47 ideas
| 326 | For relaxation one can consider the world of change, instead of eternal things [Plato] |
| Full Idea: If, for relaxation, one gives up discussing eternal things, it is pleasant to consider likely accounts of the world of change. | |
| From: Plato (Timaeus [c.349 BCE], 59c) | |
| A reaction: To understand this, examine Plato's example of the Line at 'Republic' 509d. |
| 315 | Philosophy is the supreme gift of the gods to mortals [Plato] |
| Full Idea: Philosophy is the greatest gift the gods have ever given or ever will give to mortals. | |
| From: Plato (Timaeus [c.349 BCE], 47b) | |
| A reaction: I wonder why they gave it to us? |
| 306 | Nothing can come to be without a cause [Plato] |
| Full Idea: Nothing can come to be without a cause. | |
| From: Plato (Timaeus [c.349 BCE], 28a) |
| 19463 | Induction assumes some uniformity in nature, or that in some respects the future is like the past [Ayer] |
| Full Idea: In all inductive reasoning we make the assumption that there is a measure of uniformity in nature; or, roughly speaking, that the future will, in the appropriate respects, resemble the past. | |
| From: A.J. Ayer (The Problem of Knowledge [1956], 2.viii) | |
| A reaction: I would say that nature is 'stable'. Nature changes, so a global assumption of total uniformity is daft. Do we need some global uniformity assumptions, if the induction involved is local? I would say yes. Are all inductions conditional on this? |
| 8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
| Full Idea: Intuitionists in mathematics deny excluded middle, because it is symptomatic of faith in the transcendent existence of mathematical objects and/or the truth of mathematical statements. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2) | |
| A reaction: There are other problems with excluded middle, such as vagueness, but on the whole I, as a card-carrying 'realist', am committed to the law of excluded middle. |
| 8763 | The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro] |
| Full Idea: It is surely wise to identify the positions in the natural numbers structure with their counterparts in the integer, rational, real and complex number structures. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2) | |
| A reaction: The point is that this might be denied, since 3, 3/1, 3.00.., and -3*i^2 are all arrived at by different methods of construction. Natural 3 has a predecessor, but real 3 doesn't. I agree, intuitively, with Shapiro. Russell (1919) disagreed. |
| 18249 | Cauchy gave a formal definition of a converging sequence. [Shapiro] |
| Full Idea: A sequence a1,a2,... of rational numbers is 'Cauchy' if for each rational number ε>0 there is a natural number N such that for all natural numbers m, n, if m>N and n>N then -ε < am - an < ε. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 7.2 n4) | |
| A reaction: The sequence is 'Cauchy' if N exists. |
| 8764 | Categories are the best foundation for mathematics [Shapiro] |
| Full Idea: There is a dedicated contingent who hold that the category of 'categories' is the proper foundation for mathematics. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.3 n7) | |
| A reaction: He cites Lawvere (1966) and McLarty (1993), the latter presenting the view as a form of structuralism. I would say that the concept of a category will need further explication, and probably reduce to either sets or relations or properties. |
| 8762 | Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro] |
| Full Idea: Zermelo said that for each number n, its successor is the singleton of n, so 3 is {{{null}}}, and 1 is not a member of 3. Von Neumann said each number n is the set of numbers less than n, so 3 is {null,{null},{null,{null}}}, and 1 is a member of 3. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2) | |
| A reaction: See Idea 645 - Zermelo could save Plato from the criticisms of Aristotle! These two accounts are cited by opponents of the set-theoretical account of numbers, because it seems impossible to arbitrate between them. |
| 8760 | Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro] |
| Full Idea: The structuralist vigorously rejects any sort of ontological independence among the natural numbers; the essence of a natural number is its relations to other natural numbers. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1) | |
| A reaction: This seems to place the emphasis on ordinals (what order?) rather than on cardinality (how many?). I am strongly inclined to think that this is the correct view, though you can't really have relations if there is nothing to relate. |
| 8761 | A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro] |
| Full Idea: A 'system' is a collection of objects with certain relations among them; a 'pattern' or 'structure' is the abstract form of a system, highlighting the interrelationships and ignoring any features they do not affect how they relate to other objects. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1) | |
| A reaction: Note that 'ignoring' features is a psychological account of abstraction, which (thanks to Frege and Geach) is supposed to be taboo - but which I suspect is actually indispensable in any proper account of thought and concepts. |
| 8744 | Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro] |
