18 ideas
| 3338 | Numbers have been defined in terms of 'successors' to the concept of 'zero' [Peano, by Blackburn] |
| Full Idea: Dedekind and Peano define the number series as the series of successors to the number zero, according to five postulates. | |
| From: report of Giuseppe Peano (works [1890]) by Simon Blackburn - Oxford Dictionary of Philosophy p.279 |
| 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 |
| 5897 | 0 is a non-successor number, all successors are numbers, successors can't duplicate, if P(n) and P(n+1) then P(all-n) [Peano, by Flew] |
| Full Idea: 1) 0 is a number; 2) The successor of any number is a number; 3) No two numbers have the same successor; 4) 0 is not the successor of any number; 5) If P is true of 0, and if P is true of any number n and of its successor, P is true of every number. | |
| From: report of Giuseppe Peano (works [1890]) by Antony Flew - Pan Dictionary of Philosophy 'Peano' | |
| A reaction: Devised by Dedekind and proposed by Peano, these postulates were intended to avoid references to intuition in specifying the natural numbers. I wonder if they could define 'successor' without reference to 'number'. |
| 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 |
| 13267 | Temporal parts is a crazy doctrine, because it entails constantly creating stuff ex nihilo [Thomson, by Koslicki] |
| Full Idea: Thomson famously objects that the doctrine of temporal parts is 'a crazy metaphysic - obviously false', since it entails that material objects are constantly being generated ex nihilo (or, at least, the stuff of which they are composed is). | |
| From: report of Judith (Jarvis) Thomson (Parthood and Identity across Time [1983], p.210) by Kathrin Koslicki - The Structure of Objects 2.2 | |
| A reaction: The related objections are to ask what the temporal 'width' of a part is, and whether the joins are visible. |
| 16209 | How can point-duration slices of people have beliefs or desires? [Thomson] |
| Full Idea: Can one really think that point-duration temporal slices of bodies believe things or want things? | |
| From: Judith (Jarvis) Thomson (People and Their Bodies [1997], p.211), quoted by Katherine Hawley - How Things Persist 2.9 n21 | |
| A reaction: There is a problem with a slice doing anything long-term. The bottom line is that things are said to 'endure', but that is precisely what time-slices are unable to do. Hawley rejects this idea. |
| 3061 | Anaxarchus said that he was not even sure that he knew nothing [Anaxarchus, by Diog. Laertius] |
| Full Idea: Anaxarchus said that he was not even sure that he knew nothing. | |
| From: report of Anaxarchus (fragments/reports [c.340 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 09.10.1 |
| 4058 | Is someone's right to life diminished if they were conceived by a rape? [Thomson] |
| Full Idea: Can we say that a person has a right to life only if they didn't come into existence through rape, or that the latter have less right to life? | |
| From: Judith (Jarvis) Thomson (A Defense of Abortion [1971], p.126) | |
| A reaction: This would clearly be an inconsistency for some opponents of abortion who allow rape as an exception. |
| 4060 | The right to life does not bestow the right to use someone else's body to support that life [Thomson] |
| Full Idea: Having a right to life does not guarantee having either a right to be given the use of or a right to be allowed continued use of another person's body. | |
| From: Judith (Jarvis) Thomson (A Defense of Abortion [1971], p.131) | |
| A reaction: A very nice point. You have a right to your life once you are the sole owner of it. |
| 4062 | No one is morally required to make huge sacrifices to keep someone else alive for nine months [Thomson] |
| Full Idea: No one is morally required to make large sacrifices, of health, and other interests and commitments, for nine months, in order to keep another person alive. | |
| From: Judith (Jarvis) Thomson (A Defense of Abortion [1971], p.135) | |
| A reaction: It is a trade-off. It might become a duty if society (or even a husband) urgently needed the baby. |
| 4061 | The right to life is not a right not to be killed, but not to be killed unjustly [Thomson] |
| Full Idea: Maybe the right to life consists not in the right not to be killed, but in the right not to be killed unjustly. | |
| From: Judith (Jarvis) Thomson (A Defense of Abortion [1971], p.131) | |
| A reaction: Sounds tautological. There is no right to life, then, but just the requirement that people behave justly? |
| 4057 | A newly fertilized ovum is no more a person than an acorn is an oak tree [Thomson] |
| Full Idea: A newly fertilized ovum, a newly implanted clump of cells, is no more a person than an acorn is an oak tree. | |
| From: Judith (Jarvis) Thomson (A Defense of Abortion [1971], p.125) | |
| A reaction: This relies heavily on the philosopher's concept of a 'person', but it seems right to me. |
| 4695 | Maybe abortion can be justified despite the foetus having full human rights [Thomson, by Foot] |
| Full Idea: Thomson suggests that abortion can be justified without the need to deny that the foetus has the moral rights of a human person. | |
| From: report of Judith (Jarvis) Thomson (A Defense of Abortion [1971]) by Philippa Foot - Killing and Letting Die p.86 | |
| A reaction: Thomson uses a dubious analogy between pregnancy and being hooked up to someone for life-support. Presumably killing an innocent person is occasionally justifiable, but the situation would normally be more abnormal than pregnancy. |
| 4059 | It can't be murder for a mother to perform an abortion on herself to save her own life [Thomson] |
| Full Idea: It cannot seriously be thought to be murder if a mother performs an abortion on herself to save her own life (if, say, she had a serious heart condition). | |
| From: Judith (Jarvis) Thomson (A Defense of Abortion [1971], p.127) | |
| A reaction: An extreme view might condemn such an action, but it can hardly be based on the 'sanctity of life'. |
| 4696 | The foetus is safe in the womb, so abortion initiates its death, with the mother as the agent. [Foot on Thomson] |
| Full Idea: A fetus is not in jeopardy because it is in the womb, so an abortion originates the fatal sequence, and the mother is the agent. Hence Thomson's argument is invalid, and we must return to question of the moral status of the foetus. | |
| From: comment on Judith (Jarvis) Thomson (A Defense of Abortion [1971]) by Philippa Foot - Killing and Letting Die p.86 | |
| A reaction: The problem would be if a 'person' was safe, but only if I continue some sustained effort which is not required of me by normal duties. |