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Single Idea 7731
[filed under theme 6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
]
Full Idea
If the number one is a property of external things, how can one pair of boots be the same as two boots? ...but if the number one is subjective, then the number a thing has for me need not be the same number the object has for you.
Gist of Idea
How can numbers be external (one pair of boots is two boots), or subjective (and so relative)?
Source
report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Joan Weiner - Frege Ch.4
Book Ref
Weiner,Joan: 'Frege' [OUP 1999], p.54
A Reaction
This nicely captures the initial dilemma over the nature of numbers. It is the commonest dilemma in all of philosophy, struggling between subjective and objective accounts of things. Hence Putnam's nice definition of philosophy (Idea 2352).
Related Idea
Idea 2352
The job of the philosopher is to distinguish facts about the world from conventions [Putnam]
The
174 ideas
from 'Grundlagen der Arithmetik (Foundations)'
15948
|
Frege developed formal systems to avoid unnoticed assumptions
[Frege, by Lavine]
|
10804
|
Thoughts have a natural order, to which human thinking is drawn
[Frege, by Yablo]
|
9832
|
Frege sees no 'intersubjective' category, between objective and subjective
[Dummett on Frege]
|
13876
|
The syntactic category is primary, and the ontological category is derivative
[Frege, by Wright,C]
|
9841
|
Frege was the first to give linguistic answers to non-linguistic questions
[Frege, by Dummett]
|
9844
|
Originally Frege liked contextual definitions, but later preferred them fully explicit
[Frege, by Dummett]
|
13881
|
We need to grasp not number-objects, but the states of affairs which make number statements true
[Frege, by Wright,C]
|
9154
|
Frege agreed with Euclid that the axioms of logic and mathematics are known through self-evidence
[Frege, by Burge]
|
9157
|
The null set is only defensible if it is the extension of an empty concept
[Frege, by Burge]
|
9835
|
It is because a concept can be empty that there is such a thing as the empty class
[Frege, by Dummett]
|
9854
|
We can introduce new objects, as equivalence classes of objects already known
[Frege, by Dummett]
|
18104
|
Frege, unlike Russell, has infinite individuals because numbers are individuals
[Frege, by Bostock]
|
9834
|
A class is, for Frege, the extension of a concept
[Frege, by Dummett]
|
16891
|
Despite Gödel, Frege's epistemic ordering of all the truths is still plausible
[Frege, by Burge]
|
16906
|
The primitive simples of arithmetic are the essence, determining the subject, and its boundaries
[Frege, by Jeshion]
|
9838
|
Treating 0 as a number avoids antinomies involving treating 'nobody' as a person
[Frege, by Dummett]
|
9564
|
For Frege 'concept' and 'extension' are primitive, but 'zero' and 'successor' are defined
[Frege, by Chihara]
|
10551
|
If objects exist because they fall under a concept, 0 is the object under which no objects fall
[Frege, by Dummett]
|
17427
|
Frege's 'isolation' could be absence of overlap, or drawing conceptual boundaries
[Frege, by Koslicki]
|
17437
|
Non-arbitrary division means that what falls under the concept cannot be divided into more of the same
[Frege, by Koslicki]
|
17438
|
Our concepts decide what is countable, as in seeing the leaves of the tree, or the foliage
[Frege, by Koslicki]
|
10034
|
The number of natural numbers is not a natural number
[Frege, by George/Velleman]
|
16883
|
Arithmetical statements can't be axioms, because they are provable
[Frege, by Burge]
|
10029
|
Numbers need to be objects, to define the extension of the concept of each successor to n
[Frege, by George/Velleman]
|
9973
|
The number of F's is the extension of the second level concept 'is equipollent with F'
[Frege, by Tait]
|
16500
|
Frege showed that numbers attach to concepts, not to objects
