38 ideas
10888 | Sets can be defined by 'enumeration', or by 'abstraction' (based on a property) [Zalabardo] |
Full Idea: We can define a set by 'enumeration' (by listing the items, within curly brackets), or by 'abstraction' (by specifying the elements as instances of a property), pretending that they form a determinate totality. The latter is written {x | x is P}. | |
From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §1.3) |
10889 | The 'Cartesian Product' of two sets relates them by pairing every element with every element [Zalabardo] |
Full Idea:
The 'Cartesian Product' of two sets, written A x B, is the relation which pairs every element of A with every element of B. So A x B = { |
|
From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §1.6) |
10890 | A 'partial ordering' is reflexive, antisymmetric and transitive [Zalabardo] |
Full Idea: A binary relation in a set is a 'partial ordering' just in case it is reflexive, antisymmetric and transitive. | |
From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §1.6) |
10886 | Determinacy: an object is either in a set, or it isn't [Zalabardo] |
Full Idea: Principle of Determinacy: For every object a and every set S, either a is an element of S or a is not an element of S. | |
From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §1.2) |
10887 | Specification: Determinate totals of objects always make a set [Zalabardo] |
Full Idea: Principle of Specification: Whenever we can specify a determinate totality of objects, we shall say that there is a set whose elements are precisely the objects that we have specified. | |
From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §1.3) | |
A reaction: Compare the Axiom of Specification. Zalabardo says we may wish to consider sets of which we cannot specify the members. |
10897 | A first-order 'sentence' is a formula with no free variables [Zalabardo] |
Full Idea: A formula of a first-order language is a 'sentence' just in case it has no free variables. | |
From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §3.2) |
10899 | Γ |= φ if φ is true when all of Γ is true, for all structures and interpretations [Zalabardo] |
Full Idea: A formula is the 'logical consequence' of a set of formulas (Γ |= φ) if for every structure in the language and every variable interpretation of the structure, if all the formulas within the set are true and the formula itself is true. | |
From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §3.5) |
10893 | Γ |= φ for sentences if φ is true when all of Γ is true [Zalabardo] |
Full Idea: A propositional logic sentence is a 'logical consequence' of a set of sentences (written Γ |= φ) if for every admissible truth-assignment all the sentences in the set Γ are true, then φ is true. | |
From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §2.4) | |
A reaction: The definition is similar for predicate logic. |
8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
Full Idea: Intuitionists in mathematics deny excluded middle, because it is symptomatic of faith in the transcendent existence of mathematical objects and/or the truth of mathematical statements. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2) | |
A reaction: There are other problems with excluded middle, such as vagueness, but on the whole I, as a card-carrying 'realist', am committed to the law of excluded middle. |
10896 | Propositional logic just needs ¬, and one of ∧, ∨ and → [Zalabardo] |
Full Idea: In propositional logic, any set containing ¬ and at least one of ∧, ∨ and → is expressively complete. | |
From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §2.8) |
10898 | The semantics shows how truth values depend on instantiations of properties and relations [Zalabardo] |
Full Idea: The semantic pattern of a first-order language is the ways in which truth values depend on which individuals instantiate the properties and relations which figure in them. ..So we pair a truth value with each combination of individuals, sets etc. | |
From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §3.3) | |
A reaction: So truth reduces to a combination of 'instantiations', which is rather like 'satisfaction'. |
10902 | We can do semantics by looking at given propositions, or by building new ones [Zalabardo] |
Full Idea: We can look at semantics from the point of view of how truth values are determined by instantiations of properties and relations, or by asking how we can build, using the resources of the language, a proposition corresponding to a given semantic pattern. | |
From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §3.6) | |
A reaction: The second version of semantics is model theory. |
10892 | We make a truth assignment to T and F, which may be true and false, but merely differ from one another [Zalabardo] |
Full Idea: A truth assignment is a function from propositions to the set {T,F}. We will think of T and F as the truth values true and false, but for our purposes all we need to assume about the identity of these objects is that they are different from each other. | |
From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §2.4) | |
A reaction: Note that T and F are 'objects'. This remark is important in understanding modern logical semantics. T and F can be equated to 1 and 0 in the language of a computer. They just mean as much as you want them to mean. |
10895 | 'Logically true' (|= φ) is true for every truth-assignment [Zalabardo] |
Full Idea: A propositional logic sentence is 'logically true', written |= φ, if it is true for every admissible truth-assignment. | |
From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §2.4) |
10900 | Logically true sentences are true in all structures [Zalabardo] |
Full Idea: In first-order languages, logically true sentences are true in all structures. | |
From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §3.5) |
