You write "One of the premises of [the skeptic's] argument is something to the effect of:I don't know that I'm not dreaming." And yes, as you imply, it would be rather odd for a skeptic to start by being too dogmatic about what he can or can't know! But perhaps he doesn't need to be. Perhaps it is better to think of the argumentative situation like this. You are cheerfully going about your business, thinking that you know perfectly well that you are seeing a computer screen right now (and the like). Your friendly neighbourhood skeptic then issues a challenge: how do you know that? You appeal to the evidence of your senses. Your friendly skeptic chides you: how do you know they don't lead you astray all the time? Perhaps it is all just a vivid dream. Or perhaps an evil demon is making it seem as if you are seeing a computer screen when you aren't. In modern dress: perhaps you are a brain in a vat, being stimulated by a mad scientist so you still think you are embodied and seeing a computer....

No. OK, the following more prosaic thought is true: increasing our knowledge can reveal new areas of ignorance. Before you discover Australia, you don't know that there is a wild continent still to be mapped. Before you discover that there are protons, you don't know that there's a question of what happens when we smash them together in a particle collider. And so it goes. But the fact that increasing our knowledge can reveal new areas of ignorance obviously does not imply that our new knowledge (as far as it goes) isn't knowledge after all. You can come to know Australia is there even though you've not mapped it all yet. So it would be fatuous to aver (as Miller seems to do) that expanding the field of knowledge is nothing but increasing our ignorance.

Perhaps there are two different questions here. There's a very general question about truth and proof; and there's a much more specific question about the sort of case exemplified by Euler, where a mathematician claims to know (or at least have good grounds for) a proposition even in the absence of a demonstrative proof. Let's take the specific question first, using a different and perhaps more familiar example. We don't know how to prove Goldbach's conjecture that every even number greater than two is the sum of two primes. Yet most mathematicians are pretty confident in its truth. Why? Well, it has been computer-verified for numbers up to the order of 10 16 . But so what? After all, there are other well-known cases where a property holds of numbers up to some much greater bound but then fails. [For example, the logarithmic integral function li ( n ) over-estimates the number of primes below n but eventually under-estimates, then over-estimates again, flipping back and forth, with...

The short answer is "no". It might take "experience" and "culture" (in a broad sense) to understand some sentence or other representation M . But it certainly doesn't follow from this that we can't know (as "objectively" as you like) whether M is correct. For a stark illustration of the point, take the case where M is a bit of mathematics, e.g. take the claim "There is an infinite number of prime numbers". Tounderstand this statement you have to call on some background experience in elementary mathematics (the use of numbers in counting and so on). We might say that to understand thissentence requires being inducted into a bit of mathematical "culture". But the fact that your understanding of this claim might besaid to rely on your mathematical "culture" in some broad sense,doesn't mean that (once you've got the understanding) you can'tacquire objective knowledge whether the claim is true. In fact,the claim is a theorem, and you can get knowledge of its truth-- objective...