Combining Texts

All the ideas for 'Mahaprajnaparamitashastra', 'What Required for Foundation for Maths?' and 'Why the Universe Exists'

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73 ideas

2. Reason / D. Definition / 2. Aims of Definition
Definitions make our intuitions mathematically useful [Mayberry]
2. Reason / E. Argument / 6. Conclusive Proof
Proof shows that it is true, but also why it must be true [Mayberry]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
Set theory can't be axiomatic, because it is needed to express the very notion of axiomatisation [Mayberry]
There is a semi-categorical axiomatisation of set-theory [Mayberry]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
The misnamed Axiom of Infinity says the natural numbers are finite in size [Mayberry]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
The set hierarchy doesn't rely on the dubious notion of 'generating' them [Mayberry]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
Limitation of size is part of the very conception of a set [Mayberry]
5. Theory of Logic / A. Overview of Logic / 2. History of Logic
The mainstream of modern logic sees it as a branch of mathematics [Mayberry]
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
First-order logic only has its main theorems because it is so weak [Mayberry]
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
Only second-order logic can capture mathematical structure up to isomorphism [Mayberry]
5. Theory of Logic / G. Quantification / 2. Domain of Quantification
Big logic has one fixed domain, but standard logic has a domain for each interpretation [Mayberry]
5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
No Löwenheim-Skolem logic can axiomatise real analysis [Mayberry]
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
'Classificatory' axioms aim at revealing similarity in morphology of structures [Mayberry]
Axiomatiation relies on isomorphic structures being essentially the same [Mayberry]
'Eliminatory' axioms get rid of traditional ideal and abstract objects [Mayberry]
5. Theory of Logic / K. Features of Logics / 6. Compactness
No logic which can axiomatise arithmetic can be compact or complete [Mayberry]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
Real numbers can be eliminated, by axiom systems for complete ordered fields [Mayberry]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / b. Quantity
Greek quantities were concrete, and ratio and proportion were their science [Mayberry]
Real numbers were invented, as objects, to simplify and generalise 'quantity' [Mayberry]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
Cantor's infinite is an absolute, of all the sets or all the ordinal numbers [Mayberry]
Cantor extended the finite (rather than 'taming the infinite') [Mayberry]
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
If proof and definition are central, then mathematics needs and possesses foundations [Mayberry]
The ultimate principles and concepts of mathematics are presumed, or grasped directly [Mayberry]
Foundations need concepts, definition rules, premises, and proof rules [Mayberry]
Axiom theories can't give foundations for mathematics - that's using axioms to explain axioms [Mayberry]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
1st-order PA is only interesting because of results which use 2nd-order PA [Mayberry]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
It is only 2nd-order isomorphism which suggested first-order PA completeness [Mayberry]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
Set theory is not just first-order ZF, because that is inadequate for mathematics [Mayberry]
We don't translate mathematics into set theory, because it comes embodied in that way [Mayberry]
Set theory is not just another axiomatised part of mathematics [Mayberry]
9. Objects / A. Existence of Objects / 2. Abstract Objects / a. Nature of abstracta
Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry]
23. Ethics / C. Virtue Theory / 3. Virtues / a. Virtues
The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna]
27. Natural Reality / A. Classical Physics / 1. Mechanics / d. Gravity
Gravity is unusual, in that it always attracts and never repels [New Sci.]
27. Natural Reality / B. Modern Physics / 1. Relativity / b. General relativity
In the Big Bang general relativity fails, because gravity is too powerful [New Sci.]
