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Single Idea 9838
[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / l. Zero
]
Full Idea
Frege's point was that by treating 0 as a number, we run into none of the antinomies that result from treating 'never' as the name of a time, or 'nobody' as the name of a person.
Gist of Idea
Treating 0 as a number avoids antinomies involving treating 'nobody' as a person
Source
report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Michael Dummett - Frege philosophy of mathematics Ch.8
Book Ref
Dummett,Michael: 'Frege: philosophy of mathematics' [Duckworth 1991], p.96
A Reaction
I don't think that is a good enough reason. Daft problems like that are solved by settling the underlying proposition or logical form (of a sentence containing 'nobody') before one begins to reason. Other antinomies arise with zero.
The
174 ideas
from 'Grundlagen der Arithmetik (Foundations)'
15916
|
Frege's one-to-one correspondence replaces well-ordering, because infinities can't be counted
[Frege, by Lavine]
|
10034
|
The number of natural numbers is not a natural number
[Frege, by George/Velleman]
|
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]
|
16883
|
Arithmetical statements can't be axioms, because they are provable
[Frege, by Burge]
|
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]
|
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]
|
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]
|
17636
|
A cardinal number may be defined as a class of similar classes
[Frege, by Russell]
|
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]
|
17820
|
If you can subdivide objects many ways for counting, you can do that to set-elements too
[Yourgrau on Frege]
|
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]
|
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]
|
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]
|
9945
|
Logicism shows that no empirical truths are needed to justify arithmetic
[Frege, by George/Velleman]
|
7739
|
Arithmetic is analytic
[Frege, by Weiner]
|
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]
|
16905
|
Arithmetic must be based on logic, because of its total generality
[Frege, by Jeshion]
|
5658
|
Numbers are definable in terms of mapping items which fall under concepts
[Frege, by Scruton]
|
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]
|
10642
|
Second-order quantifiers are committed to concepts, as first-order commits to objects
[Frege, by Linnebo]
|
17431
|
Vagueness is incomplete definition
[Frege, by Koslicki]
|
13879
|
For Frege, ontological questions are to be settled by reference to syntactic structures
[Frege, by Wright,C]
|
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]
|
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]
|
10606
|
Frege treats properties as a kind of function, and maybe a property is its characteristic function
[Frege, by Smith,P]
|
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]
|
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]
|
10526
|
Fregean abstraction creates concepts which are equivalences between initial items
[Frege, by Fine,K]
|
10525
|
Frege put the idea of abstraction on a rigorous footing
[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]
|
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]
|
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]
|
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]
|
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]
|
8414
|
Keep the psychological and subjective separate from the logical and objective
[Frege]
|
8415
|
Never lose sight of the distinction between concept and object
[Frege]
|
20295
|
All analytic truths can become logical truths, by substituting definitions or synonyms
[Frege, by Rey]
|
17495
|
Proof aims to remove doubts, but also to show the interdependence of truths
[Frege]
|
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]
|
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]
|
16900
|
Intuitions cannot be communicated
[Frege, by Burge]
|
8634
|
The equator is imaginary, but not fictitious; thought is needed to recognise it
[Frege]
|
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]
|
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]
|
14236
|
Each horse doesn't fall under the concept 'horse that draws the carriage', because all four are needed
[Oliver/Smiley on 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]
|
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]
|
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]
|
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]
|
9999
|
For science, we can translate adjectival numbers into noun form
[Frege]
|
8645
|
Convert "Jupiter has four moons" into "the number of Jupiter's moons is four"
[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]
|
9881
|
From basing 'parallel' on identity of direction, Frege got all abstractions from identity statements
[Frege, by Dummett]
|
9882
|
You can't simultaneously fix the truth-conditions of a sentence and the domain of its variables
[Dummett on Frege]
|
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]
|