If what you are saying is that there must be something to count before one counts, then... well, sure, but I don't see the relevance. — Banno
One must make certain assumptions about this something: namely that it stays put as what it is, that it is res extensia, that it has duration. But nothing in the world stays put as what it is. Moment to moment it transforms itself ever so subtly. Self-identicality is an illusion of sorts. This is my point, as stated differently by Husserl and Heidegger:
For Husserl, extension, duration and magnitude are all implied by the idealizing thinking of self-identical objects. The ideal geometry of a line made possible the empirical intuitions pertaining to various characteristics of number.
“A true object in the sense of logic is an object which is absolutely identical "with itself," that is, which is, absolutely identically, what it is; or, to express it in another way: an object is through its determinations, its quiddities , its predicates, and it is identical if these quiddities are identical as belonging to it or when their belonging absolutely excludes their not belonging. Purely mathematical thinking is related to possible objects which are thought determinately through ideal-"exact" mathematical (limit-) concepts, e.g., spatial shapes of natural objects which, as experienced, stand in a vague way under shape-concepts and [thus] have their
shape-determinations; but it is of the nature of these experiential data that one can and by rights must posit, beneath the identical object which exhibits itself in harmonious experience as existing, an ideally identical object which is ideal in all its determinations; all [its]
determinations are exact —that is, whatever [instances] fall under their generality are equal—and this equality excludes inequality; or, what is the same thing, an exact determination, in belonging to an object, excludes the possibility that this determination not belong to the
“ In this sphere of magnitudes, and initially of spatial magnitudes—first of all in classes of privileged cases (straight lines, limited plane figures, and the corresponding cases of spatial magnitudes), first of all in the empirical intuition that magnitudes divide into equal parts and are composed again of equal parts—or of aggregates of like elements which decompose into
partial aggregates and can be expanded into new aggregates through the addition of elements or
of aggregates of such elements—in this sphere, there arose the "exact" comparisons of magnitudes which led back to the comparison of numbers. Upon the vague "greater," "smaller," "more," 'less," and the vague "equal" one could determinately superimpose the exact "so much" greater or less, or "how many times" greater or less, and the exact "equal."
Every such exact consideration presupposed the possibility of stipulating an equality which excluded the greater and the smaller and of stipulating units of magnitude which were strictly substitutable for one another, were identical as magnitudes, i.e., which stood under an identical concept or essence of magnitude.”
“Thus it was possible to conceive of processes converging idealiter through which an absolute
equal could be constructed ideally as the limit of the constant approach to equality, provided that one member [of the system] was thought of as absolutely fixed, as absolutely identical with itself in magnitude. In this exact thinking with ideas one operated with ideal concepts of the unchanging, of rest, of lack of qualitative change, with ideal concepts of equality and of the
general (magnitude, shape) that gives absolute equalities in any number of ideally unchanged and thus qualitatively identical instances; every change was constructed out of phases which were looked upon as momentary, exact, and unchanging, having exact magnitudes, etc.”
Husserl and Heidegger share a focus on Galileo as originator of modern mathematical science based on an idealization of geometric spatio-temporality as objective bodies in causal interaction. Heidegger traces the origin of empirical science to the concept of enduring substance.
“Mathematical knowledge is regarded as the one way of apprehending beings which can always be certain of the secure possession of the being of the beings which it apprehends. Whatever has the kind of being adequate to the being accessible in mathematical knowledge is in the true sense. This being is what always is what it is. Thus what can be shown to have the character of constantly remaining, as remanens capax mutationem, constitutes the true being of beings which can be experienced in the world. What enduringly remains truly is. This is the sort of thing that mathematics knows. What mathematics makes accessible in beings constitutes their being. Thus the being of the "world" is, so to speak, dictated to it in terms of a definite idea of being which is embedded in the concept of substantiality and in terms of an idea of knowledge which cognizes beings in this way. Descartes does not allow the kind of being of innerworldly beings to present itself, but rather prescribes to the world, so to speak, its "true" being on the basis of an idea of being (being = constant objective presence) the source of which has not been revealed and the justification of which has not been demonstrated. Thus it is not primarily his dependence upon a science, mathematics, which just happens to be especially esteemed, that determines his
ontology of the world, rather his ontology is determined by a basic ontological orientation toward being as constant objective presence, which mathematical knowledge is exceptionally well suited to grasp.”