38 ideas
| 17651 | Without words or other symbols, we have no world [Goodman] |
| Full Idea: We can have words without a world but no world without words or other symbols. | |
| From: Nelson Goodman (Ways of Worldmaking [1978], 1.3) | |
| A reaction: Goodman seems to have a particularly extreme version of the commitment to philosophy as linguistic. Non-human animals have no world, it seems. |
| 17652 | Truth is irrelevant if no statements are involved [Goodman] |
| Full Idea: Truth pertains solely to what is said ...For nonverbal versions and even for verbal versions without statements, truth is irrelevant. | |
| From: Nelson Goodman (Ways of Worldmaking [1978], 1.5) | |
| A reaction: Goodman is a philosopher of language (like Dummett), but I am a philosopher of thought (like Evans). The test, for me, is whether truth is applicable to the thought of non-human animals. I take it to be obvious that it is applicable. |
| 10859 | A set is 'well-ordered' if every subset has a first element [Clegg] |
| Full Idea: For a set to be 'well-ordered' it is required that every subset of the set has a first element. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.13) |
| 10857 | Set theory made a closer study of infinity possible [Clegg] |
| Full Idea: Set theory made a closer study of infinity possible. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.13) |
| 10864 | Any set can always generate a larger set - its powerset, of subsets [Clegg] |
| Full Idea: The idea of the 'power set' means that it is always possible to generate a bigger one using only the elements of that set, namely the set of all its subsets. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.14) |
| 10872 | Extensionality: Two sets are equal if and only if they have the same elements [Clegg] |
| Full Idea: Axiom of Extension: Two sets are equal if and only if they have the same elements. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10875 | Pairing: For any two sets there exists a set to which they both belong [Clegg] |
| Full Idea: Axiom of Pairing: For any two sets there exists a set to which they both belong. So you can make a set out of two other sets. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10876 | Unions: There is a set of all the elements which belong to at least one set in a collection [Clegg] |
| Full Idea: Axiom of Unions: For every collection of sets there exists a set that contains all the elements that belong to at least one of the sets in the collection. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10878 | Infinity: There exists a set of the empty set and the successor of each element [Clegg] |
| Full Idea: Axiom of Infinity: There exists a set containing the empty set and the successor of each of its elements. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) | |
| A reaction: This is rather different from the other axioms because it contains the notion of 'successor', though that can be generated by an ordering procedure. |
| 10877 | Powers: All the subsets of a given set form their own new powerset [Clegg] |
| Full Idea: Axiom of Powers: For each set there exists a collection of sets that contains amongst its elements all the subsets of the given set. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) | |
| A reaction: Obviously this must include the whole of the base set (i.e. not just 'proper' subsets), otherwise the new set would just be a duplicate of the base set. |
| 10879 | Choice: For every set a mechanism will choose one member of any non-empty subset [Clegg] |
| Full Idea: Axiom of Choice: For every set we can provide a mechanism for choosing one member of any non-empty subset of the set. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) | |
| A reaction: This axiom is unusual because it makes the bold claim that such a 'mechanism' can always be found. Cohen showed that this axiom is separate. The tricky bit is choosing from an infinite subset. |
| 10871 | Axiom of Existence: there exists at least one set [Clegg] |
| Full Idea: Axiom of Existence: there exists at least one set. This may be the empty set, but you need to start with something. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10874 | Specification: a condition applied to a set will always produce a new set [Clegg] |
| Full Idea: Axiom of Specification: For every set and every condition, there corresponds a set whose elements are exactly the same as those elements of the original set for which the condition is true. So the concept 'number is even' produces a set from the integers. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) | |
| A reaction: What if the condition won't apply to the set? 'Number is even' presumably won't produce a set if it is applied to a set of non-numbers. |
| 10880 | Mathematics can be 'pure' (unapplied), 'real' (physically grounded); or 'applied' (just applicable) [Clegg] |
| Full Idea: Three views of mathematics: 'pure' mathematics, where it doesn't matter if it could ever have any application; 'real' mathematics, where every concept must be physically grounded; and 'applied' mathematics, using the non-real if the results are real. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.17) | |
