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Single Idea 18255
[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
]
Full Idea
It is essential to a quantitative domain of any kind that there should be an operation of adding its elements; that this is more fundamental thaat that they should be linearly ordered by magnitude is apparent from cyclic domains like that of angles.
Gist of Idea
Addition of quantities is prior to ordering, as shown in cyclic domains like angles
Source
Michael Dummett (Frege philosophy of mathematics [1991], 22 'Quantit')
Book Ref
Dummett,Michael: 'Frege: philosophy of mathematics' [Duckworth 1991], p.279
Related Idea
Idea 18256
Quantity is inconceivable without the idea of addition [Frege]
The
27 ideas
from 'Frege philosophy of mathematics'
|
9848
|
Content is replaceable if identical, so replaceability can't define identity
[Dummett, by Dummett]
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|
9993
|
There is no reason why abstraction by equivalence classes should be called 'logical'
[Dummett, by Tait]
|
|
18257
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Why should the limit of measurement be points, not intervals?
[Dummett]
|
|
18255
|
Addition of quantities is prior to ordering, as shown in cyclic domains like angles
[Dummett]
|
|
9896
|
A prime number is one which is measured by a unit alone
[Dummett]
|
|
9895
|
A number is a multitude composed of units
[Dummett]
|
|
9820
|
In classical logic, logical truths are valid formulas; in higher-order logics they are purely logical
[Dummett]
|
|
9829
|
The identity of a number may be fixed by something outside structure - by counting
[Dummett]
|
|
9828
|
Numbers aren't fixed by position in a structure; it won't tell you whether to start with 0 or 1
[Dummett]
|
|
9833
|
To abstract from spoons (to get the same number as the forks), the spoons must be indistinguishable too
[Dummett]
|
|
9836
|
Fregean semantics assumes a domain articulated into individual objects
[Dummett]
|
|
9842
|
Frege introduced criteria for identity, but thought defining identity was circular
[Dummett]
|
|
9847
|
A contextual definition permits the elimination of the expression by a substitution
[Dummett]
|
|
9852
|
We understand 'there are as many nuts as apples' as easily by pairing them as by counting them
[Dummett]
|
|
9849
|
Maybe a concept is 'prior' to another if it can be defined without the second concept
[Dummett]
|
|
9850
|
An argument for conceptual priority is greater simplicity in explanation
[Dummett]
|
|
9857
|
We arrive at the concept 'suicide' by comparing 'Cato killed Cato' with 'Brutus killed Brutus'
[Dummett]
|
|
9869
|
Realism is just the application of two-valued semantics to sentences
[Dummett]
|
|
9858
|
Abstract objects nowadays are those which are objective but not actual
[Dummett]
|
|
9859
|
It is absurd to deny the Equator, on the grounds that it lacks causal powers
[Dummett]
|
|
9860
|
'We've crossed the Equator' has truth-conditions, so accept the Equator - and it's an object
[Dummett]
|
|
9872
|
Abstract objects need the context principle, since they can't be encountered directly
[Dummett]
|
|
9873
|
Abstract terms are acceptable as long as we know how they function linguistically
[Dummett]
|
|
9876
|
Set theory isn't part of logic, and why reduce to something more complex?
[Dummett]
|
|
9884
|
The distinction of concrete/abstract, or actual/non-actual, is a scale, not a dichotomy
[Dummett]
|
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9880
|
Nominalism assumes unmediated mental contact with objects
[Dummett]
|
|
9885
|
The existence of abstract objects is a pseudo-problem
[Dummett]
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