Frege takes this observation to be the fundamental thought of "Grundlagen". For example, the sentence "the number of horses in the barn is four" means that four objects fall under the concept "horse in the barn". Frege attempts to explain our grasp of numbers through a contextual definition of the cardinality operation ('the number of...', or formula_1). He attempts to construct the content of a judgment involving numerical identity by relying on Hume's principle (which states that the number of Fs equals the number of Gs if and only if F and G are equinumerous, i.e. in one-one correspondence). He rejects this definition because it doesn't fix the truth value of identity statements when a singular term not of the form 'the number of Fs' flanks the identity sign. Frege goes on to give an explicit definition of number in terms of extensions of concepts, but expresses some hesitation.