| A reaction: Very helpful. No one can deny the activities of 'pure' mathematics, but I think it is undeniable that the origins of the subject are 'real' (rather than platonic). We do economics by pretending there are concepts like the 'average family'. |
| 10861 | Beyond infinity cardinals and ordinals can come apart [Clegg] |
| Full Idea: With ordinary finite numbers ordinals and cardinals are in effect the same, but beyond infinity it is possible for two sets to have the same cardinality but different ordinals. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.13) |
| 10860 | An ordinal number is defined by the set that comes before it [Clegg] |
| Full Idea: You can think of an ordinal number as being defined by the set that comes before it, so, in the non-negative integers, ordinal 5 is defined as {0, 1, 2, 3, 4}. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.13) |
| 10854 | Transcendental numbers can't be fitted to finite equations [Clegg] |
| Full Idea: The 'transcendental numbers' are those irrationals that can't be fitted to a suitable finite equation, of which π is far and away the best known. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch. 6) |
| 10858 | By adding an axis of imaginary numbers, we get the useful 'number plane' instead of number line [Clegg] |
| Full Idea: The realisation that brought 'i' into the toolkit of physicists and engineers was that you could extend the 'number line' into a new dimension, with an imaginary number axis at right angles to it. ...We now have a 'number plane'. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.12) |
| 10853 | Either lack of zero made early mathematics geometrical, or the geometrical approach made zero meaningless [Clegg] |
| Full Idea: It is a chicken-and-egg problem, whether the lack of zero forced forced classical mathematicians to rely mostly on a geometric approach to mathematics, or the geometric approach made 0 a meaningless concept, but the two remain strongly tied together. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch. 6) |
| 10866 | Cantor's account of infinities has the shaky foundation of irrational numbers [Clegg] |
| Full Idea: As far as Kronecker was concerned, Cantor had built a whole structure on the irrational numbers, and so that structure had no foundation at all. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10869 | The Continuum Hypothesis is independent of the axioms of set theory [Clegg] |
| Full Idea: Paul Cohen showed that the Continuum Hypothesis is independent of the axioms of set theory. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10862 | The 'continuum hypothesis' says aleph-one is the cardinality of the reals [Clegg] |
| Full Idea: The 'continuum hypothesis' says that aleph-one is the cardinality of the rational and irrational numbers. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.14) |
| 17656 | Being primitive or prior always depends on a constructional system [Goodman] |
| Full Idea: Nothing is primitive or derivationally prior to anything apart from a constructional system. | |
| From: Nelson Goodman (Ways of Worldmaking [1978], 1.4c) | |
| A reaction: Something may be primitive not just because we can't be bothered to analyse it any further, but because even God couldn't analyse it. Maybe. |
| 17661 | We don't recognise patterns - we invent them [Goodman] |
| Full Idea: Recognising patterns is very much a matter of inventing or imposing them. | |
| From: Nelson Goodman (Ways of Worldmaking [1978], 1.7) | |
| A reaction: I take this to be false. |
| 17659 | Reality is largely a matter of habit [Goodman] |
| Full Idea: Reality in a world, like realism in a picture, is largely a matter of habit. | |
| From: Nelson Goodman (Ways of Worldmaking [1978], 1.6) | |
| A reaction: I'm a robust realist, me, but I sort of see what he means. We become steeped in unspoken conventions about how we take our world to be, and filter out anything that conflicts with it. |
| 17657 | We build our world, and ignore anything that won't fit [Goodman] |
| Full Idea: We dismiss as illusory or negligible what cannot be fitted into the architecture of the world we are building. | |
| From: Nelson Goodman (Ways of Worldmaking [1978], 1.4d) | |
| A reaction: I'm trying to think of an example of this, but can't. Maybe poor people are invisible to the rich? |
| 17654 | A world can be full of variety or not, depending on how we sort it [Goodman] |
| Full Idea: A world may be unmanageably heterogeneous or unbearably monotonous according to how events are sorted into kinds. | |
| From: Nelson Goodman (Ways of Worldmaking [1978], 1.4a) | |
| A reaction: We might expect this from the man who invented 'grue', which allows you to classify things that change colour with things that don't. Could you describe a bird as 'might have been a fish', and classify it with fish? ('Projectible'?) |
| 17653 | Things can only be judged the 'same' by citing some respect of sameness [Goodman] |
| Full Idea: Identification rests upon organization into entities and kinds. The response to the question 'Same or not the same?' must always be 'Same what?'. ...Identity or constancy in a world is identity with respect to what is within that world as organised. | |
| From: Nelson Goodman (Ways of Worldmaking [1978], 1.4a) | |