10894 | A sentence-set is 'satisfiable' if at least one truth-assignment makes them all true [Zalabardo] |
Full Idea: A propositional logic set of sentences Γ is 'satisfiable' if there is at least one admissible truth-assignment that makes all of its sentences true. | |
From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §2.4) |
10901 | Some formulas are 'satisfiable' if there is a structure and interpretation that makes them true [Zalabardo] |
Full Idea: A set of formulas of a first-order language is 'satisfiable' if there is a structure and a variable interpretation in that structure such that all the formulas of the set are true. | |
From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §3.5) |
10903 | A structure models a sentence if it is true in the model, and a set of sentences if they are all true in the model [Zalabardo] |
Full Idea: A structure is a model of a sentence if the sentence is true in the model; a structure is a model of a set of sentences if they are all true in the structure. | |
From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §3.6) |
8763 | The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro] |
Full Idea: It is surely wise to identify the positions in the natural numbers structure with their counterparts in the integer, rational, real and complex number structures. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2) | |
A reaction: The point is that this might be denied, since 3, 3/1, 3.00.., and -3*i^2 are all arrived at by different methods of construction. Natural 3 has a predecessor, but real 3 doesn't. I agree, intuitively, with Shapiro. Russell (1919) disagreed. |
18249 | Cauchy gave a formal definition of a converging sequence. [Shapiro] |
Full Idea: A sequence a1,a2,... of rational numbers is 'Cauchy' if for each rational number ε>0 there is a natural number N such that for all natural numbers m, n, if m>N and n>N then -ε < am - an < ε. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 7.2 n4) | |
A reaction: The sequence is 'Cauchy' if N exists. |
8764 | Categories are the best foundation for mathematics [Shapiro] |
Full Idea: There is a dedicated contingent who hold that the category of 'categories' is the proper foundation for mathematics. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 10.3 n7) | |
A reaction: He cites Lawvere (1966) and McLarty (1993), the latter presenting the view as a form of structuralism. I would say that the concept of a category will need further explication, and probably reduce to either sets or relations or properties. |
10891 | If a set is defined by induction, then proof by induction can be applied to it [Zalabardo] |
Full Idea: Defining a set by induction enables us to use the method of proof by induction to establish that all the elements of the set have a certain property. | |
From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §2.3) |
8762 | Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro] |
Full Idea: Zermelo said that for each number n, its successor is the singleton of n, so 3 is {{{null}}}, and 1 is not a member of 3. Von Neumann said each number n is the set of numbers less than n, so 3 is {null,{null},{null,{null}}}, and 1 is a member of 3. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2) | |
A reaction: See Idea 645 - Zermelo could save Plato from the criticisms of Aristotle! These two accounts are cited by opponents of the set-theoretical account of numbers, because it seems impossible to arbitrate between them. |
8760 | Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro] |
Full Idea: The structuralist vigorously rejects any sort of ontological independence among the natural numbers; the essence of a natural number is its relations to other natural numbers. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1) | |
A reaction: This seems to place the emphasis on ordinals (what order?) rather than on cardinality (how many?). I am strongly inclined to think that this is the correct view, though you can't really have relations if there is nothing to relate. |
8761 | A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro] |
Full Idea: A 'system' is a collection of objects with certain relations among them; a 'pattern' or 'structure' is the abstract form of a system, highlighting the interrelationships and ignoring any features they do not affect how they relate to other objects. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1) | |
A reaction: Note that 'ignoring' features is a psychological account of abstraction, which (thanks to Frege and Geach) is supposed to be taboo - but which I suspect is actually indispensable in any proper account of thought and concepts. |
8744 | Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro] |
Full Idea: The thesis that principles of arithmetic are derivable from the laws of logic runs against a now common view that logic itself has no ontology. There are no particular logical objects. From this perspective logicism is a non-starter. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 5.1) | |
A reaction: This criticism strikes me as utterly devastating. There are two routes to go: prove that logic does have an ontology of objects (what would they be?), or - better - deny that arithmetic contains any 'objects'. Or give up logicism. |
8749 | Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro] |
Full Idea: Term Formalism is the view that mathematics is just about characters or symbols - the systems of numerals and other linguistic forms. ...This will cover integers and rational numbers, but what are real numbers supposed to be, if they lack names? | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.1) | |
A reaction: Real numbers (such as pi and root-2) have infinite decimal expansions, so we can start naming those. We could also start giving names like 'Harry' to other reals, though it might take a while. OK, I give up. |
