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All the ideas for 'Parmenides', 'Lectures 1930-32 (student notes)' and 'Philosophy of Mathematics'

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135 ideas

1. Philosophy / C. History of Philosophy / 1. History of Philosophy
The history of philosophy only matters if the subject is a choice between rival theories [Wittgenstein]
1. Philosophy / D. Nature of Philosophy / 5. Aims of Philosophy / d. Philosophy as puzzles
Philosophy tries to be rid of certain intellectual puzzles, irrelevant to daily life [Wittgenstein]
1. Philosophy / D. Nature of Philosophy / 7. Despair over Philosophy
Philosophers express puzzlement, but don't clearly state the puzzle [Wittgenstein]
1. Philosophy / F. Analytic Philosophy / 5. Linguistic Analysis
We don't need a theory of truth, because we use the word perfectly well [Wittgenstein]
1. Philosophy / F. Analytic Philosophy / 7. Limitations of Analysis
We already know what we want to know, and analysis gives us no new facts [Wittgenstein]
2. Reason / A. Nature of Reason / 1. On Reason
When questions are doubtful we should concentrate not on objects but on ideas of the intellect [Plato]
2. Reason / A. Nature of Reason / 6. Coherence
Coherence is a primitive, intuitive notion, not reduced to something formal [Shapiro]
2. Reason / B. Laws of Thought / 5. Opposites
Opposites are as unlike as possible [Plato]
2. Reason / C. Styles of Reason / 1. Dialectic
Plato's 'Parmenides' is the greatest artistic achievement of the ancient dialectic [Hegel on Plato]
2. Reason / D. Definition / 7. Contextual Definition
An 'implicit definition' gives a direct description of the relations of an entity [Shapiro]
2. Reason / F. Fallacies / 8. Category Mistake / a. Category mistakes
Words of the same kind can be substituted in a proposition without producing nonsense [Wittgenstein]
2. Reason / F. Fallacies / 8. Category Mistake / b. Category mistake as syntactic
Talking nonsense is not following the rules [Wittgenstein]
Grammar says that saying 'sound is red' is not false, but nonsense [Wittgenstein]
3. Truth / A. Truth Problems / 2. Defining Truth
There is no theory of truth, because it isn't a concept [Wittgenstein]
3. Truth / C. Correspondence Truth / 1. Correspondence Truth
All thought has the logical form of reality [Wittgenstein]
4. Formal Logic / D. Modal Logic ML / 1. Modal Logic
Modal operators are usually treated as quantifiers [Shapiro]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
Axiom of Choice: some function has a value for every set in a given set [Shapiro]
The Axiom of Choice seems to license an infinite amount of choosing [Shapiro]
4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
Anti-realists reject set theory [Shapiro]
5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
In logic nothing is hidden [Wittgenstein]
5. Theory of Logic / B. Logical Consequence / 2. Types of Consequence
The two standard explanations of consequence are semantic (in models) and deductive [Shapiro]
5. Theory of Logic / B. Logical Consequence / 5. Modus Ponens
Intuitionism only sanctions modus ponens if all three components are proved [Shapiro]
5. Theory of Logic / C. Ontology of Logic / 1. Ontology of Logic
Either logic determines objects, or objects determine logic, or they are separate [Shapiro]
5. Theory of Logic / C. Ontology of Logic / 4. Logic by Convention
Laws of logic are like laws of chess - if you change them, it's just a different game [Wittgenstein]
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
The law of excluded middle might be seen as a principle of omniscience [Shapiro]
5. Theory of Logic / D. Assumptions for Logic / 3. Contradiction
Contradiction is between two rules, not between rule and reality [Wittgenstein]
5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / a. Logical connectives
Classical connectives differ from their ordinary language counterparts; '∧' is timeless, unlike 'and' [Shapiro]
5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / c. not
We may correctly use 'not' without making the rule explicit [Wittgenstein]
5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / d. and
Saying 'and' has meaning is just saying it works in a sentence [Wittgenstein]
5. Theory of Logic / E. Structures of Logic / 5. Functions in Logic
A function is just an arbitrary correspondence between collections [Shapiro]
5. Theory of Logic / F. Referring in Logic / 1. Naming / a. Names
