40 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? |
| 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. |
| 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. |
| 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. |
| 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. |
| 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. |
| 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. |
| 23262 | Experience, sympathy and history are sensible grounds for laying claim to rights [Grayling] |
| Full Idea: Personal experience, social sympathies, and history together licence laying claim to rights …which we see to make good mutual as well as individual sense. | |
| From: A.C. Grayling (The Good State [2020], 6) | |
| A reaction: There are no such thing as natural rights, but there are clearly natural grounds on which it is very reasonable to base a claim for legal rights. If positive rights are just arbitrary, or expressions of power struggles, that is crazy. |
| 23263 | Politics is driven by power cliques [Grayling] |
| Full Idea: What drives political history is power cliques. | |
| From: A.C. Grayling (The Good State [2020], Conc) | |
| A reaction: A simple ideas which strikes me as accurate. Alternative views are that power is universally distributed (Foucault), or that power resides in a social class (Marx). Grayling's idea strikes me as more accurate. Each class has its cliques. |
| 23254 | Democracies should require a supermajority for major questions [Grayling] |
| Full Idea: A threshhold or supermajority bar (such as 60%) is the appropriate way to deal with highly consequential questions. | |
| From: A.C. Grayling (The Good State [2020], p.23) | |
| A reaction: This seems to be a very conservative view, because rejection of a major change is a decision in favour of the status quo. Would this rule apply equally to abolishing capital punishment and to reintroducing it? |
| 23255 | It is essential for democracy that voting is free and well informed [Grayling] |
| Full Idea: A necessary condition for democracy to be realised is that the act of voting should be free and informed. | |
| From: A.C. Grayling (The Good State [2020], p.25) | |
| A reaction: The requirement that voters should be well informed has become an increasing modern problem, because the media are owned by the wealthy, and false rumours can spread at lightning speed. |
| 23260 | A cap on time of service would restrict party control and career ambitions [Grayling] |
| Full Idea: A method by which legislators can be rendered independent of both party control and career ambitions is a cap on the amount of time they can serve as legislators. | |
| From: A.C. Grayling (The Good State [2020], 4) | |
| A reaction: The time of service must allow for learning the job, and then using the wisdom of experience. Presumably some career ambitions are needed if we are to have leaders. Not all party discipline is bad; great achievements are hard without it. |
| 23253 | Majority decisions are only acceptable if the minority interests are not vital [Grayling] |
| Full Idea: A majority being in favour of some course of action is the acceptable means of reaching decisions when no vital interest of a minority is endangered. | |
| From: A.C. Grayling (The Good State [2020], 1) | |
| A reaction: This is generally accepted in extreme cases, such as the majority voting to exterminate the minority. The difficulty is to decide what is a 'vital' interest, and to get the majority to care about it. |
| 23256 | Liberty and equality cannot be reconciled [Grayling] |
| Full Idea: Liberty and equality appear to be irresolvable contradictions. | |
| From: A.C. Grayling (The Good State [2020], 2) | |
| A reaction: [He particularly cites Isaiah Berlin for this view] Hm. The liberty of one is the liberty of all. I don't think I would feel that my liberty was unreasonably infringed if I lived in a society of imposed equality. The greedy hate equality the most. |
| 23258 | The very concept of democracy entails a need for justice [Grayling] |
| Full Idea: The concept of democracy - embodying the principles of participation and equal concern - entails that social justice is a mandatory aim. | |
| From: A.C. Grayling (The Good State [2020], 2) | |
| A reaction: The idea that democracy entails participation in any direct way is what the right wing reject. Sustained participation would presumably entail various sorts of justice. |
| 23259 | There should be separate legislative, executive and judicial institutions [Grayling] |
| Full Idea: The obvious solution is where the legislative, executive and judicial powers are exercised by different institutions, distinguished by function. The executive is answerable to the legislative, and the judicial is controlled by neither. | |
| From: A.C. Grayling (The Good State [2020], 3) | |
| A reaction: Separation by institution, rather than merely by separate individuals exercising the powers. I agree (with Popper etc) that institutions are the way to secure long-term success and justice. Grayling says the judiciary must not paralyse government. |
| 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. |
| 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. |
| 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. |