44 ideas
17774 | Definitions make our intuitions mathematically useful [Mayberry] |
8623 | Proof reveals the interdependence of truths, as well as showing their certainty [Euclid, by Frege] |
17773 | Proof shows that it is true, but also why it must be true [Mayberry] |
13907 | If you pick an arbitrary triangle, things proved of it are true of all triangles [Euclid, by Lemmon] |
17795 | Set theory can't be axiomatic, because it is needed to express the very notion of axiomatisation [Mayberry] |
17796 | There is a semi-categorical axiomatisation of set-theory [Mayberry] |
17800 | The misnamed Axiom of Infinity says the natural numbers are finite in size [Mayberry] |
17801 | The set hierarchy doesn't rely on the dubious notion of 'generating' them [Mayberry] |
17803 | Limitation of size is part of the very conception of a set [Mayberry] |
17786 | The mainstream of modern logic sees it as a branch of mathematics [Mayberry] |
17788 | First-order logic only has its main theorems because it is so weak [Mayberry] |
17791 | Only second-order logic can capture mathematical structure up to isomorphism [Mayberry] |
17787 | Big logic has one fixed domain, but standard logic has a domain for each interpretation [Mayberry] |
17790 | No Löwenheim-Skolem logic can axiomatise real analysis [Mayberry] |
17778 | Axiomatiation relies on isomorphic structures being essentially the same [Mayberry] |
17779 | 'Classificatory' axioms aim at revealing similarity in morphology of structures [Mayberry] |
17780 | 'Eliminatory' axioms get rid of traditional ideal and abstract objects [Mayberry] |
17789 | No logic which can axiomatise arithmetic can be compact or complete [Mayberry] |
6297 | Euclid's geometry is synthetic, but Descartes produced an analytic version of it [Euclid, by Resnik] |
9603 | An assumption that there is a largest prime leads to a contradiction [Euclid, by Brown,JR] |
17784 | Real numbers can be eliminated, by axiom systems for complete ordered fields [Mayberry] |
9894 | A unit is that according to which each existing thing is said to be one [Euclid] |
17782 | Greek quantities were concrete, and ratio and proportion were their science [Mayberry] |
17781 | Real numbers were invented, as objects, to simplify and generalise 'quantity' [Mayberry] |
17799 | Cantor's infinite is an absolute, of all the sets or all the ordinal numbers [Mayberry] |
8738 | Postulate 2 says a line can be extended continuously [Euclid, by Shapiro] |
17797 | Cantor extended the finite (rather than 'taming the infinite') [Mayberry] |
17775 | If proof and definition are central, then mathematics needs and possesses foundations [Mayberry] |
17776 | The ultimate principles and concepts of mathematics are presumed, or grasped directly [Mayberry] |
17777 | Foundations need concepts, definition rules, premises, and proof rules [Mayberry] |
17804 | Axiom theories can't give foundations for mathematics - that's using axioms to explain axioms [Mayberry] |
10302 | Euclid says we can 'join' two points, but Hilbert says the straight line 'exists' [Euclid, by Bernays] |
22278 | Euclid relied on obvious properties in diagrams, as well as on his axioms [Potter on Euclid] |
8673 | Euclid's parallel postulate defines unique non-intersecting parallel lines [Euclid, by Friend] |
10250 | Euclid needs a principle of continuity, saying some lines must intersect [Shapiro on Euclid] |
14157 | Modern geometries only accept various parts of the Euclid propositions [Russell on Euclid] |
17792 | 1st-order PA is only interesting because of results which use 2nd-order PA [Mayberry] |
17793 | It is only 2nd-order isomorphism which suggested first-order PA completeness [Mayberry] |
1600 | Euclid's common notions or axioms are what we must have if we are to learn anything at all [Euclid, by Roochnik] |
17794 | Set theory is not just first-order ZF, because that is inadequate for mathematics [Mayberry] |
17802 | We don't translate mathematics into set theory, because it comes embodied in that way [Mayberry] |
17805 | Set theory is not just another axiomatised part of mathematics [Mayberry] |
17785 | Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry] |
16764 | The soul conserves the body, as we see by its dissolution when the soul leaves [Toletus] |