25 ideas
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) |
19722 | We could know the evidence for our belief without knowing why it is such evidence [Mittag] |
Full Idea: While one might understand the proposition entailed by one's evidence, one might have no idea how or why one's evidence entails it. This seems to imply one is not justified in believing the proposition on the basis of one's evidence. | |
From: Daniel M. Mittag (Evidentialism [2011], 'Evidential') | |
A reaction: An example might be seen if a layman tours a physics lab. This looks like a serious problem for evidentialism. Once you see why the evidence entails the proposition, you are getting closer to understanding than to knowledge. Explanation. |
19723 | Evidentialism can't explain that we accept knowledge claims if the evidence is forgotten [Mittag] |
Full Idea: If one came to believe p with good evidence, but has since forgotten that evidence, we might think one can continue to believe justifiably, but evidentialism appears unable to account for this. | |
From: Daniel M. Mittag (Evidentialism [2011], 'Forgotten') | |
A reaction: We would still think that the evidence was important, and we would need to trust the knower's claim that the forgotten evidence was good. So it doesn't seem to destroy the evidentialist thesis. |
19720 | Evidentialism concerns the evidence for the proposition, not for someone to believe it [Mittag] |
Full Idea: Evidentialism is not a theory about when one's believing is justified; it is a theory about what makes one justified in believing a proposition. It is a thesis regarding 'propositional justification', not 'doxastic justification'. | |
From: Daniel M. Mittag (Evidentialism [2011], 'Preliminary') | |
A reaction: Thus it is entirely about whether the evidence supports the proposition, and has no interest in who believes it or why. Knowledge is when you believe a true proposition which has good support. This could be internalist or externalist? |
19721 | Coherence theories struggle with the role of experience [Mittag] |
Full Idea: Traditional coherence theories seem unable to account for the role experience plays in justification. | |
From: Daniel M. Mittag (Evidentialism [2011], 'Evidence') | |
A reaction: I'm inclined to say that experience only becomes a justification when it has taken propositional (though not necessarily lingistic) form. That is, when you see it 'as' something. Uninterpreted shape and colour can justify virtually nothing. |
3061 | Anaxarchus said that he was not even sure that he knew nothing [Anaxarchus, by Diog. Laertius] |
Full Idea: Anaxarchus said that he was not even sure that he knew nothing. | |
From: report of Anaxarchus (fragments/reports [c.340 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 09.10.1 |