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

[reasons for believing maths entities exists]

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