Jan 21, 2018 02:09 AM
The Metaphysical Status of Quantities
http://thebjps.typepad.com/my-blog/2017/...wolff.html
EXCERPT: Quantities pose peculiar epistemological and metaphysical challenges. A natural way to describe what is special about quantities is to say that quantities, in contrast to other attributes, come in degrees. Dogs may be ranked by how fast they can run or how big they are, but there is no ranking of them by how much they are dogs. Being a dog is a sortal, whereas speed and size are quantities. A quantity’s ‘coming in degrees’ can be understood as saying that quantities have (at least) one dimension of variation. For many paradigmatic physical attributes, we find a range of possible ‘amounts’ of that attribute, which we typically express as numerical values in terms of some unit. Having a range of possible amounts seems to be required by the idea that a quantity is an attribute that comes in degrees: gradations are possible in virtue of there being different amounts of the same quantity. To understand the metaphysical status of quantities, we need some account of how a gradable property like mass relates to specific amounts of mass. In the metaphysics literature, this question is often formulated in terms of determinables and determinates. But since this terminology comes with a specific understanding of the relationship between quantities and magnitudes, I will not use these terms here. In fact, I argue that the model of determinables and determinates is ultimately a poor fit for quantities, despite superficially appealing features. A second intuitive way of characterizing what is different about quantities, when compared to other attributes, is that only quantities involve numbers....
MORE: http://thebjps.typepad.com/my-blog/2017/...wolff.html
Platonism in the Philosophy of Mathematics --substantive revision Thu Jan 18, 2018
https://plato.stanford.edu/entries/plato...thematics/
INTRO: Platonism about mathematics (or mathematical platonism) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices. Just as electrons and planets exist independently of us, so do numbers and sets. And just as statements about electrons and planets are made true or false by the objects with which they are concerned and these objects’ perfectly objective properties, so are statements about numbers and sets. Mathematical truths are therefore discovered, not invented.
The most important argument for the existence of abstract mathematical objects derives from Gottlob Frege and goes as follows (Frege 1953). The language of mathematics purports to refer to and quantify over abstract mathematical objects. And a great number of mathematical theorems are true. But a sentence cannot be true unless its sub-expressions succeed in doing what they purport to do. So there exist abstract mathematical objects that these expressions refer to and quantify over.
Frege’s argument notwithstanding, philosophers have developed a variety of objections to mathematical platonism. Thus, abstract mathematical objects are claimed to be epistemologically inaccessible and metaphysically problematic. Mathematical platonism has been among the most hotly debated topics in the philosophy of mathematics over the past few decades....
MORE: https://plato.stanford.edu/entries/plato...thematics/
http://thebjps.typepad.com/my-blog/2017/...wolff.html
EXCERPT: Quantities pose peculiar epistemological and metaphysical challenges. A natural way to describe what is special about quantities is to say that quantities, in contrast to other attributes, come in degrees. Dogs may be ranked by how fast they can run or how big they are, but there is no ranking of them by how much they are dogs. Being a dog is a sortal, whereas speed and size are quantities. A quantity’s ‘coming in degrees’ can be understood as saying that quantities have (at least) one dimension of variation. For many paradigmatic physical attributes, we find a range of possible ‘amounts’ of that attribute, which we typically express as numerical values in terms of some unit. Having a range of possible amounts seems to be required by the idea that a quantity is an attribute that comes in degrees: gradations are possible in virtue of there being different amounts of the same quantity. To understand the metaphysical status of quantities, we need some account of how a gradable property like mass relates to specific amounts of mass. In the metaphysics literature, this question is often formulated in terms of determinables and determinates. But since this terminology comes with a specific understanding of the relationship between quantities and magnitudes, I will not use these terms here. In fact, I argue that the model of determinables and determinates is ultimately a poor fit for quantities, despite superficially appealing features. A second intuitive way of characterizing what is different about quantities, when compared to other attributes, is that only quantities involve numbers....
MORE: http://thebjps.typepad.com/my-blog/2017/...wolff.html
Platonism in the Philosophy of Mathematics --substantive revision Thu Jan 18, 2018
https://plato.stanford.edu/entries/plato...thematics/
INTRO: Platonism about mathematics (or mathematical platonism) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices. Just as electrons and planets exist independently of us, so do numbers and sets. And just as statements about electrons and planets are made true or false by the objects with which they are concerned and these objects’ perfectly objective properties, so are statements about numbers and sets. Mathematical truths are therefore discovered, not invented.
The most important argument for the existence of abstract mathematical objects derives from Gottlob Frege and goes as follows (Frege 1953). The language of mathematics purports to refer to and quantify over abstract mathematical objects. And a great number of mathematical theorems are true. But a sentence cannot be true unless its sub-expressions succeed in doing what they purport to do. So there exist abstract mathematical objects that these expressions refer to and quantify over.
Frege’s argument notwithstanding, philosophers have developed a variety of objections to mathematical platonism. Thus, abstract mathematical objects are claimed to be epistemologically inaccessible and metaphysically problematic. Mathematical platonism has been among the most hotly debated topics in the philosophy of mathematics over the past few decades....
MORE: https://plato.stanford.edu/entries/plato...thematics/
