1984 | The Nature of Mathematical Knowledge |
p.64 | 6298 | Kitcher says maths is an idealisation of the world, and our operations in dealing with it |
Intro | p.5 | 12387 | Mathematical knowledge arises from basic perception |
Intro | p.12 | 12412 | My constructivism is mathematics as an idealization of collecting and ordering objects |
01.2 | p.17 | 12413 | A 'warrant' is a process which ensures that a true belief is knowledge |
01.3 | p.22 | 12389 | Knowledge is a priori if the experience giving you the concepts thus gives you the knowledge |
01.4 | p.24 | 12390 | A priori knowledge comes from available a priori warrants that produce truth |
01.6 | p.29 | 12416 | We have some self-knowledge a priori, such as knowledge of our own existence |
02.2 | p.45 | 12418 | In long mathematical proofs we can't remember the original a priori basis |
02.3 | p.46 | 12392 | Mathematical a priorism is conceptualist, constructivist or realist |
03.1 | p.50 | 12421 | Kant's intuitions struggle to judge relevance, impossibility and exactness |
03.1 | p.50 | 12420 | If mathematics comes through intuition, that is either inexplicable, or too subjective |
03.2 | p.53 | 12393 | Intuition is no basis for securing a priori knowledge, because it is fallible |
03.3 | p.61 | 18061 | Mathematical intuition is not the type platonism needs |
04.1 | p.65 | 18063 | Conceptualists say we know mathematics a priori by possessing mathematical concepts |
04.6 | p.86 | 18064 | If meaning makes mathematics true, you still need to say what the meanings refer to |
04.I | p.68 | 12423 | Analyticity avoids abstract entities, but can there be truth without reference? |
05.2 | p.92 | 18065 | We derive limited mathematics from ordinary things, and erect powerful theories on their basis |
06.1 | p.105 | 18066 | The old view is that mathematics is useful in the world because it describes the world |
06.1 | p.107 | 18067 | Abstract objects were a bad way of explaining the structure in mathematics |
06.2 | p.108 | 18068 | Arithmetic is made true by the world, but is also made true by our constructions |
06.2 | p.109 | 18069 | Arithmetic is an idealizing theory |
06.2 | p.111 | 18070 | We develop a language for correlations, and use it to perform higher level operations |
06.3 | p.112 | 18071 | A one-operation is the segregation of a single object |
06.4 | p.124 | 12395 | Real numbers stand to measurement as natural numbers stand to counting |
06.5 | p.142 | 18073 | Dummett says classical logic rests on meaning as truth, while intuitionist logic rests on assertability |
06.5 | p.142 | 18072 | Constructivism is ontological (that it is the work of an agent) and epistemological (knowable a priori) |
06.5 | p.143 | 18074 | Intuitionists rely on assertability instead of truth, but assertability relies on truth |
06.5 | p.144 | 18075 | Idealisation trades off accuracy for simplicity, in varying degrees |
07.5 | p.176 | 12425 | Complex numbers were only accepted when a geometrical model for them was found |
07.5 | p.176 | 18077 | The defenders of complex numbers had to show that they could be expressed in physical terms |
09.3 | p.206 | 18078 | The interest or beauty of mathematics is when it uses current knowledge to advance undestanding |
09.4 | p.212 | 12426 | The 'beauty' or 'interest' of mathematics is just explanatory power |
10.2 | p.238 | 18083 | With infinitesimals, you divide by the time, then set the time to zero |
2000 | A Priori Knowledge Revisited |
§II | p.69 | 12429 | Knowing our own existence is a priori, but not necessary |
§II | p.69 | 12428 | Many necessities are inexpressible, and unknowable a priori |
§VII | p.86 | 12430 | Classical logic is our preconditions for assessing empirical evidence |
§VII | p.87 | 12431 | I believe classical logic because I was taught it and use it, but it could be undermined |