14 ideas
| 3035 | Dialectic involves conversations with short questions and brief answers [Diog. Laertius] |
| Full Idea: Dialectic is when men converse by putting short questions and giving brief answers to those who question them. | |
| From: Diogenes Laertius (Lives of Eminent Philosophers [c.250], 3.1.52) |
| 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 |
| 1816 | Sceptics say demonstration depends on self-demonstrating things, or indemonstrable things [Diog. Laertius] |
| Full Idea: Sceptics say that every demonstration depends on things which demonstrates themselves, or on things which can't be demonstrated. | |
| From: Diogenes Laertius (Lives of Eminent Philosophers [c.250], 9.Py.11) | |
| A reaction: This refers to two parts of Agrippa's Trilemma (the third being that demonstration could go on forever). He makes the first option sound very rationalist, rather than experiential. |
| 1819 | Scepticism has two dogmas: that nothing is definable, and every argument has an opposite argument [Diog. Laertius] |
| Full Idea: Sceptics actually assert two dogmas: that nothing should be defined, and that every argument has an opposite argument. | |
| From: Diogenes Laertius (Lives of Eminent Philosophers [c.250], 9.Py.11) |
| 3064 | When sceptics say that nothing is definable, or all arguments have an opposite, they are being dogmatic [Diog. Laertius] |
| Full Idea: When sceptics say that they define nothing, and that every argument has an opposite argument, they here give a positive definition, and assert a positive dogma. | |
| From: Diogenes Laertius (Lives of Eminent Philosophers [c.250], 9.11.11) |
| 3033 | Induction moves from some truths to similar ones, by contraries or consequents [Diog. Laertius] |
| Full Idea: Induction is an argument which by means of some admitted truths establishes naturally other truths which resemble them; there are two kinds, one proceeding from contraries, the other from consequents. | |
| From: Diogenes Laertius (Lives of Eminent Philosophers [c.250], 3.1.23) |
| 1838 | Cyrenaic pleasure is a motion, but Epicurean pleasure is a condition [Diog. Laertius] |
| Full Idea: Cyrenaics place pleasure wholly in motion, whereas Epicurus admits it as a condition. | |
| From: Diogenes Laertius (Lives of Eminent Philosophers [c.250], 10.28) | |
| A reaction: Not a distinction we meet in modern discussions. Do events within the mind count as 'motion'? If so, these two agree. If not, I'd vote for Epicurus. |
| 1769 | Cynics believe that when a man wishes for nothing he is like the gods [Diog. Laertius] |
| Full Idea: Cynics believe that when a man wishes for nothing he is like the gods. | |
| From: Diogenes Laertius (Lives of Eminent Philosophers [c.250], 6.Men.3) |
| 1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
| Full Idea: Archelaus was the first person to say that the universe is boundless. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.Ar.3 |
| 5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |
| Full Idea: Archelaus wrote that life on Earth began in a primeval slime. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Malcolm Schofield - Archelaus | |
| A reaction: This sounds like a fairly clearcut assertion of the production of life by evolution. Darwin's contribution was to propose the mechanism for achieving it. We should honour the name of Archelaus for this idea. |