| 9569 | The origin of geometry started in sensation, then moved to calculation, and then to reason [Proclus] |
| Full Idea: It is unsurprising that geometry was discovered in the necessity of Nile land measurement, since everything in the world of generation goes from imperfection to perfection. They would naturally pass from sense-perception to calculation, and so to reason. | |
| From: Proclus (Commentary on Euclid's 'Elements' [c.452]), quoted by Charles Chihara - A Structural Account of Mathematics 9.12 n55 | |
| A reaction: The last sentence is the core of my view on abstraction, that it proceeds by moving through levels of abstraction, approaching more and more general truths. |
| 10503 | We abstract forms from appearances, and acquire knowledge of immaterial things [Aquinas] |
| Full Idea: To cognize that which is in individual matter, not as it is in such matter, is to abstract the form from the individual matter that the phantasms represents. Thus we come to a degree of cognition of immaterial things. | |
| From: Thomas Aquinas (Summa Theologicae [1265], Q85 1 Reply) | |
| A reaction: This offers abstraction as a kind of inference to best explanation which takes us beyond immediate empirical experience to what is behind it. Aquinas thinks the concepts and explanation are spiritual, but they may be generalities and essences. |
| 10509 | Understanding consists entirely of grasping abstracted species [Aquinas] |
| Full Idea: Of the thing understood all that is within the actually understanding intellect is the abstracted intelligible species. | |
| From: Thomas Aquinas (Summa Theologicae [1265], Q85 Art2) | |
| A reaction: Abstraction is never supposed to be a luxury bolt-on, but is always seen (in this tradition, and presumably in the modern one), as essential to the intellect, and its way of coming to understand the world. Aristotelian definition is behind this idea. |
| 5827 | A species of thing is an abstract idea, and a word is a sign that refers to the idea [Locke] |
| Full Idea: That which general words signify is a sort of things; and does it by being a sign of an abstract idea in the mind; ..so that the essences (or species) of things are nothing else but these abstract ideas. | |
| From: John Locke (Essay Conc Human Understanding (2nd Ed) [1694]), quoted by Stephen P. Schwartz - Intro to Naming,Necessity and Natural Kinds §II | |
| A reaction: This has come in for a lot of criticism, culminating in Putnam saying that meanings 'ain't in the head' (Idea 4099). Wittgenstein's 'beetle in the box' problem is also partly aimed at it (Idea 4147). Locke misses the social aspect of language. |
| 23648 | First we notice and name attributes ('abstracting'); then we notice that subjects share them ('generalising') [Reid] |
| Full Idea: First we resolve or analyse a subject into its known attributes, and give a name to each attribute. Then we observe one or more attributes to be common to many subjects. The first philosophers call 'abstraction', and the second is 'generalising'. | |
| From: Thomas Reid (Essays on Intellectual Powers 5: Abstraction [1785], 3) | |
| A reaction: It is very unfashionable in analytic philosophy to view universals in this way, but it strikes me as obviously correct. There are not weird abstract entities awaiting a priori intuition. There are just features of the world to be observed and picked out. |
| 6925 | The new philosophy thinks of the concrete in a concrete (not a abstract) manner [Feuerbach] |
| Full Idea: The new philosophy is the philosophy that thinks of the concrete not in an abstract, but in a concrete manner. | |
| From: Ludwig Feuerbach (Principles of Philosophy of the Future [1843], §31) | |
| A reaction: This leads to placing a high value on art, and on virtuous action through particulars rather than principles, and on empirical science. The only problem is that what he proposes is impossible. To think 'about' is to abstract from the particulars. |
| 9078 | The study of the nature of Abstract Ideas does not belong to logic, but to a different science [Mill] |
| Full Idea: The metaphysical inquiry into the nature and composition of what have been called Abstract Ideas, or in other words, of the notions which answer in the mind to classes and to general names, belongs not to Logic, but to a different science. | |
| From: John Stuart Mill (System of Logic [1843], 4.2.1) | |
| A reaction: He doesn't name the science, but the point here seems to be precisely what Frege so vigorously disagreed with. I would say that the state of being 'abstract' has logical aspects, and can be partly described by logic, but that Mill is basically right. |
