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Single Idea 12412

[filed under theme 6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism ]

Full Idea

The constructivist position I defend claims that mathematics is an idealized science of operations which can be performed on objects in our environment. It offers an idealized description of operations of collecting and ordering.

Gist of Idea

My constructivism is mathematics as an idealization of collecting and ordering objects

Source

Philip Kitcher (The Nature of Mathematical Knowledge [1984], Intro)

Book Ref

Kitcher,Philip: 'The Nature of Mathematical Knowledge' [OUP 1984], p.12

A Reaction

I think this is right. What is missing from Kitcher's account (and every other account I've met) is what is meant by 'idealization'. How do you go about idealising something? Hence my interest in the psychology of abstraction.

The 32 ideas from 'The Nature of Mathematical Knowledge'

6298 Kitcher says maths is an idealisation of the world, and our operations in dealing with it [Kitcher, by Resnik]
12387 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 If mathematics comes through intuition, that is either inexplicable, or too subjective [Kitcher]
12393 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 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 Abstract objects were a bad way of explaining the structure in mathematics [Kitcher]
18068 Arithmetic is made true by the world, but is also made true by our constructions [Kitcher]
18069 Arithmetic is an idealizing theory [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 The interest or beauty of mathematics is when it uses current knowledge to advance undestanding [Kitcher]
12426 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]
20473 If experiential can defeat a belief, then its justification depends on the defeater's absence [Kitcher, by Casullo]