A person's name doesn't mean their body; bodies don't sit down, and their existence can be denied [Wittgenstein]
5. Theory of Logic / G. Quantification / 6. Plural Quantification
Maybe plural quantifiers should be understood in terms of classes or sets [Shapiro]
5. Theory of Logic / I. Semantics of Logic / 4. Satisfaction
A sentence is 'satisfiable' if it has a model [Shapiro]
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
Model theory deals with relations, reference and extensions [Shapiro]
The central notion of model theory is the relation of 'satisfaction' [Shapiro]
5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
Theory ontology is never complete, but is only determined 'up to isomorphism' [Shapiro]
The set-theoretical hierarchy contains as many isomorphism types as possible [Shapiro]
5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
Any theory with an infinite model has a model of every infinite cardinality [Shapiro]
5. Theory of Logic / L. Paradox / 3. Antinomies
Plato found antinomies in ideas, Kant in space and time, and Bradley in relations [Plato, by Ryle]
Plato's 'Parmenides' is perhaps the best collection of antinomies ever made [Russell on Plato]
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
Virtually all of mathematics can be modeled in set theory [Shapiro]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
We don't get 'nearer' to something by adding decimals to 1.1412... (root-2) [Wittgenstein]
Real numbers are thought of as either Cauchy sequences or Dedekind cuts [Shapiro]
Understanding the real-number structure is knowing usage of the axiomatic language of analysis [Shapiro]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
Cuts are made by the smallest upper or largest lower number, some of them not rational [Shapiro]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
Infinity is not a number, so doesn't say how many; it is the property of a law [Wittgenstein]
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
There is no grounding for mathematics that is more secure than mathematics [Shapiro]
6. Mathematics / B. Foundations for Mathematics / 2. Proof in Mathematics
For intuitionists, proof is inherently informal [Shapiro]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
Natural numbers just need an initial object, successors, and an induction principle [Shapiro]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / b. Greek arithmetic
Mathematics originally concerned the continuous (geometry) and the discrete (arithmetic) [Shapiro]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory
Mathematical foundations may not be sets; categories are a popular rival [Shapiro]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
Baseball positions and chess pieces depend entirely on context [Shapiro]
The even numbers have the natural-number structure, with 6 playing the role of 3 [Shapiro]
Could infinite structures be apprehended by pattern recognition? [Shapiro]
The 4-pattern is the structure common to all collections of four objects [Shapiro]
The main mathematical structures are algebraic, ordered, and topological [Shapiro]
Some structures are exemplified by both abstract and concrete [Shapiro]
Mathematical structures are defined by axioms, or in set theory [Shapiro]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / b. Varieties of structuralism
The main versions of structuralism are all definitionally equivalent [Shapiro]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / c. Nominalist structuralism
Is there is no more to structures than the systems that exemplify them? [Shapiro]
Number statements are generalizations about number sequences, and are bound variables [Shapiro]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / d. Platonist structuralism
Because one structure exemplifies several systems, a structure is a one-over-many [Shapiro]
There is no 'structure of all structures', just as there is no set of all sets [Shapiro]
Shapiro's structuralism says model theory (comparing structures) is the essence of mathematics [Shapiro, by Friend]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
Does someone using small numbers really need to know the infinite structure of arithmetic? [Shapiro]
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
One is, so numbers exist, so endless numbers exist, and each one must partake of being [Plato]
We distinguish realism 'in ontology' (for objects), and 'in truth-value' (for being either true or false) [Shapiro]
