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

13876

The syntactic category is primary, and the ontological category is derivative [Wright,C]

1. Philosophy / F. Analytic Philosophy / 4. Conceptual Analysis

8415	Never lose sight of the distinction between concept and object

1. Philosophy / F. Analytic Philosophy / 5. Linguistic Analysis

9841	Frege was the first to give linguistic answers to non-linguistic questions [Dummett]

9840	Frege initiated linguistic philosophy, studying number through the sense of sentences [Dummett]

1. Philosophy / F. Analytic Philosophy / 6. Logical Analysis

15948

Frege developed formal systems to avoid unnoticed assumptions [Lavine]

2. Reason / A. Nature of Reason / 3. Pure Reason

10804

Thoughts have a natural order, to which human thinking is drawn [Yablo]

2. Reason / A. Nature of Reason / 5. Objectivity

9832	Frege sees no 'intersubjective' category, between objective and subjective [Dummett]

8414	Keep the psychological and subjective separate from the logical and objective

2. Reason / D. Definition / 7. Contextual Definition

9844	Originally Frege liked contextual definitions, but later preferred them fully explicit [Dummett]

9822	Nothing should be defined in terms of that to which it is conceptually prior [Dummett]

2. Reason / E. Argument / 6. Conclusive Proof

17495

Proof aims to remove doubts, but also to show the interdependence of truths

2. Reason / F. Fallacies / 9. Category Mistake / a. Category mistakes

8632	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

13881

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

9154	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

9157	The null set is only defensible if it is the extension of an empty concept [Burge]

9835	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

9854	We can introduce new objects, as equivalence classes of objects already known [Dummett]

9883	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

18104

Frege, unlike Russell, has infinite individuals because numbers are individuals [Bostock]

4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / c. Logical sets

9834	A class is, for Frege, the extension of a concept [Dummett]

5. Theory of Logic / E. Structures of Logic / 1. Logical Form

8645	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

16891

Despite Gödel, Frege's epistemic ordering of all the truths is still plausible [Burge]

16906

The primitive simples of arithmetic are the essence, determining the subject, and its boundaries [Jeshion]

5. Theory of Logic / G. Quantification / 6. Plural Quantification

14236

Each horse doesn't fall under the concept 'horse that draws the carriage', because all four are needed [Oliver/Smiley]

5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models

22294

We can show that a concept is consistent by producing something which falls under it

5. Theory of Logic / K. Features of Logics / 1. Axiomatisation

17624

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

8640	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

9838	Treating 0 as a number avoids antinomies involving treating 'nobody' as a person [Dummett]

9564	For Frege 'concept' and 'extension' are primitive, but 'zero' and 'successor' are defined [Chihara]

10551

If objects exist because they fall under a concept, 0 is the object under which no objects fall [Dummett]

8653	Nought is the number belonging to the concept 'not identical with itself'

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / m. One

8654	One is the Number which belongs to the concept "identical with 0"

8636	We can say 'a and b are F' if F is 'wise', but not if it is 'one'

6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / a. Units

8641	You can abstract concepts from the moon, but the number one is not among them

9989	Units can be equal without being identical [Tait]

17429

Frege says only concepts which isolate and avoid arbitrary division can give units [Koslicki]

6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / d. Counting via concepts

17427

Frege's 'isolation' could be absence of overlap, or drawing conceptual boundaries [Koslicki]

17437

Non-arbitrary division means that what falls under the concept cannot be divided into more of the same [Koslicki]

17438

Our concepts decide what is countable, as in seeing the leaves of the tree, or the foliage [Koslicki]

17426

A concept creating a unit must isolate and unify what falls under it

17428

Frege says counting is determining what number belongs to a given concept [Koslicki]

6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / e. Counting by correlation

15916

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

10034

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

16883

Arithmetical statements can't be axioms, because they are provable [Burge]

6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers

10625

Frege had a motive to treat numbers as objects, but not a justification [Hale/Wright]

13871

Frege claims that numbers are objects, as opposed to them being Fregean concepts [Wright,C]

