# Use of the verb "induct" in proofs by mathematical induction

Occasionally, in a proof by mathematical induction, the writer will say something like, "We induct on $$n$$" or "We induct on the number of vertices." This usage of the verb induct has always sounded strange to me. My first question is:

When did mathematicians start using the verb induct in this way?

The earliest instance that I could find using Google Books was in Mathematics 281: Algebraic Topology I by Spanier in 1955, but I suspect that there are earlier instances. Florian Cajori's paper, Origin of the Name “Mathematical Induction” demonstrates that the history of the term mathematical induction goes back much further, but Cajori's article does not discuss the use of the verb induct. It looks to me like an instance of back-formation; i.e., the term induction came first, and then people decided they wanted a verb, so they came up with induct. If this hypothesis is correct, then it prompts my second question:

Has any mathematician used the term induce on instead of induct on?

The term induction is also used in the philosophy of science; the process of inferring a general principle from many instances is often called inductive reasoning, and is contrasted with deductive reasoning. Now, in logic, the verb form of deductive is deduce; nobody ever uses the verb deduct to refer to logical inference. Therefore, one might naïvely expect (by analogy) that people would use the verb induce rather than induct in the context of mathematical induction. But I don't recall ever encountering that usage. Maybe I'm just uninformed? A possible explanation for avoiding the word induce in this context is that induce has other meanings in mathematics (e.g., if $$X$$ is a subspace of a topological space $$Y$$, then the topology on $$Y$$ induces the subspace topology on $$X$$).

• From Etymonline. Induct: late 14c., "introduce, initiate, especially into office or employment"... from Latin inductus, past participle of inducere "to lead into, introduce". Induction: As a term in logic (early 15c.). Induce: sense of "to infer by reasoning" is from 1560s. So yes, the use of "induct" in logic came much later than the use of "induce" and "induction", although both derive from Latin precursors. Commented Mar 3, 2023 at 6:13
• @Conifold Interesting fact about induce. I wonder if one can find an example from the 1560s? Commented Mar 3, 2023 at 13:40

If we search 19th century Google books, "induct on" does not exist with "mathematical induction". Perhaps, the term "induct on N" means to "form an induction on N". Again this is a rare usage as per the OED.

The unabridged Oxford English Dictionary gives a usage with a quotation on

absol. To form an induction; to infer by induction. rare. 1832 W. Whewell in I. Todhunter William Whewell (1876) II. 141 The conceptions which must exist in the mind in order to get by induction a law from a collection of facts; and the impossibility of inducting or even of collecting without this.

Another author (Investigations in Algebra By Albert Cuoco, 1990, MIT Press) puts "induct on" in quotations. So I agree with the OP, that induct on does sound unfamiliar to a non-native speaker (including myself) but a lot of mathematical words have not made sense of their choices, such as functional, fields, space, kernel etc. Perhaps this is the beauty of languages.

Chapter 5 $$\mathrm{G}_N(M)=\frac{M(M+1)(M+2)(M+3) \cdots(M+N-1)}{N !}$$ The numerator of $$\mathrm{G}_N(M)$$ is the product of all the integers between $$M$$ and $$M+N-1$$

We want to prove that for every positive integer $$N, \mathrm{~F}_N=\mathrm{G}_N$$ on $$\mathbf{Z}^{+}$$. We will prove this (of course) by the principle of mathematical induction; we will "induct on $$N . "$$ If $$N=1$$, we have that $$F_1(M)=M$$

German Wikipedia on Vollständige Induktion has some good information too

Die Bezeichnung Induktion leitet sich ab von lat. inductio, wörtlich „Hineinführung“. Der Zusatz vollständig signalisiert, dass es sich hier im Gegensatz zur philosophischen Induktion, die aus Spezialfällen ein allgemeines Gesetz erschließt und kein exaktes Schlussverfahren ist, um ein anerkanntes deduktives Beweisverfahren handelt. Das Induktionsprinzip steckt latent bereits in der von Euklid überlieferten pythagoreischen Zahlendefinition: „Zahl ist die aus Einheiten zusammengesetzte Menge.“ Euklid führte aber noch keine Induktionsbeweise, sondern begnügte sich mit intuitiven, exemplarischen Induktionen, die sich aber vervollständigen lassen. Auch andere bedeutende Mathematiker der Antike und des Mittelalters hatten noch kein Bedürfnis nach präzisen Induktionsbeweisen. Vereinzelte Ausnahmen im hebräischen und arabischen Sprachraum blieben ohne Nachfolge.

Machine Translation:

"The term induction is derived from Latin inductio, literally " introduction ". The addition of the word "complete" indicates that this is a recognized deductive method of proof, in contrast to philosophical induction, which deduces a general law from special cases and is not an exact conclusion.

The principle of induction is already latent in the Pythagorean definition of number handed down by Euclid: "Number is the quantity composed of units." Euclid, however, did not yet carry out induction proofs, but contented himself with intuitive, exemplary inductions, which, however, can be completed. Also other important mathematicians of the antiquity and the Middle Ages had no need for precise induction proofs yet. Isolated exceptions in the Hebrew and Arabic language areas remained without succession."

• To reassure you, "induct on" sounds quite strange to a native speaker too. Commented Mar 7, 2023 at 16:43