47 ideas
3123 | Science is in the business of carving nature at the joints [Segal] |
Full Idea: Science is in the business of carving nature at the joints. | |
From: Gabriel M.A. Segal (A Slim Book about Narrow Content [2000], 5) |
3125 | Psychology studies the way rationality links desires and beliefs to causality [Segal] |
Full Idea: A person's desires and beliefs tend to cause what they tend to rationalise. This coordination of causality and rationalisation lies at the heart of psychology. | |
From: Gabriel M.A. Segal (A Slim Book about Narrow Content [2000], 5.3) |
18996 | A statement S is 'partly true' if it has some wholly true parts [Yablo] |
Full Idea: A statement S is 'partly true' insofar as it has wholly true parts: wholly true implications whose subject matter is included in that of S. | |
From: Stephen Yablo (Aboutness [2014], 01.6) | |
A reaction: He suggests that if we have rival theories, we agree that it is one or the other. And 'we may have pork for dinner, or human flesh' is partly true. |
19006 | An 'enthymeme' is an argument with an indispensable unstated assumption [Yablo] |
Full Idea: An 'enthymeme' is a deductive argument with an unstated assumption that must be true for the premises to lead to the conclusion. | |
From: Stephen Yablo (Aboutness [2014], 11.1) |
18999 | y is only a proper part of x if there is a z which 'makes up the difference' between them [Yablo] |
Full Idea: The principle of Supplementation says that y is properly part of x, only if a z exists that 'makes up the difference' between them. [note: that is, z is disjoint from y and sums with y to form x] | |
From: Stephen Yablo (Aboutness [2014], 03.2) |
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. |
19001 | 'Pegasus doesn't exist' is false without Pegasus, yet the absence of Pegasus is its truthmaker [Yablo] |
Full Idea: 'Pegasus does not exist' has a paradoxical, self-undermining flavour. On the one hand, the empty name makes it untrue. But now, why is the name empty? Because Pegasus does not exist. 'Pegasus does not exist' is untrue because Pegasus does not exist. | |
From: Stephen Yablo (Aboutness [2014], 05.7 n20) | |
A reaction: Beautiful! This is Yablo's reward for continuing to ask 'why?' after everyone else has stopped in bewilderment at the tricky phenomenon. |
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. |
19002 | A nominalist can assert statements about mathematical objects, as being partly true [Yablo] |
Full Idea: If I am a nominalist non-Platonist, I think it is false that 'there are primes over 10', but I want to be able to say it like everyone else. I argue that this because the statement has a part that I do believe, a part that remains interestingly true. | |
From: Stephen Yablo (Aboutness [2014], 05.8) | |
A reaction: This is obviously a key motivation for Yablo's book, as it reinforces his fictional view of abstract objects, but aims to capture the phenomena, by investigating what such sentences are 'about'. Admirable. |
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. |
18998 | Parthood lacks the restriction of kind which most relations have [Yablo] |
Full Idea: Most relations obtain only between certain kinds of thing. To learn that x is a part of y, however, tells you nothing about x and y taken individually. | |
From: Stephen Yablo (Aboutness [2014], 03.2) | |
A reaction: Too sweeping. To be a part of crowd you have to be a person. To be part of the sea you have to be wet. It might depend on whether composition is unrestricted. |
3105 | Is 'Hesperus = Phosphorus' metaphysically necessary, but not logically or epistemologically necessary? [Segal] |
Full Idea: It is metaphysically necessary that Hesperus is Phosphorus, but not logically necessary, since logical deduction could not reveal its truth, and it is not epistemologically necessary, as the ancient Greeks didn't know the identity. (Natural necessity?) | |
From: Gabriel M.A. Segal (A Slim Book about Narrow Content [2000], 1.6) |
3106 | If claims of metaphysical necessity are based on conceivability, we should be cautious [Segal] |
Full Idea: Since conceivability is the chief method of assessing the claims of metaphysical necessity, I think such claims are incautious. | |
From: Gabriel M.A. Segal (A Slim Book about Narrow Content [2000], 1.6) |
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. |
19004 | Gettier says you don't know if you are confused about how it is true [Yablo] |
Full Idea: We know from Gettier that if you are right to regard Q as true, but you are sufficiently confused about HOW it is true - about how things stand with respect to its subject matter - then you don't know that Q. | |
From: Stephen Yablo (Aboutness [2014], 07.4) | |
A reaction: I'm inclined to approach Gettier by focusing on the propositions being expressed, where his cases tend to focus on the literal wording of the sentences. What did the utterer mean by the sentences - not what did they appear to say. |
19007 | A theory need not be true to be good; it should just be true about its physical aspects [Yablo] |
Full Idea: A physical theory need not be true to be good, Field has argued, and I agree. All we ask of it truth-wise is that its physical implications should be true, or, in my version, that it should be true about the physical. | |
From: Stephen Yablo (Aboutness [2014], 12.5) | |
