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Single Idea 18116
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
]
Full Idea
There is no ground for saying that a number IS a position, if the truth is that there is nothing to determine which number is which position.
Gist of Idea
Numbers can't be positions, if nothing decides what position a given number has
Source
David Bostock (Philosophy of Mathematics [2009], 6.4)
Book Ref
Bostock,David: 'Philosophy of Mathematics: An Introduction' [Wiley-Blackwell 2009], p.190
A Reaction
If numbers lose touch with the empirical ability to count physical objects, they drift off into a mad world where they crumble away.
The
121 ideas
from David Bostock
13821
|
Extensionality is built into ordinary logic semantics; names have objects, predicates have sets of objects
[Bostock]
|
13346
|
Truth is the basic notion in classical logic
[Bostock]
|
13347
|
Validity is a conclusion following for premises, even if there is no proof
[Bostock]
|
13348
|
It seems more natural to express |= as 'therefore', rather than 'entails'
[Bostock]
|
13349
|
Γ|=φ is 'entails'; Γ|= is 'is inconsistent'; |=φ is 'valid'
[Bostock]
|
13350
|
'Assumptions' says that a formula entails itself (φ|=φ)
[Bostock]
|
13351
|
'Thinning' allows that if premisses entail a conclusion, then adding further premisses makes no difference
[Bostock]
|
13352
|
'Cutting' allows that if x is proved, and adding y then proves z, you can go straight to z
[Bostock]
|
13353
|
'Negation' says that Γ,¬φ|= iff Γ|=φ
[Bostock]
|
13354
|
'Conjunction' says that Γ|=φ∧ψ iff Γ|=φ and Γ|=ψ
[Bostock]
|
13355
|
'Disjunction' says that Γ,φ∨ψ|= iff Γ,φ|= and Γ,ψ|=
[Bostock]
|
13356
|
The 'conditional' is that Γ|=φ→ψ iff Γ,φ|=ψ
[Bostock]
|
13421
|
'Disjunctive Normal Form' is ensuring that no conjunction has a disjunction within its scope
[Bostock]
|
13422
|
'Conjunctive Normal Form' is ensuring that no disjunction has a conjunction within its scope
[Bostock]
|
13357
|
Truth-functors are usually held to be defined by their truth-tables
[Bostock]
|
13359
|
Complete induction assumes for all numbers less than n, then also for n, and hence for all numbers
[Bostock]
|
13358
|
Ordinary or mathematical induction assumes for the first, then always for the next, and hence for all
[Bostock]
|
13361
|
An expression is only a name if it succeeds in referring to a real object
[Bostock]
|
13360
|
In logic, a name is just any expression which refers to a particular single object
[Bostock]
|
13362
|
If an object has two names, truth is undisturbed if the names are swapped; this is Extensionality
[Bostock]
|
13363
|
A (modern) predicate is the result of leaving a gap for the name in a sentence
[Bostock]
|
13364
|
Interpretation by assigning objects to names, or assigning them to variables first
[Bostock, by PG]
|
13438
|
'Prenex normal form' is all quantifiers at the beginning, out of the scope of truth-functors
[Bostock]
|
13439
|
Venn Diagrams map three predicates into eight compartments, then look for the conclusion
[Bostock]
|
13611
|
Tableau proofs use reduction - seeking an impossible consequence from an assumption
[Bostock]
|
13612
|
Non-branching rules add lines, and branching rules need a split; a branch with a contradiction is 'closed'
[Bostock]
|
13540
|
A set of formulae is 'inconsistent' when there is no interpretation which can make them all true
[Bostock]
|
13541
|
For 'negation-consistent', there is never |-(S)φ and |-(S)¬φ
[Bostock]
|
13542
|
A proof-system is 'absolutely consistent' iff we don't have |-(S)φ for every formula
[Bostock]
|
13613
|
A completed open branch gives an interpretation which verifies those formulae
[Bostock]
|
13543
|
A relation is not reflexive, just because it is transitive and symmetrical
[Bostock]
|
13545
|
Elementary logic cannot distinguish clearly between the finite and the infinite
[Bostock]
|
13544
|
Inconsistency or entailment just from functors and quantifiers is finitely based, if compact
[Bostock]
|
13623
|
The syntactic turnstile |- φ means 'there is a proof of φ' or 'φ is a theorem'
[Bostock]
|
13610
|
A logic with ¬ and → needs three axiom-schemas and one rule as foundation
[Bostock]
|
13615
|
'Conditonalised' inferences point to the Deduction Theorem: If Γ,φ|-ψ then Γ|-φ→ψ
[Bostock]
|
13616
|
The Deduction Theorem greatly simplifies the search for proof
[Bostock]
|
13614
|
MPP: 'If Γ|=φ and Γ|=φ→ψ then Γ|=ψ' (omit Γs for Detachment)
[Bostock]
|
13617
|
MPP is a converse of Deduction: If Γ |- φ→ψ then Γ,φ|-ψ
[Bostock]
|
13618
|
Compactness means an infinity of sequents on the left will add nothing new
[Bostock]
|
13620
|
Proof by Assumptions can always be reduced to Proof by Axioms, using the Deduction Theorem
[Bostock]
|
13619
|
Quantification adds two axiom-schemas and a new rule
[Bostock]
|
13621
|
The Deduction Theorem and Reductio can 'discharge' assumptions - they aren't needed for the new truth
[Bostock]
|
13622
|
Axiom systems from Frege, Russell, Church, Lukasiewicz, Tarski, Nicod, Kleene, Quine...
