 Problems arose because of the way that Hilbert and others described what they were doing. The history is complicated, but roughly the following happened. Around the middle of the nineteenth century people noticed, for example, that in an abelian group the minus function is definable in terms of 0 and + (namely: \(-a\) is the element \(b\) such that \(a + b = 0)\). Since this description of minus is in fact one of the axioms defining abelian groups, we can say (using a term taken from J. D. Gergonne, who should not be held responsible for the later use made of it) that the axioms for abelian groups implicitly define minus. In the jargon of the time, one said not that the axioms define the function minus, but that they define the concept minus. Now suppose we switch around and try to define plus in terms of minus and 0. This way round it can’t be done, since one can have two abelian groups with the same 0 and minus but different plus functions. Rather than say this, the nineteenth century mathematicians concluded that the axioms only partially define plus in terms of minus and 0. Having swallowed that much, they went on to say that the axioms together form an implicit definition of the concepts plus, minus and 0 together, and that this implicit definition is only partial but it says about these concepts precisely as much as we need to know.