With the turn of the century, Husserl's attention turned from and to one; that is, away from the mental activity of combining, and toward that which is reliably there to be combined. He wanted to show that mental activity is not the source of the latter. Chapter 8 of LU I exposes and refutes the three premises or "prejudices" of psychologism. In short, "psychologism" for Husserl is the error of collapsing the normative or regulative discipline of logic down onto the merely descriptive discipline of psychology. It would make mental operations (such as combination) the source of their own regulation. The "should" of logic, that utter necessity inhering in logical inference, would become no more than the "is" or facticity of our customary thinking processes, empirically described.
Husserl's formulation and refutation of the three psychologistic premises is wickedly clever, but cannot be treated in detail here. (See # 43-49 of LU I.) One example must suffice. Psychologism, Husserl charges, would place logical inferences on the same plane with mental operations (# 44), and this would make even mathematics into a branch of psychology (# 45). Indeed, math and logic do have structures that are isomorphic to those of mental operations, such as combination and distinction. But given that similarity, how then would one distinguish the regulation of any of these processes from the description of it? Under psychologism, there's no way. But Husserl makes the distinction in a way that also shows how regulation (i.e., the laws of logic) comes from elsewhere than the plane of mental activity.
And he does this by virtue of one. In # 46 Husserl agrees with his opponents that arithmetical operations occur in patterns that refer back to mental acts for their origin and also for their meaning. However, there's a difference between them as well. Mental acts transpire in time: they begin and end, and they can be repeated and individually counted. Numbers, in contrast, are timeless. While they can be represented in mental acts, this representation is not a fresh production of the number but rather an instantiation of its form. There is only one five. Any time we count five things, it isn't a production of a new five but merely a deja vu for the same old five, eternal five. We can't count numbers themselves, for there's only one of each. (A similar argument is made in #22 of Ideas I.)
The same goes for logic, Husserl says. Concepts comprising the laws of pure logic can have no empirical range. Their range or sphere is ideal singulars, not mental generalizations from multiple instantiations. The operators of logic are other than those mental acts that happen to share the same names: "and," "not," "is," "or," "implies," "may," "must," "should." Psychologically, there can be many factual acts of combining, negating, etc. Logically, there is only one "and," one "not," etc. Husserl concedes here, as he did for arithmetic, that the logical operators take their origin and meaning from the mental acts. This accounts for the equivocal character of logical terms, which refer both to ideal singulars, and to mental states and acts. But if you fail to notice this equivocation, you become ensnared in psychologism, losing the possibility of pure logic and unified science.
The danger of equivocation extends over judgments as well. On the one hand, we can count multiple apperceptive events of affirmation, occurring psychologically, which proceed in time, begin and end, and recur as often as we like, in happenings that can be distinguished one from another. On the other hand, the judgment thus reached remains the same throughout each act accessing it. It seems to persist and to be called back for encore appearances; it seems even to have pre-existed its first appearance to me (# 47). In this latter sense, the judgment is not the same as the mental act that reaches it. Moreover, the truth of the judgment is neither equivalent to nor dependent upon the psychological experience of clear evidence that accompanies the mental act embracing it. Husserl easily shows this by recalling that in both logic and arithmetic, there are truths that have never been entertained in any human consciousness, and indeed could never be humanly conceived (# 50). (Cases of truth without the possibility of psychological evidence would include the computation of very large numbers, and decisions about membership in sets that are uncountably large. The arithmetical and logical operations connected with such determinations could never be "done" by a human mind or a computer. Their truth cannot be "factual.")
The number one, then, has become Husserl's touchstone for discriminating between psychological processes and logical laws. It is his reality detector. What is psychological (or empirical) comes on in discrete individual instances–ones–and you can examine their edges. What is logical (or ideal) comes on as a seamless oceanic unity without temporal edges, reliably persisting even when not attended to. Husserl's sensitivity to the modes of unity, first expressed in the Habilitationsschrift and developed in LU, provides the launching pad for transcendental phenomenology.