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In "Number: The Language of Science" (1930), Tobias Dantzig refers to what we call the base case of mathematical induction as "the induction step" (and refers to what we call the induction step as "the recurrence step" or "the proof of the hereditary property"). Was this standard terminology a century ago, or was Dantzig confused?

Note that he wrote this way back when mathematical induction was commonly called complete induction as opposed to Baconian or incomplete induction. Since verification of a single base case could be viewed as a minimalist version of Baconian induction, Dantzig's terminology does not seem totally illogical to me. Perhaps his use of the phrase "induction step" was standard a century ago, and over time its meaning shifted so that it now has the "opposite" meaning (that is, it now refers to the other component of proof by mathematical induction).

I'd be grateful for comments by those who know more history of mathematics than I do, as well as those who can bring a multicultural perspective to this question (what sort of terminology for mathematical induction is used in other languages?).

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  • $\begingroup$ As your tagging and body indicate, this is probably better suited for HSMSE. $\endgroup$
    – LSpice
    Commented Feb 1, 2023 at 21:49
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    $\begingroup$ I’ve posted it to hsm.stackexchange.com/questions/15133/… (for those who are curious) and I’ll give Ira’s answer my approval because I like Ira. :-) $\endgroup$
    – James Propp
    Commented Feb 1, 2023 at 23:44

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Not really an answer (too long for a comment) but I hope that this will be helpful:

In Jeff Miller's "Earliest Uses of Some Words of Mathematics" (https://mathshistory.st-andrews.ac.uk/Miller/mathword/) we find the following:

The term MATHEMATICAL INDUCTION was introduced by Augustus de Morgan (1806-1871) in 1838 in the article Induction (Mathematics) which he wrote for the Penny Cyclopedia. De Morgan had suggested the name successive induction in the same article and only used the term mathematical induction incidentally. The expression complete induction attained popularity in Germany after Dedekind used it in a paper of 1887 (Burton, page 440; Boyer, page 404).

and

COMPLETE INDUCTION (vollständige Induktion) was the term employed by Dedekind in his Was sind und Was sollen die Zahlen? (1887) for what is nowadays called "mathematical induction", and whose "scientific basis" ("wissenschaftliche grundlage") he claimed to have established with his "Theorem of complete induction" (§59). Dedekind also used occasionally the phrase "inference from n to n + 1", but nowhere in his booklet did he try to justify the adjective "complete".

In Concerning the axiom of infinity and mathematical induction (Bull. Amer. Math. Soc. 1903, pp. 424-434) C. J. Keyser referred to "complete induction" as a form of procedure unknown to the Aristotelian system, for this latter allows apodictic certainty in case of deduction only, while it is just characteristic of complete induction that it yields such certainty by the reverse process, a movement from the particular to the general, from the finite to the infinite.

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