13872

Numbers are second-level, ascribing properties to concepts rather than to objects [Wright,C]

9816	For Frege, successor was a relation, not a function [Dummett]

17460

A statement of number contains a predication about a concept

9953	Numbers are more than just 'second-level concepts', since existence is also one [George/Velleman]

9954	"Number of x's such that ..x.." is a functional expression, yielding a name when completed [George/Velleman]

10139

Frege gives an incoherent account of extensions resulting from abstraction [Fine,K]

10028

For Frege the number of F's is a collection of first-level concepts [George/Velleman]

10029

Numbers need to be objects, to define the extension of the concept of each successor to n [George/Velleman]

9973	The number of F's is the extension of the second level concept 'is equipollent with F' [Tait]

16500

Frege showed that numbers attach to concepts, not to objects [Wiggins]

9990	Frege replaced Cantor's sets as the objects of equinumerosity attributions with concepts [Tait]

7738	Zero is defined using 'is not self-identical', and one by using the concept of zero [Weiner]

23456

Frege said logical predication implies classes, which are arithmetical objects [Morris,M]

17819

A set doesn't have a fixed number, because the elements can be seen in different ways [Yourgrau]

17820

If you can subdivide objects many ways for counting, you can do that to set-elements too [Yourgrau]

16890

Frege's problem is explaining the particularity of numbers by general laws [Burge]

8630	Individual numbers are best derived from the number one, and increase by one

11029

'Exactly ten gallons' may not mean ten things instantiate 'gallon' [Rumfitt]

10013

Numerical statements have first-order logical form, so must refer to objects [Hodes]

18181

The Number for F is the extension of 'equal to F' (or maybe just F itself)

18103

Numbers are objects because they partake in identity statements [Bostock]

13887

Frege started with contextual definition, but then switched to explicit extensional definition [Wright,C]

13897

Each number, except 0, is the number of the concept of all of its predecessors [Wright,C]

9856	Frege's account of cardinals fails in modern set theory, so they are now defined differently [Dummett]

9902	Frege's incorrect view is that a number is an equivalence class [Benacerraf]

17814

The natural number n is the set of n-membered sets [Yourgrau]

17636

A cardinal number may be defined as a class of similar classes [Russell]

6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle

9956	'The number of Fs' is the extension (a collection of first-level concepts) of the concept 'equinumerous with F' [George/Velleman]

13527

Frege's cardinals (equivalences of one-one correspondences) is not permissible in ZFC [Wolf,RS]

22292

Hume's Principle fails to implicitly define numbers, because of the Julius Caesar [Potter]

17442

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

11030

The words 'There are exactly Julius Caesar moons of Mars' are gibberish [Rumfitt]

10030

'Julius Caesar' isn't a number because numbers inherit properties of 0 and successor [George/Velleman]

8690	From within logic, how can we tell whether an arbitrary object like Julius Caesar is a number? [Friend]

10219

Frege said 2 is the extension of all pairs (so Julius Caesar isn't 2, because he's not an extension) [Shapiro]

13889

Fregean numbers are numbers, and not 'Caesar', because they correlate 1-1 [Wright,C]

18142

One-one correlations imply normal arithmetic, but don't explain our concept of a number [Bostock]

9046	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

16896

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

8639	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

13874

Numbers seem to be objects because they exactly fit the inference patterns for identities

13875

Frege's platonism proposes that objects are what singular terms refer to [Wright,C]

7731	How can numbers be external (one pair of boots is two boots), or subjective (and so relative)? [Weiner]

7737	Identities refer to objects, so numbers must be objects [Weiner]

8635	Numbers are not physical, and not ideas - they are objective and non-sensible

8652	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

17816

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

8633	There is no physical difference between two boots and one pair of boots

6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival

9951	It appears that numbers are adjectives, but they don't apply to a single object [George/Velleman]

9952	Numerical adjectives are of the same second-level type as the existential quantifier [George/Velleman]

11031

'Jupiter has many moons' won't read as 'The number of Jupiter's moons equals the number many' [Rumfitt]