| Full Idea: The thesis that principles of arithmetic are derivable from the laws of logic runs against a now common view that logic itself has no ontology. There are no particular logical objects. From this perspective logicism is a non-starter. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 5.1) | |
| A reaction: This criticism strikes me as utterly devastating. There are two routes to go: prove that logic does have an ontology of objects (what would they be?), or - better - deny that arithmetic contains any 'objects'. Or give up logicism. |
| 8749 | Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro] |
| Full Idea: Term Formalism is the view that mathematics is just about characters or symbols - the systems of numerals and other linguistic forms. ...This will cover integers and rational numbers, but what are real numbers supposed to be, if they lack names? | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.1) | |
| A reaction: Real numbers (such as pi and root-2) have infinite decimal expansions, so we can start naming those. We could also start giving names like 'Harry' to other reals, though it might take a while. OK, I give up. |
| 8750 | Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro] |
| Full Idea: Game Formalism likens mathematics to chess, where the 'content' of mathematics is exhausted by the rules of operating with its language. ...This, however, leaves the problem of why the mathematical games are so useful to the sciences. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.2) | |
| A reaction: This thought pushes us towards structuralism. It could still be a game, but one we learned from observing nature, which plays its own games. Chess is, after all, modelled on warfare. |
| 8752 | Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro] |
| Full Idea: The Deductivist version of formalism (sometimes called 'if-thenism') says that the practice of mathematics consists of determining logical consequences of otherwise uninterpreted axioms. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.2) | |
| A reaction: [Hilbert is the source] More plausible than Term or Game Formalism (qv). It still leaves the question of why it seems applicable to nature, and why those particular axioms might be chosen. In some sense, though, it is obviously right. |
| 8753 | Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro] |
| Full Idea: Critics commonly complain that the intuitionist restrictions cripple the mathematician. On the other hand, intuitionist mathematics allows for many potentially important distinctions not available in classical mathematics, and is often more subtle. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 7.1) | |
| A reaction: The main way in which it cripples is its restriction on talk of infinity ('Cantor's heaven'), which was resented by Hilbert. Since high-level infinities are interesting, it would be odd if we were not allowed to discuss them. |
| 8731 | Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro] |
| Full Idea: I classify conceptualists according to what they say about properties or concepts. If someone classified properties as existing independent of language I would classify her as a realist in ontology of mathematics. Or they may be idealists or nominalists. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 2.2.1) | |
| A reaction: In other words, Shapiro wants to eliminate 'conceptualist' as a useful label in philosophy of mathematics. He's probably right. All thought involves concepts, but that doesn't produce a conceptualist theory of, say, football. |
| 8730 | 'Impredicative' definitions refer to the thing being described [Shapiro] |
| Full Idea: A definition of a mathematical entity is 'impredicative' if it refers to a collection that contains the defined entity. The definition of 'least upper bound' is impredicative as it refers to upper bounds and characterizes a member of this set. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2) | |
| A reaction: The big question is whether mathematics can live with impredicative definitions, or whether they threaten to be viciously circular, and undermine the whole enterprise. |
| 324 | Before the existence of the world there must have been being, space and becoming [Plato] |
| Full Idea: There were, before the world came into existence, being, space, and becoming, three distinct realities. | |
| From: Plato (Timaeus [c.349 BCE], 52d) |
| 20364 | The apprehensions of reason remain unchanging, but reasonless sensation shows mere becoming [Plato] |
| Full Idea: That which is apprehended by intelligence and reason is always in the same state, but that which is conceived by opinion with the help of sensation and without reason is always in a process of becoming and perishing, and never really is. | |
| From: Plato (Timaeus [c.349 BCE], 28a) | |
| A reaction: Lots of problems with this, of which I take the main one to be the idea that sensation is 'without reason', as if there were a sharp dichotomy in our ways of evaluating reality. Laws of nature seem to be laws of change, not of stasis. |
| 12042 | Plato's Forms were seen as part of physics, rather than of metaphysics [Plato, by Annas] |
| Full Idea: In the ancient world Plato's Theory of Forms was mostly seen as one aspect of Plato's 'physics' or theory of the world (rather than as 'metaphysics'). | |
| From: report of Plato (Timaeus [c.349 BCE]) by Julia Annas - Ancient Philosophy: very short introduction Ch.5 | |
