2000 | Structures and Structuralism in Phil of Maths |
§2 | p.343 | 10165 | 'Analysis' is the theory of the real numbers |
Full Idea: 'Analysis' is the theory of the real numbers. | |||
From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2) | |||
A reaction: 'Analysis' began with the infinitesimal calculus, which later built on the concept of 'limit'. A continuum of numbers seems to be required to make that work. |
§2 | p.343 | 10164 | Peano Arithmetic can have three second-order axioms, plus '1' and 'successor' |
Full Idea: A common formulation of Peano Arithmetic uses 2nd-order logic, the constant '1', and a one-place function 's' ('successor'). Three axioms then give '1 is not a successor', 'different numbers have different successors', and induction. | |||
From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2) | |||
A reaction: This is 'second-order' Peano Arithmetic, though it is at least as common to formulate in first-order terms (only quantifying over objects, not over properties - as is done here in the induction axiom). I like the use of '1' as basic instead of '0'! |
§2 | p.344 | 10166 | ZFC set theory has only 'pure' sets, without 'urelements' |
Full Idea: In standard ZFC ('Zermelo-Fraenkel with Choice') set theory we deal merely with pure sets, not with additional urelements. | |||
From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2) | |||
A reaction: The 'urelements' would the actual objects that are members of the sets, be they physical or abstract. This idea is crucial to understanding philosophy of mathematics, and especially logicism. Must the sets exist, just as the urelements do? |
§2 | p.346 | 10167 | Structuralism emerged from abstract algebra, axioms, and set theory and its structures |
Full Idea: Structuralism has emerged from the development of abstract algebra (such as group theory), the creation of axiom systems, the introduction of set theory, and Bourbaki's encyclopaedic survey of set theoretic structures. | |||
From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2) | |||
A reaction: In other words, mathematics has gradually risen from one level of abstraction to the next, so that mathematical entities like points and numbers receive less and less attention, with relationships becoming more prominent. |
§3 | p.348 | 10168 | Formalist Structuralism says the ontology is vacuous, or formal, or inference relations |
Full Idea: Formalist Structuralism endorses structural methodology in mathematics, but rejects semantic and metaphysical problems as either meaningless, or purely formal, or as inference relations. | |||
From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §3) | |||
A reaction: [very compressed] I find the third option fairly congenial, certainly in preference to rather platonist accounts of structuralism. One still needs to distinguish the mathematical from the non-mathematical in the inference relations. |
§4 | p.349 | 10169 | Relativist Structuralism just stipulates one successful model as its arithmetic |
Full Idea: Relativist Structuralism simply picks one particular model of axiomatised arithmetic (i.e. one particular interpretation that satisfies the axioms), and then stipulates what the elements, functions and quantifiers refer to. | |||
From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |||
A reaction: The point is that a successful model can be offered, and it doesn't matter which one, like having any sort of aeroplane, as long as it flies. I don't find this approach congenial, though having a model is good. What is the essence of flight? |
§4 | p.350 | 10171 | The existence of an infinite set is assumed by Relativist Structuralism |
Full Idea: Relativist Structuralism must first assume the existence of an infinite set, otherwise there would be no model to pick, and arithmetical terms would have no reference. | |||
From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |||
A reaction: See Idea 10169 for Relativist Structuralism. They point out that ZFC has an Axiom of Infinity. |
§4 | p.350 | 10170 | While true-in-a-model seems relative, true-in-all-models seems not to be |
Full Idea: While truth can be defined in a relative way, as truth in one particular model, a non-relative notion of truth is implied, as truth in all models. | |||
From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |||
A reaction: [The article is actually discussing arithmetic] This idea strikes me as extremely important. True-in-all-models is usually taken to be tautological, but it does seem to give a more universal notion of truth. See semantic truth, Tarski, Davidson etc etc. |
§4 | p.351 | 10172 | Set-theory gives a unified and an explicit basis for mathematics |
Full Idea: The merits of basing an account of mathematics on set theory are that it allows for a comprehensive unified treatment of many otherwise separate branches of mathematics, and that all assumption, including existence, are explicit in the axioms. | |||
From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |||
A reaction: I am forming the impression that set-theory provides one rather good model (maybe the best available) for mathematics, but that doesn't mean that mathematics is set-theory. The best map of a landscape isn't a landscape. |
§4 | p.352 | 10173 | A nominalist might avoid abstract objects by just appealing to mereological sums |
Full Idea: One way for a nominalist to reject appeal to all abstract objects, including sets, is to only appeal to nominalistically acceptable objects, including mereological sums. | |||
