29 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 |
| 458 | Nothing could come out of nothing, and existence could never completely cease [Empedocles] |
| Full Idea: From what in no wise exists, it is impossible for anything to come into being; for Being to perish completely is incapable of fulfilment and unthinkable. | |
| From: Empedocles (fragments/reports [c.453 BCE], B012), quoted by Anon (Lyc) - On Melissus 975b1-4 | |
| A reaction: The first statement of a conservation law. Modern physics is wonderful, but hasn't offered a flicker of an answer to this puzzle. Quantum fluctuations are a mode of Being. |
| 5112 | Empedocles says things are at rest, unless love unites them, or hatred splits them [Empedocles, by Aristotle] |
| Full Idea: Empedocles claims that things are alternately changing and at rest - that they are changing whenever love is creating a unity out of plurality, or hatred is creating plurality out of unity, and they are at rest in the times in between. | |
| From: report of Empedocles (fragments/reports [c.453 BCE]) by Aristotle - Physics 250b26 | |
| A reaction: I suppose one must say that this an example of Ruskin's 'pathetic fallacy' - reading human emotions into the cosmos. Being constructive little creatures, we think goodness leads to construction. I'm afraid Empedocles is just wrong. |
| 13209 | There is no coming-to-be of anything, but only mixing and separating [Empedocles, by Aristotle] |
| Full Idea: Empedocles says there is no coming-to-be of anything, but only a mingling and a divorce of what has been mingled. | |
| From: report of Empedocles (fragments/reports [c.453 BCE]) by Aristotle - Coming-to-be and Passing-away (Gen/Corr) 314b08 | |
| A reaction: Aristotle comments that this prevents Empedocleans from distinguishing between superficial alteration and fundamental change of identity. Presumably, though, that wouldn't bother them. |
| 457 | Substance is not created or destroyed in mortals, but there is only mixing and exchange [Empedocles] |
| Full Idea: There is no creation of substance in any one of mortal existence, nor any end in execrable death, but only mixing and exchange of what has been mixed. | |
| From: Empedocles (fragments/reports [c.453 BCE], B008), quoted by Plutarch - 74: Reply to Colotes 1111f | |
| A reaction: also Aristotle 314b08 |
| 462 | One vision is produced by both eyes [Empedocles] |
| Full Idea: One vision is produced by both eyes | |
| From: Empedocles (fragments/reports [c.453 BCE], B088), quoted by Strabo - works 8.364.3 |
| 22765 | Wisdom and thought are shared by all things [Empedocles] |
| Full Idea: Wisdom and power of thought, know thou, are shared in by all things. | |
| From: Empedocles (fragments/reports [c.453 BCE]), quoted by Sextus Empiricus - Against the Logicians (two books) II.286 | |
| A reaction: Sextus quotes this, saying that it is 'still more paradoxical', and that it explicitly includes plants. This may mean that Empedocles was not including inanimate matter. |
| 1524 | For Empedocles thinking is almost identical to perception [Empedocles, by Theophrastus] |
| Full Idea: Empedocles assumes that thinking is either identical to or very similar to sense-perception. | |
| From: report of Empedocles (fragments/reports [c.453 BCE], A86) by Theophrastus - On the Senses 9 | |
| A reaction: Not to be sniffed at. We can, of course, control our thinking (though we can't control the controller) and we contemplate abstractions, but that might be seen as a sort of perception. Vision is not as visual as we think. |
| 18552 | Forget about beauty; just concentrate on the virtues of delicacy and discernment admired in critics [Hume, by Scruton] |
| Full Idea: Hume suggest we get away from the fruitless discussion of beauty, and simply concentrate on the qualities we admire, and ought to admire, in a critic - qualities such as delicacy and discernment. | |
| From: report of David Hume (Of the standard of taste [1757]) by Roger Scruton - Beauty: a very short introduction 6 | |
