| 9789 | You can't abstract natural properties to make Forms - objects and attributes are defined together [Aristotle] |
| Full Idea: Those who say there are Forms abstract natural properties, even though they are less separable than mathematical properties. This is clear if you try to define both the objects themselves and their attributes. | |
| From: Aristotle (Physics [c.337 BCE], 193b36) | |
| A reaction: (Compare Idea 9788) This is Frege's black and white cats, where you cannot abstract the black without thinking of the cat, but Aristotle thinks mathematical abstraction is more feasible. |
| 9070 | We learn primitives and universals by induction from perceptions [Aristotle] |
| Full Idea: We must get to know the primitives by induction; for this is the way in which perception instils universals. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 100b04) | |
| A reaction: This statement is so strongly empirical it could have come from John Stuart Mill. The modern post-Fregean view of universals is essentially platonist - that they have a life and logic of their own, and their method of acquisition is irrelevant. |
| 10506 | Mathematics can be abstracted from sensible matter, and from individual intelligible matter [Aquinas] |
| Full Idea: Intellect can abstract mathematical species from sensible matter, both individual and common. Yet it cannot abstract such species from common intelligible matter, but only from individual intelligible matter. | |
| From: Thomas Aquinas (Summa Theologicae [1265], Q85 Ad2) | |
| A reaction: The idea is that common intelligible matter lacks underlying substance, which is where quantity is to be found. |
| 9104 | A universal is the result of abstraction, which is only a kind of mental picturing [William of Ockham] |
| Full Idea: A universal is not the result of generation, but of abstraction, which is only a kind of mental picturing. | |
| From: William of Ockham (Ordinatio [1320], DII Qviii prima redactio) | |
| A reaction: The phrase 'mental picturing' works very plausibly for the universal 'giraffe', but not so well for 'multiplication' or 'contradiction'. Though we might broaden 'picturing' to being a much less visual concept. Mapping seems basic. |
| 23640 | Only mature minds can distinguish the qualities of a body [Reid] |
| Full Idea: I think it requires some ripeness of understanding to distinguish the qualities of a body from the body; perhaps this distinction is not made by brutes, or by infants. | |
| From: Thomas Reid (Essays on Intellectual Powers 2: Senses [1785], 19) | |
| A reaction: I'm glad the brutes get a mention in his assessment of these questions. I take such thinking to arise from what can be labelled the faculty of abstraction, which presumably only appears in a mature brain. It is second-level thinking. |
| 23653 | If you can't distinguish the features of a complex object, your notion of it would be a muddle [Reid] |
| Full Idea: If you perceive an object, white, round, and a foot in diameter, if you had not been able to distinguish the colour from the figure, and both from the magnitude, your senses would only give you one complex and confused notion of all these mingled together | |
| From: Thomas Reid (Essays on Intellectual Powers 6: Judgement [1785], 1) | |
| A reaction: His point is that if you reject the 'abstraction' of these qualities, you still cannot deny that distinguishing them is an essential aspect of perceiving complex things. Does this mean that animals distinguish such things? |
| 19577 | Everything is a chaotic unity, then we abstract, then we reunify the world into a free alliance [Novalis] |
| Full Idea: Before abstraction everything is one - but one as chaos is - after abstraction everything is again unified - but in a free alliance of independent, self-determined beings. A crowd has become a society - a chaos is transformed into a manifold world. | |
| From: Novalis (Miscellaneous Observations [1798], 094) | |
| A reaction: Personally I take (unfashionably) psychological abstraction to one of the key foundations of human thought, so I love this idea, which gives a huge picture of how the abstracting mind relates to reality. |
| 9145 | We form the image of a cardinal number by a double abstraction, from the elements and from their order [Cantor] |
| Full Idea: We call 'cardinal number' the general concept which, by means of our active faculty of thought, arises when we make abstraction from an aggregate of its various elements, and of their order. From this double abstraction the number is an image in our mind. | |
| From: George Cantor (Beitrage [1915], §1), quoted by Kit Fine - Cantorian Abstraction: Recon. and Defence Intro | |
| A reaction: [compressed] This is the great Cantor, creator of set theory, endorsing the traditional abstractionism which Frege and his followers so despise. Fine offers a defence of it. The Frege view is platonist, because it refuses to connect numbers to the world. |
