Question:

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 for this at least seemingly existent difference with regards to human imagination)

Topics:

Mathematics

Accepted:

November 11, 2016

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 math in my head, though some people can. But once again, I think I understand it. And once again, any limitations I might have are facts about me, and not about the number itself.

Someone might say that I don't have a specific concept of 532,740,029 in the way that have a specific concept of 5, and in a sense that may be true. Someone might add that I, Allen Stairs, am not capable of having the same sort of concept of 532,740,029 that I have of 5. And that, again, is true in a sense. But again: that doesn't mark a distinction among kinds of numbers.

Now i, the root of -1, is in a somewhat different category. It's not a number we use to count or measure things. But then, anyone who went around trying to find 5i objects doesn't understand i. But we can define complex numbers such as 3+i4 as pairs of real numbers wth a certain arithmetic, and any middle school child can master the arithmetic. That arithmetic is an extension of ordinary arithmetic. And it's an arithmetic that has uses. For example: in quantum theory, states are vectors in vector spaces where the coefficients can be complex numbers. This has various useful consequences. We don't need to be able to picture i for all of this to be true.

There

isa sort of scale of abstraction, of course. 0 is a number, and recognizing that called for an act of abstraction. Likewise the negative numbers; likewise non-integer rational numbers; likewise algebraic and non-algebraic real numbers; likewise complex numbers; likewise transfinite cardinals and ordinals. We can make distinctions among numbers in terms of how they are related to other sorts of numbers. But the distinctions here are a matter of internal features of the numbers themselves and not our psychology.I hope that's at least somewhat useful!