33 ideas
| 1642 | We must fight fiercely for knowledge, understanding and intelligence [Plato] |
| Full Idea: We need to use every argument we can to fight against anyone who does away with knowledge, understanding, and intelligence, but at the same time asserts anything at all about anything. | |
| From: Plato (The Sophist [c.359 BCE], 249c) | |
| A reaction: Thus showing that reason is only central if you want to put a high value on it? |
| 1645 | The desire to split everything into its parts is unpleasant and unphilosophical [Plato] |
| Full Idea: To try to set apart everything from everything is not only especially jangling, but it is the mark of someone altogether unmusical and unphilosophic. | |
| From: Plato (The Sophist [c.359 BCE], 259e) |
| 287 | Good analysis involves dividing things into appropriate forms without confusion [Plato] |
| Full Idea: It takes expertise in dialectic to divide things by kinds and not to think that the same form is a different one or that a different form is the same. | |
| From: Plato (The Sophist [c.359 BCE], 253d) |
| 1644 | Dialectic should only be taught to those who already philosophise well [Plato] |
| Full Idea: The dialectical capacity - you won't give it to anyone else, I suspect, except to whoever philosophises purely and justly. | |
| From: Plato (The Sophist [c.359 BCE], 253e) |
| 20478 | In discussion a person's opinions are shown to be in conflict, leading to calm self-criticism [Plato] |
| Full Idea: They collect someone's opinions together during the discussion, put them side by side, and show that they conflict with each other at the same time on the same subjects.... The person sees this, gets angry at themselves, and calmer towards others. | |
| From: Plato (The Sophist [c.359 BCE], 230b) | |
| A reaction: He goes on to say that the process is like a doctor purging a patient of internal harms. If anyone talks for long enough (even a good philosopher), their opinions will probably be seen to be in conflict. But which opinions do you abandon? |
| 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. |
| 11278 | What does 'that which is not' refer to? [Plato] |
| Full Idea: What should the name 'that which is not' be applied to? | |
| From: Plato (The Sophist [c.359 BCE], 237c) | |
| A reaction: This leads into a discussion of the problem, in The Sophist. It became a large issue when modern logic was being developed by Frege and Russell. |
| 1643 | If statements about non-existence are logically puzzling, so are statements about existence [Plato] |
| Full Idea: When the question was put to us as to the name of 'that which is not', to whatever one must apply it, we got stuck in every kind of perplexity. Are we now in any less perplexity about 'that which is'? | |
| From: Plato (The Sophist [c.359 BCE], 250d) | |
| A reaction: Nice. This precapitulates the whole story of modern philosophy of language. What started as a nagging doubt about reference to non-existents ends as bewilderment about everything we say. |
| 7022 | To be is to have a capacity, to act on other things, or to receive actions [Plato] |
| Full Idea: A thing really is if it has any capacity, either by nature to do something to something else or to have even the smallest thing done to it by the most trivial thing, even if it only happens once. I'll define those which are as nothing other than capacity. | |
| From: Plato (The Sophist [c.359 BCE], 247e) | |
| A reaction: If philosophy is footnotes to Plato, this should be the foundational remark in all discussions of existence (though Parmenides might claim priority). It seems to say 'to be is to have a causal role (active or passive)'. It also seems essentialist. |
| 1641 | Some alarming thinkers think that only things which you can touch exist [Plato] |
| Full Idea: One group drags everything down to earth, insisting that only what offers tangible contact is, since they define being as the same as body, despising anyone who says that something without a body is. These are frightening men. | |
| From: Plato (The Sophist [c.359 BCE], 246b) |
| 10784 | Whenever there's speech it has to be about something [Plato] |
| Full Idea: Whenever there's speech it has to be about something. It's impossible for it not to be about something. | |
| From: Plato (The Sophist [c.359 BCE], 262e) | |
| A reaction: [Quoted by Marcus about ontological commitment] The interesting test case would be speech about the existence of circular squares. |
| 16122 | Good thinkers spot forms spread through things, or included within some larger form [Plato] |
| Full Idea: It takes dialectic to divide things by kinds...such a person can discriminate a single form spread through a lot of separate things…and forms included in a single outside form…or a form connected as a unit through many wholes. | |
| From: Plato (The Sophist [c.359 BCE], 253d) | |
| A reaction: [compressed] This is very helpful in indicating the complex structure of the Forms that Plato envisages. If you talk of the meanings of words (other than names), though, it comes to the same thing. Wise people fully understand their language. |
| 10422 | The not-beautiful is part of the beautiful, though opposed to it, and is just as real [Plato] |
| Full Idea: So 'the not beautiful' turns out to be ..both marked off within one kind of those that are, and also set over against one of those that are, ..and the beautiful is no more a being than the not beautiful. | |
| From: Plato (The Sophist [c.359 BCE], 257d) | |
| A reaction: [dialogue eliminated] This is a highly significant passage, for two reasons. It suggests that the Form of the beautiful can have parts, and also that the negations of Forms are Forms themselves (both of which come as a surprise). |
| 15855 | If we see everything as separate, we can then give no account of it [Plato] |
| Full Idea: To dissociate each thing from everything else is to destroy totally everything there is to say. The weaving together of forms is what makes speech [logos] possible for us. | |
| From: Plato (The Sophist [c.359 BCE], 259e) | |
| A reaction: This I take to be the lynchpin of metaphysics. We are forced to see the world in a way which enables us to give some sort of account of it. Our metaphysics is 'inference to the best logos'. |
| 16285 | A possible world can be seen as a complete and consistent novel [Jeffrey] |
| Full Idea: A novel describes a possible world in as much detail as is possible without exceeding the resources of the agent's language. But if talk of possible worlds seems dangerously metaphysical, focus on the novels themselves, when complete and consistent. | |
| From: Richard Jeffrey (The Logic of Decision [1965], 12.8), quoted by David Lewis - On the Plurality of Worlds | |
| A reaction: Lewis seems to cite this remark from Jeffrey as the source of the idea that ersatz linguistic worlds are like novels. Why won't a novel with one tiny inconsistency count as a possible world? People seem to live in it. |
| 1637 | A soul without understanding is ugly [Plato] |
| Full Idea: The soul that lacks understanding must be set down as ugly. | |
| From: Plato (The Sophist [c.359 BCE], 228d) | |
| A reaction: The teleological view of things understands their nature in things of their perfection. and the essence of beauty is perfection. It is the mind's nature to know. Failing to know is as ugly as allowing your crops to die. |
| 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. |
| 19155 | Instead of gambling, Jeffrey made the objects of Bayesian preference to be propositions [Jeffrey, by Davidson] |
| Full Idea: Jeffrey produced a version of Bayesianism that made no direct use of gambling (as Ramsey had), but treats the objects of preference ...as propositions. | |
| From: report of Richard Jeffrey (The Logic of Decision [1965]) by Donald Davidson - Truth and Predication 3 | |
| A reaction: I'm guessing that Jeffreys launched modern Bayesian theory with this idea. It suggest that one can consider degrees of truth, rather than mere winning or losing. |
| 1636 | Wickedness is an illness of the soul [Plato] |
| Full Idea: Wickedness is a sedition and illness of the soul. | |
| From: Plato (The Sophist [c.359 BCE], 228b) |
| 1638 | Didactic education is hard work and achieves little [Plato] |
| Full Idea: With a lot of effort the admonitory species of education accomplishes little. | |
| From: Plato (The Sophist [c.359 BCE], 230a) |