| 13454 | Cantor says (vaguely) that we abstract numbers from equal sized sets [Hart,WD on Cantor] |
| Full Idea: Cantor thought that we abstract a number as something common to all and only those sets any one of which has as many members as any other. ...However one wants to see the logic of the inference. The irony is that set theory lays out this logic. | |
| From: comment on George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: The logic Hart has in mind is the notion of an equivalence relation between sets. This idea sums up the older and more modern concepts of abstraction, the first as psychological, the second as logical (or trying very hard to be!). Cf Idea 9145. |
| 9976 | Frege accepts abstraction to the concept of all sets equipollent to a given one [Tait on Frege] |
| Full Idea: Frege's own conception of abstraction (although he disapproves of the term) is in agreement with the view that abstracting from the particular nature of the elements of M would yield the concept under which fall all sets equipollent to M. | |
| From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by William W. Tait - Frege versus Cantor and Dedekind III | |
| A reaction: Nice! This shows how difficult it is to slough off the concept of abstractionism and live with purely logical concepts of it. If we 'construct' a set, then there is a process of creation to be explained; we can't just think of platonic givens. |
| 9361 | We have to separate the mathematical from physical phenomena by abstraction [Lewis,CI] |
| Full Idea: Physical processes present us with phenomena in which the purely mathematical has to be separated out by abstraction. | |
| From: C.I. Lewis (A Pragmatic Conception of the A Priori [1923], p.367) | |
| A reaction: This is the father of modal logic endorsing traditional abstractionism, it seems. He is also, though, endorsing the view that a priori knowledge is created by us, with pragmatic ends in view. |
| 10644 | A 'felt familiarity' with universals is more primitive than abstraction [Price,HH] |
| Full Idea: A 'felt familiarity' with universals seems to be more primitive than explicit abstraction. | |
| From: H.H. Price (Review of Aron 'Our Knowledge of Universals' [1946], p.188) | |
| A reaction: This I take to be part of the 'given' of the abstractionist view, which is quite well described in the first instance by Aristotle. Price says that it is 'pre-verbal'. |
| 10646 | Our understanding of 'dog' or 'house' arises from a repeated experience of concomitances [Price,HH] |
| Full Idea: Whether you call it inductive or not, our understanding of such a word as 'dog' or 'house' does arise from a repeated experience of concomitances. | |
| From: H.H. Price (Review of Aron 'Our Knowledge of Universals' [1946], p.191) | |
| A reaction: Philosophers don't use phrases like that last one any more. How else could we form the concept of 'dog' - if we are actually allowed to discuss the question of concept-formation, instead of just the logic of concepts. |
| 9031 | The basic concepts of conceptual cognition are acquired by direct abstraction from instances [Price,HH] |
| Full Idea: Basic concepts are acquired by direct abstraction from instances; unless there were some concepts acquired in this way by direct abstraction, there would be no conceptual cognition at all. | |
| From: H.H. Price (Thinking and Experience [1953], Ch.II) | |
| A reaction: This seems to me to be correct. A key point is that not only will I acquire the concept of 'dog' in this direct way, from instances, but also the concept of 'my dog Spot' - that is I can acquire the abstract concept of an instance from an instance. |
| 10557 | Abstract objects are captured by second-order modal logic, plus 'encoding' formulas [Zalta] |
| Full Idea: My object theory is formulated in a 'syntactically second-order' modal predicate calculus modified only so as to admit a second kind of atomic formula ('xF'), which asserts that object x 'encodes' property F. | |
| From: Edward N. Zalta (Deriving Kripkean Claims with Abstract Objects [2006], p.2) | |
| A reaction: This is summarising Zalta's 1983 theory of abstract objects. See Idea 10558 for Zalta's idea in plain English. |
| 12657 | Abstractionism claims that instances provide criteria for what is shared [Fodor] |
| Full Idea: In the idea of learning concepts by 'abstraction', experiences of the instances provide evidence about which of the shared properties of things in a concept's extension are 'criterial' for being in the concept's extension. | |
| From: Jerry A. Fodor (LOT 2 [2008], Ch.5.2 n6) | |
| A reaction: Fodor is fairly sceptical of this approach, and his doubts are seen in the scare-quotes around 'criterial'. He is defending the idea that only a certain degree of innateness in the concepts can get such a procedure off the ground. |