| A reaction: We might wonder how you can admire 'discernment' without some view of the thing being discern, which is in danger of being beauty. How do you judge delicacy and discernment without judging the objects of the critic's taste? Mere authority? |
| 6608 | Strong sense, delicate sentiment, practice, comparisons, and lack of prejudice, are all needed for good taste [Hume] |
| Full Idea: Strong sense, united to delicate sentiment, improved by practice, perfected by comparison, and cleared of all prejudice, can alone entitle critics to the valuable character of having 'taste'. | |
| From: David Hume (Of the standard of taste [1757]), quoted by Robert Fogelin - Walking the Tightrope of Reason Ch.6 | |
| A reaction: I agree entirely with this, but then I am a very politically incorrect elitist when it comes to taste. It just seems screamingly obvious that professional wine-tasters have a better appreciation of wine than me, and so on for the rest of the arts. |
| 552 | Empedocles said good and evil were the basic principles [Empedocles, by Aristotle] |
| Full Idea: Empedocles was the first to give evil and good as principles. | |
| From: report of Empedocles (fragments/reports [c.453 BCE]) by Aristotle - Metaphysics 985a | |
| A reaction: Once you start to think that good and evil will only matter if they have causal powers, it is an easy step to the idea of a benevolent god, and a satanic anti-god. Otherwise the 'principles' could be ignored. |
| 589 | 'Nature' is just a word invented by people [Empedocles] |
| Full Idea: Nature is but a word of human framing. | |
| From: Empedocles (fragments/reports [c.453 BCE], B008), quoted by Aristotle - Metaphysics 1015a |
| 21823 | The principle of 'Friendship' in Empedocles is the One, and is bodiless [Empedocles, by Plotinus] |
| Full Idea: In Empedocles we have a dividing principle, 'Strife', set against 'Friendship' - which is the One and is to him bodiless, while the elements represent matter. | |
| From: report of Empedocles (fragments/reports [c.453 BCE]) by Plotinus - The Enneads 5.1.09 | |
| A reaction: The first time I've seen the principle of Love in Empedocles identified with the One of Parmenides. Plotinus is a trustworthy reporter, I think, because he was well read, and had access to lost texts. |
| 6002 | Empedocles says bone is water, fire and earth in ratio 2:4:2 [Empedocles, by Inwood] |
| Full Idea: Empedocles used numerical ratios to explain different kinds of matter; for example, bone is two parts water, four parts fire, two parts earth; and blood is an equal blend of all four elements. | |
| From: report of Empedocles (fragments/reports [c.453 BCE]) by Brad Inwood - Empedocles | |
| A reaction: Why isn't the ration 1:2:1? This presumably shows the influence of Pythagoras (who had also been based in Italy, like Empedocles), as well as that of the earlier naturalistic philosophers. It was a very good theory, though wrong. |
| 13207 | Fire, Water, Air and Earth are elements, being simple as well as homoeomerous [Empedocles, by Aristotle] |
| Full Idea: Empedocles says that Fire, Water, Air and Earth are four elements, and are thus 'simple' rather than flesh, bone and bodies which, like these, are 'homoeomeries'. | |
| From: report of Empedocles (fragments/reports [c.453 BCE]) by Aristotle - Coming-to-be and Passing-away (Gen/Corr) 314a26 | |
| A reaction: The translation is not quite clear. I take it that flesh and bone may look simple, because they are homoeomerous, but they are not really - but what is his evidence for that? Compare Idea 13208. |
| 459 | All change is unity through love or division through hate [Empedocles] |
| Full Idea: These elements never cease their continuous exchange, sometimes uniting under the influence of Love, so that all become One, at other times again moving apart through the hostile force of Hate. | |
| From: Empedocles (fragments/reports [c.453 BCE], B017), quoted by Simplicius - On Aristotle's 'Physics' 158.1- |
| 13218 | The elements combine in coming-to-be, but how do the elements themselves come-to-be? [Aristotle on Empedocles] |
