Suppose someone is doing a bunch of really long sums e.g. adding 12 digit numbers, with a blunt pencil and in a hurry. Under these circumstances they are quite likely to make at least one mistake during the course of each of the sums, so (as they learn when they check over their answers) overall they get only about one sum in ten correct. Now after doing this for a while, suppose they do one more sum and, being the confident person they are, they believe that the answer to it is in fact 1789200056911 as their calculations suggest. And suppose that this is, in fact the right answer, and in this case they have been lucky enough not to make any mistakes along the way. Then do you think that they know that whatever + whatever = 1789200056911 or not?/ Is their belief justified?
On the one hand, it seems like they know since they have gone through and been convinced by a correct process of reasoning which entails that this is the right answer. On the other hand, it seems like they don’t know that the answer is that because they usually make so many mistakes that the mere fact of their computing a certain result is very little evidence that that result is correct.
My impulse would be to say that this shows that we two different standards – for mathematical knowledge and for knowledge in general which are clashing in this case. Maybe one can also get a conflict between these standards for knowledge in the opposite direction: setting computers to check the first few billion cases of goldbach’s conjecture (that every number greater than two can be written as the sum of two primes) could eventually give you very strong justification for believing it (and hence perhaps knowledge in the ordinary sense) but it would be strange to say that you know the conjecture was true if you didn’t have a proof.
Also this case seems similar to the familiar lottery example 'do you know that you won't win the lottery, when you have evidence that that your chances of loosing are overwhealmingly good?' so maybe the lottery example is further evidence that our conception of knowledge is fragmented/highly context dependent. Read more!