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 / 5. Metaphysics as Conceptual
Syntactic category which is primary and ontological category derivative.
1. Philosophy / F. Analytic Philosophy / 1. Analysis
Frege initiated linguistic philosophy, studying number through the sense of sentences
Frege developed formal systems to avoid unnoticed assumptions
1. Philosophy / F. Analytic Philosophy / 2. Conceptual Analysis
Never lose sight of the distinction between concept and object
2. Reason / A. Nature of Reason / 5. Objectivity
Frege sees no 'intersubjective' category, between objective and subjective
Keep the psychological and subjective separate from the logical and objective
2. Reason / D. Definition / 7. Contextual Definition
Originally Frege liked contextual definitions, but later preferred them fully explicit
Nothing should be defined in terms of that to which it is conceptually prior
2. Reason / E. Argument / 6. Conclusive Proof
Proof reveals the interdependence of truths, as well as showing their certainty
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
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
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
It is because a concept can be empty that there is such a thing as the empty class
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
Frege introduced the standard device, of defining logical objects with equivalence classes
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / c. Logical sets
A class is, for Frege, the extension of a concept
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
The primitive simples of arithmetic are the essence, determining the subject, and its boundaries
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
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
To understand axioms you must grasp their logical power and priority
6. Mathematics / A. Nature of Mathematics / 3. 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. Numbers / l. Zero
Treating 0 as a number avoids antinomies involving treating 'nobody' as a person
For Frege 'concept' and 'extension' are primitive, but 'zero' and 'successor' are defined
If objects exist because they fall under a concept, 0 is the object under which no objects fall
Nought is the number belonging to the concept 'not identical with itself'
6. Mathematics / A. Nature of Mathematics / 3. 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 / 3. Numbers / o. Units
You can abstract concepts from the moon, but the number one is not among them
Units can be equal without being identical
A concept creating a unit must isolate and unify what falls under it
Frege says only concepts which isolate and avoid arbitrary division can give units
6. Mathematics / A. Nature of Mathematics / 3. Numbers / p. Counting
Frege's one-to-one correspondence replaces well-ordering, because infinities can't be counted
Non-arbitrary division means that what falls under the concept cannot be divided into more of the same
Our concepts decide what is countable, as in seeing the leaves of the tree, or the foliage
Frege says counting is determining what number belongs to a given concept
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / h. Ordinal infinity
The number of natural numbers is not a natural number
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / a. Axioms for numbers
Arithmetical statements can't be axioms, because they are provable
6. Mathematics / B. Foundations for Mathematics / 4. Definitions of Number / c. Fregean numbers
Frege claims that numbers are objects, as opposed to them being Fregean concepts
Numbers are second-level, ascribing properties to concepts rather than to objects
For Frege, successor was a relation, not a function
Numbers are more than just 'second-level concepts', since existence is also one
"Number of x's such that ..x.." is a functional expression, yielding a name when completed
Frege gives an incoherent account of extensions resulting from abstraction
For Frege the number of F's is a collection of first-level concepts
Numbers need to be objects, to define the extension of the concept of each successor to n
The number of F's is the extension of the second level concept 'is equipollent with F'
Frege had a motive to treat numbers as objects, but not a justification
Frege showed that numbers attach to concepts, not to objects
Frege replaced Cantor's sets as the objects of equinumerosity attributions with concepts
Zero is defined using 'is not self-identical', and one by using the concept of zero
Frege started with contextual definition, but then switched to explicit extensional definition
The natural number n is the set of n-membered sets
A set doesn't have a fixed number, because the elements can be seen in different ways
If you can subdivide objects many ways for counting, you can do that to set-elements too
Frege's problem is explaining the particularity of numbers by general laws
Numerical statements have first-order logical form, so must refer to objects
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
Individual numbers are best derived from the number one, and increase by one
A statement of number contains a predication about a concept
'Exactly ten gallons' may not mean ten things instantiate 'gallon'
Each number, except 0, is the number of the concept of all of its predecessors
A cardinal number may be defined as a class of similar classes
Frege's account of cardinals fails in modern set theory, so they are now defined differently
Frege's incorrect view is that a number is an equivalence class
6. Mathematics / B. Foundations for Mathematics / 4. 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'
Frege thinks number is fundamentally bound up with one-one correspondence
6. Mathematics / B. Foundations for Mathematics / 4. Definitions of Number / e. Caesar problem
'Julius Caesar' isn't a number because numbers inherit properties of 0 and successor
From within logic, how can we tell whether an arbitrary object like Julius Caesar is a number?
