Combining Texts

All the ideas for 'Defending the Axioms', 'Introduction to Mathematical Logic' and 'Coherence: The Price is Right'

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33 ideas

2. Reason / A. Nature of Reason / 6. Coherence
Coherence problems have positive and negative restraints; solutions maximise constraint satisfaction [Thagard]
Coherence is explanatory, deductive, conceptual, analogical, perceptual, and deliberative [Thagard]
Explanatory coherence needs symmetry,explanation,analogy,data priority, contradiction,competition,acceptance [Thagard]
3. Truth / A. Truth Problems / 6. Verisimilitude
Verisimilitude comes from including more phenomena, and revealing what underlies [Thagard]
4. Formal Logic / B. Propositional Logic PL / 1. Propositional Logic
Post proved the consistency of propositional logic in 1921 [Walicki]
Propositional language can only relate statements as the same or as different [Walicki]
4. Formal Logic / B. Propositional Logic PL / 3. Truth Tables
Boolean connectives are interpreted as functions on the set {1,0} [Walicki]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
The empty set is useful for defining sets by properties, when the members are not yet known [Walicki]
The empty set avoids having to take special precautions in case members vanish [Walicki]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres [Maddy]
4. Formal Logic / F. Set Theory ST / 6. Ordering in Sets
Ordinals play the central role in set theory, providing the model of well-ordering [Walicki]
5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
To determine the patterns in logic, one must identify its 'building blocks' [Walicki]
5. Theory of Logic / C. Ontology of Logic / 3. If-Thenism
Critics of if-thenism say that not all starting points, even consistent ones, are worth studying [Maddy]
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
A 'model' of a theory specifies interpreting a language in a domain to make all theorems true [Walicki]
5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
The L-S Theorem says no theory (even of reals) says more than a natural number theory [Walicki]
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
Hilbert's geometry and Dedekind's real numbers were role models for axiomatization [Maddy]
If two mathematical themes coincide, that suggest a single deep truth [Maddy]
A compact axiomatisation makes it possible to understand a field as a whole [Walicki]
Axiomatic systems are purely syntactic, and do not presuppose any interpretation [Walicki]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
Members of ordinals are ordinals, and also subsets of ordinals [Walicki]
Ordinals are the empty set, union with the singleton, and any arbitrary union of ordinals [Walicki]
The union of finite ordinals is the first 'limit ordinal'; 2ω is the second... [Walicki]
Two infinite ordinals can represent a single infinite cardinal [Walicki]
Ordinals are transitive sets of transitive sets; or transitive sets totally ordered by inclusion [Walicki]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
Every infinite set of reals is either countable or of the same size as the full set of reals [Maddy]
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
In non-Euclidean geometry, all Euclidean theorems are valid that avoid the fifth postulate [Walicki]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
Inductive proof depends on the choice of the ordering [Walicki]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
Set-theory tracks the contours of mathematical depth and fruitfulness [Maddy]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
The connection of arithmetic to perception has been idealised away in modern infinitary mathematics [Maddy]
10. Modality / A. Necessity / 2. Nature of Necessity
Scotus based modality on semantic consistency, instead of on what the future could allow [Walicki]
14. Science / B. Scientific Theories / 1. Scientific Theory
Neither a priori rationalism nor sense data empiricism account for scientific knowledge [Thagard]
14. Science / C. Induction / 6. Bayes's Theorem
Bayesian inference is forced to rely on approximations [Thagard]
14. Science / D. Explanation / 3. Best Explanation / a. Best explanation
The best theory has the highest subjective (Bayesian) probability? [Thagard]