Ideas from 'Grundlagen der Arithmetik (Foundations)' by Gottlob Frege [1884], by Theme Structure

[found in 'The Foundations of Arithmetic (Austin)' by Frege,Gottlob (ed/tr Austin,J.L.) [Blackwell 1980,0-631-12694-5]].

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1. Philosophy / E. Nature of Metaphysics / 6. Metaphysics as Conceptual
The syntactic category is primary, and the ontological category is derivative [Wright,C]
1. Philosophy / F. Analytic Philosophy / 4. Conceptual Analysis
Never lose sight of the distinction between concept and object
1. Philosophy / F. Analytic Philosophy / 5. Linguistic Analysis
Frege was the first to give linguistic answers to non-linguistic questions
Frege initiated linguistic philosophy, studying number through the sense of sentences [Dummett]
1. Philosophy / F. Analytic Philosophy / 6. Logical Analysis
Frege developed formal systems to avoid unnoticed assumptions [Lavine]
2. Reason / A. Nature of Reason / 3. Pure Reason
Thoughts have a natural order, to which human thinking is drawn. [Yablo]
2. Reason / A. Nature of Reason / 5. Objectivity
Frege sees no 'intersubjective' category, between objective and subjective [Dummett]
Keep the psychological and subjective separate from the logical and objective
2. Reason / D. Definition / 7. Contextual Definition
Nothing should be defined in terms of that to which it is conceptually prior [Dummett]
Originally Frege liked contextual definitions, but later preferred them fully explicit [Dummett]
2. Reason / E. Argument / 6. Conclusive Proof
Proof aims to remove doubts, but also to show the interdependence of truths
2. Reason / F. Fallacies / 8. Category Mistake / a. Category mistakes
You can't transfer external properties unchanged to apply to ideas
3. Truth / B. Truthmakers / 5. What Makes Truths / c. States of affairs make truths
We need to grasp not number-objects, but the states of affairs which make number statements true [Wright,C]
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / e. Axioms of PL
Frege agreed with Euclid that the axioms of logic and mathematics are known through self-evidence [Burge]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
The null set is only defensible if it is the extension of an empty concept [Burge]
It is because a concept can be empty that there is such a thing as the empty class [Dummett]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / e. Equivalence classes
We can introduce new objects, as equivalence classes of objects already known [Dummett]
Frege introduced the standard device, of defining logical objects with equivalence classes [Dummett]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
Frege, unlike Russell, has infinite individuals because numbers are individuals
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / c. Logical sets
A class is, for Frege, the extension of a concept [Dummett]
5. Theory of Logic / E. Structures of Logic / 1. Logical Form
Convert "Jupiter has four moons" into "the number of Jupiter's moons is four"
5. Theory of Logic / E. Structures of Logic / 8. Theories in Logic
Despite Gödel, Frege's epistemic ordering of all the truths is still plausible [Burge]
The primitive simples of arithmetic are the essence, determining the subject, and its boundaries [Jeshion]
5. Theory of Logic / G. Quantification / 6. Plural Quantification
Each horse doesn't fall under the concept 'horse that draws the carriage', because all four are needed [Oliver/Smiley]
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
To understand axioms you must grasp their logical power and priority [Burge]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
We cannot define numbers from the idea of a series, because numbers must precede that
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / l. Zero
Treating 0 as a number avoids antinomies involving treating 'nobody' as a person [Dummett]
For Frege 'concept' and 'extension' are primitive, but 'zero' and 'successor' are defined [Chihara]
If objects exist because they fall under a concept, 0 is the object under which no objects fall [Dummett]
Nought is the number belonging to the concept 'not identical with itself'
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / m. One
We can say 'a and b are F' if F is 'wise', but not if it is 'one'
One is the Number which belongs to the concept "identical with 0"
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / a. Units
Units can be equal without being identical [Tait]
Frege says only concepts which isolate and avoid arbitrary division can give units [Koslicki]
You can abstract concepts from the moon, but the number one is not among them
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / d. Counting via concepts
A concept creating a unit must isolate and unify what falls under it
Frege says counting is determining what number belongs to a given concept [Koslicki]
Frege's 'isolation' could be absence of overlap, or drawing conceptual boundaries [Koslicki]
