I used to laugh about (early) Pen Maddy claiming that we could see sets. But now I think that's almost right- though not in the way that Maddy intended it.
I can see that my program doesn't infinitely loop, or that the 1000th prime is 7919 by pressing enter, waiting a few seconds and then looking above the command line prompt on my computer. These are all claims about mathematical objects, yet (given suitable equipment and background knowledge, we would ordinarily say that I can see these things to be true).
This seems just as literally true as the claim that I can see that the electricity is on, when I look at the lit windows of the house next door.
In both cases I immediately form the belief, probably am justified, am depending on a lot of contingent assumptions about electronic wiring etc.
But maybe we should distinguish seeing Xs from seeing that some fact about Xs obtains? Maybe there's something especially problematic about believing in objects which you can't see?
-If seeing x = seeing that x exists, then I can see that there is a 1000th prime in the above example (suppose I wrote the program but had never seen the proof that there are infinitely many primes)
-If we take a more intuitive approach to seeing xs (i.e. is it awkward to say I am now looking an X) then:
a) certainly it is awkward to say `I am now looking at a number'...hmm though we might say `I am now seeing the line of the program that causes the crash (and lines in programs are abstract objects, just like lines in poems),'.
b) it's also pretty awkward to say `I am now seeing a drought', or `I am now seeing North America' or 'I am now seeing a proton'.
If you can see a drought when you look at a color map of precipitation, why can't you see a pair of twin primes by looking at a chart?
Seeing that P really means little more than having some visual experience which causes you to immediately believe that P, which you might cite as part of your justification for believing that p. So if you can know things (e.g. all the background mathematical beliefs involved in the program case) about numbers, then it's not too hard to arrange to see things about them.
Of course, the anti-platonist won't think that you can know things about numbers either - well that's where my thesis comes in. But if we can know some things about the numbers, its not hard to arrange things so that we can see further things about them ie rig up reliable methods for forming beliefs about them whose last step involves visual experience.