9 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 |
| 7876 | Even if we identify pain with neural events, we can't explain why those neurons cause that feeling [Levine, by Papineau] |
| Full Idea: Materialists identify pain with the firing of nociceptive-specific neurons in the parietal cortex. Even so, Levine argues, we will still lack any explanation of why nociceptive-specific neurons yield pain. | |
| From: report of Joseph Levine (Purple Haze [2001]) by David Papineau - Thinking about Consciousness 5.1 | |
| A reaction: [Proposed by Levine in 1983] I don't think we need to instantly go dualist when faced with this, but we may all eventually have to concede a bit of mysterianism. The explanation may be holistic (and hence hopelessly complex). |
| 7877 | Only phenomenal states have an explanatory gap; water is fully explained by H2O [Levine, by Papineau] |
| Full Idea: Levine says the explanatory gap is peculiar to phenomenal states. Once water has been identified with H2O, or temperature with mean kinetic energy, we do not continue to ask why H2O yields water, or why mean kinetic energy yields temperature. | |
| From: report of Joseph Levine (Purple Haze [2001]) by David Papineau - Thinking about Consciousness 5.1 | |
| A reaction: Everything is mysterious if you think about if for long enough. What about a representational gap? Why do those neurons represent that tree (if the neurons aren't tree-shaped)? To understand qualia, we must understand the whole brain, I suspect. |
| 7878 | Materialism won't explain phenomenal properties, because the latter aren't seen in causal roles [Papineau on Levine] |
| Full Idea: We cannot give materialist explanations of why brain yields phenomenal properties because phenomenal concepts are not associated with descriptions of causal roles in the same way as pre-theoretical terms in other areas of science. | |
| From: comment on Joseph Levine (Purple Haze [2001]) by David Papineau - Thinking about Consciousness 5.1 | |
| A reaction: I think Papineau has part of the answer, and I certainly like his notion of Conceptual Dualism, but if qualia are physical, there must be a physical account of how they acquire their properties. I think the whole brain needs to be understood first. |
| 7903 | The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna] |
| Full Idea: The six perfections are of giving, morality, patience, vigour, meditation, and wisdom. | |
| From: Nagarjuna (Mahaprajnaparamitashastra [c.120], 88) | |
| A reaction: What is 'morality', if giving is not part of it? I like patience and vigour being two of the virtues, which immediately implies an Aristotelian mean (which is always what is 'appropriate'). |