On the theory of the infinite in modern thought

by Eleanor F. Jourdain

He starts from the aggregate, which he analyses into the Finite and the Infinite, and the latter he analyses into the Transfinite and the Absolute. Of these two elements, one only has till just lately been the subject of mathematical treatment—​the one called the Transfinite. It is the transfinite subdivision of the Infinite to which the idea of number is applicable, and which is, therefore, in a sense inseparable from the Finite. Infinite numbers or series ought then to be more correctly described as Transfinite. But the processes of mathematics do not end here; they reach up to the idea of the Absolute Infinite, the conception of which has been attained in recent years by mathematical work. The results of this work may now be briefly summarised.

I. The Absolute appears to have the same relation to the Transfinite as the Transfinite to the Finite. If the Finite deals with numbers, and the Transfinite with series of numbers, the Absolute deals with series of series. Thus there are at least two examples of the Infinite within our grasp which lead up to the idea of the Absolute. One is the class of all classes of propositions; the other is the series of all worlds of thought, in Dedekind’s sense.

II. The Finite, Transfinite, and Absolute can be further defined in this way. There is no greatest finite number, but there is a least transfinite number, which has been called Aleph 0, and which can be proved to be greater than any possible finite number, however large, because if there were a last number it must be smaller than the sum of the whole series. There are unending series of Alephs or infinite numbers, which are as distinct from one another in idea as 1 is from 0, and which can no more be derived from one another by a mathematical process than 1 can be derived from 0, but can be reached in the same way by induction. Beyond the Transfinite we cannot discover in the Absolute the idea of least or of greatest.

III. The relation of cardinal and ordinal number also throws some light on the Finite, Transfinite, and Absolute. In the Finite, cardinals and ordinals are parallel to one another; in the Transfinite they strikingly diverge; in the Absolute we cannot trace any connection between cardinals and ordinals, i.e., it is possible to have an ordinal series to which there can be no corresponding cardinal number or type.[9]

IV. If arithmetical processes are applied to the Finite, Transfinite, or Absolute, we get interesting results. We know the effect of addition, multiplication, and raising to a power, on the Finite. The first two processes have been applied to the Alephs; the last has been formulated, but the mathematical results have not yet been brought to a satisfactory conclusion. Broadly speaking, we may say that the raising of an Aleph to a power may make it transcend the Finite and the Transfinite and melt into the Absolute. Thus all mathematical processes which find their goal in the Absolute would find their annihilation there. No finite mathematical conception would be applicable to it.

Now the conception of this Absolute Infinite, of which the aggregate of all ordinal numbers is perhaps a symbol,[10] has been subjected to criticism. Some mathematicians[11] think that it exists, but has no number. It is discovered by a logical process, but defies analysis and the application to it of the notion of number. All mathematical conceptions find in it their aim and conclusion. The importance of this theory, its practical importance, lies in the very much simpler mathematical formulæ that can be produced now that the logical process is shown to extend from the Finite to the Absolute Infinite (in the same way that the labour of summing a series arithmetically by statement and addition is shortened by the application of algebraical principles which depend on larger knowledge). Its philosophical importance is great: the Absolute is here, as elsewhere, the goal of human thought, and is the mathematician’s name for the highest power discoverable by human reason.

It would be very interesting to discuss the probable attitude of a Pascal or a Hegel to these mathematical conceptions, if they had been aware of them. Take Pascal’s puzzle of the Finite and the Infinite. He thought that if the Finite could be subtracted from the Infinite, the Infinite would thereby lose some of its quality of infinity. How differently would it have appeared to him had he realised that an aggregate infinite cardinal can have subtracted from it either finite or transfinite terms: if transfinite terms, many different answers result, giving different degrees of transfinity: if only finite terms are taken away, the Infinite remains in its entirety.

How, again, would Hegel have rejoiced in a definition of thought and existence which would bridge over the logical gulf in his system! Hegel asserted that thought and existence were one. He is objected to by many philosophers, who ask where is the tertium quid which makes it possible to reach from one to the other, or predicate their essential unity? But the mathematician defines existence as something which is not self-contradictory. Thought, then, to him is a form of existence, for thought is not self-contradictory; but the two, thought and existence, are not necessarily conterminous.[12] Hence, to say that non-contradiction is a fundamental condition of true thinking is as much as to say that it is a fundamental characteristic of real existence, and he identifies thought with reality.

Dr. Caird remarks that the secular conscience conceives of the Infinite as opposed to the Finite; the religious conscience treats the Infinite as real, presupposed by the illusory Finite. Where does the truth lie? Mathematics does not admit the necessity of adopting either view at the expense of the other.

Metaphysics standing alone produces results that may be disproved, but cannot be proved. Mathematics standing alone produces results that are susceptible of proof. Both are based on logic, and rest on the prerequisites of thought. Together they are a field for the best powers of human reason: metaphysics supplies insight, intuition, imagination; mathematics offers the indubitable proof and translates the ideal into the actual.