Last time, Fichte had reached the concept of allness, by which he meant a certain determined collection of ones. Such all, he continues, is then externally limited: it is these ones we are talking about, not those others. Internally the ones collected together are still left undetermined, but they are still distinguished from other ones through an external limitation within the whole field of quantity. Indeed, the essential character of such an external limitation, Fichte says, is that the limiting distinction of quantities does not arise from any internal qualities. Just like all quantities, this quantitative limitation can also be described as continuous, and then it could be called, Fichte suggests, a measure. This measure, he insists, should be taken as a determination of something: measure as the determination gives something its limitation, but the limitation is here understood just externally or quantitatively.
Measure, Fichte begins, is at first completely indifferent toward any qualitative distinctions of what is measured. In other words, the measured content is in general just continuous and undistinguished. Fichte's examples of such a measure are an hour of time and a quarter of a foot, where it is indifferent what period of time or length is measured and the distinction of this time and length from others seems completely arbitrary. Measure is thus an arbitrary quantitative limit for something that remains similar or continuous, because any possible qualitative distinctions have been ignored. Measure, Fichte continues, is one of the most comprehensive, but also most abstract determinations of thinking, because it leaves the measured content undetermined. It is a comprehensive determination, he explains, because having a limited magnitude is an essential condition for being something. Thus, just like everything is numerical in the sense of being at least one, everything has also essentially a limit or measure and having this measure makes something into this. At this stage, Fichte notes, all determinations appear completely quantitative, thus, here all qualitative remains still beyond.
Fichte reminds us that all previous quantitative determinations have shared an indifference toward content and thus did not just limit things, but also immediately cancelled this limit. In other words, all quantities could be increased or decreased. Now, Fichte notes, same is true also of measure: every given measure could again be cancelled or made more or less, just because there is no internal limitation through a specific content. When such an abstractly thought measure is increased or decreased without any limit, or more precisely, by cancelling every given limit, Fichte says, it becomes its own opposite or measureless. Thus, it becomes more and more evident that the concept of measure must be complemented with some internal determination.
In the superficial increasing or decreasing of measure, Fichte thinks, is still found the contradiction that a relation to the content is ignored. Measure is cancelled through such external increasing or decreasing and reduces to more abstract determination of magnitude in general. On the other hand, determination of content would limit the increasing and decreasing and make measure more stable. This means, Fichte explains, that only a specific something should have a determined measure that cancels all increasing and decreasing. So, in the concept of measure, we for the first time have determined and not abstract magnitudes.
A determined measure cannot then be arbitrarily changed without any limit. This does not mean that it cannot be changed at all, Fichte explains, but only that it cannot be increased or decreased without changing the measured content. Change is still an essential moment of this concept, because it remains a quantity and the positing and cancelling of limits forms the character of quantitative. Compared with the previous levels, here the change of measure is linked to something lying beyond merely quantitative or to the content. This content, Fichte thinks, itself cancels its measure and so finds through its own internal progress another measure. In other words, change of measure is only the quantitative expression of this internal development. Indeed, Fichte points out, since quantity in itself is meaningless, the highest and most developed forms of quantity are mere expressions of something else.
By introducing determination to quantity, Fichte remarks, we encounter again the familiar relation of opposition of determined against its other and here especially against otherwise determined measures: every thesis of a determination is inevitably combined to an antithesis. This opposition means, Fichte explains, firstly, that the determined measure is determined only as a negation against another determined measure beyond it, and similarly this other measure is just a negation of the first. Secondly, as a variable measure it is also a negation of itself, that is, it has a tendency to change into a different measure and thus in a sense oppose itself. Indeed, Fichte points out, the change of measure shows both forms of opposition: a changing measure is opposed to both itself and to other determined measures.
What could distinguish a measure from another – or in case of a changing measure, from itself – is the content, Fichte says, but we still abstract here from it. Thus, we have only a quantitative difference that can appear merely by comparing quantities with one another or by putting them in relation. Hence, Fichte argues, measure is determined only in a synthetical relation to other, equally determined measures. This synthesis behind the determined measures is the common sphere of measures comparable to one another. With this comparison of determined measures, Fichte suggests, returns the distinction or the discreteness of magnitudes, because the synthetic element or the measure for comparing and so distinguishing measures is once again number (Zahl), or at this stage, amount (Anzahl). This amount, he explains, is the same measure, but now understood as a discrete collection of units: determinations and changes of measures can be compared to one another only according to numeric relations.
