Diversity of views on the infinite

Although in § 5 I cited many passages from Leibniz’s works in which he speaks out against infinite numbers by saying, amongst other things:

“Il n’y a point de nombre infini ni de ligne ou autre quantité infini, si on les prend pour des touts veritables.” (There is no infinite number, line or any other infinite quantity, if we take them to be true entire entities).

“L’infini veritable n’est pas une modification, c’est l’absolu; au contraire, des qu’on modifie on se borne ou forme un fini.” (The true infinity is not a modification, it is the absolute; on the contrary, as soon as we modify we limit ourselves or we form a finite).

Here I agree with the first part of the second statement, but not with the second part of it. I am, on the other hand, in the fortunate position of being able to demonstrate statements of the same philosopher in which, to a certain extent, he contradicts himself, when he expresses himself in the most unambiguous way in favor of the truly infinite (as opposed to the absolute infinite). He says in Erdmann p. 118:

“Je suis tellement pour l’infini actuel, qu’au lieu d’admettre que la nature l’abhorre, comme l’on dit vulgairement, je tiens qu’elle l’affecte partout, pour mieux marquer les perfections de son Autour. Ainsi je crois qu’il n’y a aucune partie de la matiere qui ne soit, je ne dis pas divisible, mais actuellement divisée; et par conséquent la moindre particelle doit être considered comme un monde plein d’une infinité de creatures différentes.” (I am so much in favor of the actual-infinite, that instead of admitting that nature abhors it, as one commonly says, I want it to affect it everywhere, so as to better indicate the perfections of its surroundings. So I believe that there is no part of matter that is not actually divided, rather I say that it is divisible; and therefore the smallest particle must be regarded as a world full of an infinity of different creatures).

But Bernhard Bolzano, a highly astute philosopher and mathematician of our century, is the most decisive defender of the actual-infinite, as can be seen, for example, in well-defined point-sets or in the constitution of bodies made of atomic points (I do not mean the chemical-physical, Democritean atoms here because I do not consider that they exist either as a concept, or in reality, regardless of how many useful things are derived to a certain extent by this fiction).

Bolzano developed his views in the beautiful and rich script, “Paradoxien des Infendlichen”, Leipzig 1851 (paradoxes of the infinite), the purpose of which is to prove that contradictions in the infinite that over the years have been sought by skeptics and explorers, do not exist at all, provided that one undertakes the difficult task of considering with complete seriousness the true substance of concepts of infinity.

In his book one can find a discussion of the mathematical non-actual-infinite that is correct in many respects, such as it occurs in the form of first and higher order differentials or in the summation of infinite series or in other limit processes. This non-actual-infinite (called syncategorematic infinite by some scholastics) is only an auxiliary relational mental concept, which by definition includes variability and for which a “value” can never be assigned in any actual sense.

Bolzano is perhaps the only one with whom the actual-infinite numbers have a certain legitimacy, at least they are often spoken of, but I totally disagree with him in the way he deals with them, since he is not able to establish a correct definition of them. For example, the §§ 29-33 of that book are unfounded and erroneous. The author lacks both the general concept of magnitude and the precise concept of cardinal-number for the real conceptualization of actual-infinite numbers.

Both appear in specific forms at individual points, but it seems to me that he does not work it through to full clarity and precision, which explains many inconsistencies and even some errors of this valuable work.

It is most remarkable that there is no essential difference of opinion even among contemporary philosophers with regard to this kind of the infinite, if I may disregard the fact that certain modern schools of so-called positivists or realists[Footnote12] or materialists, while they see the greatest concept in this syncategorematic infinite, must concede that it has no actual existence.

But the essentially correct facts can already be found in Leibniz’s work in many places. For example, the following passage relates to this non-actual-infinite (Erdmann, p. 436):

“Ego philosophice loquendo non magis statuo magnitudines infinite parvas quam infinite magnas, seu non magis infinitesimas quam infinituplas. Utrasque enim per modum loquendi compendiosum pro mentis fictionibus habeo, ad calculum aptis, quales etiam sunt radices imaginariae in Algebra. Interim demonstravi, magnum has expressiones usum habere ad compendium cogitandi adeoque ad inventionem; et in errorem ducere non posse, cum pro infinite parvo substituere sufficiat tarn parvum quam quis volet, ut error sit minor dato, unde consequitur errorem dari non posse.” (I hold that philosophically speaking, neither the infinitely small nor the infinitely large are important.

For both are mental fictions useful for calculation using approximations, in the same way as imaginary roots in algebra. In the meantime, I have pointed out the usefulness of these expressions as a mental summary as well as for discovery of the infinitely large; they cannot lead into error, as it is sufficient to substitute in it a quantity of sufficient diminutiveness, as for the infinitely small, so that any error should be less than the total, and thus it follows that there can be no resultant error.)

Without the two concepts mentioned, I am convinced that one does not progress any further in the theory of sets, and the same applies, I believe, to those areas which come under the theory of sets or which have the most intimate contact with it, such as, for example, the modern theory of functions on the one hand and from logic and epistemology on the other. To understand the infinite as it occurs to me here and in my earlier attempts gives me real pleasure.

I gratefully indulge myself in this pleasure when I see how the concept of integers, which in the finite has only the basic concept of number, but when we rise to the infinite, in some respects the concept splits into two, that of cardinal-number, which is independent of the ordering applied to a set, and that of ordinal-number, for which it is necessary that a regular order is applied to the set (when it becomes a well-ordered set).