The above answers present very interesting points, yet it seems to me that they touch mere top of the iceberg. Philosophical problems behind numbers are far more fundamental and subtle.
First of all, no competent lecturer of philosophy of science would ever make such a bold statement and give a reputedly ultimate definition of a number, or of anything else, and for sure no one would have ever done that having provided no alternative view. Philosophy is not a science, we have different purposes. Obviously, there have been hundreds of accounts of the aim of philosophy throughout the ages, but I would risk to claim that contemporary analytic philosophy of science considers itself a sort of a metadiscipline assessing what has to be taken for granted by other fields of research for them to work.
The question of the character of numbers is a perfect example of this approach.
Thus philosophers proposing well-argued terms and definitions contrary to common scientific practice are not misled. It is their duty to think out of the paradigm. And not just philosophers, the theory of mathematics or axiology are also concerned with those problems. Even more practical mathematicians do make use of them, perhaps unconciously, when switching between paradigms, like from point-based to noncommutative geometry.
The point here is not about a functional, mathematical definition of numbers and how they work in mathematical frameworks. What a philosopher is interested in is usually their ontic status. Namely, for instance, is number 2 an abstract object that can be labelled and referred to, like mental states, etc., according to some theories. Or is it rather a certain concept we use as a part of a given language-game. Or maybe it is a scheme, a pattern, non-existent on its own but nevertheless determinable? The stake here is unbelievable - the very subject and role of mathematics.
There have been many approaches to the problem. From Plato's ideas, through Kant's relations in space, Frege's extensions of concepts based on the relation of equipotency, to moderns times of Russell, Whitehead, or Wittgenstein. The topic is huge, and there is no single, nor all the less simple answer. For preliminary reading I would suggest Frege's "The Foundations of Arithmetic", Russell and Whitehead's "Principia Mathematica", and Wittgenstein's "Remarks on the Foundations of Mathematics". There are also nice, original clues in Kripke's "Wittgenstein on Rules and Private Language".
Mathematicians usually do not care about it, I have had several opportunities to find it out myself in the faculty. But hey, this is no surprise for philosophers! (;