| Full Idea: Empedocles says it is evident that all the other bodies down to the 'elements' have their coming-to-be and their passing-away: but it is not clear how the 'elements' themselves, severally in their aggregated masses, come-to-be and pass-away. | |
| From: comment on Empedocles (fragments/reports [c.453 BCE]) by Aristotle - Coming-to-be and Passing-away (Gen/Corr) 325b20 | |
| A reaction: Presumably the elements are like axioms - and are just given. How do electrons and quarks come-to-be? |
| 13225 | Love and Strife only explain movement if their effects are distinctive [Aristotle on Empedocles] |
| Full Idea: It is not an adequate explanation to say that 'Love and Strife set things moving', unless the very nature of Love is a movement of this kind and the very nature of Strife a movement of that kind. | |
| From: comment on Empedocles (fragments/reports [c.453 BCE]) by Aristotle - Coming-to-be and Passing-away (Gen/Corr) 333b23 | |
| A reaction: I take this to be of interest for showing Aristotle's quest for explanations, and his unwillingness to be fobbed off with anything superficial. I take a task of philosophy to be to push explanations further than others wish to go. |
| 460 | If the one Being ever diminishes it would no longer exist, and what could ever increase it? [Empedocles] |
| Full Idea: Besides these elements, nothing else comes into being, nor does anything cease. For if they had been perishing continuously, they would Be no more; and what could increase the Whole? And whence could it have come? | |
| From: Empedocles (fragments/reports [c.453 BCE], B017), quoted by Simplicius - On Aristotle's 'Physics' 158.1- |
| 2680 | Empedocles said that there are four material elements, and two further creative elements [Empedocles, by Aristotle] |
| Full Idea: Empedocles holds that the corporeal elements are four, but that all the elements, including those which create motion, are six in number. | |
| From: report of Empedocles (fragments/reports [c.453 BCE]) by Aristotle - Coming-to-be and Passing-away (Gen/Corr) 314a16 |
| 5090 | Maybe bodies are designed by accident, and the creatures that don't work are destroyed [Empedocles, by Aristotle] |
| Full Idea: Is it just an accident that teeth and other parts of the body seem to have some purpose, and creatures survive because they happen to be put together in a useful way? Everything else has been destroyed, as Empedocles says of his 'cow with human head'. | |
| From: report of Empedocles (fragments/reports [c.453 BCE], 61) by Aristotle - Physics 198b29 | |
| A reaction: Good grief! Has no one ever noticed that Empedocles proposed the theory of evolution? It isn't quite natural selection, because we aren't told what does the 'destroying', but it is a little flash of genius that was quietly forgotten. |
| 461 | God is a pure, solitary, and eternal sphere [Empedocles] |
| Full Idea: God is equal in all directions to himself and altogether eternal, a rounded Sphere enjoying a circular solitude. | |
| From: Empedocles (fragments/reports [c.453 BCE], B028), quoted by John Stobaeus - Anthology 1.15.2 |
| 466 | God is pure mind permeating the universe [Empedocles] |
| Full Idea: God is mind, holy and ineffable, and only mind, which darts through the whole cosmos with its swift thought. | |
| From: Empedocles (fragments/reports [c.453 BCE], B134), quoted by Ammonius - On 'De Interpretatione' 4.5.249.6 |
| 1719 | In Empedocles' theory God is ignorant because, unlike humans, he doesn't know one of the elements (strife) [Aristotle on Empedocles] |
| Full Idea: It is a consequence of Empedocles' view that God is the most unintelligent thing, for he alone is ignorant of one of the elements, namely strife, whereas mortal creatures are familiar with them all. | |
| From: comment on Empedocles (fragments/reports [c.453 BCE]) by Aristotle - De Anima 410b08 |
| 1522 | It is wretched not to want to think clearly about the gods [Empedocles] |
| Full Idea: Wretched is he who cares not for clear thinking about the gods. | |
| From: Empedocles (fragments/reports [c.453 BCE], B132), quoted by Clement - Miscellanies 5.140.5.1 |