| A reaction: This is how I also see the theory, but then I am inclined to see religion as a rather startling branch of speculative physics. Annas cites 'Timaeus' as the key text for this. |
| 307 | Something will always be well-made if the maker keeps in mind the eternal underlying pattern [Plato] |
| Full Idea: Whenever the maker of anything keeps his eye on the eternally unchanging and uses it as his pattern for the form and function of his product the result must be good. | |
| From: Plato (Timaeus [c.349 BCE], 28b) |
| 318 | In addition to the underlying unchanging model and a changing copy of it, there must also be a foundation of all change [Plato] |
| Full Idea: In addition to an eternal unchanging model and a visible and changing copy of reality, there must be a third part, the receptacle and nurse of all becoming and change. | |
| From: Plato (Timaeus [c.349 BCE], 49b) | |
| A reaction: cf Aristotle's criticism in Metaphysics |
| 321 | For knowledge and true opinion to be different there must be Forms; otherwise we are just stuck with sensations [Plato] |
| Full Idea: If intelligence and true opinion are different, then the forms must exist, but if they are the same, then what our senses perceive must be the most certain reality. | |
| From: Plato (Timaeus [c.349 BCE], 51d) |
| 317 | The universe is basically an intelligible and unchanging model, and a visible and changing copy of it [Plato] |
| Full Idea: Our basic description of the universe contained an intelligible and unchanging model, and a visible and changing copy of it. | |
| From: Plato (Timaeus [c.349 BCE], 48e) |
| 19461 | Knowing I exist reveals nothing at all about my nature [Ayer] |
| Full Idea: To know that one exists is not to know anything about oneself any more than knowing that 'this' exists is knowing anything about 'this'. | |
| From: A.J. Ayer (The Problem of Knowledge [1956], 2.iii) | |
| A reaction: Descartes proceeds to define himself as a 'thinking thing', inferring that thinking is his essence. Ayer casts nice doubt on that. |
| 19459 | To say 'I am not thinking' must be false, but it might have been true, so it isn't self-contradictory [Ayer] |
| Full Idea: To say 'I am not thinking' is self-stultifying since if it is said intelligently it must be false: but it is not self-contradictory. The proof that it is not self-contradictory is that it might have been false. | |
| From: A.J. Ayer (The Problem of Knowledge [1956], 2.iii) | |
| A reaction: If it doesn't imply a contradiction, then it is not a necessary truth, which is what it is normally taken to be. Is 'This is a sentence' necessarily true? It might not have been one, if the rules of English syntax changed recently. |
| 19460 | 'I know I exist' has no counterevidence, so it may be meaningless [Ayer] |
| Full Idea: If there is no experience at all of finding out that one is not conscious, or that one does not exist, ..it is tempting to say that sentences like 'I exist', 'I am conscious', 'I know that I exist' do not express genuine propositions. | |
| From: A.J. Ayer (The Problem of Knowledge [1956], 2.iii) | |
| A reaction: This is, of course, an application of the somewhat discredited verification principle, but the fact that strictly speaking the principle has been sort of refuted does not mean that we should not take it seriously, and be influenced by it. |
| 8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
| Full Idea: Rationalism is a long-standing school that can be characterized as an attempt to extend the perceived methodology of mathematics to all of knowledge. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.1) | |
| A reaction: Sometimes called 'Descartes's Dream', or the 'Enlightenment Project', the dream of proving everything. Within maths, Hilbert's Programme aimed for the same certainty. Idea 22 is the motto for the opposition to this approach. |
| 334 | Only bird-brained people think astronomy is entirely a matter of evidence [Plato] |
| Full Idea: Birds are empty-headed men who grew feathers instead of hair, because they were interested in astronomy but thought it was entirely a matter of physical evidence. | |
| From: Plato (Timaeus [c.349 BCE], 91d) |
| 19464 | We only discard a hypothesis after one failure if it appears likely to keep on failing [Ayer] |
| Full Idea: Why should a hypothesis which has failed the test be discarded unless this shows it to be unreliable; that is, having failed once it is likely to fail again? There is no contradiction in a hypothesis that was falsified being more likely to pass in future. | |
| From: A.J. Ayer (The Problem of Knowledge [1956], 2.viii) | |
| A reaction: People may become more likely to pass a test after they have failed at the first attempt. Birds which fail to fly at the first attempt usually achieve total mastery of it. There are different types of hypothesis here. |
| 19462 | Induction passes from particular facts to other particulars, or to general laws, non-deductively [Ayer] |
| Full Idea: Inductive reasoning covers all cases in which we pass from a particular statement of fact, or set of them, to a factual conclusion which they do not formally entail. The inference may be to a general law, or by analogy to another particular instance. | |
| From: A.J. Ayer (The Problem of Knowledge [1956], 2.viii) | |
