I’d be very interested in hearing more about why you think integers are imposed on us. Have you read Husserl’s Philosophy of Arithmetic? There he derived the ‘unit’, same thing different time, from acts of noticing within colligations.
Husserl described a method for understanding the constitution of a multiplicity or plurality composed of independent parts, which he dubbed ‘collective combination’. According to Husserl, the basis of any sort of whole of independently apprehended parts(a whole in the pregnant sense) is the collective combination, which is an abstracting act of consciousness uniting parts.
“Collective combination plays a highly significant role in our mental life as a whole. Every complex phenomenon which presupposes parts that are separately and specifically noticed, every higher mental and emotional activity, requires, in order to be able to arise at all, collective combinations of partial phenomena. There could never even be a representation of one of the more simple relations (e.g., identity, similarity, etc.) if a unitary interest and, simultaneously with it, an act of noticing did not pick out the terms of the relation and hold them together as unified. This ‘psychical’ relation is, thus, an indispensable psychological precondition of every relation and combination whatsoever.”
He conducted these researches under a psychological rubric , leading to accusations of psychologism from Frege and others. Ten years later he understood his method to be phenomenological, correcting the impressions of psychologism without affecting the substance of his description of the constitution of totality. In Experience and Judgement, he conducts a similar investigation under the heading of apprehension of plurality.
In any such whole the parts are united in a specific manner. Fundamental to the genesis of almost all totalities is that its parts initially appear as a temporal succession.
“Succession in time constitutes an insuppressible psychological precondition for the formation of by far the most number concepts and concrete multiplicities - and practically all of the more complicated concepts in general.”(Phil of Arithmetic, p.29) “Almost all representations of multiplicities - and, in any case, all representations of numbers - are results of processes, are wholes originated gradually out of their elements. Insofar as this is so, each element bears in itself a different temporal determination.”(p.33) “Temporal succession forms the only common element in all cases of multiplicity, which therefore must constitute the foundation for the abstraction of that concept.”
While the first step of constitution of a multiplicity is the awareness of the temporal succession of parts, each of which we are made aware of as elements “separately and specifically noticed” , the collective combination itself only emerges from a secondary act of consciousness. This higher order constituting sense changes what was originally a temporal succession into a simultaneity by ‘bringing’ back ‘ the previous parts via reflecting on them in memory. Husserl says that a combination of objects is similar to the continuity of a tone. In both cases, a temporal succession is perceived through reflection as a simultaneity.
“For the apprehension of each one of the colligated contents there is required a distinct psychical act. Grasping them together then requires a new act, which obviously includes those distinct acts, and thus forms a psychical act of second order.”(p.77) “It is essential that the partial representations united in the representation of the multiplicity or number be present in our consciousness simultaneously [in an act of reflection].”
The constitution of an abstract multiplicity is analogous to the creation of any whole, even though the former involves a peculiarly external form of unification in comparison to combinations unified by similarity or continuity.
A key feature of the fact that a totality is a product of a temporally unfolding series of sense acts is that prior elements of the originally apprehended series have already changed by the time we move on to the succeeding elements of that series. “In forming the representation of the totality we do not attend to the fact that changes in the contents occur as the colligation progresses.”(p.32) The secondary sense-forming act of the uniting of the pasts into the whole is not, then, ‘faithful’ to the original meaning of the parts it colligates, in that they have already changed their original sense via the passage of time at the point where we perform the uniting act of multiplicity.
Rather than a being faithful, the sense of the unification act may better be described as a moving beyond the original sense-constituting acts forming the apprehension of the parts. In forming a new dimension of sense from retentional and protentional consciousness, the unifying act of totalization idealizes the parts that it unifies. In addition to the abstractive concept of groupness (collective combination), many kinds of more intimate idealizations are constituted as wholes out of original temporal successions.
We can see this clearly in the case of the real object, an ideal totality formed out of a continuous synthetic flow of adumbrations in which what is actually experienced in the present is not the ‘faithful’, that is, actual presencing of temporally simultaneous elements but a simultaneity of retentional series, present sense and protentional anticipations.So the integer, like the spatial object, is a subjective and relative product of idealizing constitution.
