30 ideas
| 14001 | People who use science to make philosophical points don't realise how philosophical science is [Markosian] |
| Full Idea: When people give arguments from scientific theories to philosophical conclusions, there is usually a good deal of philosophy built into the relevant scientific theories. | |
| From: Ned Markosian (A Defense of Presentism [2004], 3.9) | |
| A reaction: I love this remark, being thoroughly fed up with knowledgeable scientists who are naïve about philosophy, and think their current theory demolishes long-lasting aporiai. They are up to their necks in philosophy. |
| 13991 | Presentism has the problem that if Socrates ceases to exist, so do propositions about him [Markosian] |
| Full Idea: Presentism has a problem with singular propositions about non-present objects. ...When Socrates popped out of existence, according to Presentism, all those singular propositions about him also popped out of existence. | |
| From: Ned Markosian (A Defense of Presentism [2004], 2.1) | |
| A reaction: He seems to treat propositions in a Russellian way, as things which exist independently of thinkers, which I struggle to grasp. Markosian offers various strategies for this [§3.5]. |
| 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. |
| 14002 | Possible worlds must be abstract, because two qualitatively identical worlds are just one world [Markosian] |
| Full Idea: Possible worlds are just abstract objects that play a certain role in philosophers' talk about modality. They are ways things could be. That's why there are no two abstract possible worlds which are qualitatively identical. They count as one world. | |
| From: Ned Markosian (A Defense of Presentism [2004], 3.10) | |
| A reaction: Brilliant! This looks like the best distinction between concrete and abstract. If two concreta are identical they remain two; if two abstracta are identical they are one (like numbers, or logical connectives with the same truth table). |
| 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. |
| 14000 | 'Grabby' truth conditions first select their object, unlike 'searchy' truth conditions [Markosian] |
| Full Idea: We can talk of 'grabby' truth conditions (where an object is grabbed before predication) and 'searchy' truth conditions (where the object is included in what is being asserted). | |
| From: Ned Markosian (A Defense of Presentism [2004], 3.8) | |
| A reaction: [He credits Tom Ryckman with the terminology] I am inclined to think that the whole of language is 'searchy', even when it appears to be blatantly 'grabby'. Even ostensive reference is an act of hope rather than certainty. |
| 20739 | Levinas took 'first philosophy' to begin with seeing the vulnerable faces of others [Levinas, by Aho] |
| Full Idea: Levinas forwarded a notion of 'ethics as first philosophy' that begins from the concete exposure and openness to 'the face' of the other, an experience of vulnerability and suffering that undercuts our ordinary egoistic and objectifying tendencies. | |
| From: report of Emmanuel Levinas (works [1956]) by Kevin Aho - Existentialism: an introduction 1 'Existentialism' | |
| A reaction: Iris Murdoch speaks of seeing a falcon in flight as having a similar effect of diminishing the ego. If the main focus is on potential 'suffering' does this eventually cash out as utilitarianism? I bet not! |
| 20445 | Levinas says Marxism is the replacement of individualist ethics, by solidarity and sociality [Levinas, by Critchley] |
| Full Idea: For Levinas, Marxism is the absorption of the ethical in the socioeconomic, and so it is the disappearance of the face-to-face relation and the privileging of relations of solidarity and anonymous sociality, which he calls 'socialism'. | |
| From: report of Emmanuel Levinas (works [1956]) by Simon Critchley - Impossible Objects: interviews 1 | |
| A reaction: Startling, if you are not used to this sort of thing. If you are in trouble, I should help you, not because you are you, or a human being, but because you are a member of my group? So what about the Good Samaritan? Or solidarity with humanity? Animals? |
| 13990 | Presentism is the view that only present objects exist [Markosian] |
| Full Idea: According to Presentism, if we were to make an accurate list of all the things that exist (within the range of our most unrestricted quantifiers) there would not be a single non-present object on the list. | |
| From: Ned Markosian (A Defense of Presentism [2004], 1) | |
| A reaction: An immediate problem that needs examing is what constitutes an 'object'. It had better not range over time (like an journey). It would be hard to fit a description like 'the oldest man in England'. |
| 13992 | Presentism says if objects don't exist now, we can't have attitudes to them or relations with them [Markosian] |
