9955 | Contextual definitions replace a complete sentence containing the expression |
Full Idea: A contextual definition shows how to analyse an expression in situ, by replacing a complete sentence (of a particular form) in which the expression occurs by another in which it does not. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.2) | |
A reaction: This is a controversial procedure, which (according to Dummett) Frege originally accepted, and later rejected. It might not be the perfect definition that replacing just the expression would give you, but it is a promising step. |
10031 | Impredicative definitions quantify over the thing being defined |
Full Idea: When a definition contains a quantifier whose range includes the very entity being defined, the definition is said to be 'impredicative'. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.2) | |
A reaction: Presumably they are 'impredicative' because they do not predicate a new quality in the definiens, but make use of the qualities already known. |
10098 | The 'power set' of A is all the subsets of A |
Full Idea: The 'power set' of A is all the subsets of A. P(A) = {B : B ⊆ A}. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) |
10099 | The 'ordered pair' <a, b>, for two sets a and b, is the set {{a, b},{a}} |
Full Idea: The 'ordered pair' <a, b>, for two sets a and b, is the set {{a, b},{a}}. The existence of this set is guaranteed by three applications of the Axiom of Pairing. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) | |
A reaction: See Idea 10100 for the Axiom of Pairing. |
10101 | Cartesian Product A x B: the set of all ordered pairs in which a∈A and b∈B |
Full Idea: The 'Cartesian Product' of any two sets A and B is the set of all ordered pairs <a, b> in which a ∈ A and b ∈ B, and it is denoted as A x B. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) |
10103 | Grouping by property is common in mathematics, usually using equivalence |
Full Idea: The idea of grouping together objects that share some property is a common one in mathematics, ...and the technique most often involves the use of equivalence relations. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) |
10104 | 'Equivalence' is a reflexive, symmetric and transitive relation; 'same first letter' partitions English words |
Full Idea: A relation is an equivalence relation if it is reflexive, symmetric and transitive. The 'same first letter' is an equivalence relation on the set of English words. Any relation that puts a partition into clusters will be equivalence - and vice versa. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) | |
A reaction: This is a key concept in the Fregean strategy for defining numbers. |
10096 | Even the elements of sets in ZFC are sets, resting on the pure empty set |
Full Idea: ZFC is a theory concerned only with sets. Even the elements of all of the sets studied in ZFC are also sets (whose elements are also sets, and so on). This rests on one clearly pure set, the empty set Φ. ..Mathematics only needs pure sets. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) | |
A reaction: This makes ZFC a much more metaphysically comfortable way to think about sets, because it can be viewed entirely formally. It is rather hard to disentangle a chair from the singleton set of that chair. |
10097 | Axiom of Extensionality: for all sets x and y, if x and y have the same elements then x = y |
Full Idea: The Axiom of Extensionality says that for all sets x and y, if x and y have the same elements then x = y. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) | |
A reaction: This seems fine in pure set theory, but hits the problem of renates and cordates in the real world. The elements coincide, but the axiom can't tell you why they coincide. |
10100 | Axiom of Pairing: for all sets x and y, there is a set z containing just x and y |
Full Idea: The Axiom of Pairing says that for all sets x and y, there is a set z containing x and y, and nothing else. In symbols: ∀x∀y∃z∀w(w ∈ z ↔ (w = x ∨ w = y)). | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) | |
A reaction: See Idea 10099 for an application of this axiom. |
17900 | The Axiom of Reducibility made impredicative definitions possible |
Full Idea: The Axiom of Reducibility ...had the effect of making impredicative definitions possible. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) |
10109 | ZFC can prove that there is no set corresponding to the concept 'set' |
Full Idea: Sets, unlike extensions, fail to correspond to all concepts. We can prove in ZFC that there is no set corresponding to the concept 'set' - that is, there is no set of all sets. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.4) | |
A reaction: This is rather an important point for Frege. However, all concepts have extensions, but they may be proper classes, rather than precisely defined sets. |
10108 | As a reduction of arithmetic, set theory is not fully general, and so not logical |
Full Idea: The problem with reducing arithmetic to ZFC is not that this theory is inconsistent (as far as we know it is not), but rather that is not completely general, and for this reason not logical. For example, it asserts the existence of sets. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.4) | |
A reaction: Note that ZFC has not been proved consistent. |
10111 | Asserting Excluded Middle is a hallmark of realism about the natural world |
Full Idea: A hallmark of our realist stance towards the natural world is that we are prepared to assert the Law of Excluded Middle for all statements about it. For all statements S, either S is true, or not-S is true. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.4) | |