| A reaction: My preferred definition is 'learning from experience' - which I take to be the most rational behaviour you could possibly imagine. I don't think a definition should be couched in terms of 'objects' or 'particulars'. |
| 5962 | Plato says the soul is ordered by number [Plato, by Plutarch] |
| Full Idea: Plato regards the substance of soul not as number but as being ordered by number. | |
| From: report of Plato (Timaeus [c.349 BCE]) by Plutarch - 68: Generation of the soul in 'Timaeus' 1023 | |
| A reaction: This remark points towards Plato's esoteric doctrines, which are some sort of mathematical metaphysics. The idea that order and numbers are in some way connected is one of the most powerful in western civilization, with undeniable appeal. |
| 330 | No one wants to be bad, but bad men result from physical and educational failures, which they do not want or choose [Plato] |
| Full Idea: No one wishes to be bad, but a bad man is bad because of some flaw in his physical makeup and failure in his education, neither of which he likes or chooses. | |
| From: Plato (Timaeus [c.349 BCE], 86e) |
| 316 | Music has harmony like the soul, and serves to reorder disharmony within us [Plato] |
| Full Idea: Music has harmonic motions like the orbits of the soul, and is not for irrational pleasure, but to reduce to order any disharmony in the revolutions within us. | |
| From: Plato (Timaeus [c.349 BCE], 47d) |
| 332 | One should exercise both the mind and the body, to avoid imbalance [Plato] |
| Full Idea: One should preserve a balance and avoid exercising the mind or body without the other; mathematicians should exercise physically, and athletes mentally. | |
| From: Plato (Timaeus [c.349 BCE], 88c) | |
| A reaction: Excellent, and very modern. Use it or lose it. It suggests that Plato had a fairly holistic view of a human being, and saw mind and body as closely integrated. |
| 328 | Everything that takes place naturally is pleasant [Plato] |
| Full Idea: Everything that takes place naturally is pleasant. | |
| From: Plato (Timaeus [c.349 BCE], 81e) | |
| A reaction: Not many people would agree with this. I recently watched a sparrowhawk eat a pigeon in my garden. This is the source of the stoic formula of living according to nature. |
| 322 | Intelligence is the result of rational teaching; true opinion can result from irrational persuasion [Plato] |
| Full Idea: Intelligence is produced by teaching, involves truth and reason, and cannot be moved; true opinion involves persuasion, is irrational and can be moved. | |
| From: Plato (Timaeus [c.349 BCE], 51e) |
| 331 | Bad governments prevent discussion, and discourage the study of virtue [Plato] |
| Full Idea: Under a bad government discussion, both public and private, is bad, and no courses of study are available to cure faults of character. | |
| From: Plato (Timaeus [c.349 BCE], 87b) |
| 310 | The creator of the cosmos had no envy, and so wanted things to be as like himself as possible [Plato] |
| Full Idea: This changing cosmos was made because its maker is good, and therefore lacks envy; he therefore wished all things to be as like himself as possible. | |
| From: Plato (Timaeus [c.349 BCE], 29e) |
| 311 | The cosmos must be unique, because it resembles the creator, who is unique [Plato] |
| Full Idea: So that our universe can resemble the perfect living creature in being unique, the universe was, is and will continue to be its maker's only creation. | |
| From: Plato (Timaeus [c.349 BCE], 31c) |
| 325 | We must consider the four basic shapes as too small to see, only becoming visible in large numbers [Plato] |
| Full Idea: We must think of the individual units of all four basic shapes as being far too small to be visible, and only becoming visible when massed together in large numbers. | |
| From: Plato (Timaeus [c.349 BCE], 56c) |
| 327 | There are two types of cause, the necessary and the divine [Plato] |
| Full Idea: We must distinguish two types of cause, the necessary and the divine. | |
| From: Plato (Timaeus [c.349 BCE], 68e) |
| 314 | Heavenly movements gave us the idea of time, and caused us to inquire about the heavens [Plato] |
| Full Idea: Days, months, years and solstices have caused the invention of number, given us the notion of time, and caused us to inquire into the nature of the universe. | |
| From: Plato (Timaeus [c.349 BCE], 47a) |
| 312 | Time came into existence with the heavens, so that there will be a time when they can be dissolved [Plato] |
| Full Idea: Time came into being with the heavens, so that they should be dissolved together if ever they are dissolved. | |
| From: Plato (Timaeus [c.349 BCE], 38c) |
| 309 | Clearly the world is good, so its maker must have been concerned with the eternal, not with change [Plato] |
| Full Idea: If the world is beautiful and its maker good, he had an eye on the eternal; if not, on that which is subject to change; clearly the world is the fairest of things, and he the best of causes, so it is eternal. | |
| From: Plato (Timaeus [c.349 BCE], 29a) |
| 308 | If the cosmos is an object of perception then it must be continually changing [Plato] |
| Full Idea: The cosmos is visible, tangible and corporeal, and therefore perceptible by the senses; therefore it is an object of opinion and sensation, and therefore change and coming into being. | |
| From: Plato (Timaeus [c.349 BCE], 28d) |