Combining Texts
Ideas for
'Structures and Structuralism in Phil of Maths', 'The Nature of Mathematical Knowledge' and 'Axiomatic Thought'
expand these ideas
|
start again
|
choose
another area for these texts
display all the ideas for this combination of texts
40 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 / g. Real numbers
12395
|
Real numbers stand to measurement as natural numbers stand to counting [Kitcher]
|
10165
|
'Analysis' is the theory of the real numbers [Reck/Price]
|
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 / 2. Proof in Mathematics
17967
|
To decide some questions, we must study the essence of mathematical proof itself [Hilbert]
|
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
17965
|
The whole of Euclidean geometry derives from a basic equation and transformations [Hilbert]
|
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
10174
|
Mereological arithmetic needs infinite objects, and function definitions [Reck/Price]
|
17964
|
Number theory just needs calculation laws and rules for integers [Hilbert]
|
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
10164
|
Peano Arithmetic can have three second-order axioms, plus '1' and 'successor' [Reck/Price]
|
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
10172
|
Set-theory gives a unified and an explicit basis for mathematics [Reck/Price]
|
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
10167
|
Structuralism emerged from abstract algebra, axioms, and set theory and its structures [Reck/Price]
|
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / b. Varieties of structuralism
10169
|
Relativist Structuralism just stipulates one successful model as its arithmetic [Reck/Price]
|
10179
|
There are 'particular' structures, and 'universal' structures (what the former have in common) [Reck/Price]
|
10181
|
Pattern Structuralism studies what isomorphic arithmetic models have in common [Reck/Price]
|
10182
|
There are Formalist, Relativist, Universalist and Pattern structuralism [Reck/Price]
|
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / c. Nominalist structuralism
10168
|
Formalist Structuralism says the ontology is vacuous, or formal, or inference relations [Reck/Price]
|
10178
|
Maybe we should talk of an infinity of 'possible' objects, to avoid arithmetic being vacuous [Reck/Price]
|
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / d. Platonist structuralism
10176
|
Universalist Structuralism is based on generalised if-then claims, not one particular model [Reck/Price]
|
10177
|
Universalist Structuralism eliminates the base element, as a variable, which is then quantified out [Reck/Price]
|
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
10171
|
The existence of an infinite set is assumed by Relativist Structuralism [Reck/Price]
|
6. Mathematics / C. Sources of Mathematics / 2. Intuition of Mathematics
12393
|
Intuition is no basis for securing a priori knowledge, because it is fallible [Kitcher]
|
12420
|
If mathematics comes through intuition, that is either inexplicable, or too subjective [Kitcher]
|
18061
|
Mathematical intuition is not the type platonism needs [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]
|
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 / 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]
|