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Single Idea 18145
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle
]
Full Idea
Hume's Principle gives a criterion of identity for numbers, but it is obvious that many other things satisfy that criterion. The simplest example is probably the numerals (in any notation, decimal, binary etc.), giving many different interpretations.
Gist of Idea
Many things will satisfy Hume's Principle, so there are many interpretations of it
Source
David Bostock (Philosophy of Mathematics [2009], 9.A.2)
Book Ref
Bostock,David: 'Philosophy of Mathematics: An Introduction' [Wiley-Blackwell 2009], p.271
Related Ideas
Idea 18146
If Hume's Principle is the whole story, that implies structuralism [Bostock]
Idea 18147
Neo-logicists meet the Caesar problem by saying Hume's Principle is unique to number [Bostock]
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]
|
18101
|
Each addition changes the ordinality but not the cardinality, prior to aleph-1
[Bostock]
|
18100
|
ω + 1 is a new ordinal, but its cardinality is unchanged
[Bostock]
|
18102
|
A cardinal is the earliest ordinal that has that number of predecessors
[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]
|
18106
|
Aleph-1 is the first ordinal that exceeds aleph-0
[Bostock]
|
18108
|
First-order logic is not decidable: there is no test of whether any formula is valid
[Bostock]
|
18109
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The completeness of first-order logic implies its compactness
[Bostock]
|
18110
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Infinitesimals are not actually contradictory, because they can be non-standard real numbers
[Bostock]
|
18111
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Treating numbers as objects doesn't seem like logic, since arithmetic fixes their totality
[Bostock]
|
18115
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We could add axioms to make sets either as small or as large as possible
[Bostock]
|
18114
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There is no single agreed structure for set theory
[Bostock]
|
18116
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Numbers can't be positions, if nothing decides what position a given number has
[Bostock]
|
18117
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Structuralism falsely assumes relations to other numbers are numbers' only properties
[Bostock]
|
18122
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Classical interdefinitions of logical constants and quantifiers is impossible in intuitionism
[Bostock]
|
18120
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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]
|
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
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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]
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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]
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18137
|
Impredicative definitions are wrong, because they change the set that is being defined?
[Bostock]
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18140
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The best version of conceptualism is predicativism
[Bostock]
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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]
|
18146
|
If Hume's Principle is the whole story, that implies structuralism
[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]
|
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]
|
18149
|
There are many criteria for the identity of numbers
[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
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Nominalism as based on application of numbers is no good, because there are too many applications
[Bostock]
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