[Bostock]
|
13753
|
Natural deduction takes proof from assumptions (with its rules) as basic, and axioms play no part
[Bostock]
|
13755
|
Excluded middle is an introduction rule for negation, and ex falso quodlibet will eliminate it
[Bostock]
|
13754
|
Natural deduction rules for → are the Deduction Theorem (→I) and Modus Ponens (→E)
[Bostock]
|
13756
|
A tree proof becomes too broad if its only rule is Modus Ponens
[Bostock]
|
13757
|
Unlike natural deduction, semantic tableaux have recipes for proving things
[Bostock]
|
13758
|
In natural deduction we work from the premisses and the conclusion, hoping to meet in the middle
[Bostock]
|
13759
|
Each line of a sequent calculus is a conclusion of previous lines, each one explicitly recorded
[Bostock]
|
13760
|
A sequent calculus is good for comparing proof systems
[Bostock]
|
13761
|
In a tableau proof no sequence is established until the final branch is closed; hypotheses are explored
[Bostock]
|
13762
|
Tableau rules are all elimination rules, gradually shortening formulae
[Bostock]
|
13800
|
|= α=α and α=β |= φ(α/ξ ↔ φ(β/ξ) fix identity
[Bostock]
|
13803
|
If we are to express that there at least two things, we need identity
[Bostock]
|
13801
|
An 'informal proof' is in no particular system, and uses obvious steps and some ordinary English
[Bostock]
|
13799
|
The sign '=' is a two-place predicate expressing that 'a is the same thing as b' (a=b)
[Bostock]
|
13802
|
Relations can be one-many (at most one on the left) or many-one (at most one on the right)
[Bostock]
|
13812
|
A 'zero-place' function just has a single value, so it is a name
[Bostock]
|
13811
|
A 'total' function ranges over the whole domain, a 'partial' function over appropriate inputs
[Bostock]
|
13813
|
Definite descriptions don't always pick out one thing, as in denials of existence, or errors
[Bostock]
|
13814
|
Definite desciptions resemble names, but can't actually be names, if they don't always refer
[Bostock]
|
13816
|
Because of scope problems, definite descriptions are best treated as quantifiers
[Bostock]
|
13817
|
Definite descriptions are usually treated like names, and are just like them if they uniquely refer
[Bostock]
|
13815
|
Names do not have scope problems (e.g. in placing negation), but Russell's account does have that problem
[Bostock]
|
13820
|
The idea that anything which can be proved is necessary has a problem with empty names
[Bostock]
|
13818
|
If we allow empty domains, we must allow empty names
[Bostock]
|
13822
|
Fictional characters wreck elementary logic, as they have contradictions and no excluded middle
[Bostock]
|
13846
|
A 'free' logic can have empty names, and a 'universally free' logic can have empty domains
[Bostock]
|
13847
|
If non-existent things are self-identical, they are just one thing - so call it the 'null object'
[Bostock]
|
13848
|
We are only obliged to treat definite descriptions as non-names if only the former have scope
[Bostock]
|
18095
|
Instead of by cuts or series convergence, real numbers could be defined by axioms
[Bostock]
|
18093
|
For Eudoxus cuts in rationals are unique, but not every cut makes a real number
[Bostock]
|
18097
|
The Peano Axioms describe a unique structure
[Bostock]
|
18099
|
The number of reals is the number of subsets of the natural numbers
[Bostock]
|
18100
|
ω + 1 is a new ordinal, but its cardinality is unchanged
[Bostock]
|
18101
|
Each addition changes the ordinality but not the cardinality, prior to aleph-1
[Bostock]
|
18102
|
A cardinal is the earliest ordinal that has that number of predecessors
[Bostock]
|
18106
|
Aleph-1 is the first ordinal that exceeds aleph-0
[Bostock]
|
18107
|
A 'proper class' cannot be a member of anything
[Bostock]
|
18105
|
Replacement enforces a 'limitation of size' test for the existence of sets
