Combining Texts

All the ideas for 'Leibniz', 'Logical Properties' and 'Introducing the Philosophy of Mathematics'

expand these ideas     |    start again     |     specify just one area for these texts


82 ideas

2. Reason / D. Definition / 1. Definitions
Definitions identify two concepts, so they presuppose identity [McGinn]
2. Reason / D. Definition / 8. Impredicative Definition
An 'impredicative' definition seems circular, because it uses the term being defined [Friend]
2. Reason / D. Definition / 10. Stipulative Definition
Classical definitions attempt to refer, but intuitionist/constructivist definitions actually create objects [Friend]
2. Reason / E. Argument / 5. Reductio ad Absurdum
Reductio ad absurdum proves an idea by showing that its denial produces contradiction [Friend]
2. Reason / F. Fallacies / 2. Infinite Regress
Regresses are only vicious in the context of an explanation [McGinn]
3. Truth / A. Truth Problems / 4. Uses of Truth
Truth is a method of deducing facts from propositions [McGinn]
3. Truth / A. Truth Problems / 8. Subjective Truth
Anti-realists see truth as our servant, and epistemically contrained [Friend]
3. Truth / C. Correspondence Truth / 3. Correspondence Truth critique
'Snow does not fall' corresponds to snow does fall [McGinn]
The idea of truth is built into the idea of correspondence [McGinn]
3. Truth / D. Coherence Truth / 2. Coherence Truth Critique
The coherence theory of truth implies idealism, because facts are just coherent beliefs [McGinn]
3. Truth / H. Deflationary Truth / 3. Minimalist Truth
Truth is the property of propositions that makes it possible to deduce facts [McGinn]
Without the disquotation device for truth, you could never form beliefs from others' testimony [McGinn]
4. Formal Logic / B. Propositional Logic PL / 3. Truth Tables
In classical/realist logic the connectives are defined by truth-tables [Friend]
4. Formal Logic / E. Nonclassical Logics / 2. Intuitionist Logic
Double negation elimination is not valid in intuitionist logic [Friend]
4. Formal Logic / E. Nonclassical Logics / 6. Free Logic
Free logic was developed for fictional or non-existent objects [Friend]
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
A 'proper subset' of A contains only members of A, but not all of them [Friend]
A 'powerset' is all the subsets of a set [Friend]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
Set theory makes a minimum ontological claim, that the empty set exists [Friend]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / d. Infinite Sets
Infinite sets correspond one-to-one with a subset [Friend]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
Major set theories differ in their axioms, and also over the additional axioms of choice and infinity [Friend]
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
The law of excluded middle is syntactic; it just says A or not-A, not whether they are true or false [Friend]
5. Theory of Logic / D. Assumptions for Logic / 4. Identity in Logic
In 'x is F and x is G' we must assume the identity of x in the two statements [McGinn]
Both non-contradiction and excluded middle need identity in their formulation [McGinn]
Identity is unitary, indefinable, fundamental and a genuine relation [McGinn]
5. Theory of Logic / G. Quantification / 1. Quantification
The quantifier is overrated as an analytical tool [McGinn]
Existential quantifiers just express the quantity of things, leaving existence to the predicate 'exists' [McGinn]
5. Theory of Logic / G. Quantification / 3. Objectual Quantification
'Partial quantifier' would be a better name than 'existential quantifier', as no existence would be implied [McGinn]
5. Theory of Logic / G. Quantification / 7. Unorthodox Quantification
We need an Intentional Quantifier ("some of the things we talk about.."), so existence goes into the proposition [McGinn]
Intuitionists read the universal quantifier as "we have a procedure for checking every..." [Friend]
5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / a. Set theory paradoxes
Paradoxes can be solved by talking more loosely of 'classes' instead of 'sets' [Friend]
5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / c. Burali-Forti's paradox
The Burali-Forti paradox asks whether the set of all ordinals is itself an ordinal [Friend]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
The 'integers' are the positive and negative natural numbers, plus zero [Friend]
The 'rational' numbers are those representable as fractions [Friend]
A number is 'irrational' if it cannot be represented as a fraction [Friend]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
The natural numbers are primitive, and the ordinals are up one level of abstraction [Friend]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / f. Cardinal numbers
Cardinal numbers answer 'how many?', with the order being irrelevant [Friend]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
The 'real' numbers (rationals and irrationals combined) is the Continuum, which has no gaps [Friend]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / h. Ordinal infinity
Raising omega to successive powers of omega reveal an infinity of infinities [Friend]
The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega [Friend]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / j. Infinite divisibility
Between any two rational numbers there is an infinite number of rational numbers [Friend]
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
Is mathematics based on sets, types, categories, models or topology? [Friend]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
