30 ideas
10170 | While true-in-a-model seems relative, true-in-all-models seems not to be [Reck/Price] |
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. |
10166 | ZFC set theory has only 'pure' sets, without 'urelements' [Reck/Price] |
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? |
17824 | The master science is physical objects divided into sets [Maddy] |
Full Idea: The master science can be thought of as the theory of sets with the entire range of physical objects as ur-elements. | |
From: Penelope Maddy (Sets and Numbers [1981], II) | |
A reaction: This sounds like Quine's view, since we have to add sets to our naturalistic ontology of objects. It seems to involve unrestricted mereology to create normal objects. |
10175 | Three types of variable in second-order logic, for objects, functions, and predicates/sets [Reck/Price] |
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. |
10165 | 'Analysis' is the theory of the real numbers [Reck/Price] |
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. |
10174 | Mereological arithmetic needs infinite objects, and function definitions [Reck/Price] |
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. |
10164 | Peano Arithmetic can have three second-order axioms, plus '1' and 'successor' [Reck/Price] |
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'! |
17825 | Set theory (unlike the Peano postulates) can explain why multiplication is commutative [Maddy] |
Full Idea: If you wonder why multiplication is commutative, you could prove it from the Peano postulates, but the proof offers little towards an answer. In set theory Cartesian products match 1-1, and n.m dots when turned on its side has m.n dots, which explains it. | |
From: Penelope Maddy (Sets and Numbers [1981], II) | |
A reaction: 'Turning on its side' sounds more fundamental than formal set theory. I'm a fan of explanation as taking you to the heart of the problem. I suspect the world, rather than set theory, explains the commutativity. |
17826 | Standardly, numbers are said to be sets, which is neat ontology and epistemology [Maddy] |
Full Idea: The standard account of the relationship between numbers and sets is that numbers simply are certain sets. This has the advantage of ontological economy, and allows numbers to be brought within the epistemology of sets. | |
From: Penelope Maddy (Sets and Numbers [1981], III) | |
A reaction: Maddy votes for numbers being properties of sets, rather than the sets themselves. See Yourgrau's critique. |
17828 | Numbers are properties of sets, just as lengths are properties of physical objects [Maddy] |
Full Idea: I propose that ...numbers are properties of sets, analogous, for example, to lengths, which are properties of physical objects. | |
From: Penelope Maddy (Sets and Numbers [1981], III) | |
A reaction: Are lengths properties of physical objects? A hole in the ground can have a length. A gap can have a length. Pure space seems to contain lengths. A set seems much more abstract than its members. |
10172 | Set-theory gives a unified and an explicit basis for mathematics [Reck/Price] |
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. |
17827 | Sets exist where their elements are, but numbers are more like universals [Maddy] |
Full Idea: A set of things is located where the aggregate of those things is located, ...but a number is simultaneously located at many different places (10 in my hand, and a baseball team) ...so numbers seem more like universals than particulars. | |
From: Penelope Maddy (Sets and Numbers [1981], III) | |
A reaction: My gut feeling is that Maddy's master idea (of naturalising sets by building them from ur-elements of natural objects) won't work. Sets can work fine in total abstraction from nature. |
17830 | Number theory doesn't 'reduce' to set theory, because sets have number properties [Maddy] |
Full Idea: I am not suggesting a reduction of number theory to set theory ...There are only sets with number properties; number theory is part of the theory of finite sets. | |
From: Penelope Maddy (Sets and Numbers [1981], V) |
10167 | Structuralism emerged from abstract algebra, axioms, and set theory and its structures [Reck/Price] |
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. |
10169 | Relativist Structuralism just stipulates one successful model as its arithmetic [Reck/Price] |
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? |
10179 | There are 'particular' structures, and 'universal' structures (what the former have in common) [Reck/Price] |
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. |
