| 5105 | The incommensurability of the diagonal always exists, and so it is not in time [Aristotle] |
| Full Idea: The incommensurability of the diagonal always exists, and so it is not in time. | |
| From: Aristotle (Physics [c.337 BCE], 221b36) | |
| A reaction: This must make Aristotle sympathetic to Platonism in mathematics, even though he rejects the full theory of Forms. Such a view is not uncommon among modern philosophers. Presumably the incommensurability is true in all possible worlds? 'In'? |
| 8910 | General and universal are not real entities, but useful inventions of the mind, concerning words or ideas [Locke] |
| Full Idea: It is plain that general and universal belong not to the real existence of things; but are the inventions and creatures of the understanding, made by it for its own use, and concern only signs, whether words or ideas. | |
| From: John Locke (Essay Conc Human Understanding (2nd Ed) [1694], 3.03.11) | |
| A reaction: Frege and Geach viciously attacked this view, and it seems to be discredited, but I think it is time for a revival, given that the alternative view seems to lead to platonism. I take the first step in mental abstractionism to be pre-verbal. |
| 6717 | Abstract ideas are impossible [Berkeley] |
| Full Idea: We have, I think, shown the impossibility of Abstract Ideas. | |
| From: George Berkeley (The Principles of Human Knowledge [1710], Intro §21) | |
| A reaction: He achieves this by an attack on universals, offering the nominalist view that there are only particulars. There seems to be a middle ground, where universals don't actually exist, but there are settled conventional abstraction, beyond particulars. |
| 7700 | We can't think about the abstract idea of triangles, but only of particular triangles [Hume] |
| Full Idea: Let any man try to conceive a triangle in general, which is neither Isoceles nor Scalenum, nor has any particular length or proportion of sides; and he will perceive the absurdity of all the scholastic notions with regard to abstraction and general ideas. | |
| From: David Hume (Enquiry Conc Human Understanding [1748], XII.II.122) | |
| A reaction: I think there is a basic error in this. I admit that I can only imagine a particular triangle, but it doesn't follow that I am thinking about one triangle. Ontology/epistemology confusion. I picture a shape while believing the shape to be irrelevant. |
| 8911 | If abstracta are non-mental, quarks are abstracta, and yet chess and God's thoughts are mental [Rosen on Frege] |
| Full Idea: Frege's identification of the abstract with the realm of non-mental things entails that unobservables such as quarks are abstract. The abstract nature of chess, and the possibility of abstracta in the mind of God, show they can be mind-dependent. | |
| From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Gideon Rosen - Abstract Objects 'Way of Neg' | |
| A reaction: I like the robust question 'if a is said to 'exist', what is it said to be made of?' I consider the views of Frege to have had too much influence in this area, and recognising the role of the mind (psychology!) in abstraction is a start. |
| 8634 | The equator is imaginary, but not fictitious; thought is needed to recognise it [Frege] |
| Full Idea: We speak of the equator as an imaginary line, but it is not a fictitious line; it is not a creature of thought, the product of a psychological process, but is only recognised or apprehended by thought. | |
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §26) | |
| A reaction: Nice point. The same goes for the apparently very abstract and theoretical concept of a 'circle', because a perfect circle could be imagined in a very specific location, perhaps passing through three specified points. |
| 8960 | Internal questions about abstractions are trivial, and external ones deeply problematic [Carnap, by Szabó] |
| Full Idea: Carnap's verdict is that questions regarding the existence of abstracta tend to be trivial when taken as internal and deeply problematic when taken as external. | |
| From: report of Rudolph Carnap (Empiricism, Semantics and Ontology [1950]) by Zoltán Gendler Szabó - Nominalism 6 | |
| A reaction: If the internal aspect of the problem is 'trivial', this would put Carnap in league with fictionalists, who are only committed to entities while playing the current game. What is the status of the theory? Carnap wanted flowers to bloom. |
| 10136 | Points in Euclidean space are abstract objects, but not introduced by abstraction [Fine,K] |
| Full Idea: Points in abstract Euclidean space are abstract objects, and yet are not objects of abstraction, since they are not introduced through a principle of abstraction of the sort envisaged by Frege. | |
| From: Kit Fine (The Limits of Abstraction [2002], I.1) | |
| A reaction: The point seems to be that they are not abstracted 'from' anything, but are simpy posited as basic constituents. I suggest that points are idealisations (of smallness) rather than abstractions. They are idealised 'from' substances. |
| 10144 | Postulationism says avoid abstract objects by giving procedures that produce truth [Fine,K] |
| Full Idea: A procedural form of postulationism says that instead of stipulating that certain statements are true, one specifies certain procedures for extending the domain to one in which the statement will in fact be true, without invoking an abstract ontology. | |
| From: Kit Fine (The Limits of Abstraction [2002], II.5) | |
| A reaction: The whole of philosophy might go better if it was founded on procedures and processes, rather than on objects. The Hopi Indians were right. |
| 10145 | Abstracts cannot be identified with sets [Fine,K] |
| Full Idea: It is impossible for a proponent of both sets and abstracts to identify the abstracts, in any reasonable manner, with the sets. | |
| From: Kit Fine (The Limits of Abstraction [2002], IV.1) | |
| A reaction: [This observation emerges from a proof Fine has just completed] Cf Idea 10137. The implication is that there is no compromise view available, and one must choose between abstraction or sets as one's account of numbers and groups of concepts. |
| 12212 | Just as we introduced complex numbers, so we introduced sums and temporal parts [Fine,K] |
| Full Idea: Just as one can extend the domain of discourse to include solutions to the equation 'x^2=-1' so one can extend the domain of discourse to include objects that satisfy the condition 'x is the sum of the G's' or 'x is a temporal part of the object b at t'. | |
| From: Kit Fine (The Question of Ontology [2009], p.164) | |
| A reaction: This thought lies behind Fine's 'Proceduralism'. I take it that our collection of abstracta consists entirely of items we have either deliberately or unthinkingly 'introduced' into our discourse when they seemed useful. They then submit to certain laws. |
| 4239 | Nominalists deny abstract objects, because we can have no reason to believe in their existence [Lowe] |
| Full Idea: Nominalists tend to deny the existence of abstract objects since, given their purported nature (non-causal), we can have no reason to believe in their existence. | |
| From: E.J. Lowe (A Survey of Metaphysics [2002], p.372) | |
| A reaction: A good point. Aristotle worried about the causal inadequacy of the Forms. My mind can conceive of a 'thing' with no causal powers, just sitting there. |
| 14592 | Some abstract things have a beginning and end, so may exist in time (though not space) [Swoyer] |
| Full Idea: Many things that seem to be abstract also seem to have a beginning (and ending) in time, such as a language like Urdu. It may be tempting to say that such things exist in time but not in space, but where exactly? | |
| From: Chris Swoyer (Abstract Entities [2008], 1.1) | |
| A reaction: A few distinctions might be needed. Urdu-speaking is an ability of certain people. We abstract from that their 'language'. There is nothing there apart from that ability. It has no more abstract existence than the 'weather'. |