If mathematical objects are accepted, then a number of standard principles will follow [Shapiro]
Platonists claim we can state the essence of a number without reference to the others [Shapiro]
Platonism must accept that the Peano Axioms could all be false [Shapiro]
6. Mathematics / C. Sources of Mathematics / 2. Intuition of Mathematics
Intuition is an outright hindrance to five-dimensional geometry [Shapiro]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
A stone is a position in some pattern, and can be viewed as an object, or as a location [Shapiro]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism
Can the ideal constructor also destroy objects? [Shapiro]
Presumably nothing can block a possible dynamic operation? [Shapiro]
7. Existence / A. Nature of Existence / 1. Nature of Existence
Can we discover whether a deck is fifty-two cards, or a person is time-slices or molecules? [Shapiro]
7. Existence / A. Nature of Existence / 3. Being / c. Becoming
The one was and is and will be and was becoming and is becoming and will become [Plato]
7. Existence / A. Nature of Existence / 3. Being / f. Primary being
Plato's Parmenides has a three-part theory, of Primal One, a One-Many, and a One-and-Many [Plato, by Plotinus]
7. Existence / C. Structure of Existence / 7. Abstract/Concrete / a. Abstract/concrete
The abstract/concrete boundary now seems blurred, and would need a defence [Shapiro]
Mathematicians regard arithmetic as concrete, and group theory as abstract [Shapiro]
7. Existence / D. Theories of Reality / 3. Reality
Absolute ideas, such as the Good and the Beautiful, cannot be known by us [Plato]
7. Existence / D. Theories of Reality / 7. Fictionalism
Fictionalism eschews the abstract, but it still needs the possible (without model theory) [Shapiro]
Structuralism blurs the distinction between mathematical and ordinary objects [Shapiro]
7. Existence / D. Theories of Reality / 8. Facts / b. Types of fact
There are no positive or negative facts; these are just the forms of propositions [Wittgenstein]
8. Modes of Existence / D. Universals / 2. Need for Universals
If you deny that each thing always stays the same, you destroy the possibility of discussion [Plato]
You must always mean the same thing when you utter the same name [Plato]
8. Modes of Existence / D. Universals / 5. Universals as Concepts
Using 'green' is a commitment to future usage of 'green' [Wittgenstein]
8. Modes of Existence / D. Universals / 6. Platonic Forms / a. Platonic Forms
Greatness and smallness must exist, to be opposed to one another, and come into being in things [Plato]
Plato moves from Forms to a theory of genera and principles in his later work [Plato, by Frede,M]
It would be absurd to think there were abstract Forms for vile things like hair, mud and dirt [Plato]
The concept of a master includes the concept of a slave [Plato]
If admirable things have Forms, maybe everything else does as well [Plato]
If absolute ideas existed in us, they would cease to be absolute [Plato]
8. Modes of Existence / D. Universals / 6. Platonic Forms / b. Partaking
If things partake of ideas, this implies either that everything thinks, or that everything actually is thought [Plato]
Participation is not by means of similarity, so we are looking for some other method of participation [Plato]
The whole idea of each Form must be found in each thing which participates in it [Plato]
Each idea is in all its participants at once, just as daytime is a unity but in many separate places at once [Plato]
If things are made alike by participating in something, that thing will be the absolute idea [Plato]
8. Modes of Existence / D. Universals / 6. Platonic Forms / c. Self-predication
Nothing can be like an absolute idea, because a third idea intervenes to make them alike (leading to a regress) [Plato]
If absolute greatness and great things are seen as the same, another thing appears which makes them seem great [Plato]
9. Objects / A. Existence of Objects / 1. Physical Objects
The notion of 'object' is at least partially structural and mathematical [Shapiro]
9. Objects / B. Unity of Objects / 1. Unifying an Object / b. Unifying aggregates
Parts must belong to a created thing with a distinct form [Plato]
9. Objects / B. Unity of Objects / 3. Unity Problems / e. Vague objects
A blurry border is still a border [Shapiro]
9. Objects / C. Structure of Objects / 5. Composition of an Object
In Parmenides, if composition is identity, a whole is nothing more than its parts [Plato, by Harte,V]
