25 ideas
| 6299 | Axioms are often affirmed simply because they produce results which have been accepted [Resnik] |
| Full Idea: Many axioms have been proposed, not on the grounds that they can be directly known, but rather because they produce a desired body of previously recognised results. | |
| From: Michael D. Resnik (Maths as a Science of Patterns [1997], One.5.1) | |
| A reaction: This is the perennial problem with axioms - whether we start from them, or whether we deduce them after the event. There is nothing wrong with that, just as we might infer the existence of quarks because of their results. |
| 6304 | Mathematical realism says that maths exists, is largely true, and is independent of proofs [Resnik] |
| Full Idea: Mathematical realism is the doctrine that mathematical objects exist, that much contemporary mathematics is true, and that the existence and truth in question is independent of our constructions, beliefs and proofs. | |
| From: Michael D. Resnik (Maths as a Science of Patterns [1997], Three.12.9) | |
| A reaction: As thus defined, I would call myself a mathematical realist, but everyone must hesitate a little at the word 'exist' and ask, how does it exist? What is it 'made of'? To say that it exists in the way that patterns exist strikes me as very helpful. |
| 6300 | Mathematical constants and quantifiers only exist as locations within structures or patterns [Resnik] |
| Full Idea: In maths the primary subject-matter is not mathematical objects but structures in which they are arranged; our constants and quantifiers denote atoms, structureless points, or positions in structures; they have no identity outside a structure or pattern. | |
| From: Michael D. Resnik (Maths as a Science of Patterns [1997], Three.10.1) | |
| A reaction: This seems to me a very promising idea for the understanding of mathematics. All mathematicians acknowledge that the recognition of patterns is basic to the subject. Even animals recognise patterns. It is natural to invent a language of patterns. |
| 6303 | Sets are positions in patterns [Resnik] |
| Full Idea: On my view, sets are positions in certain patterns. | |
| From: Michael D. Resnik (Maths as a Science of Patterns [1997], Three.10.5) | |
| A reaction: I have always found the ontology of a 'set' puzzling, because they seem to depend on prior reasons why something is a member of a given set, which cannot always be random. It is hard to explain sets without mentioning properties. |
| 6302 | Structuralism must explain why a triangle is a whole, and not a random set of points [Resnik] |
| Full Idea: An objection is that structuralism fails to explain why certain mathematical patterns are unified wholes while others are not; for instance, some think that an ontological account of mathematics must explain why a triangle is not a 'random' set of points. | |
| From: Michael D. Resnik (Maths as a Science of Patterns [1997], Three.10.4) | |
| A reaction: This is an indication that we are not just saying that we recognise patterns in nature, but that we also 'see' various underlying characteristics of the patterns. The obvious suggestion is that we see meta-patterns. |
| 6295 | There are too many mathematical objects for them all to be mental or physical [Resnik] |
| Full Idea: If we take mathematics at its word, there are too many mathematical objects for it to be plausible that they are all mental or physical objects. | |
| From: Michael D. Resnik (Maths as a Science of Patterns [1997], One.1) | |
| A reaction: No one, of course, has ever claimed that they are, but this is a good starting point for assessing the ontology of mathematics. We are going to need 'rules', which can deduce the multitudinous mathematical objects from a small ontology. |
| 6296 | Maths is pattern recognition and representation, and its truth and proofs are based on these [Resnik] |
| Full Idea: I argue that mathematical knowledge has its roots in pattern recognition and representation, and that manipulating representations of patterns provides the connection between the mathematical proof and mathematical truth. | |
| From: Michael D. Resnik (Maths as a Science of Patterns [1997], One.1) | |
| A reaction: The suggestion that patterns are at the basis of the ontology of mathematics is the most illuminating thought I have encountered in the area. It immediately opens up the possibility of maths being an entirely empirical subject. |
| 6301 | Congruence is the strongest relationship of patterns, equivalence comes next, and mutual occurrence is the weakest [Resnik] |
| Full Idea: Of the equivalence relationships which occur between patterns, congruence is the strongest, equivalence the next, and mutual occurrence the weakest. None of these is identity, which would require the same position. | |
| From: Michael D. Resnik (Maths as a Science of Patterns [1997], Three.10.3) | |
