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Single Idea 18150

[filed under theme 6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / b. Indispensability of mathematics ]

Full Idea

We all know that in practice no physical measurement can be 100 per cent accurate, and so it cannot require the existence of a genuinely irrational number, rather than some of the rational numbers close to it.

Gist of Idea

Actual measurement could never require the precision of the real numbers

Source

David Bostock (Philosophy of Mathematics [2009], 9.A.3)

Book Ref

Bostock,David: 'Philosophy of Mathematics: An Introduction' [Wiley-Blackwell 2009], p.280

Related Ideas

Idea 18156 Modern axioms of geometry do not need the real numbers [Bostock]

Idea 18207 Maybe applications of continuum mathematics are all idealisations [Maddy]

The 49 ideas from 'Philosophy of Mathematics'

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]
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]
18121 In logic a proposition means the same when it is and when it is not asserted [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]
18131 If abstracta only exist if they are expressible, there can only be denumerably many of them [Bostock]
18129 Many crucial logicist definitions are in fact 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]
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]
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]