The basis of amount, Fichte continues, is one or unit that is used as a common yardstick for all measures, and the different ones or units are collected into the amount. This sounds like what happened in collecting ones into a number, but Fichte finds a difference: for abstract numbers, the content is completely indifferent or anything that could be just abstractly distinguished could also be numbered. Amount, on the other hand, is expressly connected to the content, that is, its ones or units are all expressly similar to one another: only things that can be regarded as similar can be collected into an amount. Thus, Fichte emphasises, just like the predominantly continuous determined measure contains a moment of discreteness, similarly the predominantly discrete amount contains a moment of continuity – because the units comprehended in the amount are similar, we can regard the amount also as continuous or as determined measure, just like a determined measure can be thought or measured only as numbered or as amount. Determined measure and amount are thus only one another complementing expressions of the same limited magnitude.
Determined measure and amount, Fichte reminds the reader, are to be thought as determined only in relation to other equally determined measures or in a system of measures and numbers. Thus, he argues, limited magnitude is not like it appeared at first: it cannot be arbitrarily increased or decreased, because every limitation expresses a determined relation to other magnitudes, which measure or quantitatively determine one another in a reciprocal relation and only thus gain their limit or determination. Such a relation to a system of other measures, Fichte thinks, is infinite, but not anymore in the superficial sense that every posited limit can be cancelled or that every division of a quantity can be divided further. Instead, he calls it a positive infinity that lies in the relation of numbers in general: just like this is this only as opposed to an infinitely different this, similarly every determined magnitude is determined only as opposed to every other magnitude.
Fichte has argued that it is the relation between magnitudes that determines these magnitudes and makes them more than just abstractions. This means, he suggests, that determinations that the magnitudes have in this relation to one another should remain the same, although the numeric expressions of the magnitudes and their determinations would change. Fichte is evidently speaking here of the mathematical notion of a variable being a function of another variable: the function should remain the same, no matter what numeric expressions the variables should have. In an uncareful fashion he suggests that all these relations or functions are reducible to six basic calculations that form three pairs of oppositions: in the first pair, magnitudes are taken as collections of units that are either combined (in addition) or separated (in subtraction), in the second pair, either a new magnitude is assembled by using a given magnitude as a unit for the amount corresponding to another magnitude (in multiplication) or such an assembled magnitude is disassembled (in division), and finally, in the third pair, either one and the same number is taken as both a unit and an amount of multiplication (in exponentiation) or the original number for the result of such exponentiation is searched for (in taking roots).
Every determined magnitude could now be called a quantitative thesis, Fichte says, in the sense that these magnitudes as mutually measuring one another are at first placed in a positive relation with one another, expressed through some equation like x + y = 5 or 2x = y (Fichte explicitly speaks only of additions and subtractions or magnitudes having a determined result and of cases where one magnitude is multiplied with a number to form the other magnitude). Now, every thesis should lead to a further antithesis, by which Fichte here means cases where the two magnitudes are in a converse relation of the sort xy = a, where the increase of x leads to y decreasing and vice versa. Indeed, he points out, the position and negation are immediately connected, when we note that there are three magnitudes that we are speaking of: in equation xy = a, y is in a direct, “positive” relation to a (when y increases, a increases also), but in a converse, “negative” relation to x.
No thesis and antithesis without a synthesis, Fichte is eager to point out. Indeed, the synthesis should already be implicitly contained in the antithesis and thus needs only to be made explicit for the consciousness. This happens, Fichte suggests, when we make a magnitude have both a positive and negative relation to itself: in other words, when in the equation xy = a we assume that x = y. In effect, we are now moving to equations involving exponentiation, which as the phase of synthesis should then be the highest expression of numeric relations. What Fichte sees in exponentiation – and here he is closely following Hegel – is a case where a variable magnitude that despite being increased or decreased still retains a more qualitative relation to another variable magnitude (one could think here of quantities of different dimensions). We thus now enter the final stage in Fichte’s study of quantities, which is concerned of magnitudes and their variability as governed not just by their quantitative relations to other magnitudes, but by their own internal determination.
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