11 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 |
| 21733 | The right-wing conception of freedom is based on the idea of self-ownership [Cohen,GA] |
| Full Idea: The right-wing conception of freedom is, I think, founded on the idea that each person is the morally rightful owner of himself, even if existing legal systems do not acknowledge it. Let us call that the 'self-ownership' thesis. | |
| From: G.A. Cohen (Are Freedom and Equality Compatible? [1986], 1) | |
| A reaction: He cites Nozick as articulating this view. At the end Cohen rejects self-ownership, though he agrees that no one would accept that the state could be the owner of your eyes. Do I own my hair after it is cut? |
| 21739 | Plenty of people have self-ownership, but still lack autonomy [Cohen,GA] |
| Full Idea: Universal self-ownership fails to ensure autonomy, since it tends to produce proletarians, who lack it. | |
| From: G.A. Cohen (Are Freedom and Equality Compatible? [1986], 3) | |
| A reaction: The implication is that autonomy is not a property of individuals but a social phenomenon. Self-owning people can still be imprisoned. What about autonomy without self-ownership? A bright slave who is given extensive responsibility? |
| 21736 | It is doubtful whether any private property was originally acquired legitimately [Cohen,GA] |
| Full Idea: It is easy to doubt that much actually existing private property was formed in what anyone could think was a legitimating way. | |
| From: G.A. Cohen (Are Freedom and Equality Compatible? [1986], 2) | |
| A reaction: What if I created an artificial island out of unwanted raw materials? What about the first humans to reach some remote territory? |
| 21734 | It is plausible that no one has an initial right to own land and natural resources [Cohen,GA] |
| Full Idea: One may plausibly say of external things in their initial state, of raw land and natural resources, that no person has a greater right to them than any other does. | |
| From: G.A. Cohen (Are Freedom and Equality Compatible? [1986], 1) | |
| A reaction: How about if your group has lived on that plot for fifty generations, and some interlopers arrive and claim part of it. No one thought of 'owning' it till the interlopers arrived. Native Americans and Australians. |
| 21735 | Every thing which is now private started out as unowned [Cohen,GA] |
| Full Idea: In the prehistory of anything that is now private property there was at least one moment at which something privately unowned was taken into private ownership. | |
| From: G.A. Cohen (Are Freedom and Equality Compatible? [1986], 2) | |
| A reaction: He is obviously talking about land and natural resources. Presumably a table which I made and own was always private property, although the land where the trees were grown was not. Though in some communities what I make could be automatically communal. |
| 5994 | Is the cosmos open or closed, mechanical or teleological, alive or inanimate, and created or eternal? [Robinson,TM, by PG] |
| Full Idea: The four major disputes in classical cosmology were whether the cosmos is 'open' or 'closed', whether it is explained mechanistically or teleologically, whether it is alive or mere matter, and whether or not it has a beginning. | |
| From: report of T.M. Robinson (Classical Cosmology (frags) [1997]) by PG - Db (ideas) | |
| A reaction: A nice summary. The standard modern view is closed, mechanistic, inanimate and non-eternal. But philosophers can ask deeper questions than physicists, and I say we are entitled to speculate when the evidence runs out. |