[Frege, by Wiggins]
|
9990
|
Frege replaced Cantor's sets as the objects of equinumerosity attributions with concepts
[Frege, by Tait]
|
7738
|
Zero is defined using 'is not self-identical', and one by using the concept of zero
[Frege, by Weiner]
|
23456
|
Frege said logical predication implies classes, which are arithmetical objects
[Frege, by Morris,M]
|
10625
|
Frege had a motive to treat numbers as objects, but not a justification
[Hale/Wright on Frege]
|
13871
|
Frege claims that numbers are objects, as opposed to them being Fregean concepts
[Frege, by Wright,C]
|
13872
|
Numbers are second-level, ascribing properties to concepts rather than to objects
[Frege, by Wright,C]
|
9816
|
For Frege, successor was a relation, not a function
[Frege, by Dummett]
|
17636
|
A cardinal number may be defined as a class of similar classes
[Frege, by Russell]
|
9953
|
Numbers are more than just 'second-level concepts', since existence is also one
[Frege, by George/Velleman]
|
9954
|
"Number of x's such that ..x.." is a functional expression, yielding a name when completed
[Frege, by George/Velleman]
|
10139
|
Frege gives an incoherent account of extensions resulting from abstraction
[Fine,K on Frege]
|
10028
|
For Frege the number of F's is a collection of first-level concepts
[Frege, by George/Velleman]
|
13887
|
Frege started with contextual definition, but then switched to explicit extensional definition
[Frege, by Wright,C]
|
13897
|
Each number, except 0, is the number of the concept of all of its predecessors
[Frege, by Wright,C]
|
9856
|
Frege's account of cardinals fails in modern set theory, so they are now defined differently
[Dummett on Frege]
|
9902
|
Frege's incorrect view is that a number is an equivalence class
[Benacerraf on Frege]
|
17814
|
The natural number n is the set of n-membered sets
[Frege, by Yourgrau]
|
17819
|
A set doesn't have a fixed number, because the elements can be seen in different ways
[Yourgrau on Frege]
|
17431
|
Vagueness is incomplete definition
[Frege, by Koslicki]
|
9956
|
'The number of Fs' is the extension (a collection of first-level concepts) of the concept 'equinumerous with F'
[Frege, by George/Velleman]
|
13527
|
Frege's cardinals (equivalences of one-one correspondences) is not permissible in ZFC
[Frege, by Wolf,RS]
|
22292
|
Hume's Principle fails to implicitly define numbers, because of the Julius Caesar
[Frege, by Potter]
|
17442
|
Frege thinks number is fundamentally bound up with one-one correspondence
[Frege, by Heck]
|
11030
|
The words 'There are exactly Julius Caesar moons of Mars' are gibberish
[Rumfitt on Frege]
|
10030
|
'Julius Caesar' isn't a number because numbers inherit properties of 0 and successor
[Frege, by George/Velleman]
|
8690
|
From within logic, how can we tell whether an arbitrary object like Julius Caesar is a number?
[Frege, by Friend]
|
10219
|
Frege said 2 is the extension of all pairs (so Julius Caesar isn't 2, because he's not an extension)
[Frege, by Shapiro]
|
13889
|
Fregean numbers are numbers, and not 'Caesar', because they correlate 1-1
[Frege, by Wright,C]
|
18142
|
One-one correlations imply normal arithmetic, but don't explain our concept of a number
[Frege, by Bostock]
|
17820
|
If you can subdivide objects many ways for counting, you can do that to set-elements too
[Yourgrau on Frege]
|
15916
|
Frege's one-to-one correspondence replaces well-ordering, because infinities can't be counted
[Frege, by Lavine]
|
16896
|
If numbers can be derived from logic, then set theory is superfluous
[Frege, by Burge]
|
13874
|
Numbers seem to be objects because they exactly fit the inference patterns for identities
[Frege]
|
13875
|
Frege's platonism proposes that objects are what singular terms refer to
[Frege, by Wright,C]
|
7731
|
How can numbers be external (one pair of boots is two boots), or subjective (and so relative)?
