58 ideas
17774 | Definitions make our intuitions mathematically useful [Mayberry] |
17773 | Proof shows that it is true, but also why it must be true [Mayberry] |
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] |
17779 | 'Classificatory' axioms aim at revealing similarity in morphology of structures [Mayberry] |
17778 | Axiomatiation relies on isomorphic structures being essentially the same [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] |
17784 | Real numbers can be eliminated, by axiom systems for complete ordered fields [Mayberry] |
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] |
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] |
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] |
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] |
19482 | Current physics says matter and antimatter should have reduced to light at the big bang [New Sci.] |
19483 | CP violation shows a decay imbalance in matter and antimatter, leading to matter's dominance [New Sci.] |
17785 | Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry] |
19737 | A system can infer the structure of the world by making predictions about it [New Sci.] |
7295 | Maybe induction is only reliable IF reality is stable [Mitchell,A] |
19736 | Neural networks can extract the car-ness of a car, or the chair-ness of a chair [New Sci.] |
16419 | No one has yet devised a rationality test [New Sci.] |
16417 | About a third of variation in human intelligence is environmental [New Sci.] |
16418 | People can be highly intelligent, yet very stupid [New Sci.] |
19484 | Psychologists measure personality along five dimensions [New Sci.] |
19950 | Entropy is the only time-asymmetric law, so time may be linked to entropy [New Sci.] |
19478 | Light moves at a constant space-time speed, but its direction is in neither space nor time [New Sci.] |
19474 | Quantum states are measured by external time, of unknown origin [New Sci.] |
19473 | The Schrödinger equation describes the evolution of an object's wave function in Hilbert space [New Sci.] |
19953 | In string theory space-time has a grainy indivisible substructure [New Sci.] |
19476 | String theory needs at least 10 space-time dimensions [New Sci.] |
19954 | It is impossible for find a model of actuality among the innumerable models in string theory [New Sci.] |
19947 | Hilbert Space is an abstraction representing all possible states of a quantum system [New Sci.] |
19948 | Einstein's merging of time with space has left us confused about the nature of time [New Sci.] |
19475 | Relativity makes time and space jointly basic; quantum theory splits them, and prioritises time [New Sci.] |
19955 | Space-time may be a geometrical manifestation of quantum entanglement [New Sci.] |
19949 | Quantum theory relies on a clock outside the system - but where is it located? [New Sci.] |
19951 | Entropy is puzzling, so we may need to build new laws which include time directionality [New Sci.] |
19477 | General relativity predicts black holes, as former massive stars, and as galaxy centres [New Sci.] |
19952 | Black holes have entropy, but general relativity says they are unstructured, and lack entropy [New Sci.] |
16420 | 84.5 percent of the universe is made of dark matter [New Sci.] |
17604 | We are halfway to synthesising any molecule we want [New Sci.] |
17603 | Chemistry just needs the periodic table, and protons, electrons and neutrinos [New Sci.] |