31 ideas
| 13070 | If definitions must be general, and general terms can't individuate, then Socrates can't be defined [Aquinas, by Cover/O'Leary-Hawthorne] |
| Full Idea: Socrates has no definition if definitions by their nature must be in purely general terms, and if no purely general terms can succeed in uniquely singling out this signated matter. | |
| From: report of Thomas Aquinas (De Ente et Essentia (Being and Essence) [1267], 23) by Cover,J/O'Leary-Hawthorne,J - Substance and Individuation in Leibniz 1.1.2 | |
| A reaction: There seem to be two models. That general terms actually individuate the matter of Socrates, or that they cross-reference to (so to speak) define Socrates 'by elimination', as the only individual that fits. But the latter is a poor definition. |
| 11197 | The definitions expressing identity are used to sort things [Aquinas] |
| Full Idea: What sorts things into their proper genus and species are the definitions that express what they are. | |
| From: Thomas Aquinas (De Ente et Essentia (Being and Essence) [1267], p.92) | |
| A reaction: This is straight from Aristotle, though Aristotle's view is a little more complex, I think. If the definitions 'express what they are', then definitions seem to specify the essence. |
| 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. |
| 11195 | If affirmative propositions express being, we affirm about what is absent [Aquinas] |
| Full Idea: If being is what makes propositions true, then anything we can express in an affirmative proposition, however unreal, is said to be; so lacks and absences are, since we say that absences are opposed to presences, and blindness exists in an eye. | |
| From: Thomas Aquinas (De Ente et Essentia (Being and Essence) [1267], p.92) | |
| A reaction: See Idea 11194 for the alternative Aristotelian approach to being, according to categories. Do absences and lacks have real essences, or causal properties? The absence of the sentry may cause the loss of the city. |
| 11201 | Properties have an incomplete essence, with definitions referring to their subject [Aquinas] |
| Full Idea: Incidental properties have an incomplete essence, and need to refer in their definitions to their subject, lying outside their own genus. | |
| From: Thomas Aquinas (De Ente et Essentia (Being and Essence) [1267], p.93) | |
| A reaction: These are 'incidental' properties, but it is a nice question whether properties have essences. Presumably they must have if they are universals, or platonic Forms. The notion of being 'strong' can be defined without specific examples. |
| 11205 | If the form of 'human' contains 'many', Socrates isn't human; if it contains 'one', Socrates is Plato [Aquinas] |
| Full Idea: If (in the Platonic view) manyness was contained in humanness it could never be one as it is in Socrates, and if oneness was part of its definition then Socrates would be Plato and the nature couldn't be realised more than once. | |
| From: Thomas Aquinas (De Ente et Essentia (Being and Essence) [1267], p.100) | |
| A reaction: I suppose the reply is that since we are trying to explain one-over-many, then this unusual combination of both manyness and oneness is precisely what distinguishes forms from other ideas. |
| 13090 | The principle of diversity for corporeal substances is their matter [Aquinas, by Cover/O'Leary-Hawthorne] |
| Full Idea: In the view of Aquinas, while substantial form is the ultimate ground of identity and difference of angels, it is matter that provides a principle of diversity in the case of corporeal substances. | |
| From: report of Thomas Aquinas (De Ente et Essentia (Being and Essence) [1267]) by Cover,J/O'Leary-Hawthorne,J - Substance and Individuation in Leibniz 5.2.3 | |
| A reaction: This is at least as good a proposal as their apatial location. There is more chance of reidentifying matter than of precisely reidentifying a spatial location. Two indistinguishable spheres remain the classic problem case (of Max Black, Idea 10195) |
| 11202 | It is by having essence that things exist [Aquinas] |
| Full Idea: It is by having essence that things exist. | |
| From: Thomas Aquinas (De Ente et Essentia (Being and Essence) [1267], p.94) | |
| A reaction: Compare Idea 11199, which gives a fuller picture. This idea seems to suggest essence as the cause of existence, which sounds wrong. Perhaps essence is a necessary condition of existence, but it is necessary that nothing indeterminate can exist? |
| 11203 | Specific individual essence is defined by material, and generic essence is defined by form [Aquinas] |
| Full Idea: Specific essence differs from generic essence by being demarcated: individuals are demarcated within species by dimensionally defined material, but species within genus by a defining differentiation taken from the form. | |
