39 ideas
| 15063 | Some sentences depend for their truth on worldly circumstances, and others do not [Fine,K] |
| Full Idea: There is a distinction between worldly and unworldly sentences, between sentences that depend for their truth upon the worldly circumstances and those that do not. | |
| From: Kit Fine (Necessity and Non-Existence [2005], Intro) | |
| A reaction: Fine is fishing around in the area between the necessary, the a priori, truthmakers, and truth-conditions. He appears to be attempting a singlehanded reconstruction of the concepts of metaphysics. Is he major, or very marginal? |
| 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? |
| 18935 | Semantic theory should specify when an act of naming is successful [Sawyer] |
| Full Idea: A semantic theory of names should deliver a specification of the conditions under which a name names an individual, and hence a specification of the conditions under which a name is empty. | |
| From: Sarah Sawyer (Empty Names [2012], 1) | |
| A reaction: Naming can be private, like naming my car 'Bertrand', but never tell anyone. I like Plato's remark that names are 'tools'. Do we specify conditions for successful spanner-usage? The first step must be individuation, preparatory to naming. |
| 18945 | Millians say a name just means its object [Sawyer] |
| Full Idea: The Millian view of direct reference says that the meaning of a name is the object named. | |
| From: Sarah Sawyer (Empty Names [2012], 4) | |
| A reaction: Any theory that says meaning somehow is features of the physical world strikes me as totally misguided. Napoleon is a man, so he can't be part of a sentence. He delegates that job to words (such as 'Napoleon'). |
| 18934 | Sentences with empty names can be understood, be co-referential, and even be true [Sawyer] |
| Full Idea: Some empty names sentences can be understood, so appear to be meaningful ('Pegasus was sired by Poseidon'), ...some appear to be co-referential ('Santa Claus'/'Father Christmas'), and some appear to be straightforwardly true ('Pegasus doesn't exist'). | |
| From: Sarah Sawyer (Empty Names [2012], 1) | |
| A reaction: Hang on to this, when the logicians arrive and start telling you that your talk of empty names is vacuous, because there is no object in the 'domain' to which a predicate can be attached. Meaning, reference and truth are the issues around empty names. |
| 18938 | Frege's compositional account of truth-vaues makes 'Pegasus doesn't exist' neither true nor false [Sawyer] |
| Full Idea: In Frege's account sentences such as 'Pegasus does not exist' will be neither true nor false, since the truth-value of a sentence is its referent, and the referent of a complex expression is determined by the referent of its parts. | |
| From: Sarah Sawyer (Empty Names [2012], 2) | |
| A reaction: We can keep the idea of 'sense', which is very useful for dealing with empty names, but tweak his account of truth-values to evade this problem. I'm thinking that meaning is compositional, but truth-value isn't. |
| 18947 | Definites descriptions don't solve the empty names problem, because the properties may not exist [Sawyer] |
| Full Idea: If it were possible for a definite description to be empty - not in the sense of there being no object that satisfies it, but of there being no set of properties it refers to - the problem of empty names would not have been solved. | |
| From: Sarah Sawyer (Empty Names [2012], 5) | |
| A reaction: Swoyer is thinking of properties like 'is a unicorn', which are clearly just as vulnerable to being empty as 'the unicorn' was. It seems unlikely that 'horse', 'white' and 'horn' would be empty. |
| 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'! |
| 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. |
| 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. |
| 15078 | There are levels of existence, as well as reality; objects exist at the lowest level in which they can function [Fine,K] |
| Full Idea: Just as we recognise different levels of reality, so we should recognise different levels of existence. Each object will exist at the lowest level at which it can enjoy its characteristic form of life. | |
| From: Kit Fine (Necessity and Non-Existence [2005], 10) | |
| A reaction: I'm struggling with this claim, despite my sympathy for much of Fine's picture. I'm not sure that the so-called 'levels' of reality have different degrees of reality. |
| 15072 | Bottom level facts are subject to time and world, middle to world but not time, and top to neither [Fine,K] |
| Full Idea: At the bottom are tensed or temporal facts, subject to the vicissitudes of time and hence of the world. Then come the timeless though worldly facts, subject to the world but not to time. Top are transcendental facts, subject to neither world nor time. | |
| From: Kit Fine (Necessity and Non-Existence [2005], 08) | |
| A reaction: For all of Fine's awesome grasp of logic and semantics, when he divides reality up as boldly as this I start to side a bit with the sceptics about modern metaphysics (like Ladyman and Ross). I daresay Fine acknowledges that it is 'speculative'. |
| 15071 | Tensed and tenseless sentences state two sorts of fact, which belong to two different 'realms' of reality [Fine,K] |
| Full Idea: A tensed fact is stated by a tensed sentence while a tenseless fact is stated by a tenseless sentence, and they belong to two 'realms' of reality. That Socrates drank hemlock is in the temporal realm, while 2+2=4 is presumably in the timeless realm. | |
| From: Kit Fine (Necessity and Non-Existence [2005], 07) | |
| A reaction: Put so strongly, I suddenly find sales resistance to his proposal. All my instincts favour one realm, and I take 2+2=4 to be a highly general truth about that realm. It may be a truth of any possible realm, which would distinguish it. |
| 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. |
| 15075 | Modal features are not part of entities, because they are accounted for by the entity [Fine,K] |
| Full Idea: It is natural to suggest that to be a man is to have certain kind of temporal-modal profile. ...but it seems natural that being a man accounts for the profile, ...so one should not appeal to an object's modal features in stating what the object is. | |
| From: Kit Fine (Necessity and Non-Existence [2005], 09) | |
| A reaction: This strikes me as a correct and very helpful point, as I am tempted to think that the modal dispositions of a thing are intrinsic to its identity. If we accept 'powers', must they be modal in character? Fine backs a sortal approach. That's ideology. |