[Frege, by Weiner]
|
7737
|
Identities refer to objects, so numbers must be objects
[Frege, by Weiner]
|
9951
|
It appears that numbers are adjectives, but they don't apply to a single object
[Frege, by George/Velleman]
|
9952
|
Numerical adjectives are of the same second-level type as the existential quantifier
[Frege, by George/Velleman]
|
11031
|
'Jupiter has many moons' won't read as 'The number of Jupiter's moons equals the number many'
[Rumfitt on Frege]
|
7739
|
Arithmetic is analytic
[Frege, by Weiner]
|
9945
|
Logicism shows that no empirical truths are needed to justify arithmetic
[Frege, by George/Velleman]
|
8782
|
Frege offered a Platonist version of logicism, committed to cardinal and real numbers
[Frege, by Hale/Wright]
|
13608
|
Mathematics has no special axioms of its own, but follows from principles of logic (with definitions)
[Frege, by Bostock]
|
5658
|
Numbers are definable in terms of mapping items which fall under concepts
[Frege, by Scruton]
|
16905
|
Arithmetic must be based on logic, because of its total generality
[Frege, by Jeshion]
|
10831
|
Frege only managed to prove that arithmetic was analytic with a logic that included set-theory
[Quine on Frege]
|
13864
|
Frege's platonism and logicism are in conflict, if logic must dictates an infinity of objects
[Wright,C on Frege]
|
10033
|
Why should the existence of pure logic entail the existence of objects?
[George/Velleman on Frege]
|
10010
|
Frege's belief in logicism and in numerical objects seem uncomfortable together
[Hodes on Frege]
|
9631
|
Formalism fails to recognise types of symbols, and also meta-games
[Frege, by Brown,JR]
|
9875
|
Frege was completing Bolzano's work, of expelling intuition from number theory and analysis
[Frege, by Dummett]
|
17816
|
Frege's logicism aimed at removing the reliance of arithmetic on intuition
[Frege, by Yourgrau]
|
8911
|
If abstracta are non-mental, quarks are abstracta, and yet chess and God's thoughts are mental
[Rosen on Frege]
|
10309
|
Frege says singular terms denote objects, numerals are singular terms, so numbers exist
[Frege, by Hale]
|
10550
|
Frege establishes abstract objects independently from concrete ones, by falling under a concept
[Frege, by Dummett]
|
8785
|
For Frege, objects just are what singular terms refer to
[Frege, by Hale/Wright]
|
10278
|
Without concepts we would not have any objects
[Frege, by Shapiro]
|
17432
|
Frege's universe comes already divided into objects
[Frege, by Koslicki]
|
16022
|
The idea of a criterion of identity was introduced by Frege
[Frege, by Noonan]
|
11100
|
Frege's algorithm of identity is the law of putting equals for equals
[Frege, by Quine]
|
12153
|
Geach denies Frege's view, that 'being the same F' splits into being the same and being F
[Perry on Frege]
|
9853
|
Identity between objects is not a consequence of identity, but part of what 'identity' means
[Frege, by Dummett]
|
17623
|
To understand a thought you must understand its logical structure
[Frege, by Burge]
|
9158
|
For Frege a priori knowledge derives from general principles, so numbers can't be primitive
[Frege]
|
13879
|
For Frege, ontological questions are to be settled by reference to syntactic structures
[Frege, by Wright,C]
|
10642
|
Second-order quantifiers are committed to concepts, as first-order commits to objects
[Frege, by Linnebo]
|
10606
|
Frege treats properties as a kind of function, and maybe a property is its characteristic function
[Frege, by Smith,P]
|
2514
|
Frege tried to explain synthetic a priori truths by expanding the concept of analyticity
[Frege, by Katz]
|
9870
|
Early Frege takes the extensions of concepts for granted
[Frege, by Dummett]
|
13878
|
Concepts are, precisely, the references of predicates
[Frege, by Wright,C]
|
7736
|
A concept is a non-psychological one-place function asserting something of an object