From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |||
A reaction: I'm suddenly thinking that this looks very interesting and might be the way to go. The issue seems to be whether mereological sums should be seen as constrained by nature, or whether they are unrestricted. See Mereology in Ontology...|Intrinsic Identity. |
§4 | p.352 | 10174 | Mereological arithmetic needs infinite objects, and function definitions |
Full Idea: The difficulties for a nominalistic mereological approach to arithmetic is that an infinity of physical objects are needed (space-time points? strokes?), and it must define functions, such as 'successor'. | |||
From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |||
A reaction: Many ontologically austere accounts of arithmetic are faced with the problem of infinity. The obvious non-platonist response seems to be a modal or if-then approach. To postulate infinite abstract or physical entities so that we can add 3 and 2 is mad. |
§5 | p.356 | 10175 | Three types of variable in second-order logic, for objects, functions, and predicates/sets |
Full Idea: In second-order logic there are three kinds of variables, for objects, for functions, and for predicates or sets. | |||
From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5) | |||
A reaction: It is interesting that a predicate seems to be the same as a set, which begs rather a lot of questions. For those who dislike second-order logic, there seems nothing instrinsically wicked in having variables ranging over innumerable multi-order types. |
§5 | p.356 | 10176 | Universalist Structuralism is based on generalised if-then claims, not one particular model |
Full Idea: Universalist Structuralism is a semantic thesis, that an arithmetical statement asserts a universal if-then statement. We build an if-then statement (using quantifiers) into the structure, and we generalise away from any one particular model. | |||
From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5) | |||
A reaction: There remains the question of what is distinctively mathematical about the highly generalised network of inferences that is being described. Presumable the axioms capture that, but why those particular axioms? Russell is cited as an originator. |
§5 | p.358 | 10177 | Universalist Structuralism eliminates the base element, as a variable, which is then quantified out |
Full Idea: Universalist Structuralism is eliminativist about abstract objects, in a distinctive form. Instead of treating the base element (say '1') as an ambiguous referring expression (the Relativist approach), it is a variable which is quantified out. | |||
From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5) | |||
A reaction: I am a temperamental eliminativist on this front (and most others) so this is tempting. I am also in love with the concept of a 'variable', which I take to be utterly fundamental to all conceptual thought, even in animals, and not just a trick of algebra. |
§5 | p.359 | 10178 | Maybe we should talk of an infinity of 'possible' objects, to avoid arithmetic being vacuous |
Full Idea: It is tempting to take a modal turn, and quantify over all possible objects, because if there are only a finite number of actual objects, then there are no models (of the right sort) for Peano Arithmetic, and arithmetic is vacuously true. | |||
From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5) | |||
A reaction: [compressed; Geoffrey Hellman is the chief champion of this view] The article asks whether we are not still left with the puzzle of whether infinitely many objects are possible, instead of existent. |
§6 | p.362 | 10179 | There are 'particular' structures, and 'universal' structures (what the former have in common) |
Full Idea: The term 'structure' has two uses in the literature, what can be called 'particular structures' (which are particular relational systems), but also what can be called 'universal structures' - what particular systems share, or what they instantiate. | |||
From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §6) | |||
A reaction: This is a very helpful distinction, because it clarifies why (rather to my surprise) some structuralists turn out to be platonists in a new guise. Personal my interest in structuralism has been anti-platonist from the start. |
§7 | p.363 | 10181 | Pattern Structuralism studies what isomorphic arithmetic models have in common |
Full Idea: According to 'pattern' structuralism, what we study are not the various particular isomorphic models of arithmetic, but something in addition to them: a corresponding pattern. | |||
From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §7) | |||
A reaction: Put like that, we have to feel a temptation to wield Ockham's Razor. It's bad enough trying to give the structure of all the isomorphic models, without seeking an even more abstract account of underlying patterns. But patterns connect to minds.. |
§9 | p.374 | 10182 | There are Formalist, Relativist, Universalist and Pattern structuralism |
Full Idea: There are four main variants of structuralism in the philosophy of mathematics - formalist structuralism, relativist structuralism, universalist structuralism (with modal variants), and pattern structuralism. | |||
From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §9) | |||
A reaction: I'm not sure where Chihara's later book fits into this, though it is at the nominalist end of the spectrum. Shapiro and Resnik do patterns (the latter more loosely); Hellman does modal universalism; Quine does the relativist version. Dedekind? |