The justification for extending the infinite

When I spoke of traditions, this was not to understand them in the narrow sense of what was experienced, but rather to follow them back to the founders of modern philosophy and natural sciences. In order to assess the question that is at stake here, I cite only a few of the most important sources, such as:

Locke, Essay on Human Understanding, Bk II, Chs. 16 and 17.

Descartes, Letters, and the Discussions of his Meditations; also, Principia I, 26.

Spinoza, Letter 29, Cogitata Metaph., parts I and II.

Leibniz, Erdmannsche Edition. pp.138, 244, 436, 744; Pertzsche Edition II, 1 p. 209; III, 4 p. 218; III, 5 p. 307, 322, 389; III, 7 p. 273.[Footnote10]

One cannot devise even today any stronger reasons than those noted here against the introduction of infinite integers; one may therefore examine them and compare them with mine. I will reserve a detailed and exhaustive discussion of these passages, and in particular of Spinoza’s most replete and important letter to L. Meyer, for another opportunity, but I will limit myself here to the following.

As different as the theories of these writers are, in their judgment of the finite and the infinite they essentially agree that the concept of a number includes the finiteness of it, and that on the other hand the true infinite or absolute, which resides in God, is not permitted any determination.

As far as the latter point is concerned, I fully agree with it, as it cannot be otherwise, because the sentence: “omnis determinatio est negatio” (every determination is negative) is, for me, completely beyond question.

On the other hand, as I have already said above when discussing the Aristotelian reasons against the “infinitum actu” (actual-infinite), I see in the former a petitio principii (begging the question), which renders explainable many contradictions which can be found in all these authors and especially in Spinoza and Leibniz.

The assumption is that, apart from the finite, and the absolute infinite that cannot be reached by any determination, there can be no new entities which are infinite but at the same time are determinate numbers, the ones that I refer to as the actual-infinite. I do not find that this assumption is justified by anything and, in my opinion, it even stands in contradiction to certain assertions put forward by the two aforementioned philosophers.

What I maintain and believe to have proved through this work, as well as through my earlier efforts, is that after the finite there is a transfinite (which might also be called a suprafinite), i.e: it gives an unlimited scale of definitive steps, which in their nature are not finite but rather infinite, but which, like finite entities, can be determined by definite, well-defined and mutually distinguishable numbers. My conviction is therefore that the finite magnitudes do not constrict the realm of definable magnitudes and that the limits of our knowledge can be extended accordingly without our nature having to apply any limitation. Instead of the Aristotelian school sentence discussed above in Section 4, I therefore put another:

“Omnia seu finita seu infinita definita sunt et excepto Deo ab intellectu determinari possunt.” (All things finite or infinite are definite and, except for God, can be determined by the intellect).[Footnote11]

The finiteness of human understanding is very often cited as the reason why only finite numbers can be the subject of thought, but I see in this assertion again the aforementioned circular argument. In the case of the “finiteness of understanding”, it is implicitly meant that mental ability is limited to finite numbers in view of the formation of numbers.

If it turns out that the mind is also infinite, in that it can define infinite numbers and distinguish them from one another, then either the words “finiteness of understanding” must be given an expanded meaning, so that that very conclusion can readily be concluded from them, or else the predicate “infinite” must also be used in certain respects for human understanding, which in my opinion is the only correct option.

The words “finiteness of understanding”, which one hears so often, do not, I believe, in any way apply; as limited as human nature is, in truth it is much attached to the infinite and I believe that even if it were not itself infinite in many aspects, we all have a firm confidence and certainty regarding the being of the absolute, and in which we all agree is known to be unexplainable. And in particular I take the view that the human mind has an unlimited capacity for the stepwise formation of sets of integers which are related to the infinite steps and whose cardinal-numbers are of increasing magnitude.

The main difficulties in the outwardly different but internally related philosophies of Spinoza and Leibniz can, I believe, be brought closer to the solution by the path I have chosen, and some of them can already be satisfactorily solved and clarified.

These are the difficulties which gave rise to the later criticism which, for all its advantages, does not seem to me to provide a sufficient substitute for the inhibited development of the teachings of Spinoza and Leibniz.

For alongside or in place of the mechanical explanation of nature, which within its sphere has all the means and advantages of mathematical analysis at its disposal, but whose one-sidedness and inadequacy were so aptly revealed by Kant, there has hitherto not even begun any explanation with the same mathematical rigor that reaches beyond that organic natural explanation. I believe it can only be initiated by resuming and developing the work and efforts of those people.

A difficult point in Spinoza’s system is the relation of the finite steps to the infinite steps.

It remains unexplained how and under what circumstances the finite can assert itself against the infinite or the infinite assert its independence against the still stronger infinite.

My example at Section 4 indicates the solution.

If ω is the first number of the second number-class, then one has 1 + ω = ω, on the other hand ω + 1 = (ω + 1), where (ω + 1)`` is a number that is quite different from ω`.

Everything depends on the position of the finite in relation to the infinite, as we can clearly see here; if the former comes first, it appears in the infinite and disappears into it, but if it is modest and takes its place after the infinite, it is preserved and combines with it to form a new, since modified, infinite.