[Frege, by Weiner]
|
17430
|
Fregean concepts have precise boundaries and universal applicability
[Frege, by Koslicki]
|
9976
|
Frege accepts abstraction to the concept of all sets equipollent to a given one
[Tait on Frege]
|
10803
|
Frege himself abstracts away from tone and color
[Yablo on Frege]
|
9855
|
Frege's logical abstaction identifies a common feature as the maximal set of equivalent objects
[Frege, by Dummett]
|
10802
|
Frege's 'parallel' and 'direction' don't have the same content, as we grasp 'parallel' first
[Yablo on Frege]
|
10525
|
Frege put the idea of abstraction on a rigorous footing
[Frege, by Fine,K]
|
10526
|
Fregean abstraction creates concepts which are equivalences between initial items
[Frege, by Fine,K]
|
2515
|
Frege fails to give a concept of analyticity, so he fails to explain synthetic a priori truth that way
[Katz on Frege]
|
8619
|
To learn something, you must know that you don't know
[Frege]
|
8620
|
Thought is the same everywhere, and the laws of thought do not vary
[Frege]
|
8622
|
Psychological accounts of concepts are subjective, and ultimately destroy truth
[Frege]
|
8621
|
Mental states are irrelevant to mathematics, because they are vague and fluctuating
[Frege]
|
8415
|
Never lose sight of the distinction between concept and object
[Frege]
|
8414
|
Keep the psychological and subjective separate from the logical and objective
[Frege]
|
20295
|
All analytic truths can become logical truths, by substituting definitions or synonyms
[Frege, by Rey]
|
16903
|
Justifications show the ordering of truths, and the foundation is what is self-evident
[Frege, by Jeshion]
|
17443
|
Many of us find Frege's claim that truths depend on one another an obscure idea
[Heck on Frege]
|
17495
|
Proof aims to remove doubts, but also to show the interdependence of truths
[Frege]
|
9352
|
An a priori truth is one derived from general laws which do not require proof
[Frege]
|
16889
|
A truth is a priori if it can be proved entirely from general unproven laws
[Frege]
|
9370
|
A statement is analytic if substitution of synonyms can make it a logical truth
[Frege, by Boghossian]
|
8743
|
Frege considered analyticity to be an epistemic concept
[Frege, by Shapiro]
|
8624
|
Induction is merely psychological, with a principle that it can actually establish laws
[Frege]
|
22286
|
Existence is not a first-level concept (of God), but a second-level property of concepts
[Frege, by Potter]
|
22294
|
We can show that a concept is consistent by producing something which falls under it
[Frege]
|
8626
|
In science one observation can create high probability, while a thousand might prove nothing
[Frege]
|
16890
|
Frege's problem is explaining the particularity of numbers by general laws
[Frege, by Burge]
|
8630
|
Individual numbers are best derived from the number one, and increase by one
[Frege]
|
8632
|
You can't transfer external properties unchanged to apply to ideas
[Frege]
|
8633
|
There is no physical difference between two boots and one pair of boots
[Frege]
|
8634
|
The equator is imaginary, but not fictitious; thought is needed to recognise it
[Frege]
|
16900
|
Intuitions cannot be communicated
[Frege, by Burge]
|
10539
|
Frege refers to 'concrete' objects, but they are no different in principle from abstract ones
[Frege, by Dummett]
|
8635
|
Numbers are not physical, and not ideas - they are objective and non-sensible
[Frege]
|
8636
|
We can say 'a and b are F' if F is 'wise', but not if it is 'one'
[Frege]
|
8637
|
The number 'one' can't be a property, if any object can be viewed as one or not one
[Frege]
|
9988
|
If we abstract 'from' two cats, the units are not black or white, or cats
[Tait on Frege]
|
8639
|
If numbers are supposed to be patterns, each number can have many patterns
[Frege]
|
8640
|
