35 ideas
| 7910 | Pursue truth with the urgency of someone whose clothes are on fire [Ashvaghosha] |
| Full Idea: As though your turban or your clothes were on fire, so with a sense of urgency should you apply your intellect to the comprehension of the truths. | |
| From: Ashvaghosha (Saundaranandakavya [c.50], XVI) | |
| A reaction: The best philosophers need no such urging. I retain a romantic view that we should be 'natural' in these things. See Plato's views in Idea 2153 and 1638. However, maybe I should be confronted with this quotation every morning when I awake. |
| 18365 | If truths are just identical with facts, then truths will make themselves true [David] |
| Full Idea: According to the identity theory of truth, a proposition is true if and only if it is identical with a fact. ...This leads to the unacceptable claim that every true proposition makes itself true (because it is identical to its fact). | |
| From: Marian David (Truth-making and Correspondence [2009], n 14) |
| 18362 | Examples show that truth-making is just non-symmetric, not asymmetric [David] |
| Full Idea: That 'there is at least one proposition' ...is a case where something makes itself true, which generates a counterexample to the natural assumption that truth-making is asymmetric; truth-making, it seems, is merely non-symmetric. | |
| From: Marian David (Truth-making and Correspondence [2009], 4) |
| 18360 | It is assumed that a proposition is necessarily true if its truth-maker exists [David] |
| Full Idea: Friends of the truth-maker principle usually hold that the following states a crucial necessary condition on truth-making: if x makes y true, then, necessarily, if x exists then y is true. | |
| From: Marian David (Truth-making and Correspondence [2009], 2) | |
| A reaction: My objection is that the proposition y is taken to pre-exist, primly awaiting the facts that will award it 'truth'. An ontology that contains an infinity of propositions, most of which so far lack a truth-value, is incoherent. You can have x, but no y! |
| 18358 | Two different propositions can have the same fact as truth-maker [David] |
| Full Idea: Two different propositions can have the same fact as truth-maker. For example, 'L is happy or L is hungry', and 'L is happy or L is thirsty', which are both made true by the fact that L is happy. | |
| From: Marian David (Truth-making and Correspondence [2009], 1) |
| 18355 | What matters is truth-making (not truth-makers) [David] |
| Full Idea: The term 'truthmaker' just labels whatever stands in the truth-making relation to a truth. The truth-making relation is crucial. It would have been just as well to refer to the truth-'maker' principle as the truth-'making' principle. | |
| From: Marian David (Truth-making and Correspondence [2009], 1) | |
| A reaction: This is well said. The commitment of this theory is to something which makes each proposition true. There is no initial commitment to any theories about what sorts of things do the job. |
| 18354 | Correspondence is symmetric, while truth-making is taken to be asymmetric [David] |
| Full Idea: Correspondence appears to be a symmetric relation while truth-making appears to be, or is supposed to be, an asymmetric relation. | |
| From: Marian David (Truth-making and Correspondence [2009], Intro) |
| 18356 | Correspondence is an over-ambitious attempt to explain truth-making [David] |
| Full Idea: Truth-maker theory says that the attempt by correspondence to fill in the generic truth-maker principle with something more informative fails. It is too ambitious, offering a whole zoo of funny facts that are not needed. | |
| From: Marian David (Truth-making and Correspondence [2009], 1) | |
| A reaction: A typical funny fact is a disjunctive fact, which makes 'he is hungry or thirsty' true (when it can just be made true by the simple fact that he is thirsty). |
| 18363 | Correspondence theorists see facts as the only truth-makers [David] |
| Full Idea: Correspondence theorists are committed to the view that, since truth is correspondence with a fact, only facts can make true propositions true. | |
| From: Marian David (Truth-making and Correspondence [2009], 4) |
| 18364 | Correspondence theory likes ideal languages, that reveal the structure of propositions [David] |
| Full Idea: Correspondence theorists tend to promote ideal languages, ...which is intended to mirror perfectly the structure of the propositions it expresses. | |
| From: Marian David (Truth-making and Correspondence [2009], n 03) |
| 18357 | What makes a disjunction true is simpler than the disjunctive fact it names [David] |
| Full Idea: The proposition that 'L is happy or hungry' can be made true by the fact that L is happy. This does not have the same complexity or constituent structure as the proposition it makes true. | |
| From: Marian David (Truth-making and Correspondence [2009], 1) |
| 18359 | One proposition can be made true by many different facts [David] |
| Full Idea: One proposition can be made true by many different facts (such as 'there are some happy dogs'). | |
| From: Marian David (Truth-making and Correspondence [2009], 1) |
| 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) |
| 18361 | A reflexive relation entails that the relation can't be asymmetric [David] |
| Full Idea: An asymmetric relation must be irreflexive: any case of aRa will yield a reductio of the assumption that R is asymmetric. | |
| From: Marian David (Truth-making and Correspondence [2009], 4) |
| 7909 | The Eightfold Path concerns morality, wisdom, and tranquillity [Ashvaghosha] |
| Full Idea: The Eightfold Path has three steps concerning morality - right speech, right bodily action, and right livelihood; three of wisdom - right views, right intentions, and right effort; and two of tranquillity - right mindfulness and right concentration. | |
| From: Ashvaghosha (Saundaranandakavya [c.50], XVI) | |
| A reaction: Most of this translates quite comfortably into the aspirations of western philosophy. For example, 'right effort' sounds like Kant's claim that only a good will is truly good (Idea 3710). The Buddhist division is interesting for action theory. |
| 7908 | At the end of a saint, he is not located in space, but just ceases to be disturbed [Ashvaghosha] |
| Full Idea: When an accomplished saint comes to the end, he does not go anywhere down in the earth or up in the sky, nor into any of the directions of space, but because his defilements have become extinct he simply ceases to be disturbed. | |
| From: Ashvaghosha (Saundaranandakavya [c.50], XVI) | |
| A reaction: To 'cease to be disturbed' is the most attractive account of heaven I have encountered. It all sounds a bit dull though. I wonder, as usual, how they know all this stuff. |