12 ideas
| 13949 | All models of Peano axioms are isomorphic, so the models all seem equally good for natural numbers [Cartwright,R on Peano] |
| Full Idea: Peano's axioms are categorical (any two models are isomorphic). Some conclude that the concept of natural number is adequately represented by them, but we cannot identify natural numbers with one rather than another of the isomorphic models. | |
| From: comment on Giuseppe Peano (Principles of Arithmetic, by a new method [1889], 11) by Richard Cartwright - Propositions 11 | |
| A reaction: This is a striking anticipation of Benacerraf's famous point about different set theory accounts of numbers, where all models seem to work equally well. Cartwright is saying that others have pointed this out. |
| 18113 | PA concerns any entities which satisfy the axioms [Peano, by Bostock] |
| Full Idea: Peano Arithmetic is about any system of entities that satisfies the Peano axioms. | |
| From: report of Giuseppe Peano (Principles of Arithmetic, by a new method [1889], 6.3) by David Bostock - Philosophy of Mathematics 6.3 | |
| A reaction: This doesn't sound like numbers in the fullest sense, since those should facilitate counting objects. '3' should mean that number of rose petals, and not just a position in a well-ordered series. |
| 17634 | Peano axioms not only support arithmetic, but are also fairly obvious [Peano, by Russell] |
| Full Idea: Peano's premises are recommended not only by the fact that arithmetic follows from them, but also by their inherent obviousness. | |
| From: report of Giuseppe Peano (Principles of Arithmetic, by a new method [1889], p.276) by Bertrand Russell - Regressive Method for Premises in Mathematics p.276 |
| 15653 | We can add Reflexion Principles to Peano Arithmetic, which assert its consistency or soundness [Halbach on Peano] |
| Full Idea: Peano Arithmetic cannot derive its own consistency from within itself. But it can be strengthened by adding this consistency statement or by stronger axioms (particularly ones partially expressing soundness). These are known as Reflexion Principles. | |
| From: comment on Giuseppe Peano (Principles of Arithmetic, by a new method [1889], 1.2) by Volker Halbach - Axiomatic Theories of Truth (2005 ver) 1.2 |
| 17635 | Arithmetic can have even simpler logical premises than the Peano Axioms [Russell on Peano] |
| Full Idea: Peano's premises are not the ultimate logical premises of arithmetic. Simpler premises and simpler primitive ideas are to be had by carrying our analysis on into symbolic logic. | |
| From: comment on Giuseppe Peano (Principles of Arithmetic, by a new method [1889], p.276) by Bertrand Russell - Regressive Method for Premises in Mathematics p.276 |
| 16657 | Substance, Quantity and Quality are real; other categories depend on those three [Henry of Ghent] |
| Full Idea: Among creatures there are only three 'res' belong to the three first categories: Substance, Quantity and Quality. All other are aspects [rationes] and intellectual concepts with respect to them, with reality only as grounded on the res of those three. | |
| From: Henry of Ghent (Quodlibeta [1284], VII:1-2), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 12.3 | |
| A reaction: Pasnau connects with the 'arrangement of being', giving an 'ontologically innocent' structure to reality. That seems to be what we all want, if only we could work out the ontologically guilty bit. |
| 16658 | The only reality in the category of Relation is things from another category [Henry of Ghent] |
| Full Idea: There is beyond a doubt nothing real in the category of Relation, except what is a thing from another category. | |
| From: Henry of Ghent (Quodlibeta [1284], VII:1-2), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 12.3 | |
| A reaction: This seems to have been the fairly orthodox scholastic view of relations. |
| 14979 | Being alone doesn't guarantee intrinsic properties; 'being alone' is itself extrinsic [Lewis, by Sider] |
| Full Idea: The property of 'being alone in the world' is an extrinsic property, even though it has had by an object that is alone in the world. | |
| From: report of David Lewis (Extrinsic Properties [1983]) by Theodore Sider - Writing the Book of the World 01.2 | |
| A reaction: I always choke on my cornflakes whenever anyone cites a true predicate as if it were a genuine property. This is a counterexample to Idea 14978. Sider offers another more elaborate example from Lewis. |
| 15454 | Extrinsic properties come in degrees, with 'brother' less extrinsic than 'sibling' [Lewis] |
| Full Idea: Properties may be more or less intrinsic; being a brother has more of an admixture of intrinsic structure than being a sibling does, yet both are extrinsic. | |
| From: David Lewis (Extrinsic Properties [1983], I) | |
| A reaction: I suppose the point is that a brother is intrinsically male - but then a sibling is intrinsically human. A totally extrinsic relation would be one between entities which shared virtually no categories of existence. |
| 16645 | Accidents are diminished beings, because they are dispositions of substance (unqualified being) [Henry of Ghent] |
| Full Idea: Accidents are beings only in a qualified and diminished sense, because they are not called beings, nor are they beings, except because they are dispositions of an unqualified being, a substance. | |
| From: Henry of Ghent (Quodlibeta [1284], XV.5), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 10.4 | |
| A reaction: This is aimed to 'half' detach the accidents (as the Eucharist requires). Later scholastics detached them completely. Late scholastics seem to have drifted back to Henry's view. The equivocal use of 'being' here was challenged later. |
| 15455 | Total intrinsic properties give us what a thing is [Lewis] |
| Full Idea: The way something is is given by the totality of its intrinsic properties. | |
| From: David Lewis (Extrinsic Properties [1983], I) | |
| A reaction: No. Some properties are intrinsic but trivial. The 'important' ones fix the identity (if the identity is indeed 'fixed'). |
| 22012 | Kant says things-in-themselves cause sensations, but then makes causation transcendental! [Henry of Ghent, by Pinkard] |
| Full Idea: Kant claimed that things-in-themselves caused our sensations; but causality was a transcendental condition of experience, not a property of things-in-themselves, so the great Kant had contradicted himself. | |
| From: report of Henry of Ghent (Quodlibeta [1284], Supplement) by Terry Pinkard - German Philosophy 1760-1860 04 | |
| A reaction: This early objection by the conservative Jacobi (who disliked Enlightenment rational religion) is the key to the dispute over whether Kant is an idealist. Kant denied being an idealist, but how can he be, if this idea is correct? |