31 ideas
| 2666 | Carneades' pinnacles of philosophy are the basis of knowledge (the criterion of truth) and the end of appetite (good) [Carneades, by Cicero] |
| Full Idea: Carneades said the two greatest things in philosophy were the criterion of truth and the end of goods, and no man could be a sage who was ignorant of the existence of either a beginning of the process of knowledge or an end of appetition. | |
| From: report of Carneades (fragments/reports [c.174 BCE]) by M. Tullius Cicero - Academica II.09.29 | |
| A reaction: Nice, but I would want to emphasise the distinction between truth and its criterion. Admittedly we would have no truth without a good criterion, but the truth itself should be held in higher esteem than our miserable human means of grasping it. |
| 21390 | Future events are true if one day we will say 'this event is happening now' [Carneades] |
| Full Idea: We call those past events true of which at an earlier time this proposition was true: 'They are present now'; similarly, we shall call those future events true of which at some future time this proposition will be true: 'They are present now'. | |
| From: Carneades (fragments/reports [c.174 BCE]), quoted by M. Tullius Cicero - On Fate ('De fato') 9.23-8 | |
| A reaction: This is a very nice way of paraphrasing statements about the necessity of true future contingent events. It still relies, of course, on the veracity of a tensed assertion |
| 21672 | We say future things are true that will possess actuality at some following time [Carneades, by Cicero] |
| Full Idea: Just as we speak of past things as true that possessed true actuality at some former time, so we speak of future things as true that will possess true actuality at some following time. | |
| From: report of Carneades (fragments/reports [c.174 BCE]) by M. Tullius Cicero - On Fate ('De fato') 11.27 | |
| A reaction: This ducks the Aristotle problem of where it is true NOW when you say there will be a sea-fight tomorrow, and it turns out to be true. Carneades seems to be affirming a truth when it does not yet have a truthmaker. |
| 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'. |
| 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) |
| 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) |
| 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) |
| 15825 | Carneades denied the transitivity of identity [Carneades, by Chisholm] |
| Full Idea: Carneades denied the principle of the transitivity of identity. | |
| From: report of Carneades (fragments/reports [c.174 BCE], fr 41-42) by Roderick Chisholm - Person and Object 3.1 | |
| A reaction: Chisholm calls this 'extreme', but I assume Carneades wouldn't deny the principle in mathematics. I'm guessing that he just means that nothing ever stays quite the same. |
| 21389 | Carneades distinguished logical from causal necessity, when talking of future events [Long on Carneades] |
| Full Idea: From 'E will take place is true' it follows that E must take place. But 'must' here is logical not causal necessity. It is a considerable achievement of Carneades to have distinguished these two senses of necessity. | |
| From: comment on Carneades (fragments/reports [c.174 BCE]) by A.A. Long - Hellenistic Philosophy 3 | |
| A reaction: Personally I am inclined to think 'necessity' is univocal, and does not have two senses. What Carneades has nicely done is distinguish the two different grounds for the necessities. |
| 20653 | Six reduction levels: groups, lives, cells, molecules, atoms, particles [Putnam/Oppenheim, by Watson] |
| Full Idea: There are six 'reductive levels' in science: social groups, (multicellular) living things, cells, molecules, atoms, and elementary particles. | |
| From: report of H.Putnam/P.Oppenheim (Unity of Science as a Working Hypothesis [1958]) by Peter Watson - Convergence 10 'Intro' | |
| A reaction: I have the impression that fields are seen as more fundamental that elementary particles. What is the status of the 'laws' that are supposed to govern these things? What is the status of space and time within this picture? |
| 21671 | Voluntary motion is intrinsically within our power, and this power is its cause [Carneades, by Cicero] |
| Full Idea: Voluntary motion possesses the intrinsic property of being in our power and of obeying us, and its obedience is not uncaused, for its nature is itself the cause of this. | |
| From: report of Carneades (fragments/reports [c.174 BCE]) by M. Tullius Cicero - On Fate ('De fato') 11.25 | |
| A reaction: To say that actions arise from our 'intrinsic power' is not much of an explanation, but it is still informative - that you should study the intrinsic powers of humans if you want to explain it. |
| 21391 | Some actions are within our power; determinism needs prior causes for everything - so it is false [Carneades, by Cicero] |
| Full Idea: Now something is in our power; but if everything happens as a result of destiny all things happen as a result of antecedent causes; therefore what happens does not happen as a result of destiny. | |
| From: report of Carneades (fragments/reports [c.174 BCE]) by M. Tullius Cicero - On Fate ('De fato') 14.31 | |
| A reaction: This invites the question of whether some things really are 'in our power'. Carneades (as expressed by Cicero) takes that for granted. Our 'power' may be antecedent causes in disguise. |
| 21674 | Even Apollo can only foretell the future when it is naturally necessary [Carneades, by Cicero] |
| Full Idea: Carneades used to say that not even Apollo could tell any future events except those whose causes were so held together that they must necessarily happen. | |
| From: report of Carneades (fragments/reports [c.174 BCE]) by M. Tullius Cicero - On Fate ('De fato') 14.32 | |
| A reaction: Carneades is opposing the usual belief in divination, where even priests can foretell contingent future events to some extent. Careneades, of course, was defending free will. |
| 7398 | Carneades said that after a shipwreck a wise man would seize the only plank by force [Carneades, by Tuck] |
| Full Idea: Carneades argued forcefully that in the event of a shipwreck, the wise man would be prepared to seize the only plank capable of bearing him to shore, even if that meant pushing another person off it. | |
| From: report of Carneades (fragments/reports [c.174 BCE]) by Richard Tuck - Hobbes Ch.1 | |
| A reaction: [source for this?] This thought seems to have provoked great discussion in the sixteenth century (mostly sympathetic). I can't help thinking the right answer depends on assessing your rival. Die for a hero, drown a nasty fool. |
| 21392 | People change laws for advantage; either there is no justice, or it is a form of self-injury [Carneades, by Lactantius] |
| Full Idea: The same people often changed laws according to circumstances; there is no natural law. There is no such thing as justice or, if there is, it is the height of folly, since a man injures himself in taking thought for the advantage of others. | |
| From: report of Carneades (fragments/reports [c.174 BCE]) by Lactantius - Institutiones Divinae 5.16.4 | |
| A reaction: [An argument used by Carneades on his notorious 156BCE visit to Rome, where he argued both for and against justice] This is probably the right wing view of justice. Why give other people what they want, if it is at our expense? |