It might be useful to look at what happens in mathematics. What, for example, is a “number”? In antiquity, there were the whole numbers and fractions of everyday experience. You can count apples, and cut an apple in half. (BTW, I recently discovered that among the ancient Greeks, there was some dispute about whether 1 was a number. No, some said, 1 was the unit with which other things were measured. 2, 3, 4, and so on were numbers, but not 1.)
Then irrationals were discovered, and negative numbers, and the real line, and complex numbers, and octonions, and Cayley numbers, and p-adic numbers, and perhaps there are even more things that mathematicians call numbers. And there are other ways that the ways that “numbers” behave have been generalised to define such things as fields, vector spaces, rings, and many more, the elements of which are generally not called numbers. But unlike philosophers, mathematicians do not dispute which of these is the “right” concept of “number”. All of the concepts have their uses, and many of them are called “numbers”, but “number” has never been given a formal definition, and does not need one.
For another example, consider “integration”. The idea of dividing an arbitrary shape into pieces of known area and summing their areas goes back at least to Archimedes’ “method of exhaustion”. When real numbers and functions became better understood it was formalised as Riemann integration. That was later generalised to Lebesgue integration, and then to Haar measure. Stochastic processes brought in Itô integration and several other forms.
Again, no-one as far as I know has ever troubled with the question, “but what is integration, really?” There is a general, intuitive idea of “measuring the size of things”, which has been given various precise formulations in various contexts. In some of those contexts it may make sense to speak of the “right” concept of integration, when there is one that subsumes all of the others and appears to be the most general possible (e.g. Lebesgue integration on Euclidean spaces), but in other contexts there may be multiple incomparable concepts, each with its own uses (e.g. Itô and Stratonovich integration for stochastic processes).
But in philosophy, there are no theorems by which to judge the usefulness of a precisely defined concept.
I think this is a very good contrast, indeed. I agree with your view of the matter, and I think I will use “number” as a particular example next time I recount the thoughts which brought me to write the post. Thank you.
It might be useful to look at what happens in mathematics. What, for example, is a “number”? In antiquity, there were the whole numbers and fractions of everyday experience. You can count apples, and cut an apple in half. (BTW, I recently discovered that among the ancient Greeks, there was some dispute about whether 1 was a number. No, some said, 1 was the unit with which other things were measured. 2, 3, 4, and so on were numbers, but not 1.)
Then irrationals were discovered, and negative numbers, and the real line, and complex numbers, and octonions, and Cayley numbers, and p-adic numbers, and perhaps there are even more things that mathematicians call numbers. And there are other ways that the ways that “numbers” behave have been generalised to define such things as fields, vector spaces, rings, and many more, the elements of which are generally not called numbers. But unlike philosophers, mathematicians do not dispute which of these is the “right” concept of “number”. All of the concepts have their uses, and many of them are called “numbers”, but “number” has never been given a formal definition, and does not need one.
For another example, consider “integration”. The idea of dividing an arbitrary shape into pieces of known area and summing their areas goes back at least to Archimedes’ “method of exhaustion”. When real numbers and functions became better understood it was formalised as Riemann integration. That was later generalised to Lebesgue integration, and then to Haar measure. Stochastic processes brought in Itô integration and several other forms.
Again, no-one as far as I know has ever troubled with the question, “but what is integration, really?” There is a general, intuitive idea of “measuring the size of things”, which has been given various precise formulations in various contexts. In some of those contexts it may make sense to speak of the “right” concept of integration, when there is one that subsumes all of the others and appears to be the most general possible (e.g. Lebesgue integration on Euclidean spaces), but in other contexts there may be multiple incomparable concepts, each with its own uses (e.g. Itô and Stratonovich integration for stochastic processes).
But in philosophy, there are no theorems by which to judge the usefulness of a precisely defined concept.
I think this is a very good contrast, indeed. I agree with your view of the matter, and I think I will use “number” as a particular example next time I recount the thoughts which brought me to write the post. Thank you.