Combining Texts

All the ideas for 'Mathematical Methods in Philosophy', 'Nature and Meaning of Numbers' and 'Essays on Intellectual Powers 6: Judgement'

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


43 ideas

1. Philosophy / F. Analytic Philosophy / 5. Linguistic Analysis
The existence of tensed verbs shows that not all truths are necessary truths [Reid]
2. Reason / D. Definition / 9. Recursive Definition
Dedekind proved definition by recursion, and thus proved the basic laws of arithmetic [Dedekind, by Potter]
2. Reason / F. Fallacies / 7. Ad Hominem
An ad hominem argument is good, if it is shown that the man's principles are inconsistent [Reid]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / d. Infinite Sets
An infinite set maps into its own proper subset [Dedekind, by Reck/Price]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
We have the idea of self, and an idea of that idea, and so on, so infinite ideas are available [Dedekind, by Potter]
4. Formal Logic / G. Formal Mereology / 1. Mereology
Dedekind originally thought more in terms of mereology than of sets [Dedekind, by Potter]
5. Theory of Logic / A. Overview of Logic / 9. Philosophical Logic
Three stages of philosophical logic: syntactic (1905-55), possible worlds (1963-85), widening (1990-) [Horsten/Pettigrew]
5. Theory of Logic / E. Structures of Logic / 1. Logical Form
Logical formalization makes concepts precise, and also shows their interrelation [Horsten/Pettigrew]
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
Models are sets with functions and relations, and truth built up from the components [Horsten/Pettigrew]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
Numbers are free creations of the human mind, to understand differences [Dedekind]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
Dedekind defined the integers, rationals and reals in terms of just the natural numbers [Dedekind, by George/Velleman]
Ordinals can define cardinals, as the smallest ordinal that maps the set [Dedekind, by Heck]
Order, not quantity, is central to defining numbers [Dedekind, by Monk]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
Dedekind's ordinals are just members of any progression whatever [Dedekind, by Russell]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
Dedekind's axiom that his Cut must be filled has the advantages of theft over honest toil [Dedekind, by Russell]
Dedekind says each cut matches a real; logicists say the cuts are the reals [Dedekind, by Bostock]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
In counting we see the human ability to relate, correspond and represent [Dedekind]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / b. Mark of the infinite
A system S is said to be infinite when it is similar to a proper part of itself [Dedekind]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
Dedekind gives a base number which isn't a successor, then adds successors and induction [Dedekind, by Hart,WD]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
Zero is a member, and all successors; numbers are the intersection of sets satisfying this [Dedekind, by Bostock]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
Categoricity implies that Dedekind has characterised the numbers, because it has one domain [Rumfitt on Dedekind]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
Induction is proved in Dedekind, an axiom in Peano; the latter seems simpler and clearer [Dedekind, by Russell]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
Dedekind originated the structuralist conception of mathematics [Dedekind, by MacBride]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / b. Varieties of structuralism
Dedekindian abstraction talks of 'positions', where Cantorian abstraction talks of similar objects [Dedekind, by Fine,K]
7. Existence / A. Nature of Existence / 1. Nature of Existence
If 'exist' doesn't express a property, we can hardly ask for its essence [Horsten/Pettigrew]
9. Objects / A. Existence of Objects / 3. Objects in Thought
A thing is completely determined by all that can be thought concerning it [Dedekind]
10. Modality / E. Possible worlds / 1. Possible Worlds / a. Possible worlds
A Tarskian model can be seen as a possible state of affairs [Horsten/Pettigrew]
The 'spheres model' was added to possible worlds, to cope with counterfactuals [Horsten/Pettigrew]
10. Modality / E. Possible worlds / 1. Possible Worlds / b. Impossible worlds
Epistemic logic introduced impossible worlds [Horsten/Pettigrew]
10. Modality / E. Possible worlds / 1. Possible Worlds / e. Against possible worlds
Possible worlds models contain sets of possible worlds; this is a large metaphysical commitment [Horsten/Pettigrew]
Using possible worlds for knowledge and morality may be a step too far [Horsten/Pettigrew]
11. Knowledge Aims / B. Certain Knowledge / 4. The Cogito
If someone denies that he is thinking when he is conscious of it, we can only laugh [Reid]
11. Knowledge Aims / C. Knowing Reality / 1. Perceptual Realism / b. Direct realism
The existence of ideas is no more obvious than the existence of external objects [Reid]
11. Knowledge Aims / C. Knowing Reality / 4. Solipsism
We are only aware of other beings through our senses; without that, we are alone in the universe [Reid]
12. Knowledge Sources / E. Direct Knowledge / 1. Common Sense
In obscure matters the few must lead the many, but the many usually lead in common sense [Reid]
12. Knowledge Sources / E. Direct Knowledge / 4. Memory
The theory of ideas, popular with philosophers, means past existence has to be proved [Reid]
15. Nature of Minds / B. Features of Minds / 1. Consciousness / a. Consciousness
Consciousness is an indefinable and unique operation [Reid]
18. Thought / A. Modes of Thought / 8. Human Thought
The structure of languages reveals a uniformity in basic human opinions [Reid]
18. Thought / E. Abstraction / 2. Abstracta by Selection
If you can't distinguish the features of a complex object, your notion of it would be a muddle [Reid]
18. Thought / E. Abstraction / 3. Abstracta by Ignoring
Dedekind said numbers were abstracted from systems of objects, leaving only their position [Dedekind, by Dummett]
We derive the natural numbers, by neglecting everything of a system except distinctness and order [Dedekind]
18. Thought / E. Abstraction / 8. Abstractionism Critique
Dedekind has a conception of abstraction which is not psychologistic [Dedekind, by Tait]
21. Aesthetics / A. Aesthetic Experience / 3. Taste
There are axioms of taste - such as a general consensus about a beautiful face [Reid]