| From: Thomas Aquinas (De Ente et Essentia (Being and Essence) [1267], p.95) | |
| A reaction: It clearly won't be enough to define an individual just to define its material and its shape. The material might also be essential to the genus, as when defining fire. Probably not very helpful. |
| 11200 | The definition of a physical object must include the material as well as the form [Aquinas] |
| Full Idea: Form alone cannot be a composite substance's essence. For a thing's essence is expressed by its definition, and unless the definition of a physical substance included both form and material, the definition wouldn't differ from mathematical objects. | |
| From: Thomas Aquinas (De Ente et Essentia (Being and Essence) [1267], p.93) | |
| A reaction: This is the sort of thoroughly sensible remark that you only get from the greatest philosophers. Minor philosophers fall in love with things like forms, and then try to use them to explain everything. |
| 11196 | Essence is something in common between the natures which sort things into categories [Aquinas] |
| Full Idea: Since being as belonging to a category expresses the 'isness' of things, and belongs to all ten Aristotelian categories, essence must be something all the natures that sort different beings into genera and species have in common. | |
| From: Thomas Aquinas (De Ente et Essentia (Being and Essence) [1267], p.92) | |
| A reaction: I like this because it is the essence which does the sorting, not the sorting which defines the essence (which seems to me to be a deep and widespread confusion). |
| 11208 | A simple substance is its own essence [Aquinas] |
| Full Idea: A simple substance is its own essence. | |
| From: Thomas Aquinas (De Ente et Essentia (Being and Essence) [1267], p.103) | |
| A reaction: Aquinas takes complex substances to have their essences in various ways, but this thought is the basis of all essence. Presumably the Greek word 'ousia' names the key ingredient. |
| 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. |
| 11198 | Definition of essence makes things understandable [Aquinas] |
| Full Idea: It is definition of essence that makes things understandable. | |
| From: Thomas Aquinas (De Ente et Essentia (Being and Essence) [1267], p.92) | |
| A reaction: The aim of philosophy is understanding, which is achieved by successful explanation. I totally agree with this Aristotelian view, so neatly summarised by Aquinas. |
| 11206 | The mind constructs complete attributions, based on the unified elements of the real world [Aquinas] |
| Full Idea: Attribution is something mind brings to completion by constructing propositional connections and disconnections, basing itself on real-world unity possessed by the things being attributed to one another. | |
| From: Thomas Aquinas (De Ente et Essentia (Being and Essence) [1267], p.102) | |
| A reaction: This compromise story seems to me to be exactly right. I take it that we respond to the real joints of nature, but using thought and language which is riddled with convention. |
| 12165 | Romantics say music expresses ideas, or the Will, or intuitions, or feelings [Scruton] |
| Full Idea: According to the Romantic theory music was an expression of something, of an idea (Hegel), of the Will (Schopenhauer), of 'intuitions' (Croce), or of feelings (Collingwood). | |
| From: Roger Scruton (The Nature of Musical Expression [1981], p.54) | |
| A reaction: Deryck Cooke was the culmination of music as expression of feeling, and Stravinsky was the greatest rebel against the whole idea of expression in music. You can set out to create interesting music which does or does not grab the emotions. |
| 12164 | Expressing melancholy is a good thing, but arousing it is a bad thing [Scruton] |
| Full Idea: To describe a piece of music as expressive of melancholy is to give a reason for listening to it; to describe it as arousing or evoking melancholy is to give a reason for avoiding it. | |
| From: Roger Scruton (The Nature of Musical Expression [1981], p.49) | |
| A reaction: Expressing sexual desire, while avoiding arousing it, is the nice challenge for a particular type of art. Would Scruton say that expressing joy is a good thing, but arousing it is bad? It is a nice observation, though. |
| 11207 | A cause can exist without its effect, but the effect cannot exist without its cause [Aquinas] |
| Full Idea: When things are so related that one causes the other to exist, the cause can exist without what it causes but not vice versa. | |
| From: Thomas Aquinas (De Ente et Essentia (Being and Essence) [1267], p.103) | |
| A reaction: This is open to question, if causes are supposed to be sufficient for effects. Presumably Aquinas would support the view that if the cause had not been, the effect would not have happened. But the current idea indicates the priority relation. |