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It seems to me that there are two kinds of numbers: the kind that the concept of which we can grasp by imagining a case that instantiates the concept, and the kind that we cannot imagine. For example, we can grasp the concept of 1 by imagining one object. The same goes for 2, 3, 0.5 or 0, and pretty much all the most common numbers. But there is this second kind that we cannot imagine. For example, i (square root of -1) or '532,740,029'. It seems to me that nobody can really imagine what 532,740,029 objects or i object(you see, I don't even know whether I should put 'object' or 'objects' here or not because I don't know whether i is single or plural; I don't know what i is) are like. So, Q1) if I cannot imagine a case that instantiates concepts like '532,740,029', do I really know the concept, and if so, how do I know the concept? Q2) is there a fundamental difference between numbers whose instances I can imagine and those I cannot? (I lead towards there is no difference, but I don't know how to account...

It seems to me that there are two kinds of numbers: the kind that the concept of which we can grasp by imagining a case that instantiates the concept, and the kind that we cannot imagine. For example, we can grasp the concept of 1 by imagining one object. The same goes for 2, 3, 0.5 or 0, and pretty much all the most common numbers. But there is this second kind that we cannot imagine. For example, i (square root of -1) or '532,740,029'. It seems to me that nobody can really imagine what 532,740,029 objects or i object(you see, I don't even know whether I should put 'object' or 'objects' here or not because I don't know whether i is single or plural; I don't know what i is) are like. So, Q1) if I cannot imagine a case that instantiates concepts like '532,740,029', do I really know the concept, and if so, how do I know the concept? Q2) is there a fundamental difference between numbers whose instances I can imagine and those I cannot? (I lead towards there is no difference, but I don't know how to account...

Response from Allen Stairs on :

I'd suggest that while there may be differences in how easy it is for us to "picture" or "imagine" different numbers, this isn't a difference in the numbers themselves; it's a rather variable fact about us. I can mentally picture 5 things with no trouble. If I try for ten, it's harder (I have to think of five pairs of things.) If I try for 100, it's pretty hopeless, though you might be better at it than me. But I'm pretty sure that there's no interesting mathematical difference behind that. I'm also pretty sure that I understand the number 100 quite well. I don't need to be able to imagine 100 things to be able to see that 2x2x5x5 is the prime factorization of 100, for example, nor to see that 100 is a perfect square.
But that may still be misleading. I have no idea offhand whether 532,740,029 is prime. But I know what it would mean for it to be prime -- or not prime. And in fact, a bit of googling for the right calculators tells me that
532,740,029 = 43 x 1621 x 7643
I can't verify that by doing the...