structure for 'Mathematics'    |     alphabetical list of themes    |     expand these ideas

### 6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism

#### [reasons for believing maths entities exists]

33 ideas
 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 [Frege, by Wright,C]
 7731 How can numbers be external (one pair of boots is two boots), or subjective (and so relative)? [Frege, by Weiner]
 7737 Identities refer to objects, so numbers must be objects [Frege, by Weiner]
 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]