8750 | Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro] |
Full Idea: Game Formalism likens mathematics to chess, where the 'content' of mathematics is exhausted by the rules of operating with its language. ...This, however, leaves the problem of why the mathematical games are so useful to the sciences. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.2) | |
A reaction: This thought pushes us towards structuralism. It could still be a game, but one we learned from observing nature, which plays its own games. Chess is, after all, modelled on warfare. |
8752 | Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro] |
Full Idea: The Deductivist version of formalism (sometimes called 'if-thenism') says that the practice of mathematics consists of determining logical consequences of otherwise uninterpreted axioms. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 6.2) | |
A reaction: [Hilbert is the source] More plausible than Term or Game Formalism (qv). It still leaves the question of why it seems applicable to nature, and why those particular axioms might be chosen. In some sense, though, it is obviously right. |
8753 | Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro] |
Full Idea: Critics commonly complain that the intuitionist restrictions cripple the mathematician. On the other hand, intuitionist mathematics allows for many potentially important distinctions not available in classical mathematics, and is often more subtle. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 7.1) | |
A reaction: The main way in which it cripples is its restriction on talk of infinity ('Cantor's heaven'), which was resented by Hilbert. Since high-level infinities are interesting, it would be odd if we were not allowed to discuss them. |
8731 | Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro] |
Full Idea: I classify conceptualists according to what they say about properties or concepts. If someone classified properties as existing independent of language I would classify her as a realist in ontology of mathematics. Or they may be idealists or nominalists. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 2.2.1) | |
A reaction: In other words, Shapiro wants to eliminate 'conceptualist' as a useful label in philosophy of mathematics. He's probably right. All thought involves concepts, but that doesn't produce a conceptualist theory of, say, football. |
8730 | 'Impredicative' definitions refer to the thing being described [Shapiro] |
Full Idea: A definition of a mathematical entity is 'impredicative' if it refers to a collection that contains the defined entity. The definition of 'least upper bound' is impredicative as it refers to upper bounds and characterizes a member of this set. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2) | |
A reaction: The big question is whether mathematics can live with impredicative definitions, or whether they threaten to be viciously circular, and undermine the whole enterprise. |
8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
Full Idea: Rationalism is a long-standing school that can be characterized as an attempt to extend the perceived methodology of mathematics to all of knowledge. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 1.1) | |
A reaction: Sometimes called 'Descartes's Dream', or the 'Enlightenment Project', the dream of proving everything. Within maths, Hilbert's Programme aimed for the same certainty. Idea 22 is the motto for the opposition to this approach. |
7257 | All modern social systems seem to be conspiracies of the rich [More,T] |
Full Idea: When I consider any social system that prevails in the modern world, I can't see it as anything but a conspiracy of the rich to advance their own interests under the pretext of organizing society. | |
From: Thomas More (Utopia [1516], Bk 2) | |
A reaction: I'm afraid this is my own view of most conservative politics. I don't deny that there is a good case to be made for the conservative view (by Burke and Scruton, for example), but the rich will always latch onto its coat-tails. Cf. Idea 122. |
7254 | If you try to get elected, you should be permanently barred from seeking office [More,T] |
Full Idea: In Utopia, anyone who deliberately tries to get himself elected to a public office is permanently disqualified from holding one. | |
From: Thomas More (Utopia [1516], Bk 2) | |
A reaction: This echoes a thought found in Plato (Idea 2149). I've always liked this idea. Why can't we have elections were a group of the best people are invited to stand? Well, yes, it would lead to corruption... Still, the best should be pushed to the front. |
7255 | Only Utopians fail to see glory in warfare [More,T] |
Full Idea: Utopians are practically the only people on earth who fail to see anything glorious in war. | |
From: Thomas More (Utopia [1516], Bk 2) | |
A reaction: A refreshing thought for such an early date. Whatever dubious behaviour is nowadays attributed to Thomas More, you have to admire someone who writes this during the reign of Henry VIII. |
7253 | In Utopia, legal euthanasia is considered honourable [More,T] |
Full Idea: In Utopia, officially sanctioned euthanasia is regarded as an honourable death. | |
From: Thomas More (Utopia [1516], Bk 2) | |
A reaction: A bit surprising coming from a writer who is now a Catholic martyr and saint. |
7256 | In Utopia, the Supreme Being is identical with Nature [More,T] |
Full Idea: Everyone in Utopia agrees that the Supreme Being (which they call Mythras) is identical with Nature. | |
From: Thomas More (Utopia [1516], Bk 2) | |
A reaction: This sounds remarkably like full-blown Spinozean pantheism, though it should be interpreted with caution. It certainly seems to show that pantheism was a possibility in the minds of late medieval religious thinkers. |