| 9080 | General conceptions are a necessary preliminary to Induction [Mill] |
| Full Idea: Forming general conceptions is a necessary preliminary to Induction. | |
| From: John Stuart Mill (System of Logic [1843], 4.2.1) | |
| A reaction: A key link in the framework of empirical philosophies, which gets us from experience to science. Induction is the very process of generalisation. We can't bring a concept like 'evolution' to preliminary observations, so it must be formulated inductively. |
| 9846 | Defining 'direction' by parallelism doesn't tell you whether direction is a line [Dummett on Frege] |
| Full Idea: The stipulation that the direction of a line a is to be the same as that of a line b just in case a is parallel to b does not determine whether the direction of a line is itself a line or something quite different. | |
| From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §60) by Michael Dummett - Frege philosophy of mathematics Ch.11 | |
| A reaction: Nice point. Maybe not being able to say exactly what something is is either a symptom of nonsense, and simply a symptom that we are dealing with an abstract concept. If abstractions don't exist, they don't need individuation criteria. |
| 9030 | Abstractions can be interpreted dispositionally, as the ability to recognise or imagine an item [Price,HH] |
| Full Idea: An abstract idea may have a dispositional as well as an occurrent interpretation. ..A man who possesses the concept Dog, when he is actually perceiving a dog can recognize that it is one, and can think about dogs when he is not perceiving any dog. | |
| From: H.H. Price (Thinking and Experience [1953], Ch.IX) | |
| A reaction: Ryle had just popularised the 'dispositional' account of mental events. Price is obviously right. The man may also be able to use the word 'dog' in sentences, but presumably dogs recognise dogs, and probably dream about dogs too. |
| 9029 | If ideas have to be images, then abstract ideas become a paradoxical problem [Price,HH] |
| Full Idea: There used to be a 'problem of Abstract Ideas' because it was assumed that an idea ought, somehow, to be a mental image; if some of our ideas appeared not to be images, this was a paradox and some solution must be found. | |
| From: H.H. Price (Thinking and Experience [1953], Ch.VIII) | |
| A reaction: Berkeley in particular seems to be struck by the fact that we are incapable of thinking of a general triangle, simply because there is no image related to it. Most conversations go too fast for images to form even of very visual things. |
| 11102 | Apply '-ness' or 'class of' to abstract general terms, to get second-level abstract singular terms [Quine] |
| Full Idea: Applying the operator '-ness' or 'class of' to abstract general terms, we get second-level abstract singular terms. | |
| From: Willard Quine (Identity, Ostension, and Hypostasis [1950], 5) | |
| A reaction: This is the derivation of abstract concepts by naming classes, rather than by deriving equivalence classes. Any theory which doesn't allow multi-level abstraction is self-evidently hopeless. Quine says Frege and Russell get numbers this way. |
| 17664 | Each subject has an appropriate level of abstraction [Armstrong] |
| Full Idea: To every subject, its appropriate level of abstraction. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 01.2) | |
| A reaction: Mathematics rises through many levels of abstraction. Economics can be very concrete or very abstract. It think it is clearer to talk of being 'general', rather than 'abstract'. |
| 9873 | Abstract terms are acceptable as long as we know how they function linguistically [Dummett] |
| Full Idea: To recognise abstract terms as perfectly proper items of a vocabulary depends upon allowing that all that is necessary for the lawful introduction of a range of expressions into the language is a coherent account of how they are to function in sentences. | |
| From: Michael Dummett (Frege philosophy of mathematics [1991], Ch.16) | |
| A reaction: Why can't the 'coherent account' of the sentences include the fact that there must be something there for the terms to refer to? How else are we to eliminate nonsense words which obey good syntactical rules? Cf. Idea 9872. |
| 10839 | You can't infer a dog's abstract concepts from its behaviour [Dummett] |
| Full Idea: One could train a dog to bark only when a bell rang and a light shone without presupposing that it possessed the concept of conjunction. | |
