more from this thinker
|
more from this text
Single Idea 10200
[filed under theme 6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
]
Full Idea
We must distinguish between 'realism in ontology' - that mathematical objects exist - and 'realism in truth-value', which is suggested by the model-theoretic framework - that each well-formed meaningful sentence is non-vacuously either true or false.
Gist of Idea
We distinguish realism 'in ontology' (for objects), and 'in truth-value' (for being either true or false)
Source
Stewart Shapiro (Philosophy of Mathematics [1997], Intro)
Book Ref
Shapiro,Stewart: 'Philosophy of Mathematics:structure and ontology' [OUP 1997], p.4
A Reaction
My inclination is fairly strongly towards realism of the second kind, but not of the first. A view about the notion of a 'truth-maker' might therefore be required. What do the truths refer to? Answer: not objects, but abstractions from objects.
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]
|