| A reaction: And the gist of his book is that 'organised' is done by us, not by the world. He seems to be committed to the full Geachean relative identity, rather than the mere Wigginsian relative individuation. An unfashionable view! |
| 17660 | Discovery is often just finding a fit, like a jigsaw puzzle [Goodman] |
| Full Idea: Discovery often amounts, as when I place a piece in a jigsaw puzzle, not to arrival at a proposition for declaration or defense, but to finding a fit. | |
| From: Nelson Goodman (Ways of Worldmaking [1978], 1.7) | |
| A reaction: I find Goodman's views here pretty alien, but I like this bit. Coherence really rocks. |
| 17658 | Users of digital thermometers recognise no temperatures in the gaps [Goodman] |
| Full Idea: To use a digital thermometer with readings in tenths of a degree is to recognise no temperature as lying between 90 and 90.1 degrees. | |
| From: Nelson Goodman (Ways of Worldmaking [1978], 1.4d) | |
| A reaction: This appears to be nonsense, treating users of digital thermometers as if they were stupid. No one thinks temperatures go up and down in quantum leaps. We all know there is a gap between instrument and world. (Very American, I'm thinking!) |
| 17650 | We lack frames of reference to transform physics, biology and psychology into one another [Goodman] |
| Full Idea: We have no neat frames of reference, no ready rules for transforming physics, biology and psychology into one another. | |
| From: Nelson Goodman (Ways of Worldmaking [1978], 1.2) |
| 17655 | Grue and green won't be in the same world, as that would block induction entirely [Goodman] |
| Full Idea: Grue cannot be a relevant kind for induction in the same world as green, for that would preclude some of the decisions, right or wrong, that constitute inductive inference. | |
| From: Nelson Goodman (Ways of Worldmaking [1978], 1.4b) | |
| A reaction: This may make 'grue' less mad than I thought it was. I always assume we are slicing the world as 'green, blue and grue'. I still say 'green' is a basic predicate of experience, but 'grue' is amenable to analysis. |
| 18225 | If we can't reason about value, we can reason about the unconditional source of value [Korsgaard] |
| Full Idea: If you can only know what is intrinsically valuable through intuition (as Moore claims), you can still argue about what is unconditionally valuable. There must be something unconditionally valuable because there must be a source of value. | |
| From: Christine M. Korsgaard (Aristotle and Kant on the Source of Value [1986], 8 'Three') | |
| A reaction: If you only grasped the values through intuition, does that give you enough information to infer the dependence relations between values? |
| 18228 | An end can't be an ultimate value just because it is useless! [Korsgaard] |
| Full Idea: If what is final is whatever is an end but never a means, ...why should something be more valuable just because it is useless? | |
| From: Christine M. Korsgaard (Aristotle and Kant on the Source of Value [1986], 8 'Finality') | |
| A reaction: Korsgaard is offering this as a bad reading of what Aristotle intends. |
| 18224 | Goodness is given either by a psychological state, or the attribution of a property [Korsgaard] |
| Full Idea: 'Subjectivism' identifies good ends with or by reference to some psychological state. ...'Objectivism' says that something is good as an end if a property, intrinsic goodness, is attributed to it. | |
| From: Christine M. Korsgaard (Aristotle and Kant on the Source of Value [1986], 8 'Three') |
| 18233 | Contemplation is final because it is an activity which is not a process [Korsgaard] |
| Full Idea: It is because contemplation is an activity that is not also a process that Aristotle identifies it as the most final good. | |
| From: Christine M. Korsgaard (Aristotle and Kant on the Source of Value [1986], 8 'Activity') | |
| A reaction: Quite a helpful way of labelling what Aristotle has in mind. So should we not aspire to be involved in processes, except reluctantly? I take the mind itself to be a process, so that may be difficult! |
| 18226 | For Aristotle, contemplation consists purely of understanding [Korsgaard] |
| Full Idea: Contemplation, as Aristotle understand it, is not research or inquiry, but an activity that ensues on these: an activity that consists in understanding. | |
| From: Christine M. Korsgaard (Aristotle and Kant on the Source of Value [1986], 8 'Aristotle') | |
| A reaction: Fairly obvious, when you read the last part of 'Ethics', but helpful in grasping Aristotle, because understanding is the objective of 'Posterior Analytics' and 'Metaphysics', so he tells you how to achieve the ideal moral state. |
| 17649 | If the world is one it has many aspects, and if there are many worlds they will collect into one [Goodman] |
| Full Idea: If there is but one world, it embraces a multiplicity of contrasting aspects; if there are many worlds, the collection of them all is one. One world may be taken as many, or many worlds taken as one; whether one or many depends on the way of taking. | |
| From: Nelson Goodman (Ways of Worldmaking [1978], 1.2) | |
| A reaction: He cites 'The Pluralistic Universe' by William James for this idea. The idea is that the distinction 'evaporates under analysis'. Parmenides seems to have thought that no features could be distinguished in the true One. |