| From: Michael Dummett (Truth [1959], p.235) |
| 9086 | The idea of abstract objects is not ontological; it comes from the epistemological idea of abstraction [Plantinga] |
| Full Idea: The notion of an abstract object comes from the notion of abstraction; it is in origin an epistemological rather than an ontological category. | |
| From: Alvin Plantinga (Why Propositions cannot be concrete [1993], p.232) | |
| A reaction: Etymology doesn't prove anything. However, if you define abstract objects as not existing in space or time, you must recognise that this may only be because that is how humans imaginatively created them in the first place. |
| 9087 | Theists may see abstract objects as really divine thoughts [Plantinga] |
| Full Idea: Theists may find attractive a view popular among medieval philosophers from Augustine on: that abstract objects are really divine thoughts. More exactly, propositions are divine thoughts, properties divine concepts, and sets divine collections. | |
| From: Alvin Plantinga (Why Propositions cannot be concrete [1993], p.233) | |
| A reaction: Hm. I pass this on because we should be aware that there is a theological history to discussions of abstract objects, and some people have vested interests in keeping them outside of the natural world. Aren't properties natural? Does God gerrymander sets? |
| 8901 | Abstraction is usually explained either by example, or conflation, or abstraction, or negatively [Lewis] |
| Full Idea: Abstraction is usually explained in one of four ways: the Way of Example (cf. donkeys and numbers), the Way of Conflation (same as sets), the Negative Way (non-spatial and non-causal) or the Way of Abstraction (incomplete descriptions). | |
| From: David Lewis (On the Plurality of Worlds [1986], 1.7) | |
| A reaction: [Compressed; a footnote dismisses Dummett's fifth way] Example has blurred boundaries, and explains nothing. Gerrymandered sets don't produce concepts. Negative accounts explain nothing. So it's the Way of Abstraction! |
| 9144 | Fine's 'procedural postulationism' uses creative definitions, but avoids abstract ontology [Fine,K, by Cook/Ebert] |
| Full Idea: Fine says creative definitions can found mathematics. His 'procedural postulationism' says one stipulates not truths, but certain procedures for extending a domain. The procedures can be stated without invoking an abstract ontology. | |
| From: report of Kit Fine (The Limits of Abstraction [2002], 100) by R Cook / P Ebert - Notice of Fine's 'Limits of Abstraction' 4 | |
| A reaction: (For creative definitions, see Idea 9143) This sounds close in spirit to fictionalism, but with the emphasis on the procedure (which can presumably be formalized) rather than a pure act of imaginative creation. |
| 8299 | Abstractions are non-spatial, or dependent, or derived from concepts [Lowe] |
| Full Idea: There are three conceptions of abstractness: 1) non-spatial entities, the opposite of 'concrete' (e.g. numbers and universals); 2) an entity logically incapable of a separate existence (e.g. an apple's colour); 3) Fregean abstractions from concepts. | |
| From: E.J. Lowe (The Possibility of Metaphysics [1998], 10.1) | |
| A reaction: [Lowe p.218 explains the third one] Lowe rejects the third one, and it is a moot point whether the second one could actually be classed as an entity (do they have identity-conditions?), so the big issue is the first one. |
| 9609 | The older sense of 'abstract' is where 'redness' or 'group' is abstracted from particulars [Brown,JR] |
| Full Idea: The older sense of 'abstract' applies to universals, where a universal like 'redness' is abstracted from red particulars; it is the one associated with the many. In mathematics, the notion of 'group' or 'vector space' perhaps fits this pattern. | |
| From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 2) | |
| A reaction: I am currently investigating whether this 'older' concept is in fact dead. It seems to me that it is needed, as part of cognitive science, and as the crucial link between a materialist metaphysic and the world of ideas. |
| 9611 | 'Abstract' nowadays means outside space and time, not concrete, not physical [Brown,JR] |
| Full Idea: The current usage of 'abstract' simply means outside space and time, not concrete, not physical. | |
| From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 2) | |
| A reaction: This is in contrast to Idea 9609 (the older notion of being abstracted). It seems odd that our ancestors had a theory about where such ideas came from, but modern thinkers have no theory at all. Blame Frege for that. |