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Single Idea 18157

[filed under theme 6. Mathematics / C. Sources of Mathematics / 3. Mathematical Nominalism ]

Full Idea

The style of nominalism which aims to reduce statements about numbers to statements about their applications does not work for the natural numbers, because they have many applications, and it is arbitrary to choose just one of them.

Gist of Idea

Nominalism as based on application of numbers is no good, because there are too many applications

Source

David Bostock (Philosophy of Mathematics [2009], 9.B.5.iii)

Book Ref

Bostock,David: 'Philosophy of Mathematics: An Introduction' [Wiley-Blackwell 2009], p.301

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]
13358 Ordinary or mathematical induction assumes for the first, then always for the next, and hence for all [Bostock]
13359 Complete induction assumes for all numbers less than n, then also for n, and hence for all numbers [Bostock]
13362 If an object has two names, truth is undisturbed if the names are swapped; this is Extensionality [Bostock]
13360 In logic, a name is just any expression which refers to a particular single object [Bostock]
13361 An expression is only a name if it succeeds in referring to a real object [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]
13542 A proof-system is 'absolutely consistent' iff we don't have |-(S)φ for every formula [Bostock]
13541 For 'negation-consistent', there is never |-(S)φ and |-(S)¬φ [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]
13544 Inconsistency or entailment just from functors and quantifiers is finitely based, if compact [Bostock]
13545 Elementary logic cannot distinguish clearly between the finite and the infinite [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]
13617 MPP is a converse of Deduction: If Γ |- φ→ψ then Γ,φ|-ψ [Bostock]
13614 MPP: 'If Γ|=φ and Γ|=φ→ψ then Γ|=ψ' (omit Γs for Detachment) [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]
13618 Compactness means an infinity of sequents on the left will add nothing new [Bostock]
13619 Quantification adds two axiom-schemas and a new rule [Bostock]
13620 Proof by Assumptions can always be reduced to Proof by Axioms, using the Deduction Theorem [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]
13758 In natural deduction we work from the premisses and the conclusion, hoping to meet in the middle [Bostock]
13757 Unlike natural deduction, semantic tableaux have recipes for proving things [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]
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]
13803 If we are to express that there at least two things, we need identity [Bostock]
13800 |= α=α and α=β |= φ(α/ξ ↔ φ(β/ξ) fix identity [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]
13818 If we allow empty domains, we must allow empty names [Bostock]
13820 The idea that anything which can be proved is necessary has a problem with 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]
18109 The completeness of first-order logic implies its compactness [Bostock]
18108 First-order logic is not decidable: there is no test of whether any formula is valid [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]
18115 We could add axioms to make sets either as small or as large as possible [Bostock]
18114 There is no single agreed structure for set theory [Bostock]
18117 Structuralism falsely assumes relations to other numbers are numbers' only properties [Bostock]
18121 In logic a proposition means the same when it is and when it is not asserted [Bostock]
18122 Classical interdefinitions of logical constants and quantifiers is impossible in intuitionism [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]
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]
18133 The usual definitions of identity and of natural numbers are impredicative [Bostock]
18132 The predicativity restriction makes a difference with the real numbers [Bostock]
18137 Impredicative definitions are wrong, because they change the set that is being defined? [Bostock]
18140 The best version of conceptualism is predicativism [Bostock]
18138 Conceptualism fails to grasp mathematical properties, infinity, and objective truth values [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]
18144 Neo-logicists agree that HP introduces number, but also claim that it suffices for the job [Bostock]
18147 Neo-logicists meet the Caesar problem by saying Hume's Principle is unique to number [Bostock]
18146 If Hume's Principle is the whole story, that implies structuralism [Bostock]
18149 There are many criteria for the identity of numbers [Bostock]
18148 Hume's Principle is a definition with existential claims, and won't explain 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]