclude that all the people in London are women? Fine in logic and mathematics, suspect in life. |
| 13521 | Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance [Wolf,RS] |
| Full Idea: Universal Specification: from ∀xP(x) we may conclude P(t), where t is an appropriate term. If something is true for all members of a domain, then it is true for some particular one that we specify. | |
| From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 1.3) |
| 13523 | Existential Generalization (or 'proof by example'): if we can say P(t), then we can say something is P [Wolf,RS] |
| Full Idea: Existential Generalization (or 'proof by example'): From P(t), where t is an appropriate term, we may conclude ∃xP(x). | |
| From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 1.3) | |
| A reaction: It is amazing how often this vacuous-sounding principles finds itself being employed in discussions of ontology, but I don't quite understand why. |
| 13529 | Empty Set: ∃x∀y ¬(y∈x). The unique empty set exists [Wolf,RS] |
| Full Idea: Empty Set Axiom: ∃x ∀y ¬ (y ∈ x). There is a set x which has no members (no y's). The empty set exists. There is a set with no members, and by extensionality this set is unique. | |
| From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 2.3) | |
| A reaction: A bit bewildering for novices. It says there is a box with nothing in it, or a pair of curly brackets with nothing between them. It seems to be the key idea in set theory, because it asserts the idea of a set over and above any possible members. |
| 13526 | Comprehension Axiom: if a collection is clearly specified, it is a set [Wolf,RS] |
| Full Idea: The comprehension axiom says that any collection of objects that can be clearly specified can be considered to be a set. | |
| From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 2.2) | |
| A reaction: This is virtually tautological, since I presume that 'clearly specified' means pinning down exact which items are the members, which is what a set is (by extensionality). The naïve version is, of course, not so hot. |
| 13534 | In first-order logic syntactic and semantic consequence (|- and |=) nicely coincide [Wolf,RS] |
| Full Idea: One of the most appealing features of first-order logic is that the two 'turnstiles' (the syntactic single |-, and the semantic double |=), which are the two reasonable notions of logical consequence, actually coincide. | |
| From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.3) | |
| A reaction: In the excitement about the possibility of second-order logic, plural quantification etc., it seems easy to forget the virtues of the basic system that is the target of the rebellion. The issue is how much can be 'expressed' in first-order logic. |
| 13535 | First-order logic is weakly complete (valid sentences are provable); we can't prove every sentence or its negation [Wolf,RS] |
| Full Idea: The 'completeness' of first order-logic does not mean that every sentence or its negation is provable in first-order logic. We have instead the weaker result that every valid sentence is provable. | |
| From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.3) | |
| A reaction: Peter Smith calls the stronger version 'negation completeness'. |
| 13531 | Model theory reveals the structures of mathematics [Wolf,RS] |
| Full Idea: Model theory helps one to understand what it takes to specify a mathematical structure uniquely. | |
| From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.1) | |
| A reaction: Thus it is the development of model theory which has led to the 'structuralist' view of mathematics. |
| 13532 | Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants' [Wolf,RS] |
| Full Idea: A 'structure' in model theory has a non-empty set, the 'universe', as domain of variables, a subset for each 'relation', some 'functions', and 'constants'. | |
| From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.2) |
| 13519 | Model theory uses sets to show that mathematical deduction fits mathematical truth [Wolf,RS] |
| Full Idea: Model theory uses set theory to show that the theorem-proving power of the usual methods of deduction in mathematics corresponds perfectly to what must be true in actual mathematical structures. | |
| From: Robert S. Wolf (A Tour through Mathematical Logic [2005], Pref) | |
| A reaction: That more or less says that model theory demonstrates the 'soundness' of mathematics (though normal arithmetic is famously not 'complete'). Of course, he says they 'correspond' to the truths, rather than entailing them. |
| 13533 | First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem [Wolf,RS] |
| Full Idea: The three foundations of first-order model theory are the Completeness theorem, the Compactness theorem, and the Löwenheim-Skolem-Tarski theorem. | |
| From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.3) | |
| A reaction: On p.180 he notes that Compactness and LST make no mention of |- and are purely semantic, where Completeness shows the equivalence of |- and |=. All three fail for second-order logic (p.223). |
| 13537 | An 'isomorphism' is a bijection that preserves all structural components [Wolf,RS] |
| Full Idea: An 'isomorphism' is a bijection between two sets that preserves all structural components. The interpretations of each constant symbol are mapped across, and functions map the relation and function symbols. | |
| From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.4) |
| 13539 | The LST Theorem is a serious limitation of first-order logic [Wolf,RS] |
| Full Idea: The Löwenheim-Skolem-Tarski theorem demonstrates a serious limitation of first-order logic, and is one of primary reasons for considering stronger logics. | |
| From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.7) |
| 13538 | If a theory is complete, only a more powerful language can strengthen it [Wolf,RS] |
| Full Idea: It is valuable to know that a theory is complete, because then we know it cannot be strengthened without passing to a more powerful language. | |
| From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.5) |
| 13525 | Most deductive logic (unlike ordinary reasoning) is 'monotonic' - we don't retract after new givens [Wolf,RS] |
| Full Idea: Deductive logic, including first-order logic and other types of logic used in mathematics, is 'monotonic'. This means that we never retract a theorem on the basis of new givens. If T|-φ and T⊆SW, then S|-φ. Ordinary reasoning is nonmonotonic. | |
| From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 1.7) | |
