16150 | One is, so numbers exist, so endless numbers exist, and each one must partake of being [Plato] |
9863 | We aim for elevated discussion of pure numbers, not attaching them to physical objects [Plato] |
9864 | In pure numbers, all ones are equal, with no internal parts [Plato] |
8727 | Geometry is not an activity, but the study of unchanging knowledge [Plato] |
10216 | We master arithmetic by knowing all the numbers in our soul [Plato] |
13738 | It is a simple truth that the objects of mathematics have being, of some sort [Aristotle] |
13874 | Numbers seem to be objects because they exactly fit the inference patterns for identities [Frege] |
13875 | Frege's platonism proposes that objects are what singular terms refer to [Wright,C on Frege] |
7731 | How can numbers be external (one pair of boots is two boots), or subjective (and so relative)? [Weiner on Frege] |
7737 | Identities refer to objects, so numbers must be objects [Weiner on Frege] |
8635 | Numbers are not physical, and not ideas - they are objective and non-sensible [Frege] |
8652 | Numbers are objects, because they can take the definite article, and can't be plurals [Frege] |
9580 | Our concepts recognise existing relations, they don't change them [Frege] |
9589 | Numbers are not real like the sea, but (crucially) they are still objective [Frege] |
10303 | Restricted Platonism is just an ideal projection of a domain of thought [Bernays] |
10043 | Mathematical objects are as essential as physical objects are for perception [Gödel] |
10490 | Mathematics isn't surprising, given that we experience many objects as abstract [Boolos] |
12328 | Platonists like axioms and decisions, Aristotelians like definitions, possibilities and logic [Badiou] |
13869 | Number platonism says that natural number is a sortal concept [Wright,C] |
10021 | It is claimed that numbers are objects which essentially represent cardinality quantifiers [Hodes] |
10022 | Numerical terms can't really stand for quantifiers, because that would make them first-level [Hodes] |
8757 | The Indispensability Argument is the only serious ground for the existence of mathematical entities [Field,H] |
10200 | We distinguish realism 'in ontology' (for objects), and 'in truth-value' (for being either true or false) [Shapiro] |
10210 | If mathematical objects are accepted, then a number of standard principles will follow [Shapiro] |
10215 | Platonists claim we can state the essence of a number without reference to the others [Shapiro] |
10233 | Platonism must accept that the Peano Axioms could all be false [Shapiro] |
8298 | Sets are instances of numbers (rather than 'collections'); numbers explain sets, not vice versa [Lowe] |
8311 | If 2 is a particular, then adding particulars to themselves does nothing, and 2+2=2 [Lowe] |
9606 | The irrationality of root-2 was achieved by intellect, not experience [Brown,JR] |
4241 | If there are infinite numbers and finite concrete objects, this implies that numbers are abstract objects [Lowe] |
9183 | Platonism claims that some true assertions have singular terms denoting abstractions, so abstractions exist [Williamson] |
10003 | Why is arithmetic hard to learn, but then becomes easy? [Hofweber] |
13741 | If 'there are red roses' implies 'there are roses', then 'there are prime numbers' implies 'there are numbers' [Schaffer,J] |