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Single Idea 9998
[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
]
Full Idea
There are three different uses of the number words: the singular-term use (as in 'the number of moons of Jupiter is four'), the adjectival (or determiner) use (as in 'Jupiter has four moons'), and the symbolic use (as in '4'). How are they related?
Gist of Idea
What is the relation of number words as singular-terms, adjectives/determiners, and symbols?
Source
Thomas Hofweber (Number Determiners, Numbers, Arithmetic [2005], §1)
Book Ref
-: 'Philosophical Review 114' [Phil Review 2005], p.180
A Reaction
A classic philosophy of language approach to the problem - try to give the truth-conditions for all three types. The main problem is that the first one implies that numbers are objects, whereas the others do not. Why did Frege give priority to the first?
The
26 ideas
with the same theme
[general ideas concerning numbers]:
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1729
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We perceive number by the denial of continuity
[Aristotle]
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13273
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Pluralities divide into discontinous countables; magnitudes divide into continuous things
[Aristotle]
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10958
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Perhaps numbers are substances?
[Aristotle]
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13152
|
We can talk of 'innumerable number', about the infinite points on a line
[Newton]
|
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16920
|
Numbers are formed by addition of units in time
[Kant]
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9823
|
Numbers are free creations of the human mind, to understand differences
[Dedekind]
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23186
|
Numbers enable us to manage the world - to the limits of counting
[Nietzsche]
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13412
|
Obtaining numbers by abstraction is impossible - there are too many; only a rule could give them, in order
[Benacerraf]
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13413
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We must explain how we know so many numbers, and recognise ones we haven't met before
[Benacerraf]
|
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9901
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Numbers can't be sets if there is no agreement on which sets they are
[Benacerraf]
|
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9912
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There are no such things as numbers
[Benacerraf]
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12334
|
There is no single unified definition of number
[Badiou]
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12335
|
Numbers are for measuring and for calculating (and the two must be consistent)
[Badiou]
|
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10599
|
For primes we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1)))
[Smith,P]
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13861
|
Number theory aims at the essence of natural numbers, giving their nature, and the epistemology
[Wright,C]
|
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10027
|
Mathematics is higher-order modal logic
[Hodes]
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8958
|
In Field's version of science, space-time points replace real numbers
[Field,H, by Szabó]
|
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8865
|
If 'the number of Democrats is on the rise', does that mean that 50 million is on the rise?
[Yablo]
|
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9923
|
We should talk about possible existence, rather than actual existence, of numbers
[Burgess/Rosen]
|
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9622
|
'There are two apples' can be expressed logically, with no mention of numbers
[Brown,JR]
|
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17453
|
The meaning of a number isn't just the numerals leading up to it
[Heck]
|
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12238
|
The Aristotelian view is that numbers depend on (and are abstracted from) other things
[Oderberg]
|
|
9998
|
What is the relation of number words as singular-terms, adjectives/determiners, and symbols?
[Hofweber]
|
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10002
|
'2 + 2 = 4' can be read as either singular or plural
[Hofweber]
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21644
|
Numbers are used as singular terms, as adjectives, and as symbols
[Hofweber]
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21646
|
The Amazonian Piraha language is said to have no number words
[Hofweber]
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