A reaction: Yablo is, of course, writing a book here about the concept of 'about'. This seems persuasive. The internal terminology of the theory isn't committed to anything - it is only at its physical periphery (Quine) that the ontology matters. |
18993 | If sentences point to different evidence, they must have different subject-matter [Yablo] |
Full Idea: 'All crows are black' cannot say quite the same as 'All non-black things are non-crows', for the two are confirmed by different evidence. Subject matter looks to be the distinguishing feature. One is about crows, the other not. | |
From: Stephen Yablo (Aboutness [2014], Intro) | |
A reaction: You might reply that they are confirmed by the same evidence (but only in its unobtainable totality). The point, I think, is that the sentences invite you to start your search in different places. |
19003 | Most people say nonblack nonravens do confirm 'all ravens are black', but only a tiny bit [Yablo] |
Full Idea: The standard response to the raven paradox is to say that a nonblack nonraven does confirm that all ravens are black. But it confirms it just the teeniest little bit - not as much as a black raven does. | |
From: Stephen Yablo (Aboutness [2014], 06.5) | |
A reaction: It depends on the proportion between the relevant items. How do you confirm 'all the large animals in this zoo are mammals'? Check for size every animal which is obviously not a mammal? |
3113 | The success and virtue of an explanation do not guarantee its truth [Segal] |
Full Idea: The success and virtue of an explanation do not guarantee its truth. | |
From: Gabriel M.A. Segal (A Slim Book about Narrow Content [2000], 2.2) |
3112 | Folk psychology is ridiculously dualist in its assumptions [Segal] |
Full Idea: Commonsense psychology is a powerful explanatory theory, and largely correct, but it seems to be profoundly dualist, and treats minds as immaterial spirits which can transmigrate and exist disembodied. | |
From: Gabriel M.A. Segal (A Slim Book about Narrow Content [2000], 2.2) | |
A reaction: Fans of folk psychology tend to focus on central normal experience, but folk psychology also seems to range from quirky to barking mad. A 'premonition' is a widely accepted mental event. |
3108 | If 'water' has narrow content, it refers to both H2O and XYZ [Segal] |
Full Idea: My view is that the concepts of both the Earth person and the Twin Earth person refer to BOTH forms of diamonds or water (H2O and XYZ). | |
From: Gabriel M.A. Segal (A Slim Book about Narrow Content [2000], 1.7) | |
A reaction: Fair enough, though that seems to imply that my current concepts may actually refer to all sorts of items of which I am currently unaware. But that may be so. |
3110 | Humans are made of H2O, so 'twins' aren't actually feasible [Segal] |
Full Idea: Humans are largely made of H2O, so there could be no twin on Twin Earth, and (as Kuhn noted) nothing with a significantly different structure from H2O could be macroscopically very like water (but topaz and citrine will do). | |
From: Gabriel M.A. Segal (A Slim Book about Narrow Content [2000], 2.1) | |
A reaction: A small point, but one that appeals to essentialists like me (see under Natural Theory/Laws of Nature). We can't learn much metaphysics from impossible examples. |
3124 | Externalists can't assume old words refer to modern natural kinds [Segal] |
Full Idea: The question of what a pre-scientific term extends over is extremely difficult for a Putnam-style externalist to answer. …There seems no good reason to assume that they extend over natural kinds ('whale', 'cat', 'water'). | |
From: Gabriel M.A. Segal (A Slim Book about Narrow Content [2000], 5.1) | |
A reaction: The assumption seems to be that they used to extend over descriptions, and now they extend over essences, or expert references. This can't be right. They have never changed, but now contain fewer errors. |
3104 | Must we relate to some diamonds to understand them? [Segal] |
Full Idea: Is a relationship with diamonds necessary for having a concept of diamonds? | |
From: Gabriel M.A. Segal (A Slim Book about Narrow Content [2000], 1.4) | |
A reaction: Probably not, given that I have a concept of kryptonite, and that I can invent my own concepts. Suppose I was brought up to believe that diamonds are a myth? |
3109 | If content is external, so are beliefs and desires [Segal] |
Full Idea: If we accept Putnam's externalist conclusion about the meaning of a word, it is a short step to a similar conclusion about the contents of the twins' beliefs, desires and so on. | |
From: Gabriel M.A. Segal (A Slim Book about Narrow Content [2000], 2.1) | |
A reaction: This is the key step which has launched a whole new externalist view of the nature of the mind. It is one thing to say that I don't quite know what my words mean, another that I don't know my own beliefs. |
3103 | Maybe content involves relations to a language community [Segal] |
Full Idea: It has been argued (e.g. by Tyler Burge) that certain relations to other language users are determinants of content. | |
From: Gabriel M.A. Segal (A Slim Book about Narrow Content [2000], 1.4) | |
A reaction: Burge's idea (with Wittgenstein behind him) strikes me as plausible (more plausible than water and elms determining the content). Our concepts actually shift during conversations. |
