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Single Idea 18155
[filed under theme 6. Mathematics / C. Sources of Mathematics / 9. Fictional Mathematics
]
Full Idea
A common view is that although a fairy tale may provide very useful predictions, it cannot provide explanations for why things happen as they do. In order to do that a theory must also be true (or, at least, an approximation to the truth).
Gist of Idea
A fairy tale may give predictions, but only a true theory can give explanations
Source
David Bostock (Philosophy of Mathematics [2009], 9.B.5)
Book Ref
Bostock,David: 'Philosophy of Mathematics: An Introduction' [Wiley-Blackwell 2009], p.290
A Reaction
Of course, fictionalism offers an explanation of mathematics as a whole, but not of the details (except as the implications of the initial fictional assumptions).
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
|
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]
|
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]
|
18116
|
Numbers can't be positions, if nothing decides what position a given number has
[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]
|
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]
|
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
|
Nominalism as based on application of numbers is no good, because there are too many applications
[Bostock]
|