display all the ideas for this combination of texts
11 ideas
6298 | Kitcher says maths is an idealisation of the world, and our operations in dealing with it [Kitcher, by Resnik] |
Full Idea: Kitcher says maths is an 'idealising theory', like some in physics; maths idealises features of the world, and practical operations, such as segregating and matching (numbering), measuring, cutting, moving, assembling (geometry), and collecting (sets). | |
From: report of Philip Kitcher (The Nature of Mathematical Knowledge [1984]) by Michael D. Resnik - Maths as a Science of Patterns One.4.2.2 | |
A reaction: This seems to be an interesting line, which is trying to be fairly empirical, and avoid basing mathematics on purely a priori understanding. Nevertheless, we do not learn idealisation from experience. Resnik labels Kitcher an anti-realist. |
12392 | Mathematical a priorism is conceptualist, constructivist or realist [Kitcher] |
Full Idea: Proposals for a priori mathematical knowledge have three main types: conceptualist (true in virtue of concepts), constructivist (a construct of the human mind) and realist (in virtue of mathematical facts). | |
From: Philip Kitcher (The Nature of Mathematical Knowledge [1984], 02.3) | |
A reaction: Realism is pure platonism. I think I currently vote for conceptualism, with the concepts deriving from the concrete world, and then being extended by fictional additions, and shifts in the notion of what 'number' means. |
18078 | The interest or beauty of mathematics is when it uses current knowledge to advance undestanding [Kitcher] |
Full Idea: What makes a question interesting or gives it aesthetic appeal is its focussing of the project of advancing mathematical understanding, in light of the concepts and systems of beliefs already achieved. | |
From: Philip Kitcher (The Nature of Mathematical Knowledge [1984], 09.3) | |
A reaction: Kitcher defends explanation (the source of understanding, presumably) in terms of unification with previous theories (the 'concepts and systems'). I always have the impression that mathematicians speak of 'beauty' when they see economy of means. |
12426 | The 'beauty' or 'interest' of mathematics is just explanatory power [Kitcher] |
Full Idea: Insofar as we can honor claims about the aesthetic qualities or the interest of mathematical inquiries, we should do so by pointing to their explanatory power. | |
From: Philip Kitcher (The Nature of Mathematical Knowledge [1984], 09.4) | |
A reaction: I think this is a good enough account for me (but probably not for my friend Carl!). Beautiful cars are particularly streamlined. Beautiful people look particularly healthy. A beautiful idea is usually wide-ranging. |
12395 | Real numbers stand to measurement as natural numbers stand to counting [Kitcher] |
Full Idea: The real numbers stand to measurement as the natural numbers stand to counting. | |
From: Philip Kitcher (The Nature of Mathematical Knowledge [1984], 06.4) |
12425 | Complex numbers were only accepted when a geometrical model for them was found [Kitcher] |
Full Idea: An important episode in the acceptance of complex numbers was the development by Wessel, Argand, and Gauss, of a geometrical model of the numbers. | |
From: Philip Kitcher (The Nature of Mathematical Knowledge [1984], 07.5) | |
A reaction: The model was in terms of vectors and rotation. New types of number are spurned until they can be shown to integrate into a range of mathematical practice, at which point mathematicians change the meaning of 'number' (without consulting us). |
18071 | A one-operation is the segregation of a single object [Kitcher] |
Full Idea: We perform a one-operation when we perform a segregative operation in which a single object is segregated. | |
From: Philip Kitcher (The Nature of Mathematical Knowledge [1984], 06.3) | |
A reaction: This is part of Kitcher's empirical but constructive account of arithmetic, which I find very congenial. He avoids the word 'unit', and goes straight to the concept of 'one' (which he treats as more primitive than zero). |
17694 | Some non-count nouns can be used for counting, as in 'several wines' or 'fewer cheeses' [Laycock] |
Full Idea: The very words we class as non-count nouns may themselves be used for counting, of kinds or types, and phrases like 'several wines' are perfectly in order. ...Not only do we have 'less cheese', but we also have the non-generic 'fewer cheeses'. | |
From: Henry Laycock (Words without Objects [2006], Intro 4 n23) | |
A reaction: [compressed] Laycock generally endorses the thought that what can be counted is not simply distinguished by a precise class of applied vocabulary. He offers lots of borderline or ambiguous cases in his footnotes. |
17695 | Some apparent non-count words can take plural forms, such as 'snows' or 'waters' [Laycock] |
Full Idea: Some words that seem to be semantically non-count can take syntactically plural forms: 'snows', 'sands', 'waters' and the like. | |
From: Henry Laycock (Words without Objects [2006], Intro 4 n24) | |
A reaction: This seems to involve parcels of the stuff. The 'snows of yesteryear' occur at different times. 'Taking the waters' probably involves occasions. The 'Arabian sands' presumably occur in different areas. Semantics won't fix what is countable. |
18066 | The old view is that mathematics is useful in the world because it describes the world [Kitcher] |
Full Idea: There is an old explanation of the utility of mathematics. Mathematics describes the structural features of our world, features which are manifested in the behaviour of all the world's inhabitants. | |
From: Philip Kitcher (The Nature of Mathematical Knowledge [1984], 06.1) | |
A reaction: He only cites Russell in modern times as sympathising with this view, but Kitcher gives it some backing. I think the view is totally correct. The digression produced by Cantorian infinities has misled us. |
18083 | With infinitesimals, you divide by the time, then set the time to zero [Kitcher] |
Full Idea: The method of infinitesimals is that you divide by the time, and then set the time to zero. | |
From: Philip Kitcher (The Nature of Mathematical Knowledge [1984], 10.2) |