This was a paper I wrote 8 − 10 years ago while taking a philosophy of science course primarily directed at Hume and Popper. Sorry about the math, I’ll try to fix it when I have a moment.
The general point is this:
I am trying to highlight a distinction between two cases.
Case A—We say ‘All swans are white.’ and mean something like, ‘There are an infinite number of swans in the Universe and all of them are white.’.
Hume’s primary point, as I interpreted him, is that since there are an infinite number of observations that would need to be made to justify this assertion, making a single observation of a white swan doesn’t make any sort of dent in the list of observations we would need to make. If you have an infinitely long ‘to do list’, then checking off items from your list, doesn’t actually make any progress on completing your list.
Case B—We say ‘All swans are white.’ and mean something like, ‘There are a finite number of swans (n) in the Universe and all of them are white.’ (and (n) is going to be really big.).
If we mean this instead, then we can see that no matter how large (n) is, each observation makes comprehensive and calculable progress towards justifying that (n) swans are indeed white. I’m saying that, no matter how long your finite ‘to do list’ is, checking off an item is calculable progress towards the assertion that (n) swans are white.
In general, I think Hume did a great job of demonstrating why we can’t justify assertions like the one in case A. I agree with him on that. What I’m saying, is that we shouldn’t make statements like the one in case A. They are absurd (in the formal sense).
What I’m saying is that, yes, observations of instances can’t provide any justification for general claims about infinite sets, but they can provide justification of general claims about finite sets (as large as you would like to make them) and that is important to consider.
This was a paper I wrote 8 − 10 years ago while taking a philosophy of science course primarily directed at Hume and Popper. Sorry about the math, I’ll try to fix it when I have a moment.
The general point is this:
I am trying to highlight a distinction between two cases.
Case A—We say ‘All swans are white.’ and mean something like, ‘There are an infinite number of swans in the Universe and all of them are white.’.
Hume’s primary point, as I interpreted him, is that since there are an infinite number of observations that would need to be made to justify this assertion, making a single observation of a white swan doesn’t make any sort of dent in the list of observations we would need to make. If you have an infinitely long ‘to do list’, then checking off items from your list, doesn’t actually make any progress on completing your list.
Case B—We say ‘All swans are white.’ and mean something like, ‘There are a finite number of swans (n) in the Universe and all of them are white.’ (and (n) is going to be really big.).
If we mean this instead, then we can see that no matter how large (n) is, each observation makes comprehensive and calculable progress towards justifying that (n) swans are indeed white. I’m saying that, no matter how long your finite ‘to do list’ is, checking off an item is calculable progress towards the assertion that (n) swans are white.
In general, I think Hume did a great job of demonstrating why we can’t justify assertions like the one in case A. I agree with him on that. What I’m saying, is that we shouldn’t make statements like the one in case A. They are absurd (in the formal sense).
What I’m saying is that, yes, observations of instances can’t provide any justification for general claims about infinite sets, but they can provide justification of general claims about finite sets (as large as you would like to make them) and that is important to consider.
Hume’s pont might just be that theres no deductive justification of induction...one thing happening doesn’t imply that a similar one must.