A reaction: Personally I firmly subscribe to realism, so I suppose I must subscribe to Excluded Middle. ...Provided the statement is properly formulated. Or does liking excluded middle lead me to realism? |
10129 | A 'model' is a meaning-assignment which makes all the axioms true |
Full Idea: A 'model' of a theory is an assignment of meanings to the symbols of its language which makes all of its axioms come out true. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.7) | |
A reaction: If the axioms are all true, and the theory is sound, then all of the theorems will also come out true. |
10105 | Differences between isomorphic structures seem unimportant |
Full Idea: Mathematicians tend to regard the differences between isomorphic mathematical structures as unimportant. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) | |
A reaction: This seems to be a pointer towards Structuralism as the underlying story in mathematics. The intrinsic character of so-called 'objects' seems unimportant. How theories map onto one another (and onto the world?) is all that matters? |
10119 | Consistency is a purely syntactic property, unlike the semantic property of soundness |
Full Idea: Consistency is a purely syntactic property, unlike the semantic property of soundness. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.6) |
10126 | A 'consistent' theory cannot contain both a sentence and its negation |
Full Idea: If there is a sentence such that both the sentence and its negation are theorems of a theory, then the theory is 'inconsistent'. Otherwise it is 'consistent'. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.7) |
10120 | Soundness is a semantic property, unlike the purely syntactic property of consistency |
Full Idea: Soundness is a semantic property, unlike the purely syntactic property of consistency. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.6) |
10127 | A 'complete' theory contains either any sentence or its negation |
Full Idea: If there is a sentence such that neither the sentence nor its negation are theorems of a theory, then the theory is 'incomplete'. Otherwise it is 'complete'. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.7) | |
A reaction: Interesting questions are raised about undecidable sentences, irrelevant sentences, unknown sentences.... |
10106 | Rational numbers give answers to division problems with integers |
Full Idea: We can think of rational numbers as providing answers to division problems involving integers. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) | |
A reaction: Cf. Idea 10102. |
10102 | The integers are answers to subtraction problems involving natural numbers |
Full Idea: In defining the integers in set theory, our definition will be motivated by thinking of the integers as answers to subtraction problems involving natural numbers. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) | |
A reaction: Typical of how all of the families of numbers came into existence; they are 'invented' so that we can have answers to problems, even if we can't interpret the answers. It it is money, we may say the minus-number is a 'debt', but is it? Cf Idea 10106. |
10107 | Real numbers provide answers to square root problems |
Full Idea: One reason for introducing the real numbers is to provide answers to square root problems. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) | |
A reaction: Presumably the other main reasons is to deal with problems of exact measurement. It is interesting that there seem to be two quite distinct reasons for introducing the reals. Cf. Ideas 10102 and 10106. |
9946 | Logicists say mathematics is applicable because it is totally general |
Full Idea: The logicist idea is that if mathematics is logic, and logic is the most general of disciplines, one that applies to all rational thought regardless of its content, then it is not surprising that mathematics is widely applicable. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.2) | |
A reaction: Frege was keen to emphasise this. You are left wondering why pure logic is applicable to the physical world. The only account I can give is big-time Platonism, or Pythagoreanism. Logic reveals the engine-room of nature, where the design is done. |
10125 | The classical mathematician believes the real numbers form an actual set |
Full Idea: Unlike the intuitionist, the classical mathematician believes in an actual set that contains all the real numbers. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.6) |
17899 | Second-order induction is stronger as it covers all concepts, not just first-order definable ones |
Full Idea: The first-order version of the induction axiom is weaker than the second-order, because the latter applies to all concepts, but the first-order applies only to concepts definable by a formula in the first-order language of number theory. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.7 n7) |
10128 | The Incompleteness proofs use arithmetic to talk about formal arithmetic |
Full Idea: The idea behind the proofs of the Incompleteness Theorems is to use the language of Peano Arithmetic to talk about the formal system of Peano Arithmetic itself. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.7) | |
A reaction: The mechanism used is to assign a Gödel Number to every possible formula, so that all reasonings become instances of arithmetic. |
17902 | A successor is the union of a set with its singleton |
Full Idea: For any set x, we define the 'successor' of x to be the set S(x) = x U {x}. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) | |