Most mathematical theories can be translated into the language of set theory [Friend]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
The number 8 in isolation from the other numbers is of no interest [Friend]
In structuralism the number 8 is not quite the same in different structures, only equivalent [Friend]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / b. Varieties of structuralism
Are structures 'ante rem' (before reality), or are they 'in re' (grounded in physics)? [Friend]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / c. Nominalist structuralism
Structuralist says maths concerns concepts about base objects, not base objects themselves [Friend]
Structuralism focuses on relations, predicates and functions, with objects being inessential [Friend]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / d. Platonist structuralism
'In re' structuralism says that the process of abstraction is pattern-spotting [Friend]
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
The big problem for platonists is epistemic: how do we perceive, intuit, know or detect mathematical facts? [Friend]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / b. Indispensability of mathematics
Mathematics should be treated as true whenever it is indispensable to our best physical theory [Friend]
6. Mathematics / C. Sources of Mathematics / 7. Formalism
Formalism is unconstrained, so cannot indicate importance, or directions for research [Friend]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism
Constructivism rejects too much mathematics [Friend]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
Intuitionists typically retain bivalence but reject the law of excluded middle [Friend]
7. Existence / A. Nature of Existence / 1. Nature of Existence
Existence is a primary quality, non-existence a secondary quality [McGinn]
7. Existence / A. Nature of Existence / 6. Criterion for Existence
Existence can't be analysed as instantiating a property, as instantiation requires existence [McGinn]
We can't analyse the sentence 'something exists' in terms of instantiated properties [McGinn]
7. Existence / D. Theories of Reality / 3. Reality
If causal power is the test for reality, that will exclude necessities and possibilities [McGinn]
7. Existence / D. Theories of Reality / 8. Facts / b. Types of fact
Facts are object-plus-extension, or property-plus-set-of-properties, or object-plus-property [McGinn]
9. Objects / A. Existence of Objects / 2. Abstract Objects / a. Nature of abstracta
Structuralists call a mathematical 'object' simply a 'place in a structure' [Friend]
9. Objects / F. Identity among Objects / 1. Concept of Identity
Identity propositions are not always tautological, and have a key epistemic role [McGinn]
9. Objects / F. Identity among Objects / 2. Defining Identity
Identity is as basic as any concept could ever be [McGinn]
9. Objects / F. Identity among Objects / 4. Type Identity
Type-identity is close similarity in qualities [McGinn]
Qualitative identity is really numerical identity of properties [McGinn]
Qualitative identity can be analysed into numerical identity of the type involved [McGinn]
It is best to drop types of identity, and speak of 'identity' or 'resemblance' [McGinn]
9. Objects / F. Identity among Objects / 5. Self-Identity
Existence is a property of all objects, but less universal than self-identity, which covers even conceivable objects [McGinn]
Sherlock Holmes does not exist, but he is self-identical [McGinn]
9. Objects / F. Identity among Objects / 6. Identity between Objects
All identity is necessary, though identity statements can be contingently true [McGinn]
9. Objects / F. Identity among Objects / 7. Indiscernible Objects
The Identity of Indiscernibles is really the same as the verification principle [Jolley]
9. Objects / F. Identity among Objects / 8. Leibniz's Law
Leibniz's Law says 'x = y iff for all P, Px iff Py' [McGinn]
Leibniz's Law is so fundamental that it almost defines the concept of identity [McGinn]
Leibniz's Law presupposes the notion of property identity [McGinn]
10. Modality / C. Sources of Modality / 5. Modality from Actuality
Modality is not objects or properties, but the type of binding of objects to properties [McGinn]
10. Modality / E. Possible worlds / 1. Possible Worlds / b. Impossible worlds
If 'possible' is explained as quantification across worlds, there must be possible worlds [McGinn]
12. Knowledge Sources / D. Empiricism / 5. Empiricism Critique
Necessity and possibility are big threats to the empiricist view of knowledge [McGinn]
13. Knowledge Criteria / D. Scepticism / 1. Scepticism
Scepticism about reality is possible because existence isn't part of appearances [McGinn]
17. Mind and Body / E. Mind as Physical / 2. Reduction of Mind
Studying biology presumes the laws of chemistry, and it could never contradict them [Friend]
18. Thought / D. Concepts / 1. Concepts / a. Nature of concepts
Concepts can be presented extensionally (as objects) or intensionally (as a characterization) [Friend]
19. Language / C. Assigning Meanings / 5. Fregean Semantics
Semantics should not be based on set-membership, but on instantiation of properties in objects [McGinn]
19. Language / C. Assigning Meanings / 7. Extensional Semantics
Clearly predicates have extensions (applicable objects), but are the extensions part of their meaning? [McGinn]
28. God / B. Proving God / 2. Proofs of Reason / b. Ontological Proof critique
If Satan is the most imperfect conceivable being, he must have non-existence [McGinn]
I think the fault of the Ontological Argument is taking the original idea to be well-defined [McGinn]