| A reaction: The classic example of nonmonotonic reasoning is the induction that 'all birds can fly', which is retracted when the bird turns out to be a penguin. He says nonmonotonic logic is a rich field in computer science. |
| 13530 | An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive [Wolf,RS] |
| Full Idea: Less theoretically, an ordinal is an equivalence class of well-orderings. Formally, we say a set is 'transitive' if every member of it is a subset of it, and an ordinal is a transitive set, all of whose members are transitive. | |
| From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 2.4) | |
| A reaction: He glosses 'transitive' as 'every member of a member of it is a member of it'. So it's membership all the way down. This is the von Neumann rather than the Zermelo approach (which is based on singletons). |
| 13518 | Modern mathematics has unified all of its objects within set theory [Wolf,RS] |
| Full Idea: One of the great achievements of modern mathematics has been the unification of its many types of objects. It began with showing geometric objects numerically or algebraically, and culminated with set theory representing all the normal objects. | |
| From: Robert S. Wolf (A Tour through Mathematical Logic [2005], Pref) | |
| A reaction: His use of the word 'object' begs all sorts of questions, if you are arriving from the street, where an object is something which can cause a bruise - but get used to it, because the word 'object' has been borrowed for new uses. |
| 3061 | Anaxarchus said that he was not even sure that he knew nothing [Anaxarchus, by Diog. Laertius] |
| Full Idea: Anaxarchus said that he was not even sure that he knew nothing. | |
| From: report of Anaxarchus (fragments/reports [c.340 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 09.10.1 |
| 22190 | If a theory is more informative it is less probable [Gorham] |
| Full Idea: Popper's theory implies that more informative theories seem to be less probable. | |
| From: Geoffrey Gorham (Philosophy of Science [2009], 3) | |
| A reaction: [On p.75 Gorham replies to this objection] The point is that to be more testable they must be more detailed. He's not wrong. Theories are meant to be general, so they sweep up the details. But they need precise generalities and specifics. |
| 22189 | Why abandon a theory if you don't have a better one? [Gorham] |
| Full Idea: There is no sense in abandoning a successful theory if you have nothing to replace it with. | |
| From: Geoffrey Gorham (Philosophy of Science [2009], 2) | |
| A reaction: This is also a problem for infererence to the best explanation. What to do if your best explanation is not very good? The simple message is do not rush to dump a theory when faced with an anomaly. |
| 22192 | Is Newton simpler with universal simultaneity, or Einstein simpler without absolute time? [Gorham] |
| Full Idea: Is Newton's theory simpler than Einstein's, since there is only one relation of simultaneity in absolute time, or is Einstein's simpler because it dispenses with absolute time altogether? | |
| From: Geoffrey Gorham (Philosophy of Science [2009], 4) | |
| A reaction: A nice question, to which a good scientist might be willing to offer an answer. Since simultaneity is crucial but the existence of time is not, I would vote for Newton as the simpler. |
| 22194 | Structural Realism says mathematical structures persist after theory rejection [Gorham] |
| Full Idea: Structural Realists say that modern science achieves a true or 'truer' account of the world only with respect to its mathematical structure rather than its intrinsic qualities or nature. The structure carries over to new theories. | |
| From: Geoffrey Gorham (Philosophy of Science [2009], 4) | |
| A reaction: At first glance I am unconvinced that when an old theory is replaced it neverthess contains some sort of 'mathematical structure' which endures and is worth preserving. No doubt Worrall, French and co have examples. |
| 22195 | Structural Realists must show the mathematics is both crucial and separate [Gorham] |
| Full Idea: Structural Realists must show that it is the mathematical aspects of the theories, not their content, that account for their success ….and that their structure and content can be clearly separated. | |
| From: Geoffrey Gorham (Philosophy of Science [2009], 4) | |
| A reaction: Their approach certainly seems to rely on mathematical types of science, so it presumably fits biology, geology and even astronomy less well. |
| 22197 | Theories aren't just for organising present experience if they concern the past or future [Gorham] |
| Full Idea: The strangeness of interpreting theories as mere tools for organising present experience is brought out clearly in sciences like cosmology and paleontology, which largely concern events in the remote past or future. | |
| From: Geoffrey Gorham (Philosophy of Science [2009], 4) | |
| A reaction: Not conclusive. An anti-realist has to interpret those sciences in terms of the current observations that are available. |
| 22196 | For most scientists their concepts are not just useful, but are meant to be true and accurate [Gorham] |
| Full Idea: The main difficulty with instrumentalism is its implausible account ot the meaning of theoretical claims and concepts. Most scientists take them to be straightforward attempts to describe the world. Most say they are useful because they are accurate. | |
| From: Geoffrey Gorham (Philosophy of Science [2009], 4) | |
| A reaction: Instrumentalism is seen as a Pragmatist view, and Dewey is cited. |
| 22193 | Consilience makes the component sciences more likely [Gorham] |
| Full Idea: The more unification and integration is found among the modern sciences, the less likely it seems it will have all been a dream. | |
| From: Geoffrey Gorham (Philosophy of Science [2009], 4) | |
| A reaction: I believe this strongly. Ancient theories which were complex, wide ranging and false do not impress me. This is part of my coherence view of justification. |
| 22198 | Aristotelian physics has circular celestial motion and linear earthly motion [Gorham] |
| Full Idea: Aristotelian physics assumed that celestial motion is naturally circular and eternal while terrestrial motion is naturally toward the center of the earth and final. | |
| From: Geoffrey Gorham (Philosophy of Science [2009], 4) | |
| A reaction: The overthrow of this by Galileo and then Newton may have been the most dramatic revolution of the new science. It opened up the possibility of universal laws of physics. |