26 ideas
| 23367 | Even pointing a finger should only be done for a reason [Epictetus] |
| Full Idea: Philosophy says it is not right even to stretch out a finger without some reason. | |
| From: Epictetus (fragments/reports [c.57], 15) | |
| A reaction: The key point here is that philosophy concerns action, an idea on which Epictetus is very keen. He rather despise theory. This idea perfectly sums up the concept of the wholly rational life (which no rational person would actually want to live!). |
| 5745 | Quine says quantified modal logic creates nonsense, bad ontology, and false essentialism [Melia on Quine] |
| Full Idea: Quine charges quantified modal systems of logic with giving rise to unintended sense or nonsense, committing us to an incomprehensible ontology, and entailing an implausible or unsustainable Aristotelian essentialism. | |
| From: comment on Willard Quine (Existence and Quantification [1966]) by Joseph Melia - Modality Ch.3 | |
| A reaction: A nice summary. Personally I like essentialism in accounts of science (see Nature|Laws of Nature|Essentialism), so would like to save it in metaphysics. Possible worlds ontology may be very surprising, rather than 'incomprehensible'. |
| 8789 | Various strategies try to deal with the ontological commitments of second-order logic [Hale/Wright on Quine] |
| Full Idea: Quine said higher-order logic is 'set theory in sheep's clothing', and there is concern about the ontology that is involved. One approach is to deny quantificational ontological commitments, or say that the entities involved are first-order objects. | |
| From: comment on Willard Quine (Existence and Quantification [1966]) by B Hale / C Wright - Logicism in the 21st Century 8 | |
| A reaction: [compressed] The second strategy is from Boolos. This question seems to be right at the heart of the strategy of exploring our ontology through the study of our logic. |
| 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. |
| 16966 | Philosophers tend to distinguish broad 'being' from narrower 'existence' - but I reject that [Quine] |
| Full Idea: It has been fairly common in philosophy early and late to distinguish between being, as the broadest concept, and existence, as narrower. This is no distinction of mine; I mean 'exist' to cover all there is. | |
| From: Willard Quine (Existence and Quantification [1966], p.100) | |
| A reaction: I sort of agree with Quine, but 'being' has a role in philosophy that is not required in science and daily life, as the name of the central problem of ontology, which probably has to be broken down before any progress can happen. |
| 16965 | All we have of general existence is what existential quantifiers express [Quine] |
| Full Idea: Existence is what existential quantification expresses. …It is unreasonable to ask for an explication of (general) existence in simpler terms. …We may still ask what counts as evidence for existential quantifications. | |
| From: Willard Quine (Existence and Quantification [1966], p.97) | |
| A reaction: This has been orthodoxy for the last 60 years, with philosophers talking of 'quantifying over' instead of 'exists'. But are we allowed second-order logic, and plural quantification, and vague domains? |
| 16963 | Existence is implied by the quantifiers, not by the constants [Quine] |
| Full Idea: In the quantification '(∃)(x=a)', it is the existential quantifier, not the 'a' itself, which carries the existential import. | |
| From: Willard Quine (Existence and Quantification [1966], p.94) | |
| A reaction: The Fregean idea seems to be that the criterion of existence is participation in an equality, but here the equality seems not more than assigning a name. Why can't I quantify over 'sakes', in 'for the sake of the children'? |
| 16964 | Theories are committed to objects of which some of its predicates must be true [Quine] |
| Full Idea: Another way of saying what objects a theory requires is to say that they are the objects that some of the predicates of the theory have to be true of, in order for the theory to be true. | |
| From: Willard Quine (Existence and Quantification [1966], p.95) | |
| A reaction: The other was for the objects to be needed by the bound variables of the theory. This is the first-order approach, that predication is a commitment to an object. So what of predicates which have no application? |
| 4216 | Express a theory in first-order predicate logic; its ontology is the types of bound variable needed for truth [Quine, by Lowe] |
| Full Idea: According to Quine, we find the ontological commitments of a theory by expressing it in first-order predicate logic, then determining what kind of entities must be admitted as bound variables if the theory is true. | |
| From: report of Willard Quine (Existence and Quantification [1966]) by E.J. Lowe - A Survey of Metaphysics p.216 | |
| A reaction: To me this is horribly wrong. The ontological commitments of our language is not the same as ontology. What are the ontological commitments of a pocket calculator? |
| 18966 | Ontological commitment of theories only arise if they are classically quantified [Quine] |
| Full Idea: I hold that the question of the ontological commitment of a theory does not properly arise except as that theory is expressed in classical quantificational form. | |
| From: Willard Quine (Existence and Quantification [1966], p.106) | |
| A reaction: He is attacking substitutional quantification for its failure to commit. I smell circularity. If it must be quantified in the first-order classical manner, that restricts your ontology to objects before you've even started. Chicken/egg. |
| 14490 | You can be implicitly committed to something without quantifying over it [Thomasson on Quine] |
| Full Idea: Quine's test for ontological commitment ignores the fact that there are often implicit commitments to certain kinds of entities even where we are not yet quantifying over them. | |
| From: comment on Willard Quine (Existence and Quantification [1966]) by Amie L. Thomasson - Ordinary Objects 09.4 | |
| A reaction: Put this with the obvious problem (of which Quine is aware) that we don't quantify over 'sakes' in 'for the sake of the children', and quantification and commitment have been rather clearly pulled apart. |
| 16961 | In formal terms, a category is the range of some style of variables [Quine] |
| Full Idea: In terms of formalized quantification theory, each category comprises the range of some distinctive style of variable. | |
| From: Willard Quine (Existence and Quantification [1966], p.92) | |
| A reaction: I add this for those who dream of formalising everything, but be warned that even Quine thought it of little help in deciding on the categories. Presumably there would be some variable that ranged across tigers. |
| 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. |