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Single Idea 9828
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
]
Full Idea
The number 0 is not differentiated from 1 by its position in a progression, otherwise there would be no difference between starting with 0 and starting with 1. That is enough to show that numbers are not identifiable just as positions in structures.
Gist of Idea
Numbers aren't fixed by position in a structure; it won't tell you whether to start with 0 or 1
Source
Michael Dummett (Frege philosophy of mathematics [1991], Ch. 5)
Book Ref
Dummett,Michael: 'Frege: philosophy of mathematics' [Duckworth 1991], p.53
A Reaction
This sounds conclusive, but doesn't feel right. If numbers are a structure, then where you 'start' seems unimportant. Where do you 'start' in St Paul's Cathedral? Starting sounds like a constructivist concept for number theory.
The
27 ideas
from 'Frege philosophy of mathematics'
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9848
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Content is replaceable if identical, so replaceability can't define identity
[Dummett, by Dummett]
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9993
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There is no reason why abstraction by equivalence classes should be called 'logical'
[Dummett, by Tait]
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18257
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Why should the limit of measurement be points, not intervals?
[Dummett]
|
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18255
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Addition of quantities is prior to ordering, as shown in cyclic domains like angles
[Dummett]
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9896
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A prime number is one which is measured by a unit alone
[Dummett]
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9895
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A number is a multitude composed of units
[Dummett]
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9820
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In classical logic, logical truths are valid formulas; in higher-order logics they are purely logical
[Dummett]
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9829
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The identity of a number may be fixed by something outside structure - by counting
[Dummett]
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9828
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Numbers aren't fixed by position in a structure; it won't tell you whether to start with 0 or 1
[Dummett]
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9833
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To abstract from spoons (to get the same number as the forks), the spoons must be indistinguishable too
[Dummett]
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9836
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Fregean semantics assumes a domain articulated into individual objects
[Dummett]
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9842
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Frege introduced criteria for identity, but thought defining identity was circular
[Dummett]
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9847
|
A contextual definition permits the elimination of the expression by a substitution
[Dummett]
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9852
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We understand 'there are as many nuts as apples' as easily by pairing them as by counting them
[Dummett]
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9849
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Maybe a concept is 'prior' to another if it can be defined without the second concept
[Dummett]
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9850
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An argument for conceptual priority is greater simplicity in explanation
[Dummett]
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9857
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We arrive at the concept 'suicide' by comparing 'Cato killed Cato' with 'Brutus killed Brutus'
[Dummett]
|
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9869
|
Realism is just the application of two-valued semantics to sentences
[Dummett]
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9858
|
Abstract objects nowadays are those which are objective but not actual
[Dummett]
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9859
|
It is absurd to deny the Equator, on the grounds that it lacks causal powers
[Dummett]
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9860
|
'We've crossed the Equator' has truth-conditions, so accept the Equator - and it's an object
[Dummett]
|
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9872
|
Abstract objects need the context principle, since they can't be encountered directly
[Dummett]
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9873
|
Abstract terms are acceptable as long as we know how they function linguistically
[Dummett]
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9876
|
Set theory isn't part of logic, and why reduce to something more complex?
[Dummett]
|
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9884
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The distinction of concrete/abstract, or actual/non-actual, is a scale, not a dichotomy
[Dummett]
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9880
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Nominalism assumes unmediated mental contact with objects
[Dummett]
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9885
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The existence of abstract objects is a pseudo-problem
[Dummett]
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