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Single Idea 7731

[filed under theme 6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism ]

Full Idea

If the number one is a property of external things, how can one pair of boots be the same as two boots? ...but if the number one is subjective, then the number a thing has for me need not be the same number the object has for you.

Gist of Idea

How can numbers be external (one pair of boots is two boots), or subjective (and so relative)?

Source

report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Joan Weiner - Frege Ch.4

Book Ref

Weiner,Joan: 'Frege' [OUP 1999], p.54

A Reaction

This nicely captures the initial dilemma over the nature of numbers. It is the commonest dilemma in all of philosophy, struggling between subjective and objective accounts of things. Hence Putnam's nice definition of philosophy (Idea 2352).

Related Idea

Idea 2352 The job of the philosopher is to distinguish facts about the world from conventions [Putnam]

The 34 ideas with the same theme [reasons for believing maths entities exists]:

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]
23622 We can only mentally construct potential infinities, but maths needs actual infinities [Hossack]