I’m not sure the first example is really an error on the part of the commenter, unless there was an implicit shared technical usage at play. The word ‘any’ in the quote you give below is not very clear. I knew what it meant, but only because I understood what the argument was getting at.
“If, for any small positive number you give me (epsilon), I can show that the difference between A and B is less than epsilon, then I have shown A and B are equal.”
In this case, ‘any’ means ‘if whatever number is given, the following analysis applies, the conclusion is reached’
Compare:
“If, for any small positive number you can give me (epsilon), I can show that the difference between A and B is greater than epsilon, then I have shown A and B are not equal”.
Here, the natural reading is ‘if a single case is found where the following analysis applies, the conclusion is reached’
As I said, this may be a failing of technical language on my part, but I don’t think normal English is clear here.
I think you’re right. I was using prior knowledge to interpret the argument correctly. The ambiguity in the language definitely makes my example weaker. I tried empathizing with the commenter as an intuition thinker to try figuring out what the most likely mistake caused the confusion. I still think the commenter most likely didn’t pay attention to those words, but it’s also quite likely he understood the technically correct alternative interpretation.
In his situation, I’d probably read ‘any’ in the second sense simply because as a non-mathematician I can imagine the second sense being a practical test: (I give you a number, you show me that the difference between A and B is smaller, we reach a conclusion) whereas the first seems esoteric (you test every conceivable small number...)
On the other hand, the first reading is so blatantly wrong, the commenter really should have stepped back and thought ‘could this sentence be saying something that wasn’t obviously incorrect?’ Principle of charity and all that.
I’m not sure the first example is really an error on the part of the commenter, unless there was an implicit shared technical usage at play. The word ‘any’ in the quote you give below is not very clear. I knew what it meant, but only because I understood what the argument was getting at.
“If, for any small positive number you give me (epsilon), I can show that the difference between A and B is less than epsilon, then I have shown A and B are equal.”
In this case, ‘any’ means ‘if whatever number is given, the following analysis applies, the conclusion is reached’
Compare:
“If, for any small positive number you can give me (epsilon), I can show that the difference between A and B is greater than epsilon, then I have shown A and B are not equal”.
Here, the natural reading is ‘if a single case is found where the following analysis applies, the conclusion is reached’
As I said, this may be a failing of technical language on my part, but I don’t think normal English is clear here.
I think you’re right. I was using prior knowledge to interpret the argument correctly. The ambiguity in the language definitely makes my example weaker. I tried empathizing with the commenter as an intuition thinker to try figuring out what the most likely mistake caused the confusion. I still think the commenter most likely didn’t pay attention to those words, but it’s also quite likely he understood the technically correct alternative interpretation.
In his situation, I’d probably read ‘any’ in the second sense simply because as a non-mathematician I can imagine the second sense being a practical test: (I give you a number, you show me that the difference between A and B is smaller, we reach a conclusion) whereas the first seems esoteric (you test every conceivable small number...)
On the other hand, the first reading is so blatantly wrong, the commenter really should have stepped back and thought ‘could this sentence be saying something that wasn’t obviously incorrect?’ Principle of charity and all that.