3116 | Maybe experts fix content, not ordinary users [Segal] |
Full Idea: Putnam and Burge claim that there could be two words that a misinformed subject uses to express different concepts, but that express just one concept of the experts. | |
From: Gabriel M.A. Segal (A Slim Book about Narrow Content [2000], 3.2) | |
A reaction: This pushes the concept outside the mind of the user, which leaves an ontological problem of what concepts are made of, how you individuate them, and where they are located. |
3111 | Externalism can't explain concepts that have no reference [Segal] |
Full Idea: Empty terms and concepts provide the largest problem for the externalist thesis of the world dependence of concepts. | |
From: Gabriel M.A. Segal (A Slim Book about Narrow Content [2000], 2.2) | |
A reaction: A speculative concept could then become a reality (e.g. an invention). The solution seems to be to say that there is an internal and an external component to most concepts. |
3117 | Concepts can survive a big change in extension [Segal] |
Full Idea: We need to think of concepts as organic entities that can persist through changes of extension. | |
From: Gabriel M.A. Segal (A Slim Book about Narrow Content [2000], 3.3) | |
A reaction: This would be 'organic' in the sense of modifying and growing. This is exactly right, and the interesting problem becomes the extreme cases, where an individual stretches a concept a long way. |
3121 | If content is narrow, my perfect twin shares my concepts [Segal] |
Full Idea: To say that contents of my belief are narrow is to say that they are intrinsic to me, hence that any perfect twin of mine would have beliefs with the same contents. | |
From: Gabriel M.A. Segal (A Slim Book about Narrow Content [2000], 5) | |
A reaction: I personally find this more congenial than externalism. If my twin and I studied chemistry, we would reach identical conclusions about water, as long as we remained perfect twins. |
3118 | If thoughts ARE causal, we can't explain how they cause things [Segal] |
Full Idea: If we identify a psychological property with its causal role then we lose the obvious explanation of why the event has the causal role that it has. | |
From: Gabriel M.A. Segal (A Slim Book about Narrow Content [2000], 4.1) | |
A reaction: This pinpoints very nicely one of the biggest errors in modern philosophy. There are good naturalistic reasons to reduce everything to causal role, but there is a deeper layer. Essences! |
3119 | Even 'mass' cannot be defined in causal terms [Segal] |
Full Idea: We can't define mass in terms of its causal powers because massive objects do different things in different physical systems. …What an object (or concept) with a given property does depends on what it interacts with. | |
From: Gabriel M.A. Segal (A Slim Book about Narrow Content [2000], 4.1) | |
A reaction: This leaves an epistemological problem, that we believe in mass, but can only get at it within a particular gravitational or inertial system. Don't give up on ontology at this point. |
18992 | Sentence-meaning is the truth-conditions - plus factors responsible for them [Yablo] |
Full Idea: A sentence's meaning is to do with its truth-value in various possible scenarios, AND the factors responsible for that truth-value. | |
From: Stephen Yablo (Aboutness [2014], Intro) | |
A reaction: The thesis of his book, which I welcome. I'm increasingly struck by the way in which much modern philosophy settles for a theory being complete, when actually further explanation is possible. Exhibit A is functional explanations. Why that function? |
18994 | The content of an assertion can be quite different from compositional content [Yablo] |
Full Idea: Assertive content - what a sentence is heard as saying - can be at quite a distance from compositional content. | |
From: Stephen Yablo (Aboutness [2014], Intro) | |
A reaction: This is the obvious reason why semantics cannot be entirely compositional, since there is nearly always a contextual component which then has to be added. In the case of irony, the compositional content is entirely reversed. |
18997 | Truth-conditions as subject-matter has problems of relevance, short cut, and reversal [Yablo] |
Full Idea: If the subject-matter of S is how it is true, we get three unfortunate results: S has truth-value in worlds where its subject-matter draws a blank; learning what S is about tells you its truth-value; negating S changes what it's about. | |
From: Stephen Yablo (Aboutness [2014], 02.8) | |
A reaction: Together these make fairly devastating objections to the truth-conditions (in possible worlds) theory of meaning. The first-objection concerns when S is false |
19005 | Not-A is too strong to just erase an improper assertion, because it actually reverses A [Yablo] |
Full Idea: The idea that negation is, or can be, a cancellation device raises an interesting question. What does one do to wipe the slate clean after an improper assertion? Not-A is too strong; it reverses our stand on A rather than nullifying it. | |
From: Stephen Yablo (Aboutness [2014], 09.8) | |
A reaction: [He is discussing a remark of Strawson 1952] It seems that 'not' has two meanings or uses: a weak use of 'nullifying' an assertion, and a strong use of 'reversing' an assertion. One could do both: 'that's not right; in fact, it's just the opposite'. |