We cannot define numbers from the idea of a series, because numbers must precede that
[Frege]
|
8641
|
You can abstract concepts from the moon, but the number one is not among them
[Frege]
|
14236
|
Each horse doesn't fall under the concept 'horse that draws the carriage', because all four are needed
[Oliver/Smiley on Frege]
|
11029
|
'Exactly ten gallons' may not mean ten things instantiate 'gallon'
[Rumfitt on Frege]
|
17460
|
A statement of number contains a predication about a concept
[Frege]
|
8642
|
Abstraction from things produces concepts, and numbers are in the concepts
[Frege]
|
8643
|
Affirmation of existence is just denial of zero
[Frege]
|
8644
|
Because existence is a property of concepts the ontological argument for God fails
[Frege]
|
9989
|
Units can be equal without being identical
[Tait on Frege]
|
17429
|
Frege says only concepts which isolate and avoid arbitrary division can give units
[Frege, by Koslicki]
|
17426
|
A concept creating a unit must isolate and unify what falls under it
[Frege]
|
17428
|
Frege says counting is determining what number belongs to a given concept
[Frege, by Koslicki]
|
10013
|
Numerical statements have first-order logical form, so must refer to objects
[Frege, by Hodes]
|
9046
|
Our definition will not tell us whether or not Julius Caesar is a number
[Frege]
|
8645
|
Convert "Jupiter has four moons" into "the number of Jupiter's moons is four"
[Frege]
|
9999
|
For science, we can translate adjectival numbers into noun form
[Frege]
|
8646
|
Words in isolation seem to have ideas as meanings, but words have meaning in propositions
[Frege]
|
9846
|
Defining 'direction' by parallelism doesn't tell you whether direction is a line
[Dummett on Frege]
|
8647
|
Not all objects are spatial; 4 can still be an object, despite lacking spatial co-ordinates
[Frege]
|
8648
|
Ideas are not spatial, and don't have distances between them
[Frege]
|
9840
|
Frege initiated linguistic philosophy, studying number through the sense of sentences
[Frege, by Dummett]
|
9822
|
Nothing should be defined in terms of that to which it is conceptually prior
[Frege, by Dummett]
|
10556
|
We create new abstract concepts by carving up the content in a different way
[Frege]
|
17445
|
Parallelism is intuitive, so it is more fundamental than sameness of direction
[Frege, by Heck]
|
9882
|
You can't simultaneously fix the truth-conditions of a sentence and the domain of its variables
[Dummett on Frege]
|
9881
|
From basing 'parallel' on identity of direction, Frege got all abstractions from identity statements
[Frege, by Dummett]
|
9883
|
Frege introduced the standard device, of defining logical objects with equivalence classes
[Frege, by Dummett]
|
8651
|
A concept is a possible predicate of a singular judgement
[Frege]
|
18181
|
The Number for F is the extension of 'equal to F' (or maybe just F itself)
[Frege]
|
8652
|
Numbers are objects, because they can take the definite article, and can't be plurals
[Frege]
|
8653
|
Nought is the number belonging to the concept 'not identical with itself'
[Frege]
|
8654
|
One is the Number which belongs to the concept "identical with 0"
[Frege]
|
10032
|
'Ancestral' relations are derived by iterating back from a given relation
[Frege, by George/Velleman]
|
8656
|
The laws of number are not laws of nature, but are laws of the laws of nature
[Frege]
|
8655
|
Arithmetic is analytic and a priori, and thus it is part of logic
[Frege]
|
8657
|
Mathematicians just accept self-evidence, whether it is logical or intuitive
[Frege]
|
17624
|
To understand axioms you must grasp their logical power and priority
[Frege, by Burge]
|
18103
|
Numbers are objects because they partake in identity statements
[Frege, by Bostock]
|
7732
|
Never ask for the meaning of a word in isolation, but only in the context of a proposition
[Frege]
|