| 15065 | What it is is fixed prior to existence or the object's worldly features [Fine,K] |
| Full Idea: The identity of an object - what it is - is not a worldly matter; essence will precede existence in that the identity of an object may be fixed by its unworldly features even before any question of its existence or other worldly features is considered. | |
| From: Kit Fine (Necessity and Non-Existence [2005], Intro) | |
| A reaction: I'm not clear how this cashes out. If I remove the 'worldly features' of an object, what is there left which establishes identity? Fine carefully avoids talk of 'a priori' knowledge of identity. |
| 15076 | Essential features of an object have no relation to how things actually are [Fine,K] |
| Full Idea: It is the core essential features of the object that will be independent of how things turn out, and they will be independent in the sense of holding regardless of circumstances, not whatever the circumstances. | |
| From: Kit Fine (Necessity and Non-Existence [2005], 09) | |
| A reaction: The distinction at the end seems to be that 'regardless' pays no attention to circumstances, whereas 'whatever' pays attention to all circumstances. In other words, essence has no relationship to how things are. Plausible. Nice to see 'core'. |
| 15073 | Self-identity should have two components, its existence, and its neutral identity with itself [Fine,K] |
| Full Idea: The existential identity of an object with itself needs analysis into two components, one the neutral identity of the object with itself, and the other its existence. The existence of the object appears to be merely a gratuitous addition to its identity. | |
| From: Kit Fine (Necessity and Non-Existence [2005], 08) | |
| A reaction: This is at least a step towards clarification of the notion, which might be seen as just a way of asserting that something 'has an identity'. Fine likes the modern Fregean way of expressing this, as an equality relation. |
| 15074 | We would understand identity between objects, even if their existence was impossible [Fine,K] |
| Full Idea: If there were impossible objects, ones that do not possibly exist, we would have no difficulty in understanding what it is for such objects to be identical or distinct than in the case of possible objects. | |
| From: Kit Fine (Necessity and Non-Existence [2005], 08) | |
| A reaction: Thus, a 'circular square' seems to be the same as a 'square circle'. Fine is arguing for identity to be independent of any questions of existence. |
| 15064 | Proper necessary truths hold whatever the circumstances; transcendent truths regardless of circumstances [Fine,K] |
| Full Idea: We distinguish between the necessary truths proper, those that hold whatever the circumstances, and the transcendent truths, those that hold regardless of the circumstances. | |
| From: Kit Fine (Necessity and Non-Existence [2005], Intro) | |
| A reaction: Fine's project seems to be dividing the necessities which derive from essence from the necessities which tended to be branded in essentialist discussions as 'trivial'. |
| 15070 | It is the nature of Socrates to be a man, so necessarily he is a man [Fine,K] |
| Full Idea: It is of the nature of Socrates to be a man; and from this it appears to follow that necessarily he is a man. | |
| From: Kit Fine (Necessity and Non-Existence [2005], 04) | |
| A reaction: I'm always puzzled by this line of thought, because it is only the intrinsic nature of beings like Socrates which decides in the first place what a 'man' is. How can something help to create a category, and then necessarily belong to that category? |
| 15069 | Possible worlds may be more limited, to how things might actually turn out [Fine,K] |
| Full Idea: An alternative conception of a possible world says it is constituted, not by the totality of facts, or of how things might be, but by the totality of circumstances, or how things might turn out. | |
| From: Kit Fine (Necessity and Non-Existence [2005], 02) | |
| A reaction: The general idea is to make a possible world more limited than in Idea 15068. It only contains properties arising from 'engagement with the world', and won't include timeless sentences. It is a bunch of possibilities, not of actualities? |
| 15068 | The actual world is a totality of facts, so we also think of possible worlds as totalities [Fine,K] |
| Full Idea: We are accustomed think of the actual world as the totality of facts, and so we think of any possible world as being like the actual world in settling the truth-value of every single proposition. | |
| From: Kit Fine (Necessity and Non-Existence [2005], 02) | |
| A reaction: Hence it is normal to refer to a possible world as a 'maximal' set of of propositions (sentences, etc). See Idea 15069 for his proposed alternative view. |
| 15067 | A-theorists tend to reject the tensed/tenseless distinction [Fine,K] |
| Full Idea: Most A-theorists have been inclined to reject the tensed/tenseless distinction. | |
| From: Kit Fine (Necessity and Non-Existence [2005], 01) | |
| A reaction: Presumably this is because they reject the notion of 'tenseless' truths. But sentences like 'two and two make four' seem not to be very tensy. |
| 15077 | It is said that in the A-theory, all existents and objects must be tensed, as well as the sentences [Fine,K] |
| Full Idea: It is said that there is no room in the A-theorists' ontology for a realm of timeless existents. Just as there is a tendency to think that every sentence is tensed, so there is a tendency to think that every object must enjoy a tensed form of existence. | |
| From: Kit Fine (Necessity and Non-Existence [2005], 10) | |
| A reaction: Fine is arguing for certain things to exist or be true independently of time (such as arithmetic, or essential identities). I struggle with the notion of timeless existence. |
| 15066 | B-theorists say tensed sentences have an unfilled argument-place for a time [Fine,K] |
| Full Idea: B-theorists regard tensed sentences as incomplete expressions, implicitly containing an unfilled argument-place for the time at which they are to be evaluated. | |
| From: Kit Fine (Necessity and Non-Existence [2005], 01) | |
| A reaction: To distinguish past from future it looks as if you would need two argument-places, not one. Then there are 'used to be' and 'had been' to evaluate. |