27. Natural Reality / B. Modern Physics / 2. Electrodynamics / a. Electrodynamics
Quantum electrodynamics incorporates special relativity and quantum mechanics [New Sci.]
Photons have zero rest mass, so virtual photons have infinite range [New Sci.]
27. Natural Reality / B. Modern Physics / 2. Electrodynamics / b. Fields
In the standard model all the fundamental force fields merge at extremely high energies [New Sci.]
27. Natural Reality / B. Modern Physics / 2. Electrodynamics / c. Electrons
Electrons move fast, so are subject to special relativity [New Sci.]
27. Natural Reality / B. Modern Physics / 3. Chromodynamics / a. Chromodynamics
The strong force is repulsive at short distances, strong at medium, and fades at long [New Sci.]
Gluons, the particles carrying the strong force, interact because of their colour charge [New Sci.]
The strong force binds quarks tight, and the nucleus more weakly [New Sci.]
27. Natural Reality / B. Modern Physics / 3. Chromodynamics / b. Quarks
Classifying hadrons revealed two symmetry patterns, produced by three basic elements [New Sci.]
Quarks in threes can build hadrons with spin ½ or with spin 3/2 [New Sci.]
Three different colours of quark (as in the proton) can cancel out to give no colour [New Sci.]
27. Natural Reality / B. Modern Physics / 4. Standard Model / b. Standard model
The four fundamental forces (gravity, electromagnetism, weak and strong) are the effects of particles [New Sci.]
The weak force explains beta decay, and the change of type by quarks and leptons [New Sci.]
Three particles enable the weak force: W+ and W- are charged, and Z° is not [New Sci.]
The weak force particles are heavy, so the force has a short range [New Sci.]
Why do the charges of the very different proton and electron perfectly match up? [New Sci.]
The Standard Model cannot explain dark energy, survival of matter, gravity, or force strength [New Sci.]
27. Natural Reality / B. Modern Physics / 4. Standard Model / c. Particle properties
Spin is a built-in ration of angular momentum [New Sci.]
Quarks have red, green or blue colour charge (akin to electric charge) [New Sci.]
Fermions, with spin ½, are antisocial, and cannot share quantum states [New Sci.]
Spin is akin to rotation, and is easily measured in a magnetic field [New Sci.]
Particles are spread out, with wave-like properties, and higher energy shortens the wavelength [New Sci.]
27. Natural Reality / B. Modern Physics / 4. Standard Model / d. Mass
The mass of protons and neutrinos is mostly binding energy, not the quarks [New Sci.]
Gravitional mass turns out to be the same as inertial mass [New Sci.]
27. Natural Reality / B. Modern Physics / 4. Standard Model / e. Protons
Neutrons are slightly heavier than protons, and decay into them by emitting an electron [New Sci.]
Top, bottom, charm and strange quarks quickly decay into up and down [New Sci.]
27. Natural Reality / B. Modern Physics / 4. Standard Model / f. Neutrinos
Neutrinos were proposed as the missing energy in neutron beta decay [New Sci.]
Only neutrinos spin anticlockwise [New Sci.]
27. Natural Reality / B. Modern Physics / 4. Standard Model / g. Anti-matter
Standard antineutrinos have opposite spin and opposite lepton number [New Sci.]
27. Natural Reality / B. Modern Physics / 5. Unified Models / a. Electro-weak unity
The symmetry of unified electromagnetic and weak forces was broken by the Higgs field [New Sci.]
27. Natural Reality / B. Modern Physics / 5. Unified Models / b. String theory
Supersymmetric string theory can be expressed using loop quantum gravity [New Sci.]
String theory is now part of 11-dimensional M-Theory, involving p-branes [New Sci.]
String theory might be tested by colliding strings to make bigger 'stringballs' [New Sci.]
String theory offers a quantum theory of gravity, by describing the graviton [New Sci.]
27. Natural Reality / B. Modern Physics / 5. Unified Models / c. Supersymmetry
Only supersymmetry offers to incorporate gravity into the scheme [New Sci.]
Supersymmetry has extra heavy bosons and heavy fermions [New Sci.]
Supersymmetry says particles and superpartners were unities, but then split [New Sci.]
The evidence for supersymmetry keeps failing to appear [New Sci.]
27. Natural Reality / C. Space / 4. Substantival Space
The Higgs field means even low energy space is not empty [New Sci.]
27. Natural Reality / E. Cosmology / 8. Dark Matter
Dark matter must have mass, to produce gravity, and no electric charge, to not reflect light [New Sci.]