Consider the following analogy. What is a chicken? Are chickens real?
There was a time (most places in Europe, anyway) when this would have seemed an even more stupid question than it does now. Everyone knew exactly what a chicken was. Even a rich noble would have only had to walk perhaps fifteen minutes and point to an example of a chicken. It was a vibrant and noteworthy part of everyone's every-day experience. So too, our experience with numbers. That (point at a carton of six eggs) is six. That (point to an apple, and another apple which has been cut in half and one of the halves removed) is three-halves. And so on.
The fact that you can't just point at a collection of something and say "there is negative-three", "there is square-root-of-five", or "there is six plus three-i" are the reason why some people who are frustrated with those ideas feel justified in saying they aren't actual numbers. It's a fair criticism, in fact, and points to the fact that we never sit down and talk about what numbers are really meant to be. Of course, these days someone could also go their entire lives without seeing a chicken, and they accept that there is an animal which is vaguely involved in the creation of the eggs which they sometimes eat for breakfast. Certainly, for those of us who did not grow up on or near to a farm or a zoo with chickens, we accept the existence of chickens as an article of faith for some years. Similarly do we take the idea there are "numbers" which don't correspond to collections of things as a received idea.
So if numbers don't have to correspond to collections of things, what are they? Well, in the case of (positive) irrational numbers, they can correspond to lengths of lines or of areas — to continuous amounts of something, which is a nice generalisation of collection-sizes. And negative numbers can correspond to deficits or differences of such amounts. And complex numbers, er... well, they're... useful for quantum mechanics, and electrical engineering... and, um, so are quaternions... We find that we stretch the definition of number from "amount of" to "being useful for", which I think is an important thing to notice.
There's no obvious place where we should simply stop. The fact that the complex numbers can't even be ordered any more (never mind the quaternions, for which multiplication doesn't even commute) suggests that just because something solves x²+1=0 does not mean that it's a number (that complex numbers aren't 'numbers' in general). But we can say that just because something is an upper bound for a bounded sequence of numbers, that it isn't a number (real numbers aren't all 'numbers', and the square root of two or five in particular); or that just because something is the difference of two numbers, that it isn't a number (negative numbers aren't all 'numbers'); or that just because something is a ratio of two numbers, doesn't make it a number (positive rational numbers aren't all 'numbers'). But that rules out everything but the non-negative integers; and people have historically even looked askance at zero. You could even argue that one isn't a number, if you argued that by "a number" you mean a plural amount.
So it's pretty important to ask ourselves: what is a number?
What is a chicken? It's a small-ish bird which doesn't fly very well. But we don't want to include kiwis or puffins as 'chickens', so perhaps we should specify that they have short beaks and don't swim well. But what about pheasants? Even if we go on to successfully isolate chickens from all other living birds by definitions, what about the ancestors of chickens who evolved into the modern farm animal? At some point there weren't chickens, and then there were. When did things change?
The problem with chickens, and also with numbers, is that in the end we only have definitions for these words by convention, which are based on examples. We accept modern chickens as 'chickens', and don't accept kiwis as 'chickens'. Similarly, we want to include 'six' and probably 'three-halves' and maybe 'negative-two' and 'square-root-of-five' as numbers, but we don't want to include the function f: ℤ→ℤ given by f(x)=3x+2 as a number. It's not what we want to think of as a number, because it can't be used the way we want to use numbers. Numbers are tools for understanding the world.
Which birds do we accept as chickens? Those that behave in a particular way, and in particular which we can understand in a particular way. Their eggs taste a particular way, their meat tastes a particular way, and they behave in a particular way. We care about how they act and how they taste because we are interested in them as features of the environment which we will interact with (perhaps to eat them). The concept of a chicken is something we have invented to distinguish some animals from others. If we didn't care about the difference between a chicken and a pheasant, we wouldn't have separate ideas for chickens and pheasants. (Just because we have different words for things doesn't make them different, but it does mean that we do care about what differences we think they have.) The concept of 'chicken' is a tool which we use to understand some of the animals we know about.
