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Single Idea 10043
[filed under theme 6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
]
Full Idea
Classes and concepts may be conceived of as real objects, ..and are as necessary to obtain a satisfactory system of mathematics as physical bodies are necessary for a satisfactory theory of our sense perceptions, with neither case being about 'data'.
Gist of Idea
Mathematical objects are as essential as physical objects are for perception
Source
Kurt Gödel (Russell's Mathematical Logic [1944], p.456)
Book Ref
'Philosophy of Mathematics: readings (2nd)', ed/tr. Benacerraf/Putnam [CUP 1983], p.456
A Reaction
Note that while he thinks real objects are essential for mathematics, be may not be claiming the same thing for our knowledge of logic. If logic contains no objects, then how could mathematics be reduced to it, as in logicism?
The
34 ideas
with the same theme
[reasons for believing maths entities exists]:
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16150
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One is, so numbers exist, so endless numbers exist, and each one must partake of being
[Plato]
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9863
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We aim for elevated discussion of pure numbers, not attaching them to physical objects
[Plato]
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9864
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In pure numbers, all ones are equal, with no internal parts
[Plato]
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8727
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Geometry is not an activity, but the study of unchanging knowledge
[Plato]
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10216
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We master arithmetic by knowing all the numbers in our soul
[Plato]
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13738
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It is a simple truth that the objects of mathematics have being, of some sort
[Aristotle]
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13874
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Numbers seem to be objects because they exactly fit the inference patterns for identities
[Frege]
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13875
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Frege's platonism proposes that objects are what singular terms refer to
[Frege, by Wright,C]
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7731
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How can numbers be external (one pair of boots is two boots), or subjective (and so relative)?
[Frege, by Weiner]
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7737
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Identities refer to objects, so numbers must be objects
[Frege, by Weiner]
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8635
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Numbers are not physical, and not ideas - they are objective and non-sensible
[Frege]
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8652
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Numbers are objects, because they can take the definite article, and can't be plurals
[Frege]
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9580
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Our concepts recognise existing relations, they don't change them
[Frege]
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9589
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Numbers are not real like the sea, but (crucially) they are still objective
[Frege]
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10303
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Restricted Platonism is just an ideal projection of a domain of thought
[Bernays]
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10043
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Mathematical objects are as essential as physical objects are for perception
[Gödel]
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10490
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Mathematics isn't surprising, given that we experience many objects as abstract
[Boolos]
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12328
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Platonists like axioms and decisions, Aristotelians like definitions, possibilities and logic
[Badiou]
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13869
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Number platonism says that natural number is a sortal concept
[Wright,C]
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10021
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It is claimed that numbers are objects which essentially represent cardinality quantifiers
[Hodes]
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10022
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Numerical terms can't really stand for quantifiers, because that would make them first-level
[Hodes]
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8757
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The Indispensability Argument is the only serious ground for the existence of mathematical entities
[Field,H]
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10200
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We distinguish realism 'in ontology' (for objects), and 'in truth-value' (for being either true or false)
[Shapiro]
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10210
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If mathematical objects are accepted, then a number of standard principles will follow
[Shapiro]
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10215
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Platonists claim we can state the essence of a number without reference to the others
[Shapiro]
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10233
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Platonism must accept that the Peano Axioms could all be false
[Shapiro]
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8298
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Sets are instances of numbers (rather than 'collections'); numbers explain sets, not vice versa
[Lowe]
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8311
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If 2 is a particular, then adding particulars to themselves does nothing, and 2+2=2
[Lowe]
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9606
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The irrationality of root-2 was achieved by intellect, not experience
[Brown,JR]
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4241
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If there are infinite numbers and finite concrete objects, this implies that numbers are abstract objects
[Lowe]
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9183
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Platonism claims that some true assertions have singular terms denoting abstractions, so abstractions exist
[Williamson]
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10003
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Why is arithmetic hard to learn, but then becomes easy?
[Hofweber]
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13741
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If 'there are red roses' implies 'there are roses', then 'there are prime numbers' implies 'there are numbers'
[Schaffer,J]
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23622
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We can only mentally construct potential infinities, but maths needs actual infinities
[Hossack]
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