| Full Idea: If there are no non-present objects (according to Presentism), then no one can now stand in any relation to any non-present object. You cannot now 'admire' Socrates, and no present event has a causal relation to Washington crossing the Delaware. | |
| From: Ned Markosian (A Defense of Presentism [2004], 2.2) | |
| A reaction: You can have an overlapping causal chain that gets you back to Washington, and a causal chain can connect Socrates to our thoughts about him (as in baptismal reference). A simple reply needs an 'overlap' though. |
| 13994 | Presentism seems to entail that we cannot talk about other times [Markosian] |
| Full Idea: It is very natural to talk about times, ...but Presentism seems to entail that we never say anything about any such times. | |
| From: Ned Markosian (A Defense of Presentism [2004], 2.4) | |
| A reaction: I'm beginning to think that Markosian is in the grips of a false notion of proposition, as something that exists independently of thinkers, and is entailed by the facts and objects of reality. This is not what language does. |
| 13995 | Serious Presentism says things must exist to have relations and properties; Unrestricted version denies this [Markosian] |
| Full Idea: Mark Hinchliff distinguishes between 'Serious' Presentism (objects only have relations and properties when they exist) and 'Unrestricted' Presentism (objects can have relations and properties even when they don't exist). | |
| From: Ned Markosian (A Defense of Presentism [2004], 3.1) | |
| A reaction: [Hinchliff 1996:124-6] Markosian votes for the Serious version, as being the only true Presentism. I think he is muddling language and reality, predicates and properties. |
| 13996 | Maybe Presentists can refer to the haecceity of a thing, after the thing itself disappears [Markosian] |
| Full Idea: Some Presentists (such as Adams) believe that a haecceity (a property unique to some entity) continues to exist even after its object ceases to exist. A sentence about Socrates still expresses a proposition, about 'Socraticity'. | |
| From: Ned Markosian (A Defense of Presentism [2004], 3.4) | |
| A reaction: [Adams 1986] This is rather puzzling. In what sense could a haecceity 'exist' to be referred to? Existence, but not as we know it, Jim. This smacks of medieval theology. |
| 13997 | Maybe Presentists can paraphrase singular propositions about the past [Markosian] |
| Full Idea: Maybe Presentists can paraphrase singular propositions about the past, into purely general past- and future-tensed sentences. | |
| From: Ned Markosian (A Defense of Presentism [2004], 3.5) | |
| A reaction: I'm not clear why Markosian worries about singular propositions, but is happy with general ones. Surely the latter refer as much as the former to what doesn't exist? Markosian objects that the paraphrase has a different meaning. |
| 13993 | Special Relativity denies the absolute present which Presentism needs [Markosian] |
| Full Idea: The objection to Presentism from Special Relativity is this: 1) Relativity is true, 2) so there is no absolute simultaneity, 3) so there is no absolute presentness, but 4) Presentism entails absolute presentness, so 5) Presentism is false. | |
| From: Ned Markosian (A Defense of Presentism [2004], 3.9) | |
| A reaction: I don't accept this objection. There may be accounts that can give Relativity one present (Idea 12689-90). Maybe Einstein was too instrumentalist in his account. Maybe we can have Presentism with multiple present moments. |
| 13998 | Objects in the past, like Socrates, are more like imaginary objects than like remote spatial objects [Markosian] |
| Full Idea: Maybe putative non-present objects like Socrates have more in common with putative non-actual objects like Santa Claus than they have in common with objects located elsewhere in space, like Alpha Centauri. | |
| From: Ned Markosian (A Defense of Presentism [2004], 3.7) | |
| A reaction: We can see Alpha Centauri, so we need an example beyond some 'event horizon'. He credits Arthur Prior with this line of thought. He seems to me to drift towards a Descriptive Theory of Reference (shock!). Does the nature of reference change with death? |
| 13999 | People are mistaken when they think 'Socrates was a philosopher' says something [Markosian] |
| Full Idea: People sometimes think that 'Socrates was a philosopher' expresses something like a true, singular proposition about Socrates. They're making a mistake, but still, this explains why they think it is true. | |
| From: Ned Markosian (A Defense of Presentism [2004], 3.8) | |
| A reaction: A classic error theory, about our talk of the past. Personally I would say that the sentence really is true, and that needing a tangible object to refer to is a totally bogus requirement. 'I wonder if there are any scissors in the house?' |