8637	The number 'one' can't be a property, if any object can be viewed as one or not one

9999	For science, we can translate adjectival numbers into noun form

6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism

9945	Logicism shows that no empirical truths are needed to justify arithmetic [George/Velleman]

8782	Frege offered a Platonist version of logicism, committed to cardinal and real numbers [Hale/Wright]

13608

Mathematics has no special axioms of its own, but follows from principles of logic (with definitions) [Bostock]

5658	Numbers are definable in terms of mapping items which fall under concepts [Scruton]

16905

Arithmetic must be based on logic, because of its total generality [Jeshion]

8655	Arithmetic is analytic and a priori, and thus it is part of logic

7739	Arithmetic is analytic [Weiner]

6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique

10831

Frege only managed to prove that arithmetic was analytic with a logic that included set-theory [Quine]

10033

Why should the existence of pure logic entail the existence of objects? [George/Velleman]

10010

Frege's belief in logicism and in numerical objects seem uncomfortable together [Hodes]

13864

Frege's platonism and logicism are in conflict, if logic must dictates an infinity of objects [Wright,C]

6. Mathematics / C. Sources of Mathematics / 7. Formalism

9631	Formalism fails to recognise types of symbols, and also meta-games [Brown,JR]

6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism

9875	Frege was completing Bolzano's work, of expelling intuition from number theory and analysis [Dummett]

6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism

8642	Abstraction from things produces concepts, and numbers are in the concepts

6. Mathematics / C. Sources of Mathematics / 10. Constructivism / e. Psychologism

8621	Mental states are irrelevant to mathematics, because they are vague and fluctuating

7. Existence / A. Nature of Existence / 1. Nature of Existence

8643	Affirmation of existence is just denial of zero

7. Existence / A. Nature of Existence / 4. Abstract Existence

8911	If abstracta are non-mental, quarks are abstracta, and yet chess and God's thoughts are mental [Rosen]

8634	The equator is imaginary, but not fictitious; thought is needed to recognise it

7. Existence / C. Structure of Existence / 4. Ontological Dependence

17443

Many of us find Frege's claim that truths depend on one another an obscure idea [Heck]

17445

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

10539

Frege refers to 'concrete' objects, but they are no different in principle from abstract ones [Dummett]

7. Existence / D. Theories of Reality / 10. Vagueness / d. Vagueness as linguistic

17431

Vagueness is incomplete definition [Koslicki]

7. Existence / D. Theories of Reality / 11. Ontological Commitment / a. Ontological commitment

13879

For Frege, ontological questions are to be settled by reference to syntactic structures [Wright,C]

7. Existence / D. Theories of Reality / 11. Ontological Commitment / c. Commitment of predicates

10642

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

10032

'Ancestral' relations are derived by iterating back from a given relation [George/Velleman]

8. Modes of Existence / B. Properties / 1. Nature of Properties

10606

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

8647	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

10309

Frege says singular terms denote objects, numerals are singular terms, so numbers exist [Hale]

10550

Frege establishes abstract objects independently from concrete ones, by falling under a concept [Dummett]

9. Objects / A. Existence of Objects / 3. Objects in Thought

8785	For Frege, objects just are what singular terms refer to [Hale/Wright]

10278

Without concepts we would not have any objects [Shapiro]

9. Objects / A. Existence of Objects / 5. Individuation / a. Individuation

17432

Frege's universe comes already divided into objects [Koslicki]

9. Objects / F. Identity among Objects / 1. Concept of Identity

16022

The idea of a criterion of identity was introduced by Frege [Noonan]

11100

Frege's algorithm of identity is the law of putting equals for equals [Quine]

9. Objects / F. Identity among Objects / 3. Relative Identity

12153

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

9853	Identity between objects is not a consequence of identity, but part of what 'identity' means [Dummett]

11. Knowledge Aims / A. Knowledge / 2. Understanding

17623

To understand a thought you must understand its logical structure [Burge]