Non-arbitrary division means that what falls under the concept cannot be divided into more of the same [Koslicki]
Our concepts decide what is countable, as in seeing the leaves of the tree, or the foliage [Koslicki]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / e. Counting by correlation
Frege's one-to-one correspondence replaces well-ordering, because infinities can't be counted [Lavine]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / h. Ordinal infinity
The number of natural numbers is not a natural number [George/Velleman]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
Arithmetical statements can't be axioms, because they are provable [Burge]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers
Frege had a motive to treat numbers as objects, but not a justification [Hale/Wright]
Frege claims that numbers are objects, as opposed to them being Fregean concepts [Wright,C]
Numbers are second-level, ascribing properties to concepts rather than to objects [Wright,C]
For Frege, successor was a relation, not a function [Dummett]
Numbers are more than just 'second-level concepts', since existence is also one [George/Velleman]
"Number of x's such that ..x.." is a functional expression, yielding a name when completed [George/Velleman]
A cardinal number may be defined as a class of similar classes [Russell]
A statement of number contains a predication about a concept
Frege gives an incoherent account of extensions resulting from abstraction [Fine,K]
For Frege the number of F's is a collection of first-level concepts [George/Velleman]
Numbers need to be objects, to define the extension of the concept of each successor to n [George/Velleman]
Frege's account of cardinals fails in modern set theory, so they are now defined differently [Dummett]
Frege's incorrect view is that a number is an equivalence class [Benacerraf]
The natural number n is the set of n-membered sets [Yourgrau]
A set doesn't have a fixed number, because the elements can be seen in different ways [Yourgrau]
If you can subdivide objects many ways for counting, you can do that to set-elements too [Yourgrau]
Frege's problem is explaining the particularity of numbers by general laws [Burge]
Individual numbers are best derived from the number one, and increase by one
'Exactly ten gallons' may not mean ten things instantiate 'gallon' [Rumfitt]
Numerical statements have first-order logical form, so must refer to objects [Hodes]
The Number for F is the extension of 'equal to F' (or maybe just F itself)
Numbers are objects because they partake in identity statements [Bostock]
The number of F's is the extension of the second level concept 'is equipollent with F' [Tait]
Frege showed that numbers attach to concepts, not to objects [Wiggins]
Frege replaced Cantor's sets as the objects of equinumerosity attributions with concepts [Tait]
Zero is defined using 'is not self-identical', and one by using the concept of zero [Weiner]
Frege started with contextual definition, but then switched to explicit extensional definition [Wright,C]
Each number, except 0, is the number of the concept of all of its predecessors [Wright,C]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle
'The number of Fs' is the extension (a collection of first-level concepts) of the concept 'equinumerous with F' [George/Velleman]
Frege's cardinals (equivalences of one-one correspondences) is not permissible in ZFC
Frege thinks number is fundamentally bound up with one-one correspondence [Heck]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / e. Caesar problem
The words 'There are exactly Julius Caesar moons of Mars' are gibberish [Rumfitt]
'Julius Caesar' isn't a number because numbers inherit properties of 0 and successor [George/Velleman]
From within logic, how can we tell whether an arbitrary object like Julius Caesar is a number? [Friend]
Frege said 2 is the extension of all pairs (so Julius Caesar isn't 2, because he's not an extension) [Shapiro]
Fregean numbers are numbers, and not 'Caesar', because they correlate 1-1 [Wright,C]
One-one correlations imply normal arithmetic, but don't explain our concept of a number
Our definition will not tell us whether or not Julius Caesar is a number
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory
If numbers can be derived from logic, then set theory is superfluous [Burge]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
If numbers are supposed to be patterns, each number can have many patterns
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
Numbers seem to be objects because they exactly fit the inference patterns for identities
Frege's platonism proposes that objects are what singular terms refer to [Wright,C]
How can numbers be external (one pair of boots is two boots), or subjective (and so relative)? [Weiner]
Identities refer to objects, so numbers must be objects [Weiner]
Numbers are not physical, and not ideas - they are objective and non-sensible
Numbers are objects, because they can take the definite article, and can't be plurals
