Combining Texts

All the ideas for 'The Fixation of Belief', 'Public Text and Common Reader' and 'What Required for Foundation for Maths?'

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51 ideas

1. Philosophy / E. Nature of Metaphysics / 3. Metaphysical Systems
Metaphysics does not rest on facts, but on what we are inclined to believe [Peirce]
2. Reason / A. Nature of Reason / 4. Aims of Reason
Reason aims to discover the unknown by thinking about the known [Peirce]
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]
7. Existence / D. Theories of Reality / 2. Realism
Realism is basic to the scientific method [Peirce]
7. Existence / D. Theories of Reality / 4. Anti-realism
If someone doubted reality, they would not actually feel dissatisfaction [Peirce]
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]
11. Knowledge Aims / A. Knowledge / 4. Belief / c. Aim of beliefs
The feeling of belief shows a habit which will determine our actions [Peirce]
We are entirely satisfied with a firm belief, even if it is false [Peirce]
We want true beliefs, but obviously we think our beliefs are true [Peirce]
A mere question does not stimulate a struggle for belief; there must be a real doubt [Peirce]
13. Knowledge Criteria / B. Internal Justification / 2. Pragmatic justification
We need our beliefs to be determined by some external inhuman permanency [Peirce]
13. Knowledge Criteria / B. Internal Justification / 4. Foundationalism / b. Basic beliefs
Demonstration does not rest on first principles of reason or sensation, but on freedom from actual doubt [Peirce]
13. Knowledge Criteria / C. External Justification / 1. External Justification
Doubts should be satisfied by some external permanency upon which thinking has no effect [Peirce]
13. Knowledge Criteria / D. Scepticism / 6. Scepticism Critique
Once doubt ceases, there is no point in continuing to argue [Peirce]
21. Aesthetics / A. Aesthetic Experience / 3. Taste
Literary meaning emerges in comparisons, and tradition shows which comparisons are relevant [Scruton]
21. Aesthetics / B. Nature of Art / 5. Art as Language
In literature, word replacement changes literary meaning [Scruton]
21. Aesthetics / C. Artistic Issues / 1. Artistic Intentions
Without intentions we can't perceive sculpture, but that is not the whole story [Scruton]
21. Aesthetics / C. Artistic Issues / 3. Artistic Representation
In aesthetic interest, even what is true is treated as though it were not [Scruton]
21. Aesthetics / C. Artistic Issues / 5. Objectivism in Art
We can be objective about conventions, but love of art is needed to understand its traditions [Scruton]
26. Natural Theory / B. Natural Kinds / 2. Defining Kinds
What is true of one piece of copper is true of another (unlike brass) [Peirce]
27. Natural Reality / G. Biology / 3. Evolution
Natural selection might well fill an animal's mind with pleasing thoughts rather than true ones [Peirce]
28. God / B. Proving God / 2. Proofs of Reason / d. Pascal's Wager
If death is annihilation, belief in heaven is a cheap pleasure with no disappointment [Peirce]