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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    $\begingroup$ Also posted to MO. $\endgroup$
    – LSpice
    Feb 1, 2023 at 23:45
  • $\begingroup$ This is something you can probably easily research online by looking at how mathematical induction is treated in college algebra texts (and other variously named books at the same level) from the mid 1800s to the mid 1920s (ending years chosen to match what is currently freely available in google books and at archive.org). Also, the older (before 1940) references in this 15 Jan 2007 sci.math post might be useful (most, if not all, available at JSTOR). $\endgroup$ Feb 2, 2023 at 18:09
  • $\begingroup$ Thanks, Dave! Cajori's paper "Origin of the Name "Mathematical Induction"" isidore.co/misc/Physics%20papers%20and%20books/… is very informative. Having read it, I strongly suspect that Dantzig's use of the phrase "induction step" was unique to Dantzig. $\endgroup$ Feb 2, 2023 at 21:13
  • $\begingroup$ I strongly suspect that Dantzig's use of the phrase "induction step" was unique to Dantzig --- I thought this might be the case, because I don't recall having encountered his usage, but I didn't have time yesterday to look up anything (had online meetings, and documents of someone else to proof/edit and some that I had to write, relating to my "day job"), hence I mostly just told you how to fish rather than peppering a comment or two with a bunch of URL's and such (as I often tend to do in Stack Exchange). $\endgroup$ Feb 3, 2023 at 12:29


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