| 9149 | To obtain the number 2 by abstraction, we only want to abstract the distinctness of a pair of objects [Fine,K] |
| Full Idea: In abstracting from the elements of a doubleton to obtain 2, we do not wish to abstract away from all features of the objects. We wish to take account of the fact that the two objects are distinct; this alone should be preserved under abstraction. | |
| From: Kit Fine (Cantorian Abstraction: Recon. and Defence [1998], §3) | |
| A reaction: This is Fine's strategy for meeting Frege's objection to abstraction, summarised in Idea 9146. It seems to use the common sense idea that abstraction is not all-or-nothing. Abstraction has degrees (and levels). |
| 9150 | We should define abstraction in general, with number abstraction taken as a special case [Fine,K] |
| Full Idea: Number abstraction can be taken to be a special case of abstraction in general, which can then be defined without recourse to the concept of number. | |
| From: Kit Fine (Cantorian Abstraction: Recon. and Defence [1998], §3) | |
| A reaction: At last, a mathematical logician recognising that they don't have a monopoly on abstraction. It is perfectly obvious that abstractions of simple daily concepts must be chronologically and logically prior to number abstraction. Number of what? |
| 9981 | Abstraction is 'logical' if the sense and truth of the abstraction depend on the concrete [Tait] |
| Full Idea: If the sense of a proposition about the abstract domain is given in terms of the corresponding proposition about the (relatively) concrete domain, ..and the truth of the former is founded upon the truth of the latter, then this is 'logical abstraction'. | |
| From: William W. Tait (Frege versus Cantor and Dedekind [1996], V) | |
| A reaction: The 'relatively' in parentheses allows us to apply his idea to levels of abstraction, and not just to the simple jump up from the concrete. I think Tait's proposal is excellent, rather than purloining 'abstraction' for an internal concept within logic. |
| 9982 | Cantor and Dedekind use abstraction to fix grammar and objects, not to carry out proofs [Tait] |
| Full Idea: Although (in Cantor and Dedekind) abstraction does not (as has often been observed) play any role in their proofs, but it does play a role, in that it fixes the grammar, the domain of meaningful propositions, and so determining the objects in the proofs. | |
| From: William W. Tait (Frege versus Cantor and Dedekind [1996], V) | |
| A reaction: [compressed] This is part of a defence of abstractionism in Cantor and Dedekind (see K.Fine also on the subject). To know the members of a set, or size of a domain, you need to know the process or function which created the set. |
| 10141 | Many different kinds of mathematical objects can be regarded as forms of abstraction [Fine,K] |
| Full Idea: Many different kinds of mathematical objects (natural numbers, the reals, points, lines, figures, groups) can be regarded as forms of abstraction, with special theories having their basis in a general theory of abstraction. | |
| From: Kit Fine (The Limits of Abstraction [2002], I.4) | |
| A reaction: This result, if persuasive, would be just the sort of unified account which the whole problem of abstact ideas requires. |
| 10229 | Simple types can be apprehended through their tokens, via abstraction [Shapiro] |
| Full Idea: Some realists argue that simple types can be apprehended through their tokens, via abstraction. | |
| From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.2) | |
| A reaction: One might rephrase that to say that types are created by abstraction from tokens (and then preserved in language). |
| 9919 | The old debate classified representations as abstract, not entities [Burgess/Rosen] |
| Full Idea: The original debate was over abstract ideas; thus it was mental (or linguistic) representations that were classified as abstract or otherwise, and not the entities represented. | |
| From: JP Burgess / G Rosen (A Subject with No Object [1997], I.A.1.b) | |
| A reaction: This seems to beg the question of whether there are any such entities. It is equally plausible to talk of the entities that are 'constructed', rather than 'represented'. |
| 8917 | The Way of Abstraction used to say an abstraction is an idea that was formed by abstracting [Rosen] |
| Full Idea: The simplest version of the Way of Abstraction would be to say that an object is abstract if it is a referent of an idea that was formed by abstraction, but this is wedded to an outmoded philosophy of mind. | |
| From: Gideon Rosen (Abstract Objects [2001], 'Way of Abs') | |
| A reaction: This presumably refers to Locke, who wields the highly ambiguous term 'idea'. But if we sort out that ambiguity (by using modern talk of mental events, concepts and content?) we might reclaim the view. But do we have a 'genetic fallacy' here? |