6. Mathematics / C. Sources of Mathematics / 2. Intuition of Mathematics
Frege's logicism aimed at removing the reliance of arithmetic on intuition [Yourgrau]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
There is no physical difference between two boots and one pair of boots
6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
It appears that numbers are adjectives, but they don't apply to a single object [George/Velleman]
Numerical adjectives are of the same second-level type as the existential quantifier [George/Velleman]
'Jupiter has many moons' won't read as 'The number of Jupiter's moons equals the number many' [Rumfitt]
The number 'one' can't be a property, if any object can be viewed as one or not one
For science, we can translate adjectival numbers into noun form
6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
Logicism shows that no empirical truths are needed to justify arithmetic [George/Velleman]
Arithmetic is analytic [Weiner]
Frege offered a Platonist version of logicism, committed to cardinal and real numbers [Hale/Wright]
Mathematics has no special axioms of its own, but follows from principles of logic (with definitions) [Bostock]
Arithmetic must be based on logic, because of its total generality [Jeshion]
Numbers are definable in terms of mapping items which fall under concepts [Scruton]
Arithmetic is analytic and a priori, and thus it is part of logic
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
Frege only managed to prove that arithmetic was analytic with a logic that included set-theory [Quine]
Frege's belief in logicism and in numerical objects seem uncomfortable together [Hodes]
Frege's platonism and logicism are in conflict, if logic must dictates an infinity of objects [Wright,C]
Why should the existence of pure logic entail the existence of objects? [George/Velleman]
6. Mathematics / C. Sources of Mathematics / 7. Formalism
Formalism fails to recognise types of symbols, and also meta-games [Brown,JR]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
Frege was completing Bolzano's work, of expelling intuition from number theory and analysis
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism
Abstraction from things produces concepts, and numbers are in the concepts
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / e. Psychologism
Mental states are irrelevant to mathematics, because they are vague and fluctuating
7. Existence / A. Nature of Existence / 1. Nature of Existence
Affirmation of existence is just denial of zero
7. Existence / A. Nature of Existence / 5. Abstract Existence
If abstracta are non-mental, quarks are abstracta, and yet chess and God's thoughts are mental [Rosen]
The equator is imaginary, but not fictitious; thought is needed to recognise it
7. Existence / C. Structure of Existence / 4. Ontological Dependence
Many of us find Frege's claim that truths depend on one another an obscure idea [Heck]
Parallelism is intuitive, so it is more fundamental than sameness of direction [Heck]
7. Existence / C. Structure of Existence / 7. Abstract/Concrete / a. Abstract/concrete
Frege refers to 'concrete' objects, but they are no different in principle from abstract ones [Dummett]
7. Existence / D. Theories of Reality / 9. Vagueness / d. Vagueness as semantic
Vagueness is incomplete definition [Koslicki]
7. Existence / D. Theories of Reality / 10. Ontological Commitment / a. Ontological commitment
For Frege, ontological questions are to be settled by reference to syntactic structures [Wright,C]
7. Existence / D. Theories of Reality / 10. Ontological Commitment / c. Commitment of predicates
Second-order quantifiers are committed to concepts, as first-order commits to objects [Linnebo]
8. Modes of Existence / A. Relations / 4. Formal Relations / c. Ancestral relation
'Ancestral' relations are derived by iterating back from a given relation [George/Velleman]
8. Modes of Existence / B. Properties / 1. Nature of Properties
Frege treats properties as a kind of function, and maybe a property is its characteristic function [Smith,P]
9. Objects / A. Existence of Objects / 2. Abstract Objects / a. Nature of abstracta
Not all objects are spatial; 4 can still be an object, despite lacking spatial co-ordinates
9. Objects / A. Existence of Objects / 2. Abstract Objects / c. Modern abstracta
Frege says singular terms denote objects, numerals are singular terms, so numbers exist [Hale]
Frege establishes abstract objects independently from concrete ones, by falling under a concept [Dummett]
9. Objects / A. Existence of Objects / 3. Objects in Thought
For Frege, objects just are what singular terms refer to [Hale/Wright]
Without concepts we would not have any objects [Shapiro]
9. Objects / A. Existence of Objects / 5. Individuation / a. Individuation
Frege's universe comes already divided into objects [Koslicki]
9. Objects / F. Identity among Objects / 1. Concept of Identity
The idea of a criterion of identity was introduced by Frege [Noonan]
Frege's algorithm of identity is the law of putting equals for equals [Quine]