| A reaction: This gives some indication of how an account of mathematics as a science of patterns might be built up. Presumably the recognition of these 'degrees of strength' cannot be straightforward observation, but will need an a priori component? |
| 1212 | Replacing timbers on Theseus' ship was the classic illustration of the problem of growth and change [Plutarch] |
| Full Idea: At intervals they removed old timbers from the preserved ship and replaced them with sound ones, so the ship became a classic illustration for the philosophers of the disputed question of growth and change, some saying it was the same, others different. | |
| From: Plutarch (Life of Theseus [c.85], 23) |
| 5958 | The sun is always bright; it doesn't become bright when it emerges [Plutarch] |
| Full Idea: The sun doesn't become bright when it emerges from the clouds; it always is bright. | |
| From: Plutarch (26: Oracles in Decline [c.85], §39) | |
| A reaction: Not an argument, but a nice appeal to common sense, like Russell's example of the cat that disappears behind the furniture and then reappears. To disagree with Plutarch here strikes me as the road to philosophical absurdity. |
| 22110 | Demonstration provides depth of understanding and explanation (rather than foundations) [Kretzmann/Stump] |
| Full Idea: According to Aquinas, what demonstration provides is not so much knowledge as conceived by foundationalists as depth of understanding and explanatory insight. | |
| From: Kretzmann/Stump (Aquinas, Thomas [2005]), quoted by Kretzmann/Stump - Aquinas, Thomas 11 | |
| A reaction: It was noticeable that Aristotle didn't make clear what demonstration aims to achieve, and he didn't employ it elsewhere in his writings. We aim for understanding, not for well grounded propositions. Understanding needs implications and mechanisms. |
| 5959 | Some philosophers say the soul is light [Plutarch] |
| Full Idea: Some philosophers hold that the soul is in its essence light. | |
| From: Plutarch (75: Is 'Live Unknown' a Wise Precept? [c.85], §6) | |
| A reaction: A nice idea, to rival the stoic view that the soul is fire. It is understandable to propose that the soul is some sort of lightweight and fast moving matter. How else could thought be achieved physically? Nowadays, parallel processing is our only model. |
| 5960 | When the soul is intelligent and harmonious, it is part of god and derives from god [Plutarch] |
| Full Idea: The soul, when it has partaken of intelligence and reason and concord, is not merely a work but also a part of god and has come to be not by his agency but both from him as source and out of his substance. | |
| From: Plutarch (67: Platonic Questions [c.85], II.1001) | |
| A reaction: A most intriguing shift of view from earlier concepts of the psuché. How did this come about? This man is a pagan. The history is in the evolution of Platonism. See 'The Middle Platonists' by John Dillon. Davidson is also very impressed by reason. |
| 5952 | Rather than being the whole soul, maybe I am its chief part? [Plutarch] |
| Full Idea: Is each of us not the soul, but the chief part of the soul, by which we think and reason and act, all the other parts of soul as well as of body being mere instruments of its power? | |
| From: Plutarch (74: Reply to Colotes [c.85], §1119) | |
| A reaction: Socrates is associated with the idea that I am my whole soul (Idea 1650). Plutarch represents an interesting development, which may lead both to the Christian 'soul' and to the Cartesian 'ego'. I think Plutarch is right, but what is the 'soul'? |
| 5951 | If atoms have no qualities, they cannot possibly produce a mind [Plutarch] |
| Full Idea: Perception, mind, intelligence and thought cannot so much as be conceived, even with the best will, as arising among void and atoms, things which taken separately have no quality. | |
| From: Plutarch (74: Reply to Colotes [c.85], §1112) | |
| A reaction: A nice articulation of the intuition of all anti-physicalists. Plutarch would have to rethink his position carefully if he learned of the sheer number of connections in the brain, and of the theory of natural selection. His challenge remains, though. |
| 5963 | Some say emotion is a sort of reason, and others say virtue concerns emotion [Plutarch] |
| Full Idea: Some philosophers make the emotions varieties of reason, on the ground that all desire and grief and anger are judgments, while others declare that the virtues have to do with emotions, as when fear is the province of courage. | |
| From: Plutarch (68: Generation of the soul in 'Timaeus' [c.85], 1025d) | |
| A reaction: The second idea comes from Aristotle, but the second is interesting, and corresponds to the views coming from modern neuroscience, where even the most basic thought seems to involve emotion. What could be the motivation for 'pure' reason? |