12. Knowledge Sources / A. A Priori Knowledge / 1. Nature of the A Priori

9158	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

8657	Mathematicians just accept self-evidence, whether it is logical or intuitive

12. Knowledge Sources / A. A Priori Knowledge / 4. A Priori as Necessities

9352	An a priori truth is one derived from general laws which do not require proof

16889

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

2514	Frege tried to explain synthetic a priori truths by expanding the concept of analyticity [Katz]

12. Knowledge Sources / E. Direct Knowledge / 2. Intuition

16900

Intuitions cannot be communicated [Burge]

13. Knowledge Criteria / B. Internal Justification / 4. Foundationalism / d. Rational foundations

16903

Justifications show the ordering of truths, and the foundation is what is self-evident [Jeshion]

14. Science / C. Induction / 1. Induction

8624	Induction is merely psychological, with a principle that it can actually establish laws

8626	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

8648	Ideas are not spatial, and don't have distances between them

18. Thought / A. Modes of Thought / 1. Thought

8620	Thought is the same everywhere, and the laws of thought do not vary

18. Thought / D. Concepts / 1. Concepts / a. Nature of concepts

9870	Early Frege takes the extensions of concepts for granted [Dummett]

18. Thought / D. Concepts / 3. Ontology of Concepts / c. Fregean concepts

7736	A concept is a non-psychological one-place function asserting something of an object [Weiner]

17430

Fregean concepts have precise boundaries and universal applicability [Koslicki]

8622	Psychological accounts of concepts are subjective, and ultimately destroy truth

13878

Concepts are, precisely, the references of predicates [Wright,C]

18. Thought / D. Concepts / 5. Concepts and Language / b. Concepts are linguistic

8651	A concept is a possible predicate of a singular judgement

18. Thought / E. Abstraction / 1. Abstract Thought

9846	Defining 'direction' by parallelism doesn't tell you whether direction is a line [Dummett]

18. Thought / E. Abstraction / 2. Abstracta by Selection

9976	Frege accepts abstraction to the concept of all sets equipollent to a given one [Tait]

18. Thought / E. Abstraction / 3. Abstracta by Ignoring

10803

Frege himself abstracts away from tone and color [Yablo]

9988	If we abstract 'from' two cats, the units are not black or white, or cats [Tait]

18. Thought / E. Abstraction / 7. Abstracta by Equivalence

9855	Frege's logical abstaction identifies a common feature as the maximal set of equivalent objects [Dummett]

10802

Frege's 'parallel' and 'direction' don't have the same content, as we grasp 'parallel' first [Yablo]

10526

Fregean abstraction creates concepts which are equivalences between initial items [Fine,K]

10525

Frege put the idea of abstraction on a rigorous footing [Fine,K]

10556

We create new abstract concepts by carving up the content in a different way

9882	You can't simultaneously fix the truth-conditions of a sentence and the domain of its variables [Dummett]

9881	From basing 'parallel' on identity of direction, Frege got all abstractions from identity statements [Dummett]

19. Language / A. Nature of Meaning / 7. Meaning Holism / a. Sentence meaning

8646	Words in isolation seem to have ideas as meanings, but words have meaning in propositions

7732	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

9370	A statement is analytic if substitution of synonyms can make it a logical truth [Boghossian]

8743	Frege considered analyticity to be an epistemic concept [Shapiro]

19. Language / E. Analyticity / 2. Analytic Truths

20295

All analytic truths can become logical truths, by substituting definitions or synonyms [Rey]

19. Language / E. Analyticity / 4. Analytic/Synthetic Critique

2515	Frege fails to give a concept of analyticity, so he fails to explain synthetic a priori truth that way [Katz]

25. Social Practice / E. Policies / 5. Education / b. Education principles

8619	To learn something, you must know that you don't know

26. Natural Theory / D. Laws of Nature / 6. Laws as Numerical

8656	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

22286

Existence is not a first-level concept (of God), but a second-level property of concepts [Potter]

8644	Because existence is a property of concepts the ontological argument for God fails