CONTINUITY AND PASSAGE TO THE LIMIT
We can now return to our examination of the 'law of continuity', or, to be more exact, to the aspect of the law that we had to momentarily lay aside, and which is precisely that aspect by which Leibnitz believed 'passage to the limit' could be justified. For him what follows from it is
that with continuous quantities, the extreme exclusive case may be treated as inclusive, and that such a case, although totally different in nature, is thus as if contained in a latent state in the general law of the other cases. [1]
Although Leibnitz himself does not appear to have suspected it, it is precisely here that the principal logical error in his conception of continuity lies, which one may quite easily recognize in the consequences he draws from it and in the ways in which he applies it. Here are a few examples:
In accordance with my law of continuity, one is allowed to consider rest to be an infinitely small motion, that is, to be equivalent to a species of its contradictory, and coincidence to be an infinitely small distance, equality the last of inequalities, etc. [2] [Or again]: In accordance with this law of continuity, which excludes all sudden changes, the case of rest can be regarded as a
special case of motion, namely as a vanishing or minimum motion, and the case of equality as a case of vanishing inequality. It follows that the laws of motion must be established in such a way that there be no need for special rules for bodies in equilibrium and at rest, but that the latter should themselves arise from the rules concerning bodies in disequilibrium and in motion; or, if one does wish to set forth particular rules for rest and equilibrium, one must take care that they not be such as to disagree with the hypothesis that holds rest to be an incipient motion or equality the final inequality. [3]
Let us add one more quotation on the subject, in which we find a new example, of a somewhat different kind from the preceding, but no less contestable from the logical point of view:
Although it is not rigorously true that rest is a species of motion, or that equality is a species of inequality, just as it is not true that the circle is a species of regular polygon, one can nevertheless say that rest, equality, and the circle are the terminations of motion, inequality, and the regular polygon, which, by continual change, arrive at the former by vanishing. And although these terminations are exclusive, that is, not rigorously included within the varieties they limit, they nevertheless have the same properties as they would if they were so included, in accordance with the language of infinites or infinitesimals, which takes the circle, for example, as a regular polygon with an infinite number of sides. Otherwise the law of continuity would be violated, that is to say that because one passes from polygons to the circle by a continual change, without any break, there must likewise be no break in the passage from the attributes of polygons to those of the circle. [4]
It is worth pointing out that, as is indicated in the beginning of the last passage cited above, Leibnitz considers these assertions to be
of the same kind as those that are merely toleranter verae, which, he says elsewhere,
above all serve the art of invention, although, in my opinion, they contain something of the fictional and imaginary which however can easily be rectified by reducing them to ordinary expressions, in order that they not produce error. [5]
But are they not precisely that already, and in reality do they not rather contain contradictions pure and simple? No doubt Leibnitz recognized that the extreme case, or ultimus casus, is exclusivus, which obviously implies that it falls outside of the series of cases that are naturally included in the general law; but then with what right can it be included in this law in spite of it, and be treated ut inclusivum [as inclusive], that is, as if it were only one particular case contained within the series? It is true that the circle is the limit of a regular polygon with an indefinitely increasing number of sides, but its definition is essentially other than that of polygons; and in such an example one can see quite clearly that there exists a qualitative difference between the limit itself and that of which it is the limit, as we have said before. Rest is in no way a particular case of motion, nor equality a particular case of inequality, nor coincidence a particular case of distance, nor parallelism a particular case of convergence; besides, Leibnitz does not suppose that they are so in a rigorous sense, but he nonetheless maintains that they can in some way be regarded as such, with the result that 'the genus terminates in the opposed quasi-species,' and that something can be 'equivalent to a species of its contradictory.' [6] Moreover, let us note in passing that Leibnitz's notion of 'virtuality' seems to be linked to this same order of ideas, as he gives it the special sense of potentiality viewed as incipient actuality, [7] which again is no less contradictory than the other examples just cited.
Whatever the point of view from which things are envisaged, it is not in the least clear that a certain species could be a 'borderline case' of the opposite species or genus, for it is not in this way that opposed things limit each other reciprocally, but definitely to the contrary in that they exclude one another, and it is impossible for one contradictory to be reduced to another; for example, can inequality have any significance apart from the degree to which it is opposed to equality and is its negation? We certainly cannot say that assertions such as these are even toleranter verae, for even if one does not accept the existence of absolutely separate genuses, it is nonetheless true that any genus, defined as such, can never become an integral part of another equally defined genus when the definition of this latter does not include its own, even if it does not exclude it formally as in the case of contradictories; and if a connection can be established between different genuses, this is not in virtue of that in which they effectively differ, but only in virtue of a higher genus, which includes both. Such a conception of continuity, which ends up abolishing not only all separation, but even all effective distinction, in allowing direct passage from one genus to another without reducing the two to a higher or more general genus, is in fact the very negation of every true logical principle; and from this to the Hegelian affirmation of the 'identity of contradictories' is then but one step which is all too easy to take.