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Single Idea 12423
[filed under theme 6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
]
Full Idea
Philosophers who hope to avoid commitment to abstract entities by claiming that mathematical statements are analytic must show how analyticity is, or provides a species of, truth not requiring reference.
Gist of Idea
Analyticity avoids abstract entities, but can there be truth without reference?
Source
Philip Kitcher (The Nature of Mathematical Knowledge [1984], 04.I)
Book Ref
Kitcher,Philip: 'The Nature of Mathematical Knowledge' [OUP 1984], p.68
A Reaction
[the last part is a quotation from W.D. Hart] Kitcher notes that Frege has a better account, because he provides objects to which reference can be made. I like this idea, which seems to raise a very large question, connected to truthmakers.
The
32 ideas
from 'The Nature of Mathematical Knowledge'
6298
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Kitcher says maths is an idealisation of the world, and our operations in dealing with it
[Kitcher, by Resnik]
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12387
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Mathematical knowledge arises from basic perception
[Kitcher]
|
12412
|
My constructivism is mathematics as an idealization of collecting and ordering objects
[Kitcher]
|
12413
|
A 'warrant' is a process which ensures that a true belief is knowledge
[Kitcher]
|
12389
|
Knowledge is a priori if the experience giving you the concepts thus gives you the knowledge
[Kitcher]
|
12390
|
A priori knowledge comes from available a priori warrants that produce truth
[Kitcher]
|
12416
|
We have some self-knowledge a priori, such as knowledge of our own existence
[Kitcher]
|
12418
|
In long mathematical proofs we can't remember the original a priori basis
[Kitcher]
|
12392
|
Mathematical a priorism is conceptualist, constructivist or realist
[Kitcher]
|
12420
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If mathematics comes through intuition, that is either inexplicable, or too subjective
[Kitcher]
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12393
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Intuition is no basis for securing a priori knowledge, because it is fallible
[Kitcher]
|
18061
|
Mathematical intuition is not the type platonism needs
[Kitcher]
|
18063
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Conceptualists say we know mathematics a priori by possessing mathematical concepts
[Kitcher]
|
18064
|
If meaning makes mathematics true, you still need to say what the meanings refer to
[Kitcher]
|
12423
|
Analyticity avoids abstract entities, but can there be truth without reference?
[Kitcher]
|
18065
|
We derive limited mathematics from ordinary things, and erect powerful theories on their basis
[Kitcher]
|
18066
|
The old view is that mathematics is useful in the world because it describes the world
[Kitcher]
|
18067
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Abstract objects were a bad way of explaining the structure in mathematics
[Kitcher]
|
18069
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Arithmetic is an idealizing theory
[Kitcher]
|
18068
|
Arithmetic is made true by the world, but is also made true by our constructions
[Kitcher]
|
18070
|
We develop a language for correlations, and use it to perform higher level operations
[Kitcher]
|
18071
|
A one-operation is the segregation of a single object
[Kitcher]
|
12395
|
Real numbers stand to measurement as natural numbers stand to counting
[Kitcher]
|
18074
|
Intuitionists rely on assertability instead of truth, but assertability relies on truth
[Kitcher]
|
18072
|
Constructivism is ontological (that it is the work of an agent) and epistemological (knowable a priori)
[Kitcher]
|
18075
|
Idealisation trades off accuracy for simplicity, in varying degrees
[Kitcher]
|
12425
|
Complex numbers were only accepted when a geometrical model for them was found
[Kitcher]
|
18077
|
The defenders of complex numbers had to show that they could be expressed in physical terms
[Kitcher]
|
18078
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The interest or beauty of mathematics is when it uses current knowledge to advance undestanding
[Kitcher]
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12426
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The 'beauty' or 'interest' of mathematics is just explanatory power
[Kitcher]
|
18083
|
With infinitesimals, you divide by the time, then set the time to zero
[Kitcher]
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20473
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If experiential can defeat a belief, then its justification depends on the defeater's absence
[Kitcher, by Casullo]
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