10181 | Pattern Structuralism studies what isomorphic arithmetic models have in common [Reck/Price] |
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.. |
10182 | There are Formalist, Relativist, Universalist and Pattern structuralism [Reck/Price] |
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? |
10168 | Formalist Structuralism says the ontology is vacuous, or formal, or inference relations [Reck/Price] |
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. |
10178 | Maybe we should talk of an infinity of 'possible' objects, to avoid arithmetic being vacuous [Reck/Price] |
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. |
10176 | Universalist Structuralism is based on generalised if-then claims, not one particular model [Reck/Price] |
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. |
10177 | Universalist Structuralism eliminates the base element, as a variable, which is then quantified out [Reck/Price] |
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. |
10171 | The existence of an infinite set is assumed by Relativist Structuralism [Reck/Price] |
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. |
17823 | If mathematical objects exist, how can we know them, and which objects are they? [Maddy] |
Full Idea: The popular challenges to platonism in philosophy of mathematics are epistemological (how are we able to interact with these objects in appropriate ways) and ontological (if numbers are sets, which sets are they). | |
From: Penelope Maddy (Sets and Numbers [1981], I) | |
A reaction: These objections refer to Benacerraf's two famous papers - 1965 for the ontology, and 1973 for the epistemology. Though he relied too much on causal accounts of knowledge in 1973, I'm with him all the way. |
17829 | Number words are unusual as adjectives; we don't say 'is five', and numbers always come first [Maddy] |
Full Idea: Number words are not like normal adjectives. For example, number words don't occur in 'is (are)...' contexts except artificially, and they must appear before all other adjectives, and so on. | |
From: Penelope Maddy (Sets and Numbers [1981], IV) | |
A reaction: [She is citing Benacerraf's arguments] |
10173 | A nominalist might avoid abstract objects by just appealing to mereological sums [Reck/Price] |
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. |
17402 | Mendeleev saw three principles in nature: matter, force and spirit (where the latter seems to be essence) [Mendeleev, by Scerri] |
Full Idea: Mendeleev rejected one unifying principles in favour of three basic components of nature: matter (substance), force (energy), and spirit (soul). 'Spirit' is said to be what we now mean by essentialism - what is irreducibly peculiar to the object. | |
From: report of Dmitri Mendeleev (The Principles of Chemistry [1870]) by Eric R. Scerri - The Periodic Table 04 'Making' |
17399 | Elements don't survive in compounds, but the 'substance' of the element does [Mendeleev] |
Full Idea: Neither mercury as a metal nor oxygen as a gas is contained in mercury oxide; it only contains the substance of the elements, just as steam only contains the substance of ice. | |
From: Dmitri Mendeleev (The Principles of Chemistry [1870], I:23), quoted by Eric R. Scerri - The Periodic Table 04 'Nature' | |
A reaction: [1889 edn] Scerri glosses the word 'substance' as meaning essence. |
17400 | Mendeleev focused on abstract elements, not simple substances, so he got to their essence [Mendeleev, by Scerri] |
Full Idea: Because he was attempting to classify abstract elements, not simple substances, Mendeleev was not misled by nonessential chemical properties. | |
From: report of Dmitri Mendeleev (The Principles of Chemistry [1870]) by Eric R. Scerri - The Periodic Table 04 'Making' | |
A reaction: I'm not fully clear about this, but I take it that Mendeleev stood back from the messy observations, and tried to see the underlying simpler principles. 'Simple substances' were ones that had not so far been decomposed. |
17401 | Mendeleev had a view of elements which allowed him to overlook some conflicting observations [Mendeleev] |
Full Idea: His view of elements allowed Mendeleev to maintain the validity of the periodic table even in instances where observational evidence seemed to point against it. | |
From: Dmitri Mendeleev (The Principles of Chemistry [1870]), quoted by Eric R. Scerri - The Periodic Table 04 'Making' | |
A reaction: Mendeleev seems to have focused on abstract essences of elements, rather than on the simplest substances they had so far managed to isolate. |