9. Objects / C. Structure of Objects / 8. Parts of Objects / a. Parts of objects
Plato says only a one has parts, and a many does not [Plato, by Harte,V]
Anything which has parts must be one thing, and parts are of a one, not of a many [Plato]
9. Objects / C. Structure of Objects / 8. Parts of Objects / c. Wholes from parts
It seems that the One must be composed of parts, which contradicts its being one [Plato]
9. Objects / F. Identity among Objects / 6. Identity between Objects
Two things relate either as same or different, or part of a whole, or the whole of the part [Plato]
10. Modality / A. Necessity / 6. Logical Necessity
Logical modalities may be acceptable, because they are reducible to satisfaction in models [Shapiro]
10. Modality / C. Sources of Modality / 3. Necessity by Convention
For each necessity in the world there is an arbitrary rule of language [Wittgenstein]
10. Modality / E. Possible worlds / 1. Possible Worlds / a. Possible worlds
Why does the 'myth' of possible worlds produce correct modal logic? [Shapiro]
11. Knowledge Aims / A. Knowledge / 2. Understanding
Understanding is translation, into action or into other symbols [Wittgenstein]
12. Knowledge Sources / B. Perception / 4. Sense Data / a. Sense-data theory
We live in sense-data, but talk about physical objects [Wittgenstein]
12. Knowledge Sources / B. Perception / 4. Sense Data / d. Sense-data problems
Part of what we mean by stating the facts is the way we tend to experience them [Wittgenstein]
12. Knowledge Sources / E. Direct Knowledge / 4. Memory
If you remember wrongly, then there must be some other criterion than your remembering [Wittgenstein]
14. Science / D. Explanation / 1. Explanation / b. Aims of explanation
Explanation and understanding are the same [Wittgenstein]
Explanation gives understanding by revealing the full multiplicity of the thing [Wittgenstein]
14. Science / D. Explanation / 2. Types of Explanation / i. Explanations by mechanism
A machine strikes us as being a rule of movement [Wittgenstein]
14. Science / D. Explanation / 3. Best Explanation / a. Best explanation
If an explanation is good, the symbol is used properly in the future [Wittgenstein]
15. Nature of Minds / C. Capacities of Minds / 3. Abstraction by mind
We apprehend small, finite mathematical structures by abstraction from patterns [Shapiro]
18. Thought / A. Modes of Thought / 1. Thought
Thought is an activity which we perform by the expression of it [Wittgenstein]
18. Thought / E. Abstraction / 2. Abstracta by Selection
Simple types can be apprehended through their tokens, via abstraction [Shapiro]
18. Thought / E. Abstraction / 3. Abstracta by Ignoring
We can apprehend structures by focusing on or ignoring features of patterns [Shapiro]
We can focus on relations between objects (like baseballers), ignoring their other features [Shapiro]
18. Thought / E. Abstraction / 7. Abstracta by Equivalence
Abstract objects might come by abstraction over an equivalence class of base entities [Shapiro]
19. Language / A. Nature of Meaning / 4. Meaning as Truth-Conditions
A proposition draws a line around the facts which agree with it [Wittgenstein]
19. Language / A. Nature of Meaning / 5. Meaning as Verification
The meaning of a proposition is the mode of its verification [Wittgenstein]
19. Language / A. Nature of Meaning / 7. Meaning Holism / a. Sentence meaning
Words function only in propositions, like levers in a machine [Wittgenstein]
19. Language / D. Propositions / 1. Propositions
A proposition is any expression which can be significantly negated [Wittgenstein]
25. Social Practice / E. Policies / 5. Education / c. Teaching
Only a great person can understand the essence of things, and an even greater person can teach it [Plato]
26. Natural Theory / A. Speculations on Nature / 6. Early Matter Theories / d. The unlimited
The unlimited has no shape and is endless [Plato]
26. Natural Theory / A. Speculations on Nature / 6. Early Matter Theories / e. The One
Some things do not partake of the One [Plato]
The only movement possible for the One is in space or in alteration [Plato]
Everything partakes of the One in some way [Plato]
26. Natural Theory / D. Laws of Nature / 11. Against Laws of Nature
Laws of nature are an aspect of the phenomena, and are just our mode of description [Wittgenstein]
28. God / B. Proving God / 2. Proofs of Reason / a. Ontological Proof
We couldn't discuss the non-existence of the One without knowledge of it [Plato]