Similarly, the concept of 'number' is a tool which we use to understand the relationships between objects. But it goes beyond just the concept of 'number' itself: each number is a concept which we use to distinguish from other numbers. We rarely think that there is just "a number" of something, to denote that there is more than zero or one or two; we care about which number. The difference between six eggs and seven eggs is important to us.
But there is another difference with chickens: we may see small chickens or large chickens (a single sort of chicken with different properties), but we never see egg sixes or apple sixes (a single sort of number with different attributes). We do see six eggs or six apples. In this case, the number is not playing the role of a noun, but an adjective. So all this talk of 'chickens', which are objects, has been misleading. What we should have been thinking is something like: "Is red real"? "Is big real"?
Well, colours are real and sizes are real, but what makes a colour 'red'? We can invent an arbitrary definition based on frequencies of light, but then we're making the definition of colour depend on numbers, which is no way to solve the problem of how to understand numbers. In the end we again end up having conventions based on examples. But surely the things which we call numbers must exist? That there really is a number three? We see it all the time, of course. Similarly, there must exist a colour red, mustn't there?
The colour red depends on our sensory apparatus, and the way our brains process the signals sent to us by our eyes. The colour red is an emergent experience, resulting from how our brains and sensory organs are structured. The notion of the colour red is a useful way of understanding our world, based on how we experience it. There's no reasonable way to deny that there are things which shine red light (light which we percieve as red); things which reflect red light (which preferentially reflect light which we percieve as red); and that red light falls roughly within some frequencies of light (we have constructed an entire theoretical apparatus for describing electromagnetism which is useful enough to build radio towers, lightning rods, x-ray machines, NMR machines, and lasers, and in this theory the light which we tend to percieve as red affects certain photosensitive apparatus in a specific way, and these predictions are borne out by experiment). The concept of 'red' is an extremely useful and robust way of describing how we experience the world.
You might even say that the world is described "unreasonably effectively" by the notion of colour; there's no particular reason why so much of our experience should be describable in terms of colour. We don't talk every day about the scent of steel, the sound of plastic, the taste of granite. Somehow, the world is shaped in such a way that our dominant mode of sensory perception happens to be extremely useful for describing a lot of the world. Surely coloured light, in exactly the range of frequencies which we are able to see with our eyes, must play a fundamental role in how the universe works! Surely 'red' has a fundamental reality beyond our own existence; surely the colour red has an unchanging, even Platonic, nature to it!
I disagree. The colour red is indeed a very useful thing to sense, and to understand, because it is how we perceive some useful physical phenomena. But if we percieved a somewhat broader spectrum which included what we call the infra-red, that would be useful too; why don't we? For accidental reasons, I suppose. Perhaps in warm climates there is too much noise in those frequencies; though this doesn't explain why some species of snakes can sense them while we cannot. The reason why we can percieve red among other colours is ultimately because it was a useful accident.
If the number three seems to us to have an extremely vital existence, this could be because the concept of number is a useful one to be able to formulate when reacting to the world around us, and so much so that it is wired into our brains at a very deep level. This means that there are really amounts of things in the world, and that some notions of 'amount' are so simple and important that you can evolve creatures who believe that the notion of amount is so vitally important, that it can exist independently of anything to have an amount of.
The non-negative integers — the "natural numbers" — are just what we call our simplest tools for measuring amount. But they are our tools, extended well beyond our ability to immediately apprehend quantity, off into the dozens, hundreds, and billions — just as we have tools to help us to sense the infra-red, though we cannot directly percieve it.
Numbers are concepts. They are our tools to help us to understand useful things about the world. They are very, very, very useful tools; and versatile enough that we have every reason to believe that they can be used to describe any pattern that we can comprehend (and many that we cannot comprehend) regardless of whether that pattern is ever realized in the material world. But there is no more reason to believe that numbers (such as Three) exist independently, any more than there is to think that there is a Platonic Red which exists independently of any red object.