[Bostock]
|
18108
|
First-order logic is not decidable: there is no test of whether any formula is valid
[Bostock]
|
18109
|
The completeness of first-order logic implies its compactness
[Bostock]
|
18110
|
Infinitesimals are not actually contradictory, because they can be non-standard real numbers
[Bostock]
|
18111
|
Treating numbers as objects doesn't seem like logic, since arithmetic fixes their totality
[Bostock]
|
18116
|
Numbers can't be positions, if nothing decides what position a given number has
[Bostock]
|
18114
|
There is no single agreed structure for set theory
[Bostock]
|
18115
|
We could add axioms to make sets either as small or as large as possible
[Bostock]
|
18117
|
Structuralism falsely assumes relations to other numbers are numbers' only properties
[Bostock]
|
18122
|
Classical interdefinitions of logical constants and quantifiers is impossible in intuitionism
[Bostock]
|
18121
|
In logic a proposition means the same when it is and when it is not asserted
[Bostock]
|
18120
|
The Deduction Theorem is what licenses a system of natural deduction
[Bostock]
|
18123
|
Substitutional quantification is just standard if all objects in the domain have a name
[Bostock]
|
18125
|
Berry's Paradox considers the meaning of 'The least number not named by this name'
[Bostock]
|
18127
|
Simple type theory has 'levels', but ramified type theory has 'orders'
[Bostock]
|
18129
|
Many crucial logicist definitions are in fact impredicative
[Bostock]
|
18131
|
If abstracta only exist if they are expressible, there can only be denumerably many of them
[Bostock]
|
18132
|
The predicativity restriction makes a difference with the real numbers
[Bostock]
|
18133
|
The usual definitions of identity and of natural numbers are impredicative
[Bostock]
|
18134
|
Predicativism makes theories of huge cardinals impossible
[Bostock]
|
18135
|
If mathematics rests on science, predicativism may be the best approach
[Bostock]
|
18136
|
If we can only think of what we can describe, predicativism may be implied
[Bostock]
|
18137
|
Impredicative definitions are wrong, because they change the set that is being defined?
[Bostock]
|
18138
|
Conceptualism fails to grasp mathematical properties, infinity, and objective truth values
[Bostock]
|
18140
|
The best version of conceptualism is predicativism
[Bostock]
|
18139
|
The Axiom of Choice relies on reference to sets that we are unable to describe
[Bostock]
|
18141
|
Nominalism about mathematics is either reductionist, or fictionalist
[Bostock]
|
18158
|
Ordinals are mainly used adjectively, as in 'the first', 'the second'...
[Bostock]
|
18159
|
Higher cardinalities in sets are just fairy stories
[Bostock]
|
18147
|
Neo-logicists meet the Caesar problem by saying Hume's Principle is unique to number
[Bostock]
|
18144
|
Neo-logicists agree that HP introduces number, but also claim that it suffices for the job
[Bostock]
|
18146
|
If Hume's Principle is the whole story, that implies structuralism
[Bostock]
|
18148
|
Hume's Principle is a definition with existential claims, and won't explain numbers
[Bostock]
|
18149
|
There are many criteria for the identity of numbers
[Bostock]
|
18145
|
Many things will satisfy Hume's Principle, so there are many interpretations of it
[Bostock]
|
18143
|
Frege makes numbers sets to solve the Caesar problem, but maybe Caesar is a set!
[Bostock]
|
18150
|
Actual measurement could never require the precision of the real numbers
[Bostock]
|
18155
|
A fairy tale may give predictions, but only a true theory can give explanations
[Bostock]
|
18156
|
Modern axioms of geometry do not need the real numbers
[Bostock]
|
18157
|
Nominalism as based on application of numbers is no good, because there are too many applications
[Bostock]
|