9. Objects / F. Identity among Objects / 3. Relative Identity
Geach denies Frege's view, that 'being the same F' splits into being the same and being F [Perry]
9. Objects / F. Identity among Objects / 6. Identity between Objects
Identity between objects is not a consequence of identity, but part of what 'identity' means [Dummett]
11. Knowledge Aims / A. Knowledge / 2. Understanding
To understand a thought you must understand its logical structure [Burge]
12. Knowledge Sources / A. A Priori Knowledge / 1. Nature of the A Priori
For Frege a priori knowledge derives from general principles, so numbers can't be primitive
12. Knowledge Sources / A. A Priori Knowledge / 2. Self-Evidence
Mathematicians just accept self-evidence, whether it is logical or intuitive
12. Knowledge Sources / A. A Priori Knowledge / 4. A Priori as Necessities
An a priori truth is one derived from general laws which do not require proof
A truth is a priori if it can be proved entirely from general unproven laws
12. Knowledge Sources / A. A Priori Knowledge / 8. A Priori as Analytic
Frege tried to explain synthetic a priori truths by expanding the concept of analyticity
12. Knowledge Sources / E. Direct Knowledge / 1. Intuition
Intuitions cannot be communicated [Burge]
13. Knowledge Criteria / B. Internal Justification / 4. Foundationalism / d. Rational foundations
Justifications show the ordering of truths, and the foundation is what is self-evident [Jeshion]
14. Science / C. Induction / 1. Induction
Induction is merely psychological, with a principle that it can actually establish laws
In science one observation can create high probability, while a thousand might prove nothing
15. Nature of Minds / A. Nature of Mind / 1. Mind / c. Features of mind
Ideas are not spatial, and don't have distances between them
18. Thought / A. Modes of Thought / 1. Thought
Thought is the same everywhere, and the laws of thought do not vary
18. Thought / D. Concepts / 1. Concepts / a. Nature of concepts
Early Frege takes the extensions of concepts for granted [Dummett]
18. Thought / D. Concepts / 3. Ontology of Concepts / c. Fregean concepts
Concepts are, precisely, the references of predicates [Wright,C]
A concept is a non-psychological one-place function asserting something of an object [Weiner]
Fregean concepts have precise boundaries and universal applicability [Koslicki]
Psychological accounts of concepts are subjective, and ultimately destroy truth
18. Thought / D. Concepts / 5. Concepts and Language / b. Concepts are linguistic
A concept is a possible predicate of a singular judgement
18. Thought / E. Abstraction / 1. Abstract Thought
Defining 'direction' by parallelism doesn't tell you whether direction is a line [Dummett]
18. Thought / E. Abstraction / 2. Abstracta by Selection
Frege accepts abstraction to the concept of all sets equipollent to a given one [Tait]
18. Thought / E. Abstraction / 3. Abstracta by Ignoring
If we abstract 'from' two cats, the units are not black or white, or cats [Tait]
Frege himself abstracts away from tone and color [Yablo]
18. Thought / E. Abstraction / 7. Abstracta by Equivalence
Frege's logical abstaction identifies a common feature as the maximal set of equivalent objects [Dummett]
Frege's 'parallel' and 'direction' don't have the same content, as we grasp 'parallel' first [Yablo]
Fregean abstraction creates concepts which are equivalences between initial items [Fine,K]
Frege put the idea of abstraction on a rigorous footing [Fine,K]
We create new abstract concepts by carving up the content in a different way
From basing 'parallel' on identity of direction, Frege got all abstractions from identity statements [Dummett]
You can't simultaneously fix the truth-conditions of a sentence and the domain of its variables [Dummett]
19. Language / A. Nature of Meaning / 7. Meaning Holism / a. Sentence meaning
Words in isolation seem to have ideas as meanings, but words have meaning in propositions
Never ask for the meaning of a word in isolation, but only in the context of a proposition
19. Language / E. Analyticity / 1. Analytic Propositions
A statement is analytic if substitution of synonyms can make it a logical truth [Boghossian]
Frege considered analyticity to be an epistemic concept [Shapiro]
19. Language / E. Analyticity / 2. Analytic Truths
All analytic truths can become logical truths, by substituting definitions or synonyms [Rey]
19. Language / E. Analyticity / 4. Analytic/Synthetic Critique
Frege fails to give a concept of analyticity, so he fails to explain synthetic a priori truth that way [Katz]
25. Society / E. State Functions / 4. Education / a. Education principles
To learn something, you must know that you don't know
26. Natural Theory / D. Laws of Nature / 6. Laws as Numerical
The laws of number are not laws of nature, but are laws of the laws of nature
28. God / B. Proving God / 2. Proofs of Reason / b. Ontological Proof critique
Because existence is a property of concepts the ontological argument for God fails