The essential characteristic of reasoning by recurrence [induction] is that it contains, condensed, so to speak, in a single formula, an infinity of syllogisms.

This procedure is the demonstration by recurrence. We first establish a theorem for n = 1; then we show that if it is true of n - 1, it is true of n, and thence conclude that it is true for all the whole numbers. ..Here then we have the mathematical reasoning par excellence, and we must examine it more closely.
...The essential characteristic of reasoning by recurrence is that it contains, condensed, so to speak, in a single formula, an infinity of syllogisms.
...to arrive at the smallest theorem [we] can not dispense with the aid of reasoning by recurrence, for this is an instrument which enables us to pass from the finite to the infinite.
This instrument is always useful, for, allowing us to overleap at a bound as many stages as we wish, it spares us verifications, long, irksome and monotonous, which would quickly become impracticable. But it becomes indispensable as soon as we aim at the general theorem...
In this domain of arithmetic,.. the mathematical infinite already plays a preponderant rôle, and without it there would be no science, because there would be nothing general.

We can not... escape the conclusion that the rule of reasoning by recurrence is irreducible to the principle of contradiction. ...Neither can this rule come to us from experience... This rule, inaccessible to analytic demonstration and to experience, is the veritable type of the synthetic a priori judgment. On the other hand, we can not think of seeing in it a convention, as in some of the postulates of geometry. ...it is only the affirmation of the power of the mind which knows itself capable of conceiving the indefinite repetition of the same act when once this act is possible. The mind has a direct intuition of this power, and experience can only give occasion for using it and thereby becoming conscious of it.

Supperational thinkers, by recursive definition, include in their calculations the fact that they are in a group of superrational thinkers.

"We have seen recursion in the grammars of languages, we have seen recursive geometrical trees, which grow upwards forever, and we have seen one way in which recursion enters the theory of solid state physics. Now we are going to see yet another way in which the whole world is built out of recursion. This has to do with the structure of elementary particles: electrons, protons, neutrons, and the tiny quanta of electromagnetic radiation called "photons". We are going to see that particles are - in a certain sense which can only be defined rigorously in relativistic quantum mechanics- nested inside each other in a way which can be described recursively, perhaps even by some sort of "grammar"."

The repetition of the orphanage drama year after year, echoing the Nietzschean idea of eternal recurrence — that everything that has happened will happen again and again — seems to rob the little world of any real meaning. Why should the repetition of the fire inspector's lament make it sound so hollow?

All acts of reasoning seem to me to be different cases of one uniform process which may perhaps be best described as the substitution of similars. ..The chief difficulty consists in showing that all the forms of the old logic, as well as the fundamental rules of mathematical reasoning, may be explained upon the same principle; and it is to this difficult task I have devoted the most attention.

Repeating then is in every one, in every one their being and their feeling and their way of realizing everything and every one comes out of them in repeating. More and more then every one comes to be clear to some one.

The ideal reasoner would, when he had once been shown a single fact in all its bearings, deduce from it not only all the chain of events which led up to it but also the results which would follow from it.

any idea, plan, or purpose may be placed in the mind through repetition of thought.

Any idea, plan, or purpose may be placed in the mind through repetition of thought. p. 54

If one basic axiom controls the cosmos, it must be this: In a situation of infinity every possible condition occurs, not once, but an infinite number of times.

All ritual has a notable tendency to reduce itself to a rehearsal of formulas.

Formal logic and the logical syllogism encapsulate connectedness in reasoning.