Combining Texts
Ideas for
'Thinking About Mathematics', 'On Husserl' and 'The Nature of Mathematical Knowledge'
expand these ideas
|
start again
|
choose
another area for these texts
display all the ideas for this combination of texts
36 ideas
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 [Kitcher, by Resnik]
|
12392
|
Mathematical a priorism is conceptualist, constructivist or realist [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]
|
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
8763
|
The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro]
|
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 [Kitcher]
|
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / h. Reals from Cauchy
18249
|
Cauchy gave a formal definition of a converging sequence. [Shapiro]
|
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 [Kitcher]
|
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / a. Units
18071
|
A one-operation is the segregation of a single object [Kitcher]
|
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 [Kitcher]
|
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 [Kitcher]
|
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
8764
|
Categories are the best foundation for mathematics [Shapiro]
|
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / f. Zermelo numbers
8762
|
Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro]
|
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
8760
|
Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro]
|
8761
|
A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro]
|
6. Mathematics / C. Sources of Mathematics / 2. Intuition of Mathematics
18061
|
Mathematical intuition is not the type platonism needs [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]
|
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
12387
|
Mathematical knowledge arises from basic perception [Kitcher]
|
12412
|
My constructivism is mathematics as an idealization of collecting and ordering objects [Kitcher]
|
18065
|
We derive limited mathematics from ordinary things, and erect powerful theories on their basis [Kitcher]
|
18077
|
The defenders of complex numbers had to show that they could be expressed in physical terms [Kitcher]
|
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
12423
|
Analyticity avoids abstract entities, but can there be truth without reference? [Kitcher]
|
8744
|
Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro]
|
6. Mathematics / C. Sources of Mathematics / 7. Formalism
8749
|
Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro]
|
8750
|
Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro]
|
8752
|
Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro]
|
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism
18069
|
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]
|
18072
|
Constructivism is ontological (that it is the work of an agent) and epistemological (knowable a priori) [Kitcher]
|
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
8753
|
Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro]
|
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism
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]
|
8731
|
Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro]
|
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
8730
|
'Impredicative' definitions refer to the thing being described [Shapiro]
|