| 20796 | Action needs an affinity for a presentation, and an impulse toward the affinity [Plutarch] |
| Full Idea: The sceptics say there are three movements of the soul: presentation, impulse and assent. …And action requires two things: a presentation of something to which one has an affinity, and an impulse toward what is presented as an object of affinity. | |
| From: Plutarch (74: Reply to Colotes [c.85], 1122c) | |
| A reaction: Not much reasoning involved in this account, which the sceptics say is compatible with suspension of judgement. |
| 1477 | Being manly and brave is the result of convention, not of human nature [Plutarch] |
| Full Idea: Manliness is not a natural human attribute, otherwise women would be just as brave. It is due to pressure from laws, and this pressure has no free will, but is a slave of convention and criticism. | |
| From: Plutarch (64: Gryllus - on Rationality in Animals [c.85], 988c) | |
| A reaction: This is the first glimmerings of seeing gender as a cultural creation, rather than as a fact. Presumably he takes the same view of some of the supposed feminine virtues. |
| 1478 | Animals don't value pleasure, as they cease sexual intercourse after impregnation [Plutarch] |
| Full Idea: Animals of both sexes cease to have intercourse after impregnation; that shows how little animals value pleasure, and that nature is all that counts. | |
| From: Plutarch (64: Gryllus - on Rationality in Animals [c.85], 990d) | |
| A reaction: A famous monkey had an implant to stimulate pleasure, and a button to trigger it. It apparently would have starved to death rather than release the button. Animal sex is dull? |
| 5948 | The good life involves social participation, loyalty, temperance and honesty [Plutarch] |
| Full Idea: To live the good life is to live a life of participation in society, of loyalty to friends, of temperance and honest dealing. | |
| From: Plutarch (74: Reply to Colotes [c.85], §1108) | |
| A reaction: 'Participation in society' is the interesting one. This might translate as 'doing your duty', or as 'leading a well-rounded life'. Solitude is wrong if you are indebted to others, and it is unhealthy if you are not. Is solitude really immoral, though? |
| 1479 | Animals have not been led into homosexuality, because they value pleasure very little [Plutarch] |
| Full Idea: Because animals value pleasure very little, they have not been led into sex between males or between females. | |
| From: Plutarch (64: Gryllus - on Rationality in Animals [c.85], 990d) |
| 5950 | If only atoms exist, how do qualities arise when the atoms come together? [Plutarch] |
| Full Idea: If you accept atomism, you must show how bodies without quality have given rise to qualities of every kind by the mere fact of coming together. For example, how has the quality called 'hot' been imposed on the atoms? | |
| From: Plutarch (74: Reply to Colotes [c.85], §1111) | |
| A reaction: This argument is still significant in current philosophy of mind. If temperature is 'mean kinetic energy', you are left wondering where the energy came from, and why minds experience the heat. This is the 'Hard Question'. |
| 5974 | People report seeing through rocks, or over the horizon, or impossibly small works [Plutarch] |
| Full Idea: It is said that Lynceus could see through rock and tree, and a lookout in Sicily saw Carthaginian ships a day and a half away, and Callicrates and Myrmecides are said to fashion carriages canopied with the wings of fly, and write on sesame seeds. | |
| From: Plutarch (72: Against Stoics on common Conceptions [c.85], 1083e) | |
| A reaction: A warning from Plutarch against believing everything you hear! |
| 5957 | Absurd superstitions make people atheist, not disharmony in nature [Plutarch] |
| Full Idea: Men have never thought the universe godless on the ground of detecting some fault in stars or seasons; ..it is the ridiculous things that superstition does that makes people say it would be better if there were no gods at all. | |
| From: Plutarch (14: Superstition [c.85], §12) | |
| A reaction: Not true, I would say. Absurd superstitions do discredit belief in the supernatural, but earthquakes are a disharmony in nature, and a nasty one at that. Nowadays we have other explanations to rival those of religion. |
| 5955 | No one will ever find a city that lacks religious practices [Plutarch] |
| Full Idea: A city without holy places and gods, without any observance of prayers, oaths, oracles, sacrifices for blessings received or rites to avert evils, no traveller has ever seen or will ever see. | |
| From: Plutarch (74: Reply to Colotes [c.85], §1125) | |
| A reaction: The nearest you might get would be Soviet Moscow, but in 1973 I saw a man there jeering at a woman who was kneeling in the street outside a closed church. Plutarch would be stunned at the decline in religious practices in modern Europe. |