Frege said 2 is the extension of all pairs (so Julius Caesar isn't 2, because he's not an extension)
The words 'There are exactly Julius Caesar moons of Mars' are gibberish
Fregean numbers are numbers, and not 'Caesar', because they correlate 1-1
Our definition will not tell us whether or not Julius Caesar is a number
6. Mathematics / B. Foundations for Mathematics / 5. Mathematics as Set Theory / b. Mathematics is not set theory
If numbers can be derived from logic, then set theory is superfluous
6. Mathematics / B. Foundations for Mathematics / 6. 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
How can numbers be external (one pair of boots is two boots), or subjective (and so relative)?
Identities refer to objects, so numbers must be objects
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
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
Numerical adjectives are of the same second-level type as the existential quantifier
'Jupiter has many moons' won't read as 'The number of Jupiter's moons equals the number many'
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
Arithmetic must be based on logic, because of its total generality
Frege offered a Platonist version of logicism, committed to cardinal and real numbers
Numbers are definable in terms of mapping items which fall under concepts
Arithmetic is analytic and a priori, and thus it is part of logic
Arithmetic is analytic
Logicism shows that no empirical truths are needed to justify arithmetic
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
Frege's belief in logicism and in numerical objects seem uncomfortable together
Why should the existence of pure logic entail the existence of objects?
Frege only managed to prove that arithmetic was analytic with a logic that included set-theory
6. Mathematics / C. Sources of Mathematics / 7. Formalism
Formalism fails to recognise types of symbols, and also meta-games
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 / 6. Abstract Existence
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
Parallelism is intuitive, so it is more fundamental than sameness of direction
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
7. Existence / D. Theories of Reality / 9. Vagueness / c. Vagueness as semantic
Vagueness is incomplete definition
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
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
8. Modes of Existence / A. Relations / 4. Formal Relations / c. Ancestral relation
'Ancestral' relations are derived by iterating back from a given relation
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
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 establishes abstract objects independently from concrete ones, by falling under a concept
Frege says singular terms denote objects, numerals are singular terms, so numbers exist
9. Objects / A. Existence of Objects / 3. Objects in Thought
For Frege, objects just are what singular terms refer to
Without concepts we would not have any objects
9. Objects / A. Existence of Objects / 4. Individuation / a. Individuation
Frege's universe comes already divided into objects
9. Objects / F. Identity among Objects / 1. Concept of Identity
The idea of a criterion of identity was introduced by Frege
Frege's algorithm of identity is the law of putting equals for equals
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
11. Knowledge Aims / A. Knowledge / 2. Understanding
To understand a thought you must understand its logical structure
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 / E. Direct Knowledge / 1. Intuition
Intuitions cannot be communicated
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
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
18. Thought / D. Concepts / 3. Ontology of Concepts / c. Fregean concepts
Concepts are, precisely, the references of predicates
A concept is a non-psychological one-place function asserting something of an object
Fregean concepts have precise boundaries and universal applicability
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
18. Thought / E. Abstraction / 2. Abstracta by Selection
Frege accepts abstraction to the concept of all sets equipollent to a given one
18. Thought / E. Abstraction / 3. Abstracta by Ignoring
Frege himself abstracts away from tone and color
If we abstract 'from' two cats, the units are not black or white, or cats
18. Thought / E. Abstraction / 7. Abstracta by Equivalence
Frege's logical abstaction identifies a common feature as the maximal set of equivalent objects
Fregean abstraction creates concepts which are equivalences between initial items
Frege put the idea of abstraction on a rigorous footing
We create new abstract concepts by carving up the content in a different way
You can't simultaneously fix the truth-conditions of a sentence and the domain of its variables
From basing 'parallel' on identity of direction, Frege got all abstractions from identity statements
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
Frege considered analyticity to be an epistemic concept
19. Language / E. Analyticity / 2. Analytic Truths
All analytic truths can become logical truths, by substituting definitions or synonyms
25. Society / E. State Functions / 5. 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 / C. Proofs of Reason / 2. Ontological Proof critique
Because existence is a property of concepts the ontological argument for God fails