Ideas from 'The Nature of Mathematical Knowledge' by Philip Kitcher [1984], by Theme Structure
[found in 'The Nature of Mathematical Knowledge' by Kitcher,Philip [OUP 1984,0195035410]].
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4. Formal Logic / E. Nonclassical Logics / 2. Intuitionist Logic
18074

Intuitionists rely on assertability instead of truth, but assertability relies on truth

6. Mathematics / A. Nature of Mathematics / 1. Mathematics
6298

Kitcher says maths is an idealisation of the world, and our operations in dealing with it [Resnik]

12392

Mathematical a priorism is conceptualist, constructivist or realist

18078

The interest or beauty of mathematics is when it uses current knowledge to advance undestanding

12426

The 'beauty' or 'interest' of mathematics is just explanatory power

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
12395

Real numbers stand to measurement as natural numbers stand to counting

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / j. Complex numbers
12425

Complex numbers were only accepted when a geometrical model for them was found

6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / a. Units
18071

A oneoperation is the segregation of a single object

6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / g. Applying mathematics
18066

The old view is that mathematics is useful in the world because it describes the world

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / k. Infinitesimals
18083

With infinitesimals, you divide by the time, then set the time to zero

6. Mathematics / C. Sources of Mathematics / 2. Intuition of Mathematics
18061

Mathematical intuition is not the type platonism needs

12420

If mathematics comes through intuition, that is either inexplicable, or too subjective

12393

Intuition is no basis for securing a priori knowledge, because it is fallible

6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
12387

Mathematical knowledge arises from basic perception

12412

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

18065

We derive limited mathematics from ordinary things, and erect powerful theories on their basis

18077

The defenders of complex numbers had to show that they could be expressed in physical terms

6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
12423

Analyticity avoids abstract entities, but can there be truth without reference?

6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism
18069

Arithmetic is an idealizing theory

18068

Arithmetic is made true by the world, but is also made true by our constructions

18070

We develop a language for correlations, and use it to perform higher level operations

18072

Constructivism is ontological (that it is the work of an agent) and epistemological (knowable a priori)

6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism
18063

Conceptualists say we know mathematics a priori by possessing mathematical concepts

18064

If meaning makes mathematics true, you still need to say what the meanings refer to

9. Objects / A. Existence of Objects / 2. Abstract Objects / b. Need for abstracta
18067

Abstract objects were a bad way of explaining the structure in mathematics

12. Knowledge Sources / A. A Priori Knowledge / 1. Nature of the A Priori
12390

A priori knowledge comes from available a priori warrants that produce truth

12. Knowledge Sources / A. A Priori Knowledge / 6. A Priori from Reason
12418

In long mathematical proofs we can't remember the original a priori basis

12. Knowledge Sources / A. A Priori Knowledge / 9. A Priori from Concepts
12389

Knowledge is a priori if the experience giving you the concepts thus gives you the knowledge

12. Knowledge Sources / A. A Priori Knowledge / 10. A Priori as Subjective
12416

We have some selfknowledge a priori, such as knowledge of our own existence

13. Knowledge Criteria / A. Justification Problems / 1. Justification / a. Justification issues
12413

A 'warrant' is a process which ensures that a true belief is knowledge

13. Knowledge Criteria / A. Justification Problems / 1. Justification / c. Defeasibility
20473

If experiential can defeat a belief, then its justification depends on the defeater's absence [Casullo]

15. Nature of Minds / C. Capacities of Minds / 6. Idealisation
18075

Idealisation trades off accuracy for simplicity, in varying degrees
