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6. Mathematics / A. Nature of Mathematics / 3. Numbers / a. Numbers

[general ideas concerning numbers]

23 ideas
We perceive number by the denial of continuity [Aristotle]
Pluralities divide into discontinous countables; magnitudes divide into continuous things [Aristotle]
Perhaps numbers are substances? [Aristotle]
We can talk of 'innumerable number', about the infinite points on a line [Newton]
Numbers are formed by addition of units in time [Kant]
Numbers are free creations of the human mind, to understand differences [Dedekind]
Obtaining numbers by abstraction is impossible - there are too many; only a rule could give them, in order [Benacerraf]
We must explain how we know so many numbers, and recognise ones we haven't met before [Benacerraf]
Numbers can't be sets if there is no agreement on which sets they are [Benacerraf]
There are no such things as numbers [Benacerraf]
Numbers are for measuring and for calculating (and the two must be consistent) [Badiou]
There is no single unified definition of number [Badiou]
For primes we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1))) [Smith,P]
Number theory aims at the essence of natural numbers, giving their nature, and the epistemology [Wright,C]
Mathematics is higher-order modal logic [Hodes]
In Field's version of science, space-time points replace real numbers [Szabó on Field,H]
If 'the number of Democrats is on the rise', does that mean that 50 million is on the rise? [Yablo]
We should talk about possible existence, rather than actual existence, of numbers [Burgess/Rosen]
'There are two apples' can be expressed logically, with no mention of numbers [Brown,JR]
The meaning of a number isn't just the numerals leading up to it [Heck]
The Aristotelian view is that numbers depend on (and are abstracted from) other things [Oderberg]
What is the relation of number words as singular-terms, adjectives/determiners, and symbols? [Hofweber]
'2 + 2 = 4' can be read as either singular or plural [Hofweber]