73 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] |
| 17785 | Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry] |
| 7903 | The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna] |
| 21167 | Gravity is unusual, in that it always attracts and never repels [New Sci.] |
| 21176 | In the Big Bang general relativity fails, because gravity is too powerful [New Sci.] |
| 21147 | Quantum electrodynamics incorporates special relativity and quantum mechanics [New Sci.] |
| 21155 | Photons have zero rest mass, so virtual photons have infinite range [New Sci.] |
| 21161 | In the standard model all the fundamental force fields merge at extremely high energies [New Sci.] |
| 21146 | Electrons move fast, so are subject to special relativity [New Sci.] |
| 21148 | The strong force is repulsive at short distances, strong at medium, and fades at long [New Sci.] |
| 21151 | Gluons, the particles carrying the strong force, interact because of their colour charge [New Sci.] |
| 21152 | The strong force binds quarks tight, and the nucleus more weakly [New Sci.] |
| 21142 | Classifying hadrons revealed two symmetry patterns, produced by three basic elements [New Sci.] |
| 21143 | Quarks in threes can build hadrons with spin ½ or with spin 3/2 [New Sci.] |
| 21150 | Three different colours of quark (as in the proton) can cancel out to give no colour [New Sci.] |
| 21145 | The four fundamental forces (gravity, electromagnetism, weak and strong) are the effects of particles [New Sci.] |
| 21153 | The weak force explains beta decay, and the change of type by quarks and leptons [New Sci.] |
| 21154 | Three particles enable the weak force: W+ and W- are charged, and Z° is not [New Sci.] |
| 21156 | The weak force particles are heavy, so the force has a short range [New Sci.] |
| 21164 | Why do the charges of the very different proton and electron perfectly match up? [New Sci.] |
| 21170 | The Standard Model cannot explain dark energy, survival of matter, gravity, or force strength [New Sci.] |
| 21140 | Spin is a built-in ration of angular momentum [New Sci.] |
| 21149 | Quarks have red, green or blue colour charge (akin to electric charge) [New Sci.] |
| 21158 | Fermions, with spin ½, are antisocial, and cannot share quantum states [New Sci.] |
| 21165 | Spin is akin to rotation, and is easily measured in a magnetic field [New Sci.] |
| 21157 | Particles are spread out, with wave-like properties, and higher energy shortens the wavelength [New Sci.] |
| 21163 | The mass of protons and neutrinos is mostly binding energy, not the quarks [New Sci.] |
| 21168 | Gravitional mass turns out to be the same as inertial mass [New Sci.] |
| 21138 | Neutrons are slightly heavier than protons, and decay into them by emitting an electron [New Sci.] |
| 21144 | Top, bottom, charm and strange quarks quickly decay into up and down [New Sci.] |
| 21141 | Neutrinos were proposed as the missing energy in neutron beta decay [New Sci.] |
| 21169 | Only neutrinos spin anticlockwise [New Sci.] |
| 21166 | Standard antineutrinos have opposite spin and opposite lepton number [New Sci.] |
| 21171 | The symmetry of unified electromagnetic and weak forces was broken by the Higgs field [New Sci.] |
| 21179 | Supersymmetric string theory can be expressed using loop quantum gravity [New Sci.] |
| 21178 | String theory is now part of 11-dimensional M-Theory, involving p-branes [New Sci.] |
| 21175 | String theory might be tested by colliding strings to make bigger 'stringballs' [New Sci.] |
| 21177 | String theory offers a quantum theory of gravity, by describing the graviton [New Sci.] |
| 21162 | Only supersymmetry offers to incorporate gravity into the scheme [New Sci.] |
| 21159 | Supersymmetry has extra heavy bosons and heavy fermions [New Sci.] |
| 21173 | Supersymmetry says particles and superpartners were unities, but then split [New Sci.] |
| 21172 | The evidence for supersymmetry keeps failing to appear [New Sci.] |
| 21160 | The Higgs field means even low energy space is not empty [New Sci.] |
| 21174 | Dark matter must have mass, to produce gravity, and no electric charge, to not reflect light [New Sci.] |