Combining Philosophers

All the ideas for Augustin-Louis Cauchy, Diogenes (Apoll) and Michal Walicki

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31 ideas

1. Philosophy / B. History of Ideas / 2. Ancient Thought
1542 Diogenes of Apollonia was the last natural scientist [Diogenes of Apollonia, by Simplicius]
4. Formal Logic / B. Propositional Logic PL / 1. Propositional Logic
17749 Post proved the consistency of propositional logic in 1921 [Walicki]
17765 Propositional language can only relate statements as the same or as different [Walicki]
4. Formal Logic / B. Propositional Logic PL / 3. Truth Tables
17764 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
17752 The empty set is useful for defining sets by properties, when the members are not yet known [Walicki]
17753 The empty set avoids having to take special precautions in case members vanish [Walicki]
4. Formal Logic / F. Set Theory ST / 6. Ordering in Sets
17759 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
17741 To determine the patterns in logic, one must identify its 'building blocks' [Walicki]
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
17747 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
17748 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
17761 A compact axiomatisation makes it possible to understand a field as a whole [Walicki]
17763 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
17757 Members of ordinals are ordinals, and also subsets of ordinals [Walicki]
17755 Ordinals are the empty set, union with the singleton, and any arbitrary union of ordinals [Walicki]
17756 The union of finite ordinals is the first 'limit ordinal'; 2ω is the second... [Walicki]
17760 Two infinite ordinals can represent a single infinite cardinal [Walicki]
17758 Ordinals are transitive sets of transitive sets; or transitive sets totally ordered by inclusion [Walicki]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / k. Infinitesimals
18085 Values that approach zero, becoming less than any quantity, are 'infinitesimals' [Cauchy]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / l. Limits
18084 When successive variable values approach a fixed value, that is its 'limit' [Cauchy]
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
17762 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
17754 Inductive proof depends on the choice of the ordering [Walicki]
9. Objects / F. Identity among Objects / 7. Indiscernible Objects
489 Each thing must be in some way unique [Diogenes of Apollonia]
10. Modality / A. Necessity / 2. Nature of Necessity
17742 Scotus based modality on semantic consistency, instead of on what the future could allow [Walicki]
12. Knowledge Sources / A. A Priori Knowledge / 2. Self-Evidence
483 Start a thesis with something undisputable [Diogenes of Apollonia]
12. Knowledge Sources / B. Perception / 6. Inference in Perception
1544 Perception must be an internal matter, because we can fail to perceive when we are preoccupied [Diogenes of Apollonia, by Theophrastus]
17. Mind and Body / E. Mind as Physical / 1. Physical Mind
24042 The older Diogenes said the soul is air, made of the smallest particles [Diogenes of Apollonia]
26. Natural Theory / A. Speculations on Nature / 2. Natural Purpose / a. Final purpose
5995 Diogenes of Apollonia offered the first teleological account of cosmology [Diogenes of Apollonia, by Robinson,TM]
26. Natural Theory / A. Speculations on Nature / 6. Early Matter Theories / c. Ultimate substances
488 Air is divine, because it is in and around everything, and arranges everything [Diogenes of Apollonia]
484 Everything is ultimately a variation of one underlying thing [Diogenes of Apollonia]
26. Natural Theory / A. Speculations on Nature / 6. Early Matter Theories / f. Ancient elements
486 Plants and animals can only come into existence if something fixes their species [Diogenes of Apollonia]
485 Things must retain their essential nature during change, or mixing would be impossible [Diogenes of Apollonia]