In that context, my understanding of Luke’s statement becomes “Mathematicians sometimes mistake B for A, but the degree of error involved in such a mistake depends on how closely B resembles A.”
By (A) I mean mathematician’s thoughts, and by (B) a syntactic definition. Since these things can’t possibly be confused, you must have read something different in my words… On the other hand, what these things talk about, that is whatever mathematician is thinking of, and whatever a formal definition defines, could be confused (given one possible reading, though not one I intended). Do you mean the latter?
In other words, I don’t think the phrase “the idea itself” has a precise enough meaning to be useful in this sort of conversation.
You can use a formal definition to access an idea, but separately from how the mathematician in question uses it to access the same idea. Using whatever means you have, you look both at how someone understands an idea (i.e. what thoughts occur, that have cognitive as well as mathematical explanations), and at the idea, and see how accurately it’s captured.
Ah, sorry: I understood by B what it seems you actually meant by C.
I should have said: “Mathematicians sometimes mistake C for A, but the degree of error involved in such a mistake depends on on how closely B resembles A.” (The relationship between C and B is also a potential source of mathematicians’ error, of course, but irrelevant to the mistake being described here.)
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Re: your last paragraph… I certainly agree that for any X that exists outside of our minds, if I and a mathematician both construct separate understandings in our minds of X, it’s in principle possible to compare those understandings to X to determine which of us is more accurate.
So, for example, I agree that if that mathematician and I both look at this image: http://a3.mzstatic.com/us/r1000/042/Purple/25/f7/a3/mzl.xofowuov.480x480-75.jpg and she concludes it isn’t a triangle (perhaps because the vertices are curved, perhaps because the bottom right vertex is missing, perhaps because the lines have nonzero thickness, perhaps for other reasons) while I conclude that it is, it follows that if a triangle (or an idea of a triangle, or a formal definition of a triangle) is a thing that exists outside of our minds then we can compare our different understandings to the actual triangle/idea/definition to determine which of us is more accurate.
And I agree that it further follows that, as you say, to the extent that we are interested in actual answers to a question and not in actual answers that a given faulty calculator provides when asked that question (in which case we are focusing on a different question entirely, most likely a wrong one), to that extent we care about the more accurate of the two understandings more than the less accurate of them.
That said, I am not certain how I would go about actually performing that comparison between my understanding and the actual formal definition of a triangle that exists outside of both my mind and the mathematician’s, and I’m not convinced it makes sense to talk about doing so.
Just to be clear, I’m not trying to go all postmodern on you here. I understand how I would use a bucket full of pebbles to keep track of a pen full of sheep and I’m perfectly happy to say that there’s a fact of the matter about the pebbles and the sheep and that it’s possible for me to be wrong about that and to determine the magnitude of my error through observation of the world, and how all of this depends on the fact that various things I can do with pebbles allow me to differentially predict what will happen when I count the sheep in the pen.
What I don’t understand what experience of the world I’d be differentially predicting when I say that this rather than that is the actual formal definition of a triangle.
By (A) I mean mathematician’s thoughts, and by (B) a syntactic definition. Since these things can’t possibly be confused, you must have read something different in my words… On the other hand, what these things talk about, that is whatever mathematician is thinking of, and whatever a formal definition defines, could be confused (given one possible reading, though not one I intended). Do you mean the latter?
You can use a formal definition to access an idea, but separately from how the mathematician in question uses it to access the same idea. Using whatever means you have, you look both at how someone understands an idea (i.e. what thoughts occur, that have cognitive as well as mathematical explanations), and at the idea, and see how accurately it’s captured.
Ah, sorry: I understood by B what it seems you actually meant by C.
I should have said: “Mathematicians sometimes mistake C for A, but the degree of error involved in such a mistake depends on on how closely B resembles A.” (The relationship between C and B is also a potential source of mathematicians’ error, of course, but irrelevant to the mistake being described here.)
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Re: your last paragraph… I certainly agree that for any X that exists outside of our minds, if I and a mathematician both construct separate understandings in our minds of X, it’s in principle possible to compare those understandings to X to determine which of us is more accurate.
So, for example, I agree that if that mathematician and I both look at this image:
http://a3.mzstatic.com/us/r1000/042/Purple/25/f7/a3/mzl.xofowuov.480x480-75.jpg
and she concludes it isn’t a triangle (perhaps because the vertices are curved, perhaps because the bottom right vertex is missing, perhaps because the lines have nonzero thickness, perhaps for other reasons) while I conclude that it is, it follows that if a triangle (or an idea of a triangle, or a formal definition of a triangle) is a thing that exists outside of our minds then we can compare our different understandings to the actual triangle/idea/definition to determine which of us is more accurate.
And I agree that it further follows that, as you say, to the extent that we are interested in actual answers to a question and not in actual answers that a given faulty calculator provides when asked that question (in which case we are focusing on a different question entirely, most likely a wrong one), to that extent we care about the more accurate of the two understandings more than the less accurate of them.
That said, I am not certain how I would go about actually performing that comparison between my understanding and the actual formal definition of a triangle that exists outside of both my mind and the mathematician’s, and I’m not convinced it makes sense to talk about doing so.
Just to be clear, I’m not trying to go all postmodern on you here. I understand how I would use a bucket full of pebbles to keep track of a pen full of sheep and I’m perfectly happy to say that there’s a fact of the matter about the pebbles and the sheep and that it’s possible for me to be wrong about that and to determine the magnitude of my error through observation of the world, and how all of this depends on the fact that various things I can do with pebbles allow me to differentially predict what will happen when I count the sheep in the pen.
What I don’t understand what experience of the world I’d be differentially predicting when I say that this rather than that is the actual formal definition of a triangle.