A reaction: This is the Fregean approach to successor, where the Dedekind approach takes 'successor' to be a primitive. Frege 1884:§76. |
10133 | Frege's Theorem shows the Peano Postulates can be derived from Hume's Principle |
Full Idea: The derivability of Peano's Postulates from Hume's Principle in second-order logic has been dubbed 'Frege's Theorem', (though Frege would not have been interested, because he didn't think Hume's Principle gave an adequate definition of numebrs). | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.8 n1) | |
A reaction: Frege said the numbers were the sets which were the extensions of the sets created by Hume's Principle. |
10130 | Set theory can prove the Peano Postulates |
Full Idea: The Peano Postulates can be proven in ZFC. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.7) |
10089 | Talk of 'abstract entities' is more a label for the problem than a solution to it |
Full Idea: One might well wonder whether talk of abstract entities is less a solution to the empiricist's problem of how a priori knowledge is possible than it is a label for the problem. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Intro) | |
A reaction: This pinpoints my view nicely. What the platonist postulates is remote, bewildering, implausible and useless! |
10131 | If mathematics is not about particulars, observing particulars must be irrelevant |
Full Idea: As, in the logicist view, mathematics is about nothing particular, it is little wonder that nothing in particular needs to be observed in order to acquire mathematical knowledge. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002]) | |
A reaction: At the very least we can say that no one would have even dreamt of the general system of arithmetic is they hadn't had experience of the particulars. Frege thought generality ensured applicability, but extreme generality might entail irrelevance. |
17901 | Type theory prohibits (oddly) a set containing an individual and a set of individuals |
Full Idea: If a is an individual and b is a set of individuals, then in the theory of types we cannot talk about the set {a,b}, since it is not an individual or a set of individuals, ...but it is hard to see what harm can come from it. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) |
10092 | In the unramified theory of types, the types are objects, then sets of objects, sets of sets etc. |
Full Idea: In the unramified theory of types, all objects are classified into a hierarchy of types. The lowest level has individual objects that are not sets. Next come sets whose elements are individuals, then sets of sets, etc. Variables are confined to types. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) | |
A reaction: The objects are Type 0, the basic sets Type 1, etc. |
10094 | The theory of types seems to rule out harmless sets as well as paradoxical ones. |
Full Idea: The theory of types seems to rule out harmless sets as well as paradoxical ones. If a is an individual and b is a set of individuals, then in type theory we cannot talk about the set {a,b}. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) | |
A reaction: Since we cheerfully talk about 'Cicero and other Romans', this sounds like a rather disasterous weakness. |
10095 | Type theory has only finitely many items at each level, which is a problem for mathematics |
Full Idea: A problem with type theory is that there are only finitely many individuals, and finitely many sets of individuals, and so on. The hierarchy may be infinite, but each level is finite. Mathematics required an axiom asserting infinitely many individuals. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) | |
A reaction: Most accounts of mathematics founder when it comes to infinities. Perhaps we should just reject them? |
10134 | Much infinite mathematics can still be justified finitely |
Full Idea: It is possible to use finitary reasoning to justify a significant part of infinitary mathematics. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.8) | |
A reaction: This might save Hilbert's project, by gradually accepting into the fold all the parts which have been giving a finitist justification. |
10114 | Bounded quantification is originally finitary, as conjunctions and disjunctions |
Full Idea: In the first instance all bounded quantifications are finitary, for they can be viewed as abbreviations for conjunctions and disjunctions. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.6) | |
A reaction: This strikes me as quite good support for finitism. The origin of a concept gives a good guide to what it really means (not a popular view, I admit). When Aristotle started quantifying, I suspect of he thought of lists, not totalities. |
10123 | The intuitionists are the idealists of mathematics |
Full Idea: The intuitionists are the idealists of mathematics. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.6) |
10124 | Gödel's First Theorem suggests there are truths which are independent of proof |
Full Idea: For intuitionists, truth is not independent of proof, but this independence is precisely what seems to be suggested by Gödel's First Incompleteness Theorem. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.8) | |
A reaction: Thus Gödel was worse news for the Intuitionists than he was for Hilbert's Programme. Gödel himself responded by becoming a platonist about his unprovable truths. |
10110 | Corresponding to every concept there is a class (some of them sets) |
Full Idea: Corresponding to every concept there is a class (some classes will be sets, the others proper classes). | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.4) |